Stories Data SpeakJekyll2018-09-14T02:27:23+00:00https://duttashi.github.io/Ashish Dutthttps://duttashi.github.io/ashishdutt@yahoo.com.my<![CDATA[To eat or not to eat! That's the question? Measuring the association between categorical variables]]>https://duttashi.github.io/blog/to-eat-or-not-to-eat2017-06-03T00:00:00+00:002017-06-03T00:00:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<h3 id="1-introduction">1. Introduction</h3>
<p>I serve as a reviewer to several ISI and Scopus indexed journals in Information Technology. Recently, I was reviewing an article, wherein the researchers had made a critical mistake in data analysis. They converted the original <code class="highlighter-rouge">categorical</code> data to <code class="highlighter-rouge">continuous</code> without providing a rigorous statistical treatment, nor, any justification to the loss of information if any. Thus, my motivation to develop this study, is borne out of their error.</p>
<p>We know the standard association measure between continuous variables is the product-moment correlation coefficient introduced by Karl Pearson. This measure determines the degree of linear association between continuous variables and is both normalized to lie between -1 and +1 and symmetric: the correlation between variables x and y is the same as that between y and x. <em>the best-known association measure between two categorical variables is probably the chi-square measure, also introduced by Karl Pearson. Like the product-moment correlation coefficient, this association measure is symmetric, but it is not normalized. This lack of normalization provides one motivation for Cramer’s V, defined as the square root of a normalized chi-square value; the resulting association measure varies between 0 and 1 and is conveniently available via the assocstats function in the vcd package. An interesting alternative to Cramer’s V is Goodman and Kruskal’s tau, which is not nearly as well known and is asymmetric. This asymmetry arises because the tau measure is based on the fraction of variability in the categorical variable y that can be explained by the categorical variable x.</em> <a href="https://cran.r-project.org/web/packages/GoodmanKruskal/vignettes/GoodmanKruskal.html">1</a></p>
<p>The data for this study is sourced from UCI Machine Learning <a href="http://archive.ics.uci.edu/ml/machine-learning-databases/mushroom/agaricus-lepiota.data">repository</a>. As it states in the <code class="highlighter-rouge">data information</code> section, “This data set includes descriptions of hypothetical samples corresponding to 23 species of gilled mushrooms in the Agaricus and Lepiota Family (pp. 500-525). Each species is identified as definitely edible, definitely poisonous, or of unknown edibility and not recommended. This latter class was combined with the poisonous one. The guide clearly states that there is no simple rule for determining the edibility of a mushroom;</p>
<p>Furthermore, the possible research questions, I want to explore are;</p>
<ul>
<li>Is significance test enough to justify a hypothesis?</li>
<li>How to measure associations between categorical predictors?</li>
</ul>
<h4 id="2-making-data-management-decisions">2. Making data management decisions</h4>
<p>As a first step, I imported the data in R environment as;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Import data from UCI ML repo
> theURL<- "http://archive.ics.uci.edu/ml/machine-learning-databases/mushroom/agaricus-lepiota.data"
# Explicitly adding the column headers from the data dictionary
> mushroom.data<- read.csv(file = theURL, header = FALSE, sep = ",",strip.white = TRUE,
stringsAsFactors = TRUE,
col.names = c("class","cap-shape","cap-surface","cap-color","bruises",
"odor","gill-attachment","gill-spacing","gill-size",
"gill-color","stalk-shape","stalk-root","stalk-surface-above-ring",
"stalk-surface-below-ring","stalk-color-above-ring","stalk-color-below-ring",
"veil-type","veil-color","ring-number","ring-type","spore-print-color",
"population","habitat"))
</code></pre></div></div>
<p>Next, I quickly summarize the dataset to get a brief glimpse. The reader’s should note that the data has no missing values. (<em>Thanks to Junhewk Kim for pointing out the earlier error in data levels</em>)</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Calculate number of levels for each variable
> mushroom.data.levels<-cbind.data.frame(Variable=names(mushroom.data), Total_Levels=sapply(mushroom.data,function(x){as.numeric(length(levels(x)))}))
> print(mushroom.data.levels)
Variable Total_Levels
class class 2
cap.shape cap.shape 6
cap.surface cap.surface 4
cap.color cap.color 10
bruises bruises 2
odor odor 9
gill.attachment gill.attachment 2
gill.spacing gill.spacing 2
gill.size gill.size 2
gill.color gill.color 12
stalk.shape stalk.shape 2
stalk.root stalk.root 5
stalk.surface.above.ring stalk.surface.above.ring 4
stalk.surface.below.ring stalk.surface.below.ring 4
stalk.color.above.ring stalk.color.above.ring 9
stalk.color.below.ring stalk.color.below.ring 9
veil.type veil.type 1
veil.color veil.color 4
ring.number ring.number 3
ring.type ring.type 5
spore.print.color spore.print.color 9
population population 6
habitat habitat 7
</code></pre></div></div>
<p>As we can see, the variable, <code class="highlighter-rouge">gill.attachement</code> has two levels (<em>Thanks to Prof. Antony Unwin for pointing out the earlier error in gill.attachment</em>). The variable, <code class="highlighter-rouge">veil.type</code> has one level.</p>
<p>The different levels are uninterpretable in their current format. I will use the data dictionary and recode the levels into meaningful names.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> levels(mushroom.data$class)<- c("edible","poisonous")
> levels(mushroom.data$cap.shape)<-c("bell","conical","flat","knobbed","sunken","convex")
> levels(mushroom.data$cap.surface)<- c("fibrous","grooves","smooth","scaly")
> levels(mushroom.data$cap.color)<- c("buff","cinnamon","red","gray","brown","pink","green","purple","white","yellow")
> levels(mushroom.data$bruises)<- c("bruisesno","bruisesyes")
> levels(mushroom.data$odor)<-c("almond","creosote","foul","anise","musty","nosmell","pungent","spicy","fishy")
> levels(mushroom.data$gill.attachment)<- c("attached","free")
> levels(mushroom.data$gill.spacing)<- c("close","crowded")
> levels(mushroom.data$gill.size)<-c("broad","narrow")
> levels(mushroom.data$gill.color)<- c("buff","red","gray","chocolate","black","brown","orange","pink","green","purple","white","yellow")
> levels(mushroom.data$stalk.shape)<- c("enlarging","tapering")
> table(mushroom.data$stalk.root) # has a missing level coded as ?
? b c e r
2480 3776 556 1120 192
> levels(mushroom.data$stalk.root)<- c("missing","bulbous","club","equal","rooted")
> levels(mushroom.data$stalk.surface.above.ring)<-c("fibrous","silky","smooth","scaly")
> levels(mushroom.data$stalk.surface.below.ring)<-c("fibrous","silky","smooth","scaly")
> levels(mushroom.data$stalk.color.above.ring)<- c("buff","cinnamon","red","gray","brown", "orange","pink","white","yellow")
> levels(mushroom.data$stalk.color.below.ring)<- c("buff","cinnamon","red","gray","brown", "orange","pink","white","yellow")
> levels(mushroom.data$veil.type)<-c("partial")
> levels(mushroom.data$veil.color)<- c("brown","orange","white","yellow")
> levels(mushroom.data$ring.number)<-c("none","one","two")
> levels(mushroom.data$ring.type)<- c("evanescent","flaring","large","none","pendant")
> levels(mushroom.data$spore.print.color)<- c("buff","chocolate","black","brown","orange","green","purple","white","yellow")
> levels(mushroom.data$population)<- c("abundant","clustered","numerous","scattered","several","solitary")
> levels(mushroom.data$habitat)<-c("woods","grasses","leaves","meadows","paths","urban","waste")
</code></pre></div></div>
<h4 id="3-initial-data-visualization">3. Initial data visualization</h4>
<p>Since, we are dealing with categorical data, plotting it is slightly different. Here we use bar charts/plots or mosaic plots rather than dot plots or scatter plots. (<em>Thanks to Prof. Antony Unwin for pointing it out</em>). The dot plot is useful for plotting continuous variables. It can be used, to plot categorical variables, but then such a visualization will be confusing.</p>
<h5 id="a-univariate-data-visualization-stacked-bar-plot">a. Univariate data visualization (Stacked Bar plot)</h5>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> p<- ggplot(data = mushroom.data)
> p+geom_bar(mapping = aes(x = cap.shape, fill=class), position = position_dodge())+ theme(legend.position = "top")
> table(mushroom.data$cap.shape, mushroom.data$class)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy_mushrooms_plot1.png" alt="plot1" /></p>
<p>Fig-1: Mushroom cap-shape and class</p>
<p>From Fig-1, we can easily notice, the mushrooms with a, <code class="highlighter-rouge">flat</code> cap-shape are mostly edible (<em>n=1596</em>) and an equally similar number are <code class="highlighter-rouge">poisonous</code> (<em>n=1556</em>). A majority of <code class="highlighter-rouge">bell</code>shaped mushrooms (<em>n=404</em>) are <em>edible</em>. All <code class="highlighter-rouge">conical</code> cap-shaped mushrooms are poisonous (<em>n=4</em>). And, all <code class="highlighter-rouge">sunken</code> cap-shaped mushrooms are edible (<em>n=32</em>).</p>
<h5 id="b-how-is-habitat-related-to-class-mosaic-plot">b. How is habitat related to class? (Mosaic Plot)</h5>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(vcd) # for mosaicplot()
> table(mushroom.data$habitat, mushroom.data$class) # creates a contingency table
edible poisonous
woods 1880 1268
grasses 1408 740
leaves 240 592
meadows 256 36
paths 136 1008
urban 96 272
waste 192 0
> mosaicplot(~ habitat+class, data = mushroom.data,cex.axis = 0.9, shade = TRUE,
main="Bivariate data visualization",
sub = "Relationship between mushroom habitat and class",
las=2, off=10,border="chocolate",xlab="habitat", ylab="class" )
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy_mushrooms_plot2.png" alt="plot2" /></p>
<p>Fig-2: Mushroom habitat and class</p>
<p>From Fig-2, we see a majority of mushrooms that live in <code class="highlighter-rouge">woods</code>, <code class="highlighter-rouge">grasses</code>, <code class="highlighter-rouge">leaves</code>, <code class="highlighter-rouge">meadows</code> and <code class="highlighter-rouge">paths</code> are edible. Surprisingly, the one’s living in <code class="highlighter-rouge">waste</code> areas are entirely edible.</p>
<h5 id="c-how-is-population-related-with-class">c. How is population related with class?</h5>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> table(mushroom.data$population, mushroom.data$class)
edible poisonous
abundant 384 0
clustered 288 52
numerous 400 0
scattered 880 368
several 1192 2848
solitary 1064 648
> mosaicplot(~ population+class, data = mushroom.data,
cex.axis = 0.9, shade = TRUE,
main="Bivariate data visualization",
sub = "Relationship between mushroom population and class",
las=2, off=10,border="chocolate",xlab="population", ylab="class")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy_mushrooms_plot3.png" alt="plot3" /></p>
<p>Fig-3: Mushroom population and class</p>
<p>From Fig-3, we can see a majority of mushroom population that is either, <code class="highlighter-rouge">clustered</code>, <code class="highlighter-rouge">scattered</code>, <code class="highlighter-rouge">several</code> or <code class="highlighter-rouge">solitary</code> are edible. The mushrooms that are either <code class="highlighter-rouge">abundant</code> or <code class="highlighter-rouge">numerous</code> in population are completely edible.</p>
<p>Although, there could be many other pretty visualizations but I will leave that as a future work.</p>
<p>I will now focus on exploratory data analysis.</p>
<h4 id="4-exploratory-data-analysis">4. Exploratory data analysis</h4>
<h5 id="a-correlation-detection--treatment-for-categorical-predictors">a. Correlation detection & treatment for categorical predictors</h5>
<p>If we look at the structure of the dataset, we notice that each variable has several factor levels. Moreover, these levels are <code class="highlighter-rouge">unordered</code>. Such unordered categorical variables are termed as <strong>nominal variables</strong>. The opposite of unordered is ordered, we all know that. The <code class="highlighter-rouge">ordered</code> categorical variables are called, <strong>ordinal variables</strong>.</p>
<p>“In the measurement hierarchy, interval variables are highest, ordinal variables are next, and nominal variables are lowest. Statistical methods for variables of one type can also be used with variables at higher levels but not at lower levels.”, see <a href="https://mathdept.iut.ac.ir/sites/mathdept.iut.ac.ir/files/AGRESTI.PDF">Agresti</a></p>
<p>I found this <a href="https://stats.idre.ucla.edu/other/mult-pkg/whatstat/">cheat-sheet</a> that can aid in determining the right kind of test to perform on categorical predictors (independent/explanatory variables). Also, this <a href="https://stats.stackexchange.com/questions/108007/correlations-with-categorical-variables">SO post</a> is very helpful. See the answer by user <code class="highlighter-rouge">gung</code>.</p>
<p>For categorical variables, the concept of correlation can be understood in terms of <strong>significance test</strong> and <strong>effect size (strength of association)</strong></p>
<p>The <strong>Pearson’s chi-squared test of independence</strong> is one of the most basic and common hypothesis tests in the statistical analysis of categorical data. It is a <strong>significance test</strong>. Given two categorical random variables, X and Y, the chi-squared test of independence determines whether or not there exists a statistical dependence between them. Formally, it is a hypothesis test. The chi-squared test assumes a null hypothesis and an alternate hypothesis. The general practice is, if the p-value that comes out in the result is less than a pre-determined significance level, which is <code class="highlighter-rouge">0.05</code> usually, then we reject the null hypothesis.</p>
<p><em>H0: The The two variables are independent</em></p>
<p><em>H1: The The two variables are dependent</em></p>
<p>The null hypothesis of the chi-squared test is that the two variables are independent and the alternate hypothesis is that they are related.</p>
<p>To establish that two categorical variables (or predictors) are dependent, the chi-squared statistic must have a certain cutoff. This cutoff increases as the number of classes within the variable (or predictor) increases.</p>
<p>In section 3a, 3b and 3c, I detected possible indications of dependency between variables by visualizing the predictors of interest. In this section, I will test to prove how well those dependencies are associated. First, I will apply the chi-squared test of independence to measure if the dependency is significant or not. Thereafter, I will apply the <strong>Goodman’s Kruskal Tau</strong> test to check for <strong>effect size (strength of association)</strong>.</p>
<h6 id="i-pearsons-chi-squared-test-of-independence-significance-test">i. Pearson’s chi-squared test of independence (significance test)</h6>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> chisq.test(mushroom.data$cap.shape, mushroom.data$cap.surface, correct = FALSE)
Pearson's Chi-squared test
data: mushroom.data$cap.shape and mushroom.data$cap.surface
X-squared = 1011.5, df = 15, p-value < 2.2e-16
</code></pre></div></div>
<p>since the p-value is <code class="highlighter-rouge">< 2.2e-16</code> is less than the cut-off value of <code class="highlighter-rouge">0.05</code>, we can reject the null hypothesis in favor of alternative hypothesis and conclude, that the variables, <code class="highlighter-rouge">cap.shape</code> and <code class="highlighter-rouge">cap.surface</code> are dependent to each other.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> chisq.test(mushroom.data$habitat, mushroom.data$odor, correct = FALSE)
Pearson's Chi-squared test
data: mushroom.data$habitat and mushroom.data$odor
X-squared = 6675.1, df = 48, p-value < 2.2e-16
</code></pre></div></div>
<p>Similarly, the variables <code class="highlighter-rouge">habitat</code> and <code class="highlighter-rouge">odor</code> are dependent to each other as the p-value <code class="highlighter-rouge">< 2.2e-16</code> is less than the cut-off value <code class="highlighter-rouge">0.05</code>.</p>
<h6 id="ii-effect-size-strength-of-association">ii. Effect size (strength of association)</h6>
<p>The measure of association does not indicate causality, but association–that is, whether a variable is associated with another variable. This measure of association also indicates the strength of the relationship, whether, weak or strong.</p>
<p>Since, I’m dealing with <code class="highlighter-rouge">nominal</code> categorical predictor’s, the <strong>Goodman and Kruskal’s tau</strong> measure is appropriate. Interested readers are invited to see pages 68 and 69 of the <a href="https://mathdept.iut.ac.ir/sites/mathdept.iut.ac.ir/files/AGRESTI.PDF">Agresti book</a>. More information on this test can be seen <a href="https://cran.r-project.org/web/packages/GoodmanKruskal/vignettes/GoodmanKruskal.html">here</a></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(GoodmanKruskal)
> varset1<- c("cap.shape","cap.surface","habitat","odor","class")
> mushroomFrame1<- subset(mushroom.data, select = varset1)
> GKmatrix1<- GKtauDataframe(mushroomFrame1)
> plot(GKmatrix1, corrColors = "blue")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy_mushrooms_plot4.png" alt="plot4" /></p>
<p>In Fig-4, I have shown the association plot. This plot is based on the <code class="highlighter-rouge">corrplot</code> library. In this plot the diagonal element <code class="highlighter-rouge">K</code> refers to number of unique levels for each variable. The off-diagonal elements contain the forward and backward tau measures for each variable pair. Specifically, the numerical values appearing in each row represent the association measure τ(x,y)τ(x,y) from the variable xx indicated in the row name to the variable yy indicated in the column name.</p>
<p>The most obvious feature from this plot is the fact that the variable <code class="highlighter-rouge">odor</code> is almost perfectly predictable (i.e. τ(x,y)=0.94) from <code class="highlighter-rouge">class</code> and this forward association is quite strong. The forward association suggest that <em>x=</em><strong>odor</strong> (which has levels “almond”, “creosote”, “foul”, “anise”, “musty”, “nosmell”, “pungent”, “spicy”, “fishy”) is highly predictive of <em>y=</em><strong>class</strong> (which has levels “edible”, “poisonous”). This association between <code class="highlighter-rouge">odor</code> and <code class="highlighter-rouge">class</code> is strong and indicates that if we know a mushroom’s odor than we can easily predict its class being edible or poisonous.</p>
<p>On the contrary, the reverse association <em>y=</em><strong>class</strong> and <em>x=</em><strong>odor</strong>(i.e. τ(y,x)=0.34; is a strong association and indicates that if we know the mushroom’s class being edible or poisonous than its easy to predict its odor.</p>
<p>Earlier we have found <code class="highlighter-rouge">cap.shape</code> and <code class="highlighter-rouge">cap.surface</code> are dependent to each other (chi-squared significance test). Now, let’s see if the association is strong too or not. Again, from Fig-4, both the forward and reverse association suggest that <em>x=</em><strong>cap shape</strong> is weakly associated to <em>y=</em><strong>cap surface</strong> (i.e.τ(x,y)=0.03) and (i.e.τ(y,x)=0.01). Thus, we can safely say that although these two variables are significant but they are association is weak; i.e. it will be difficult to predict one from another.</p>
<p>Similarly, many more associations can be interpreted from plot-4. I invite interested reader’s to explore it further.</p>
<h4 id="5-conclusion">5. Conclusion</h4>
<p>The primary objective of this study was to drive the message, <em>do not tamper the data without providing a credible justification</em>. The reason I chose categorical data for this study to provide an in-depth treatment of the various measures that can be applied to it. From my prior readings of statistical texts, I could recall that significance test alone was not enough justification; there had to be something more. It is then, I found about the different types of association measures, and it sure did clear my doubts. In my next post, I will continue the current work by providing inferential and predictive analysis. For interested reader’s, I have uploaded the complete code on my Github repository in <a href="https://github.com/duttashi/learnr/blob/master/scripts/Full%20Case%20Studies/CaseStudy-UCI-PoisonousMushroomPredict.R">here</a></p>
<![CDATA[Learning a classifier from census data]]>https://duttashi.github.io/blog/learning-a-classifier-from-census-data2017-03-02T00:00:00+00:002017-03-02T00:00:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<h3 id="introduction">Introduction</h3>
<p>While reading the local daily, <em>“The Star”</em>, my attention was caught by headlines discussing an ongoing political or social discussion on the country’s financial state. Often, it is interesting to know the underlying cause of a certain political debate or the factors contributing to an increase or decrease in inflation. “A large income is the best recipe for happiness I ever heard of” quotes the famous English novelist Jane Austen. Income is a primary concern that dictates the standard of living and economic status of an individual. Taking into account, its importance and impact on determining a nation’s growth, this study aims at presenting meaningful insights which can be used to serve as the basis for many wiser decisions that could be taken by the nation’s administrators.</p>
<p>This study is organized as follows;</p>
<ol>
<li>
<p>Research question</p>
</li>
<li>
<p>The dataset</p>
</li>
<li>
<p>Making data management decisions</p>
<p>A. Exploratory Data Analysis (EDA)</p>
<ul>
<li>Data preprocessing (collapse the factor levels & re-coding)</li>
<li>Missing data visualization</li>
<li>Some obvious relationships</li>
<li>Some not-so-obvious relationships</li>
</ul>
<p>B. Correlation Detection & Treatment</p>
<ul>
<li>Detecting skewed variables</li>
<li>Skewed variables treatment</li>
<li>Correlation detection</li>
</ul>
</li>
<li>
<p>Predictive data analytics</p>
<ul>
<li>Creating the train and test dataset</li>
<li>Fit a Logistic Regression Model</li>
<li>Fit a Decision Tree Model</li>
<li>Fit a Support Vector Machine (SVM) classification model</li>
<li>Fit a Random Forest (RF) classification model</li>
</ul>
</li>
<li>
<p>Conclusion</p>
</li>
</ol>
<h3 id="1-research-question">1. Research question</h3>
<p>This study is driven by the question, “<em>Predict if a person’s income is above or below 50K$/yr given certain features(both quantitative and qualitative)..</em>”</p>
<h3 id="2-the-dataset">2. The dataset</h3>
<p>The dataset used for the analysis is an extraction from the 1994 census data by Barry Becker and donated to the UCI Machine Learning <a href="http://archive.ics.uci.edu/ml/datasets/Census+Income">repository</a>. This dataset is popularly called the “Adult” data set.</p>
<h3 id="3-making-data-management-decisions">3. Making data management decisions</h3>
<p>With the research question in place and the data source identified, we begin the data storytelling journey. But wait, we first require to load the data,</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Import the data from a url
> theUrl<-"http://archive.ics.uci.edu/ml/machine-learning-databases/adult/adult.data"
> adult.data<- read.table(file = theUrl, header = FALSE, sep = ",",
strip.white = TRUE, stringsAsFactors = TRUE,
col.names=c("age","workclass","fnlwgt","education","educationnum","maritalstatus", "occupation","relationship","race","sex","capitalgain","capitalloss", "hoursperweek","nativecountry","income")
)
> dim (adult.data)
> [1] 32561 15
</code></pre></div></div>
<p><strong>A. Exploratory Data Analysis (EDA)</strong></p>
<p>The function, <code class="highlighter-rouge">col.names()</code> adds the user-supplied column names to the dataset. We also see <code class="highlighter-rouge">32,561</code> observations in <code class="highlighter-rouge">15</code> variables. As always, we look at the data structure,</p>
<p>Immediately, a few problems can be spotted. First, there are some categorical variables where the missing levels are coded as <code class="highlighter-rouge">?</code>; Second, there are more than 10 levels for some categorical variables.</p>
<ul>
<li><strong>Data preprocessing (collapse the factor levels & re-coding)</strong></li>
</ul>
<p>We begin by collapsing the factor levels to meaningful and relevant levels. We have also re-coded the missing levels denoted in the original data as <code class="highlighter-rouge">?</code> to <code class="highlighter-rouge">misLevel</code>.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> levels(adult.data$workclass)<- c("misLevel","FedGov","LocGov","NeverWorked","Private","SelfEmpNotInc","SelfEmpInc","StateGov","NoPay")
> levels(adult.data$education)<- list(presch=c("Preschool"), primary=c("1st-4th","5th-6th"),upperprim=c("7th-8th"), highsch=c("9th","Assoc-acdm","Assoc-voc","10th"),secndrysch=c("11th","12th"), graduate=c("Bachelors","Some-college"),master=c("Masters"), phd=c("Doctorate"))
> levels(adult.data$maritalstatus)<- list(divorce=c("Divorced","Separated"),married=c("Married-AF- spouse","Married-civ-spouse","Married-spouse-absent"),notmarried=c("Never-married"),widowed=c("Widowed"))
> levels(adult.data$occupation)<- list(misLevel=c("?"), clerical=c("Adm-clerical"), lowskillabr=c("Craft-repair","Handlers-cleaners","Machine-op-inspct","Other-service","Priv-house- serv","Prof-specialty","Protective-serv"),highskillabr=c("Sales","Tech-support","Transport-moving","Armed-Forces"),agricultr=c("Farming-fishing"))
> levels(adult.data$relationship)<- list(husband=c("Husband"), wife=c("Wife"), outofamily=c("Not-in-family"),unmarried=c("Unmarried"), relative=c("Other-relative"), ownchild=c("Own-child"))
levels(adult.data$nativecountry)<- list(misLevel=c("?","South"),SEAsia=c("Vietnam","Laos","Cambodia","Thailand"),Asia=c("China","India","HongKong","Iran","Philippines","Taiwan"),NorthAmerica=c("Canada","Cuba","Dominican-Republic","Guatemala","Haiti","Honduras","Jamaica","Mexico","Nicaragua","Puerto-Rico","El-Salvador","United-States"), SouthAmerica=c("Ecuador","Peru","Columbia","Trinadad&Tobago"),Europe=c("France","Germany","Greece","Holand-Netherlands","Italy","Hungary","Ireland","Poland","Portugal","Scotland","England","Yugoslavia"),PacificIslands=c("Japan","France"),Oceania=c("Outlying-US(Guam-USVI-etc)"))
</code></pre></div></div>
<p>Now, here is an interesting finding about this dataset. Although, the response (dependent) variable can be considered as binary but there are majority of predictors (independent) that are categorical with many levels.</p>
<p>According to Agresti [1], <em>“Categorical variables have two primary types of scales. Variables having categories without a natural ordering are called nominal. Example, mode of transportation to work (automobile, bicycle, bus, subway, walk). For nominal variables, the order of listing the categories is irrelevant. The statistical analysis does not depend on that ordering. Many categorical variables do have ordered categories. Such variables are called ordinal. Examples are size of automobile (subcompact, compact, midsize, large). Ordinal variables have ordered categories, but distances between categories are unknown. Although a person categorized as moderate is more liberal than a person categorized as conservative, no numerical value describes how much more liberal that person is. An interval variable is one that does have numerical distances between any two values.”</em></p>
<p>“<em>A variable’s measurement scale determines which statistical methods are
appropriate. In the measurement hierarchy, interval variables are highest,
ordinal variables are next, and nominal variables are lowest. Statistical
methods for variables of one type can also be used with variables at higher
levels but not at lower levels. For instance, statistical methods for nominal
variables can be used with ordinal variables by ignoring the ordering of
categories. Methods for ordinal variables cannot, however, be used with
nominal variables, since their categories have no meaningful ordering.”</em></p>
<p>“<em>Nominal variables are qualitative, distinct categories differ in quality, not in quantity. Interval variables are quantitative, distinct levels have differing
amounts of the characteristic of interest.</em>”</p>
<p>Therefore, we can say that all the categorical predictors in this study are nominal in nature. Also note that R will implicitly coerce the categorical variable with levels into numerical values so there is no need to explicitly do the coercion.</p>
<p>we check the data structure again and notice that predictors, <code class="highlighter-rouge">education</code>,<code class="highlighter-rouge">occupation</code> and <code class="highlighter-rouge">native.country</code> have <code class="highlighter-rouge">11077, 4066 and 20</code> missing value respectively. We show this distribution in Fig-1.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>aggr_plot <- aggr(adult.data, col=c('navyblue','red'), numbers=TRUE, sortVars=TRUE,
labels=names(adult.data), cex.axis=.7, gap=3,
ylab=c("Histogram of missing data","Pattern")
)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-UCI-IncomePredict-missplot.png" alt="missplot" /></p>
<p>Fig-1: Missing Data Visualization</p>
<p>Now, some scholars suggest that missing data imputation for categorical variables introduce bias in the data while others oppose it. From, an analytical perspective we will impute the missing data and will use the <code class="highlighter-rouge">missForest</code> library. The reason why we are imputing is because some classification algorithms will fail if they are passed with data containing missing values.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Missing data treatment
> library(missForest)
> imputdata<- missForest(adult.data)
# check imputed values
> imputdata$ximp
# assign imputed values to a data frame
> adult.cmplt<- imputdata$ximp
</code></pre></div></div>
<ul>
<li><strong>Some obvious relationships</strong></li>
</ul>
<p>A majority of the working adults are between 25 to 65 years of age. From Fig-2, we see that adults below 30 years earn <=50k a year while those above 43 years of age earn greater than fifty thousand dollars. This leads to the assumption that experience surely matters to earn more.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> boxplot (age ~ income, data = adult.cmplt,
main = "Age distribution for different income levels",
xlab = "Income Levels", ylab = "Age", col = "salmon")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-UCIncomePredict-boxplot1.png" alt="boxplot1" /></p>
<p>Fig-2: Boxplot for age and income</p>
<p>Evidently, those who invest more time at workplace tend to be earning more as depicted by Fig-3.</p>
<p>It is also interesting to note in Fig-5, that there are roughly 10% of people with doctorate degrees working in low-skilled jobs and earning greater than 50k/year.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> boxplot (hoursperweek ~ income, data = adult.cmplt,
main = "More work hours, more income",
xlab = "Income Levels", ylab = "Hours per week", col = "salmon")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-UCIncomePredict-boxplot3.png" alt="boxplot2" /></p>
<p>Fig-3: Boxplot for hours per week in office and income</p>
<ul>
<li><strong>Some not-so-obvious relationships</strong></li>
</ul>
<p>Question: Does higher skill-set (sales, technical-support, transport movers, armed forces) is a guarantor to high income?</p>
<p>Answer: We explore this question by plotting occupation against income levels. As shown in Fig-4, its evident that acquiring a high skill set does not guarantee increased income. The workers with a low skill set (craft-repair, maintenance services, cleaner, private house security) earn more as compared to those with higher skill set.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> qplot(income, data = adult.cmplt, fill = occupation) + facet_grid (. ~ occupation)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-UCIncomePredict-qplot1.png" alt="qplot1" /></p>
<p>Fig-4: Q-plot for occupation and income</p>
<p>Question: Does higher education help earn more money?</p>
<p>Answer: We explore this question by plotting education against income levels. As shown in Fig-5</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> qplot(income, data = adult.cmplt, fill = education) + facet_grid (. ~ education)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-UCIncomePredict-qplot2.png" alt="qplot2" /></p>
<p>Fig-5: Q-plot for education and income</p>
<p>From Fig-5, we can easily make out that the number of graduates earning >50K are more than the high school or upper-primary school educated. However, we also notice that they are certainly higher in number when compared to master’s or phd degree holders. It makes sense because if for example, in a given academic session, there will be say 90% graduates, 30% masters, <10% phd degree holders. It is also unfortunate to know that there are roughly 10% of people (<em>n=94</em>) with doctorate degrees working in low-skilled jobs and earning less than 50k/year!</p>
<p>We further drill down in this low income group bracket, shown in Fig-5, we realize that majority of them are white male married workers closely followed by the blacks and the Asia-Pacific islanders.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> qplot(income, data = adult.cmplt, fill = relationship) + facet_grid (. ~ race)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-UCIncomePredict-qplot4.png" alt="qplot4" /></p>
<p>Fig-5: Q-plot for race, relationship and income</p>
<ul>
<li><strong>Detecting skewed variables</strong></li>
</ul>
<p>A variable is considered, <code class="highlighter-rouge">highly skewed</code> if its absolute value is greater than 1. A variable is considered, <code class="highlighter-rouge">moderately skewed</code> if its absolute value is greater than 0.5.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> skewedVars<- NA
> library(moments) # for skewness()
> for(i in names(adult.cmplt)){
... if(is.numeric(adult.cmplt[,i])){
... if(i != "income"){
... # Enters this block if variable is non-categorical
... skewVal <- skewness(adult.cmplt[,i])
... print(paste(i, skewVal, sep = ": "))
... if(abs(skewVal) > 0.5){
... skewedVars <- c(skewedVars, i)
... }
... }
... }
... }
[1] "fnlwgt: 1.44691343514233"
[1] "capitalgain: 11.9532969981943"
[1] "capitalloss: 4.59441745643977"
[1] "age: 0.558717629239857"
[1] "educationnum: -0.311661509635468"
[1] "hoursperweek: 0.227632049774777"
</code></pre></div></div>
<p>We find that the predictors, <code class="highlighter-rouge">fnlwgt</code>,<code class="highlighter-rouge">capitalgain</code> and <code class="highlighter-rouge">capitalloss</code> are highly skewed as their absolute value is greater than 0.5.</p>
<ul>
<li><strong>Skewed variable treatment</strong></li>
</ul>
<p>Post identifying the skewed variables, we proceed to treating them by taking the log transformation. But, first we rearrange/reorder the columns for simplicity;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> adult.cmplt<- adult.cmplt[c(3,11:12,1,5,13,2,4,6:10,14:15)]
> str(adult.cmplt)
'data.frame': 32561 obs. of 15 variables:
$ fnlwgt : num 77516 83311 215646 234721 338409 ...
$ capitalgain : num 2174 0 0 0 0 ...
$ capitalloss : num 0 0 0 0 0 0 0 0 0 0 ...
$ age : num 39 50 38 53 28 37 49 52 31 42 ...
$ educationnum : num 13 13 9 7 13 14 5 9 14 13 ...
$ hoursperweek : num 40 13 40 40 40 40 16 45 50 40 ...
$ workclass : Factor w/ 9 levels "misLevel","FedGov",..: 8 7 5 5 5 5 5 7 5 5 ...
$ education : Factor w/ 8 levels "presch","primary",..: 6 6 5 5 6 7 4 6 7 6 ...
$ maritalstatus: Factor w/ 4 levels "divorce","married",..: 3 2 1 2 2 2 2 2 3 2 ...
$ occupation : Factor w/ 5 levels "misLevel","clerical",..: 2 5 3 3 3 3 3 4 3 4 ...
$ relationship : Factor w/ 6 levels "husband","wife",..: 3 1 3 1 2 2 3 1 3 1 ...
$ race : Factor w/ 5 levels "Amer-Indian-Eskimo",..: 5 5 5 3 3 5 3 5 5 5 ...
$ sex : Factor w/ 2 levels "Female","Male": 2 2 2 2 1 1 1 2 1 2 ...
$ nativecountry: Factor w/ 8 levels "misLevel","SEAsia",..: 4 4 4 4 4 4 4 4 4 4 ...
$ income : Factor w/ 2 levels "<=50K",">50K": 1 1 1 1 1 1 1 2 2 2 ...
</code></pre></div></div>
<p>We took a log transformation. Post skewed treatment, we notice that <code class="highlighter-rouge">capitalgain</code> & <code class="highlighter-rouge">capitalloss</code> have infinite values so we removed them from subsequent analysis.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> adult.cmplt.norm<- adult.cmplt
> adult.cmplt.norm[,1:3]<- log(adult.cmplt[1:3],2) # where 2 is log base 2
> adult.cmplt.norm$capitalgain<- NULL
> adult.cmplt.norm$capitalloss<-NULL
</code></pre></div></div>
<ul>
<li><strong>Correlation detection</strong></li>
</ul>
<p>We now checked for variables with high correlations to each other. Correlation measures the relationship between two variables. When two variables are so highly correlated that they explain each other (to the point that one can predict the variable with the other), then we have <em>collinearity</em> (or <em>multicollinearity</em>) problem. Therefore, its is important to treat collinearity problem. Let us now check, if our data has this problem or not.</p>
<p>Again, it is important to note that correlation works only for continuous variables. We can calculate the correlations by using the <code class="highlighter-rouge">cor()</code> as shown;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> correlat<- cor(adult.cmplt.norm[c(1:4)])
> corrplot(correlat, method = "pie")
> highlyCor <- colnames(adult.cmplt.num)[findCorrelation(correlat, cutoff = 0.7, verbose = TRUE)]
All correlations <= 0.7
> highlyCor # No high Correlations found
character(0)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-UCIncomePredict-corplot.png" alt="corplot" /></p>
<p>Fig-7: Correlation detection</p>
<p>From Fig-7, its evident that none of the predictors are highly correlated to each other. We now proceed to building the prediction model.</p>
<p>###4. Predictive data analytics</p>
<p>In this section, we will discuss various approaches applied to model building, predictive power and their trade-offs.</p>
<p><strong>A. Creating the train and test dataset</strong></p>
<p>We now divide the data into 75% training set and 25% testing set. We also created a root mean square evaluation function for model testing.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ratio = sample(1:nrow(adult.cmplt), size = 0.25*nrow(adult.cmplt))
> test.data = adult.cmplt[ratio,] #Test dataset 25% of total
> train.data = adult.cmplt[-ratio,] #Train dataset 75% of total
> dim(train.data)
[1] 24421 15
> dim(test.data)
[1] 8140 15
</code></pre></div></div>
<p>**B. Fit a Logistic Regression Model **</p>
<p>We fit a logistic regression model.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> glm.fit<- glm(income~., family=binomial(link='logit'),data = train.data)
Warning message:
glm.fit: fitted probabilities numerically 0 or 1 occurred
</code></pre></div></div>
<p>This Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred means that the data is possibly linearly separable. Let’s look at the summary for the model.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> summary(glm.fit)
Call:
glm(formula = income ~ ., family = binomial(link = "logit"),
data = train.data)
Deviance Residuals:
Min 1Q Median 3Q Max
-5.2316 -0.4639 -0.1713 -0.0311 3.4484
Coefficients: (1 not defined because of singularities)
Estimate Std. Error z value Pr(>|z|)
(Intercept) -27.1523662282 171.0342617549 -0.159 0.873863
age 0.0274845471 0.0019376261 14.185 < 2e-16 ***
workclassFedGov 0.4073987950 0.2085465221 1.954 0.050759 .
workclassLocGov -0.3070912295 0.1944447001 -1.579 0.114262
workclassNeverWorked -10.5345275621 510.1141392772 -0.021 0.983524
workclassPrivate -0.1374981405 0.1816628614 -0.757 0.449118
workclassSelfEmpNotInc -0.1132407363 0.1995129601 -0.568 0.570316
workclassSelfEmpInc -0.6270437314 0.1773500692 -3.536 0.000407 ***
workclassStateGov -0.4387629630 0.2049449847 -2.141 0.032284 *
workclassNoPay -13.9146466535 367.0432320049 -0.038 0.969759
fnlwgt 0.0000004226 0.0000002029 2.083 0.037252 *
educationprimary 18.6369757615 171.0337793366 0.109 0.913229
educationupperprim 18.6015984474 171.0337199499 0.109 0.913393
educationhighsch 19.4272321191 171.0336543439 0.114 0.909565
educationsecndrysch 18.3381423049 171.0336420490 0.107 0.914615
educationgraduate 20.1855955674 171.0336647250 0.118 0.906051
educationmaster 20.6169432260 171.0337212540 0.121 0.904053
educationphd 20.8122445845 171.0338205358 0.122 0.903149
educationnum 0.1301601416 0.0134594027 9.671 < 2e-16 ***
maritalstatusmarried 0.6518153342 0.1922871329 3.390 0.000699 ***
[ reached getOption("max.print") -- omitted 26 rows ]
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 26907 on 24420 degrees of freedom
Residual deviance: 14892 on 24376 degrees of freedom
AIC: 14982
Number of Fisher Scoring iterations: 14
</code></pre></div></div>
<p>Its evident that the significant predictors are <code class="highlighter-rouge">age</code>, <code class="highlighter-rouge">workclassSelfEmpInc</code>,<code class="highlighter-rouge">fnlwgt</code>,<code class="highlighter-rouge">educationnum</code> and <code class="highlighter-rouge">maritalstatusmarried</code>. As for the statistical significant variables, <code class="highlighter-rouge">age</code> and <code class="highlighter-rouge">educationnum</code> has the <code class="highlighter-rouge">lowest p value suggesting a strong association with the response, income</code>.
The <code class="highlighter-rouge">null deviance</code> shows how well the response is predicted by the model with nothing but an intercept. Deviance is a measure of goodness of fit of a generalized linear model. it’s a measure of badness of fit–higher numbers indicate worse fit. The residual deviance shows how well the response is predicted by the model when the predictors are included. From your example, it can be seen that the residual deviance decreases by <code class="highlighter-rouge">12115 (27001-14886)</code> when <code class="highlighter-rouge">15 predictors</code> were added to it.(note: degrees of freedom = no. of observations – no. of predictors). This decrease in deviance is evidence of significant fit. If the deviance would have increased it would indicate a significant lack of fit. The <code class="highlighter-rouge">AIC</code> is <code class="highlighter-rouge">14976</code>. The Akaike Information Criterion (AIC) provides a method for assessing the quality of your model through comparison of related models. It’s based on the Deviance, but penalizes you for making the model more complicated. Much like adjusted R-squared, it’s intent is to prevent you from including irrelevant predictors. However, unlike adjusted R-squared, the number itself is not meaningful. If you have more than one similar candidate models (where all of the variables of the simpler model occur in the more complex models), then you should select the model that has the smallest AIC. So AIC is useful for comparing models, but isn’t interpretable on its own.</p>
<p>We now create another logistic model that includes only the significant predictors.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> glm.fit1<- glm(income ~ age + workclass + educationnum + fnlwgt + maritalstatus, family=binomial(link='logit'),data = train.data)
</code></pre></div></div>
<p>Now we can run the <code class="highlighter-rouge">anova()</code> function on the improved model to analyze the table of deviance.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> anova(glm.fit, glm.fit1, test="Chisq")
Analysis of Deviance Table
Model 1: income ~ age + workclass + fnlwgt + education + educationnum +
maritalstatus + occupation + relationship + race + sex +
capitalgain + capitalloss + hoursperweek + nativecountry
Model 2: income ~ age + workclass + educationnum + fnlwgt + maritalstatus
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1 24376 14892
2 24406 18428 -30 -3536.1 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
</code></pre></div></div>
<p>By conducting the anova test, it performs the Chi-square test to compare <code class="highlighter-rouge">glm.fit</code> and <code class="highlighter-rouge">glm.fit1</code> (i.e. it tests whether reduction in the residual sum of squares are statistically significant or not). The test shows that, <code class="highlighter-rouge">Model 2 is statistically significant as the p value is less than 0.05</code>. Therefore, the predictors, <code class="highlighter-rouge">(age + workclass + educationnum + fnlwgt + maritalstatus)</code> are <code class="highlighter-rouge">relevant for the model</code>. See this links for details, <a href="http://stats.stackexchange.com/questions/172782/how-to-use-r-anova-results-to-select-best-model">1</a>, <a href="http://stats.stackexchange.com/questions/20523/difference-between-logit-and-probit-models/30909#30909">2</a> and <a href="http://stats.idre.ucla.edu/other/mult-pkg/faq/general/faq-how-do-i-interpret-odds-ratios-in-logistic-regression/">3</a>.</p>
<p>We now test the logistic model on all predictors and make predictions on unseen data.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> set.seed(1234)
> glm.pred<- predict(glm.fit, test.data, type = "response")
> hist(glm.pred, breaks=20)
> hist(glm.pred[test.data$income], col="red", breaks=20, add=TRUE)
> table(actual= test.data$income, predicted= glm.pred>0.5)
predicted
actual FALSE TRUE
<=50K 5674 482
>50K 678 1306
> (5674+1306)/8140
[1] 0.8574939
</code></pre></div></div>
<p>The classifier returns 86% accuracy when the model includes all predictors in it. Let us see, if the model accuracy increases with the inclusion of significant predictors only;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> set.seed(1234)
> glm.fit1<- glm(income ~ age + workclass + educationnum + fnlwgt + maritalstatus, family=binomial(link='logit'),data = train.data)
> glm.pred1<- predict(glm.fit, test.data, type = "response")
> table(actual= test.data$income, predicted= glm.pred1>0.5)
predicted
actual FALSE TRUE
<=50K 5683 473
>50K 997 987
> (5683+987)/8140
[1] 0.8194103
</code></pre></div></div>
<p>With the inclusion of significant predictors in the model, the classifier accuracy decreases by 4 percent to 82%.</p>
<p>Logistic Regression Inference: The model gives higher accuracy on unseen data when it has all the predictors included. The model’s accuracy decreases when some of the predictors are removed.</p>
<p><strong>C. Fit a Decision Tree Model</strong></p>
<p>We try the decision tree model.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> tree.model<- rpart(income~., data=train.data, method="class", minbucket=20)
> tree.predict<- predict(tree.model, test.data, type = "class")
> confusionMatrix(test.data$income, tree.predict) # 86% accuracy
Confusion Matrix and Statistics
Reference
Prediction <=50K >50K
<=50K 5832 324
>50K 760 1224
Accuracy : 0.8668
95% CI : (0.8593, 0.8741)
No Information Rate : 0.8098
P-Value [Acc > NIR] : < 2.2e-16
Kappa : 0.6097
Mcnemar's Test P-Value : < 2.2e-16
Sensitivity : 0.8847
Specificity : 0.7907
Pos Pred Value : 0.9474
Neg Pred Value : 0.6169
Prevalence : 0.8098
Detection Rate : 0.7165
Detection Prevalence : 0.7563
Balanced Accuracy : 0.8377
'Positive' Class : <=50K
</code></pre></div></div>
<p>The accuracy is 87% for the model with all the predictors in it and the accuracy decreases to 82 percent for a model with significant predictors only. Also, a decision tree model is no better than the logistic regression model in terms of accuracy.</p>
<p><strong>D. Fit a Support Vector Machine (SVM) classification model</strong></p>
<p>We tried the SVM model;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> svm.model<- svm(income~., data = train.data,kernel = "radial", cost = 1, gamma = 0.1)
> svm.predict <- predict(svm.model, test.data)
> confusionMatrix(test.data$income, svm.predict) # 87% accuracy
Confusion Matrix and Statistics
Reference
Prediction <=50K >50K
<=50K 5695 461
>50K 582 1402
Accuracy : 0.8719
95% CI : (0.8644, 0.8791)
No Information Rate : 0.7711
P-Value [Acc > NIR] : < 2.2e-16
Kappa : 0.6451
Mcnemar's Test P-Value : 0.0002027
Sensitivity : 0.9073
Specificity : 0.7525
Pos Pred Value : 0.9251
Neg Pred Value : 0.7067
Prevalence : 0.7711
Detection Rate : 0.6996
Detection Prevalence : 0.7563
Balanced Accuracy : 0.8299
'Positive' Class : <=50K
</code></pre></div></div>
<p>The classification accuracy of the SVM model having all predictors, increases by 1 percent to 87%, when compared to the decision tree and the logistic regression model. Again, its interesting to note that the SVM model accuracy decreases to 4 percent when only the significant predictors are included in the model.</p>
<p><strong>E. Fit a Random Forest (RF) classification model</strong></p>
<p>We finally try the RF model.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> rf.model<- randomForest(income~.,
... data = train.data,
... importance=TRUE,
... keep.forest=TRUE)
> rf.predict <- predict(rf.model, test.data)
> confusionMatrix(test.data$income, rf.predict) # 88%
Confusion Matrix and Statistics
Reference
Prediction <=50K >50K
<=50K 5809 347
>50K 567 1417
Accuracy : 0.8877
95% CI : (0.8807, 0.8945)
No Information Rate : 0.7833
P-Value [Acc > NIR] : < 2.2e-16
Kappa : 0.6835
Mcnemar's Test P-Value : 0.000000000000436
Sensitivity : 0.9111
Specificity : 0.8033
Pos Pred Value : 0.9436
Neg Pred Value : 0.7142
Prevalence : 0.7833
Detection Rate : 0.7136
Detection Prevalence : 0.7563
Balanced Accuracy : 0.8572
'Positive' Class : <=50K
</code></pre></div></div>
<p>So, it is the <strong>Random Forest model</strong> that gives the <strong>highest prediction accuracy of 88%</strong>.</p>
<p><strong>5. Conclusion</strong></p>
<p>In this study, we aimed to predict a person’s income based on variables like habitat, education, marital status, age, race, sex and others. We found in exploring this particular dataset that, <em>higher education is no guarantee to high income</em>. This pattern could be attributed the uneven sample distribution. Several classification models were tested for prediction accuracy and we determined that the Random Forest model gives the highest accuracy among others.</p>
<p>As a future work, we will extend this study to include feature engineering methods, to measure if the predictive power of the models could be increased or not.</p>
<p>The complete code is listed on my Github <a href="https://github.com/duttashi/LearningR/blob/master/scripts/Full%20Case%20Studies/CaseStudy-UCI-IncomePredict.R">repository</a></p>
<![CDATA[Predicting employment related factors in Malaysia- A regression analysis approach]]>https://duttashi.github.io/blog/predicting-employment-factors-in-malaysia-case-study2017-02-20T00:00:00+00:002017-02-20T00:00:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<h3 id="introduction">Introduction</h3>
<p>A recent news article published in the national daily, <a href="http://www.thestar.com.my/business/business-news/2017/02/14/jobless-rate-up-slightly/">The Star</a>, reported, “<em>The country’s unemployment rate has inched up by 0.1 percentage points to 3.5% in December 2016 compared to the previous month, according to the <a href="(https://www.dosm.gov.my/v1/index.php?r=column/cthemeByCat&cat=124&bul_id=VWl4c2VyZ0Q3MEUxU0NzOVBPMnlDUT09&menu_id=U3VPMldoYUxzVzFaYmNkWXZteGduZz09)">Statistics Department</a>. On a year-on-year comparison, the unemployment rate was also up 0.1 percentage point from December 2015. It said that in December 2016, 14,276,700 people were employed out of the country’s total labour force of 14,788,900, while 512,000 were unemployed.</em>” The news daily also reported that, “<em>Human Resources Minister Datuk Seri Richard Riot said the country’s unemployment rate was still “manageable” and unlikely to exceed 3.5% this year despite the global economic slowdown.</em>”</p>
<p>In this analytical study, we have made an attempt to verify this claim by regressing the employed work force in Malaysia on predictors like Outside Labor Force, Unemployment percentage, Labour Force and others.</p>
<p>This study is organized as follows;</p>
<ol>
<li>
<p>Business/Research Question</p>
</li>
<li>
<p>Data Source</p>
</li>
<li>
<p>Making data management decisions</p>
</li>
</ol>
A. Exploratory Data Analysis (EDA)
<ul>
<li>Data preprocessing (rename and replace)</li>
<li>Data preprocessing (joining the tables)</li>
<li>Data preprocessing (missing data visualization & imputation)</li>
</ul>
B. Basic Statistics
<ul>
<li>One-way table</li>
<li>Two-way table</li>
<li>Test of independence for categorical variables</li>
<li>Visualizing significant variables found in the test of independence</li>
</ul>
C. Outlier Detection & Treatment
<ul>
<li>Boxplots for outlier detection</li>
<li>Outlier Treatment</li>
<li>Data type conversion</li>
</ul>
D. Correlation Detection & Treatment
<ul>
<li>Detecting skewed variables</li>
<li>Skewed variable treatment</li>
<li>Correlation detection</li>
<li>Multicollinearity</li>
<li>Multicollinearity treatment
* Principal Component Analysis (PCA)
* Plotting the PCA (biplot) components
* Determining the contribution (%) of each parameter</li>
</ul>
<ol>
<li>
<p>Predictive Data Analytics</p>
<p>A. Creating the train and test dataset</p>
<p>B. Model Building - Evaluation Method</p>
<p>C. Model Building - Regression Analysis</p>
<p>D. Model Building - other supervised algorithms</p>
<ul>
<li>Regression Tree method</li>
<li>Random Forest method</li>
</ul>
<p>E. Model Performance comparison</p>
</li>
<li>
<p>Conclusion</p>
</li>
</ol>
<h3 id="1-businessresearch-question">1. Business/Research Question</h3>
<p>Determine the factors which contribute to accurately predicting unemployment rate from historical statistical data on labour force data in Malaysia.</p>
<h3 id="2-data-source">2. Data Source</h3>
<p>The data comes from the Department of Statistics, Malaysia. This is an open data source portal and the data files can be accessed from their official <a href="http://www.dosm.gov.my/v1/index.php?r=column3/accordion&menu_id=aHhRYUpWS3B4VXlYaVBOeUF0WFpWUT09">website</a>. Click the + sign next to “Labour Force & Social Statistics” to expand the drop down list to access the data files.</p>
<h3 id="3-making-data-management-decisions">3. Making data management decisions</h3>
<p>Initially, the dataset consisted of five comma-separated files. Each file provided data (from year 1965 to year 2014) on factors like number of rubber estates in Malaysia, total planted area, production of natural rubber, tapped area, yield per hectare and total number of paid employees in the rubber estate.</p>
<p><strong>A. Exploratory Data Analysis (EDA)</strong></p>
<p>This phase constitutes 80% of a data analytical work. We noticed that each data file consisted of 544 rows in 3 variables where the variable, <code class="highlighter-rouge">Year</code> was common for all data tables. This confirmed our assumption that the actual dataset was divided into six separate files. We first imported the data files into the R environment as given;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> df1<- read.csv("data/bptms-Employed_by_State.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
> df2<- read.csv("data/bptms-Labour_force_by_State.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
> df3<- read.csv("data/bptms-Labour_Force_Participation_rate_by_State.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
> df4<- read.csv("data/bptms-Outside_labour_force_by_State.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
> df5<- read.csv("data/bptms-Unemployment_Rate.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
> dim(df1)
[1] 544 3
> dim(df2)
[1] 544 3
> dim(df3)
[1] 544 3
> dim(df4)
[1] 544 3
> dim(df5)
[1] 544 3
</code></pre></div></div>
<p>Now that the data was imported in, we began with the initial process of data exploration. The first step was to look at the data structure for which we used the <code class="highlighter-rouge">str()</code> as given;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> str(df1)
'data.frame': 544 obs. of 3 variables:
$ Year : int 1982 1983 1984 1985 1986 1987 1988 1989 1990 1992 ...
$ State.Country : chr "Malaysia" "Malaysia" "Malaysia" "Malaysia" ...
$ Employed...000.: chr "5,249.00" "5,457.00" "5,566.70" "5,653.40" ...
</code></pre></div></div>
<p>and found that variable like, <code class="highlighter-rouge">Employed</code> was treated as a character data type by <code class="highlighter-rouge">R</code> because it’s values contained a comma in them. Thus, coercing the number to a character data type. We also need to rename the variables to short, succinct names. The variable naming convention will follow <code class="highlighter-rouge">CamelCase</code> style.</p>
<ul>
<li><strong>Data preprocessing (rename and replace)</strong></li>
</ul>
<p>We begin by renaming the variable names. We will use the <code class="highlighter-rouge">rename()</code> of the <code class="highlighter-rouge">plyr</code> package. This library needs to be loaded in the R environment first. We use the <code class="highlighter-rouge">gsub()</code> to replace the <code class="highlighter-rouge">comma</code> between the numbers in the <code class="highlighter-rouge">Employed</code> variable, followed by changing the data type to numeric. We show the data management steps as follows;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(plyr) # for the rename ()
> df1<- rename(df1, c("State.Country" = "State"))
> df1<- rename(df1, c("Employed...000." = "Employed"))
> df2<- rename(df2, c("State.Country" = "State"))
> df2<- rename(df2, c("Labour.Force...000." = "LabrFrc"))
> df3<- rename(df3, c("State.Country" = "State"))
> df3<- rename(df3, c("Labour.Force.Participation.Rate..Percentage." = "LabrFrcPerct"))
> df4<- rename(df4, c("State.Country" = "State"))
> df4<- rename(df4, c("Outside.Labour.Force...000." = "OutLabrFrc"))
> df5<- rename(df5, c("State.Country" = "State"))
> df5<- rename(df5, c("Unemployment.Rate..Percentage." = "UnempRatePerct"))
> ## Change data type
> df1$State<- as.factor(df1$State)
> df1$Employed<- as.numeric(gsub(",","", df1$Employed))
> df2$State<- as.factor(df2$State)
> df2$LabrFrc<- as.numeric(gsub(",","", df2$LabrFrc))
> df3$State<- as.factor(df3$State)
> df4$State<- as.factor(df4$State)
> df4$OutLabrFrc<- as.numeric(gsub(",","", df4$OutLabrFrc))
> df5$State<- as.factor(df5$State)
</code></pre></div></div>
<ul>
<li><strong>Data preprocessing (joining the tables)</strong></li>
</ul>
<p>Next, we apply the <code class="highlighter-rouge">inner_join()</code> of the <code class="highlighter-rouge">dplyr</code> package to join the five data frames to a single master data frame called, <code class="highlighter-rouge">df.master</code>. To check the time it takes for data table joins, we wrap the inner join function in <code class="highlighter-rouge">system.time()</code> method; Since, this is a small dataset so there are not much overheads involved in an operation like inner join but for large data tables, <code class="highlighter-rouge">system.time()</code> is a handy function.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(dplyr)
> system.time(join1<- inner_join(df1,df2))
Joining, by = c("Year", "State")
user system elapsed
0.00 0.00 0.47
> system.time(join2<- inner_join(df3,df4))
Joining, by = c("Year", "State")
user system elapsed
0 0 0
> system.time(join3<- inner_join(join1,join2))
Joining, by = c("Year", "State")
user system elapsed
0 0 0
> system.time(df.master<- inner_join(join3,df5))
Joining, by = c("Year", "State")
user system elapsed
0 0 0
</code></pre></div></div>
<p>Let us look at the structure of the data frame, <code class="highlighter-rouge">df.master</code></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> str(df.master)
'data.frame': 544 obs. of 7 variables:
$ Year : int 1982 1983 1984 1985 1986 1987 1988 1989 1990 1992 ...
$ State : Factor w/ 17 levels "Johor","Kedah",..: 4 4 4 4 4 4 4 4 4 4 ...
$ Employed : num 5249 5457 5567 5653 5760 ...
$ LabrFrc : num 5431 5672 5862 5990 6222 ...
$ LabrFrcPerct : num 64.8 65.6 65.3 65.7 66.1 66.5 66.8 66.2 66.5 65.9 ...
$ OutLabrFrc : num 2945 2969 3120 3125 3188 ...
$ UnempRatePerct: num 3.4 3.8 5 5.6 7.4 7.3 7.2 5.7 4.5 3.7 ...
</code></pre></div></div>
<ul>
<li><strong>Data preprocessing (missing data visualization & imputation)</strong></li>
</ul>
<p>Let us visualize the data now. The objective is to check for missing data patterns. For this, we will use the <code class="highlighter-rouge">aggr_plot()</code> function of the <code class="highlighter-rouge">VIM</code> package.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(VIM)
> aggr_plot <- aggr(df.master, col=c('navyblue','red'), numbers=TRUE, sortVars=TRUE, labels=names(df.master), cex.axis=.7, gap=3, ylab=c("Histogram of missing data","Pattern"))
</code></pre></div></div>
Variables sorted by number of missings:
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code> Variable Count
Employed 0.05330882
LabrFrc 0.05330882
LabrFrcPerct 0.05330882
OutLabrFrc 0.05330882
UnempRatePerct 0.05330882
Year 0.00000000
State 0.00000000
Warning message:
In plot.aggr(res, ...) : not enough horizontal space to display frequencies
</code></pre></div></div>
<p>Note: The warning message is generated because the plot size is not big enough. I’m using RStudio, where the plot size is small. You can safely ignore this message.</p>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-missplot.png" alt="missplot" /></p>
<p>Fig-1: Missing Data Visualization</p>
<p>In Fig-1, the missing data is shown in <code class="highlighter-rouge">red</code> color. Here we see that variables like <code class="highlighter-rouge">Employed</code>, <code class="highlighter-rouge">LabrFrc</code>, <code class="highlighter-rouge">LabrFrcPerct</code> and <code class="highlighter-rouge">OutLabrFrc</code> have missing data. To verify, how many instances of missing values are there, use, <code class="highlighter-rouge">colSums()</code> like</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> colSums(is.na(df.master))
Year State Employed LabrFrc LabrFrcPerct OutLabrFrc UnempRatePerct
0 0 29 29 29 29 29 There are 29 instances of missing data. In an earlier case study, we had used the `Boruta` package for missing data imputation. We tried it on this case study and it failed to impute all missing values, quite a strange phenomenon. Anyway, for this case study we have used the `missForest` method from the `missForest` package. You will have to install/load it in the `R` environment first if you do not have it. We save the imputed data in a new data frame called, `df.cmplt`.
> ## MISSING DATA IMPUTATION
> library(missForest)
> imputdata<- missForest(df.master)
missForest iteration 1 in progress...done!
missForest iteration 2 in progress...done!
# check imputed values
> imputdata$ximp
Year State Employed LabrFrc LabrFrcPerct OutLabrFrc UnempRatePerct
1 1982 Malaysia 5249.000 5431.400 64.800 2944.6000 3.400
2 1983 Malaysia 5457.000 5671.800 65.600 2969.4000 3.800
3 1984 Malaysia 5566.700 5862.500 65.300 3119.6000 5.000
4 1985 Malaysia 5653.400 5990.100 65.700 3124.9000 5.600
5 1986 Malaysia 5760.100 6222.100 66.100 3188.3000 7.400
[ reached getOption("max.print") -- omitted 530 rows ]
# assign imputed values to a data frame
> df.cmplt<- imputdata$ximp
# check for missing values in the new data frame
> colSums(is.na(df.cmplt))
Year State Employed LabrFrc LabrFrcPerct OutLabrFrc UnempRatePerct
0 0 0 0 0 0 0
</code></pre></div></div>
<p><strong>B. Basic Statistics</strong></p>
<p>We now provide few basic statistics on the data like frequency tables (one way table, two way table, proportion table and percentage table).</p>
<ul>
<li><strong>One-way table</strong></li>
</ul>
<p>Simple frequency counts can be generated using the <code class="highlighter-rouge">table()</code> function.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> mytable<- with(data=df.cmplt, table(State))
> mytable
State
Johor Kedah Kelantan Malaysia Melaka Negeri Sembilan
32 32 32 32 32 32
Pahang Perak Perlis Pulau Pinang Sabah Sarawak
32 32 32 32 32 32
Selangor Terengganu W.P Labuan W.P. Kuala Lumpur W.P.Putrajaya
32 32 32 32 32 * **Two-way table**
</code></pre></div></div>
<p>For two-way table, the format for the <code class="highlighter-rouge">table()</code> is <code class="highlighter-rouge">mytable<- table(A,B)</code> where <code class="highlighter-rouge">A</code> is the row variable and <code class="highlighter-rouge">B</code> is the column variable. Alternatively, the <code class="highlighter-rouge">xtabs()</code> function allows to create a contingency table using the formula style input. The format is <code class="highlighter-rouge">mytable<- xtabs(~ A + B, data=mydata)</code> where, <code class="highlighter-rouge">mydata</code> is a matrix of data frame. In general, the variables to be cross classified appear on the right side of the formula (i.e. to the right side of the ~) separated by + sign.
Use <code class="highlighter-rouge">prop.table(mytable)</code> to express table entries as fractions.</p>
<ul>
<li><strong>Test of independence for categorical variables</strong></li>
</ul>
<p>R provides several methods for testing the independence of the categorical variables like <em>chi-square test of independence</em>, <em>Fisher exact test</em>, <em>Cochran-Mantel-Haenszel test</em>.</p>
<p>For this report, we applied the <code class="highlighter-rouge">chisq.test()</code> to a two-way table to produce the <em>chi square test of independence</em> of the row and column variable as shown next;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(vcd) # for xtabs() and assocstats()
Loading required package: grid
> mytable<- xtabs(~State+Employed, data= df.cmplt)
> chisq.test(mytable)
Pearson's Chi-squared test
data: mytable
X-squared = 8534, df = 8368, p-value = 0.1003
Warning message:
In chisq.test(mytable) : Chi-squared approximation may be incorrect
</code></pre></div></div>
<p>Here, the p value is greater than 0.05, indicating no relationship between state & employed variable. Let’s look at another example as given below;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> mytable<- xtabs(~State+UnempRatePerct, data= df.cmplt)
> chisq.test(mytable)
Pearson's Chi-squared test
data: mytable
X-squared = 2104.2, df = 1776, p-value = 0.00000009352
Warning message:
In chisq.test(mytable) : Chi-squared approximation may be incorrect
</code></pre></div></div>
<p>Here, the p value is less than 0.05, indicating a relationship between state & Unemployed rate percent variable.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> mytable<- xtabs(~State+LabrFrcPerct, data= df.cmplt)
> chisq.test(mytable)
Pearson's Chi-squared test
data: mytable
X-squared = 3309.2, df = 2928, p-value = 0.0000008368
</code></pre></div></div>
<p>Again, the p value is less than 0.05, indicating a relationship between state & labour force in percentage variable</p>
<p>Therefore, to summarise, the significance test conducted using chi-square test of independence evaluates whether or not sufficient evidence exists to reject a null hypothesis of independence between the variables. We could not reject the null hypothesis for State vs Employed, Labour Force and Outside Labour Force variables confirming that there exists no relationship between these variables.</p>
<p>However, we were unable to reject the null hypothesis for state vs UnempRatePerct and LabrFrcPerct. This proves that there exist a relationship between these variables.</p>
<p>Unfortunately we cannot test the association between the two categorical variables <code class="highlighter-rouge">State</code> and <code class="highlighter-rouge">Year</code>, because the measures of association like Phi and Cramer’s V require the categorical variables to have at least two levels example <code class="highlighter-rouge">"Sex"</code> got two levels, <code class="highlighter-rouge">"Male"</code>, <code class="highlighter-rouge">"Female"</code>. Use the <code class="highlighter-rouge">assocstats()</code> from the <code class="highlighter-rouge">vcd</code> package to test association.</p>
<p>Now, that we have determined the variables that have relationships with each other, we continue to the next step of visualizing their distribution in the data. We have used density plots for continuous variable distribution.</p>
<ul>
<li><strong>Visualizing significant variables found in the test of independence</strong></li>
</ul>
<p>We have used the <code class="highlighter-rouge">ggplot2</code> library for data visualization. The plots are shown in Fig-2 and Fig-3 respectively.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ggplot(df.cmplt)+
... geom_density(aes(x=LabrFrcPerct, fill="red"))
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-chisq1.png" alt="chisqplot1" /></p>
<p>Fig-2: Density plot for variable, <code class="highlighter-rouge">LabrFrcPerct</code></p>
<p>In Fig-2, we see that a majority of the labor force in Malaysia lies between the 60-70 percentage bracket.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ggplot(df.cmplt)+
... geom_density(aes(x=UnempRatePerct, fill="red"))
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-chisq2.png" alt="chisqplot2" /></p>
<p>Fig-3: Density plot for variable, <code class="highlighter-rouge">UnempRatePerct</code></p>
<p>From Fig-3, its evident that a majority of unemployment rate peaks between 2.5 to 5.0 interval.</p>
<p>We now, derive a subset of the data based on the significant variation revealed in Fig-2 and Fig-3 respectively for further data analysis.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> subst.data.2<- subset(df.cmplt,
... (LabrFrcPerct>=60 & LabrFrcPerct <=70) &
... (UnempRatePerct>=2.5 & UnempRatePerct<=5.0)
... )
</code></pre></div></div>
<p>This reduces the dataset size to <code class="highlighter-rouge">269 observations</code> as given in <code class="highlighter-rouge">> dim(subst.data.2)
[1] 269 7</code></p>
<p><strong>C. Outlier Detection & Treatment</strong></p>
<p>Outlier treatment is a vital part of descriptive analytics since outliers can lead to misleading conclusions regarding our data. For continuous variables, the values that lie outside the 1.5 * IQR limits. For categorical variables, outliers are considered to be the values of which frequency is less than 10% outliers gets the extreme most observation from the mean. If you set the argument opposite=TRUE, it fetches from the other side.</p>
<ul>
<li><strong>Boxplots for outlier detection</strong></li>
</ul>
<p>When reviewing a boxplot, an outlier is defined as a data point that is located outside the fences (“whiskers”) of the boxplot (e.g: outside 1.5 times the interquartile range above the upper quartile and bellow the lower quartile).</p>
<p>Remember, ggplot2 requires both an x and y variable of a boxplot. Here is how to make a single boxplot as shown by leaving the <code class="highlighter-rouge">x</code> aesthetic <code class="highlighter-rouge">blank</code>;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>>p1<-ggplot(data= df.cmplt, aes(x="", y=Employed))+
geom_boxplot(outlier.size=2,outlier.colour="red")
>p2<-ggplot(data= df.cmplt, aes(x="", y=LabrFrc))+
geom_boxplot(outlier.size=2,outlier.colour="red")
>p3<-ggplot(data= df.cmplt, aes(x="", y=OutLabrFrc))+
geom_boxplot(outlier.size=2,outlier.colour="red")
> p1+ ggtitle("Employed in Malaysia (1982-2014)")+
xlab("")+ylab("Employed")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-bxp1.png" alt="boxplot1" /></p>
<p>Fig-4: Boxplot for outliers detected in variable <code class="highlighter-rouge">Employed</code></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> p2+ ggtitle("Labour Force in Malaysia (1982-2014)")+
xlab("")+ylab("Labour Force")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-bxp2.png" alt="boxplot2" /></p>
<p>Fig-5: Boxplot for outliers detected in variable <code class="highlighter-rouge">LabrFrc</code></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> p3+ ggtitle("Outside Labour Force in Malaysia (1982-2014)")+
xlab("")+ylab("Outside Labour Force")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-bxp3.png" alt="boxplot3" /></p>
<p>Fig-6: Boxplot for outliers detected in variable <code class="highlighter-rouge">OutLabrFrc</code></p>
<ul>
<li><strong>Outlier Treatment</strong></li>
</ul>
<p>One of the method is to derive a subset to remove the outliers. After, several trials of plotting boxplots, we found that variable <code class="highlighter-rouge">LabrFrc</code> when less than or equal to <code class="highlighter-rouge">1600</code> generates no outliers. So, we subset the data frame and call it as, <code class="highlighter-rouge">subst.data.3</code>.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> subst.data.3<- subset(df.cmplt,
(LabrFrc<=1200 & LabrFrcPerct>=60 & LabrFrcPerct <=70) &
(UnempRatePerct>=2.5 & UnempRatePerct<=5.0)
> dim(subst.data.3)
[1] 221 7
</code></pre></div></div>
<p>We then plot this new data frame devoid of outliers as shown in Fig-7,8,9.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> p1<-ggplot(data= subst.data.3, aes(x="", y=Employed))+
geom_boxplot(outlier.size=2,outlier.colour="red")
> p2<-ggplot(data= subst.data.3, aes(x="", y=LabrFrc))+
geom_boxplot(outlier.size=2,outlier.colour="red")
> p3<-ggplot(data= subst.data.3, aes(x="", y=OutLabrFrc))+
geom_boxplot(outlier.size=2,outlier.colour="red")
p1+ ggtitle("Employed in Malaysia (1982-2014)")+
xlab("")+ylab("Employed")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-bxp4.png" alt="boxplot4" /></p>
<p>Fig-7: Boxplot with outliers treated in variable <code class="highlighter-rouge">Employed</code></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> p2+ ggtitle("Labour Force in Malaysia (1982-2014)")+
xlab("")+ylab("Labour Force")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-bxp5.png" alt="boxplot5" /></p>
<p>Fig-8: Boxplot with outliers treated in variable <code class="highlighter-rouge">LabrFrc</code></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> p3+ ggtitle("Outside Labour Force in Malaysia (1982-2014)")+
xlab("")+ylab("Outside Labour Force")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-bxp6.png" alt="boxplot6" /></p>
<p>Fig-9: Boxplot with outliers treated in variable <code class="highlighter-rouge">OutLabrFrc</code></p>
<p>A simple and easy way to plot multiple plots is to adjust the <code class="highlighter-rouge">par</code> option. We show this as follows;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> par(mfrow=c(1,5),col.lab="blue", fg="indianred") # divide the screen into 1 row and five columns
> for(i in 3:7){
... boxplot(subst.data.2[,i], main=names(subst.data.3[i]))
... }
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-bxp7.png" alt="boxplot7" /></p>
<p>Fig-10: Easy alternative method to plot multiple boxplot with outliers</p>
<p>As is evident in Fig-10, the variables, <code class="highlighter-rouge">Employed</code>, <code class="highlighter-rouge">LabrFrc</code> and <code class="highlighter-rouge">OutLabrFrc</code> show clear indications of outliers. Subsequently, in Fig-11, we show multiple boxplots with outliers treated.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> par(mfrow=c(1,5)) # divide the screen into 1 row and four columns
> for(i in 3:7){
... boxplot(subst.data.3[,i], main=names(subst.data.3[i]))
... }
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-bxp8.png" alt="boxplot8" /></p>
<p>Fig-11: Multiple boxplot with outliers treated</p>
<ul>
<li><strong>Data type conversion</strong></li>
</ul>
<p>For subsequent data analytical activities, we converted the factor data type of the variable, <code class="highlighter-rouge">State</code> to numeric. Note, there were 17 levels in the <code class="highlighter-rouge">State</code> variable.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> table(df.cmplt$State)
Johor Kedah Kelantan Malaysia Melaka Negeri Sembilan
7 19 12 0 8 28
Pahang Perak Perlis Pulau Pinang Sabah Sarawak
26 12 7 7 4 11
Selangor Terengganu W.P Labuan W.P. Kuala Lumpur W.P.Putrajaya
4 13 16 20 27
> df.cmplt$State<-as.factor(gsub("W.P.Putrajaya","Putrajaya", df.cmplt$State,ignore.case=T))
> df.cmplt$State<-as.factor(gsub("W.P. Kuala Lumpur","Kuala Lumpur", df.cmplt$State,ignore.case=T))
> df.cmplt$State<-as.factor(gsub("W.P Labuan","Labuan", df.cmplt$State,ignore.case=T))
> df.cmplt$State<- as.numeric(df.cmplt$State)
</code></pre></div></div>
<p><strong>D. Correlation Detection & Treatment</strong></p>
<ul>
<li><strong>Detecting skewed variables</strong></li>
</ul>
<p>A variable is considered, <code class="highlighter-rouge">highly skewed</code> if its absolute value is greater than 1. A variable is considered, <code class="highlighter-rouge">moderately skewed</code> if its absolute value is greater than 0.5.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>skewedVars <- NA
> library(moments) # for skewness function
for(i in names(subst.data.3)){
if(is.numeric(subst.data.3[,i])){
if(i != "UnempRatePerct"){
# Enters this block if variable is non-categorical
skewVal <- skewness(subst.data.3[,i])
print(paste(i, skewVal, sep = ": "))
if(abs(skewVal) > 0.5){
skewedVars <- c(skewedVars, i)
}
}
}
}
[1] "Year: -0.0966073203178181"
[1] "State: 0"
[1] "Employed: 4.02774976187303"
[1] "LabrFrc: 4.00826453293672"
[1] "LabrFrcPerct: 0.576284963607043"
[1] "OutLabrFrc: 4.03480268085273"
</code></pre></div></div>
<p>We find that the variables, <code class="highlighter-rouge">Employed</code>, <code class="highlighter-rouge">LabrFrc</code> and <code class="highlighter-rouge">OutLabrFrc</code> are highly skewed.</p>
<ul>
<li><strong>Skewed variable treatment</strong></li>
</ul>
<p>Post identifying the skewed variables, we proceed to treating them by taking the log transformation. But, first we rearrange/reorder the columns for simplicity;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ## reorder the columns in df.cmplt data frame
> df.cmplt<- df.cmplt[c(1:2,4:5,3,6:7)]
> str(df.cmplt)
'data.frame': 544 obs. of 7 variables:
$ Year : num 1982 1983 1984 1985 1986 ...
$ State : num 6 6 6 6 6 6 6 6 6 6 ...
$ UnempRatePerct: num 3.4 3.8 5 5.6 7.4 7.3 7.2 5.7 4.5 3.7 ...
$ LabrFrcPerct : num 64.8 65.6 65.3 65.7 66.1 66.5 66.8 66.2 66.5 65.9 ...
$ Employed : num 5249 5457 5567 5653 5760 ...
$ LabrFrc : num 5431 5672 5862 5990 6222 ...
$ OutLabrFrc : num 2945 2969 3120 3125 3188 ...
</code></pre></div></div>
<p>Next, we treat the skewed variables by log base 2 transformation, given as follows;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> # Log transform the skewed variables
> df.cmplt.norm<-df.cmplt
> df.cmplt.norm[,3:7]<- log(df.cmplt[3:7],2) # where 2 is log base 2
> for(i in names(df.cmplt.norm)){
... if(is.numeric(df.cmplt.norm[,i])){
... if(i != "UnempRatePerct"){
... # Enters this block if variable is non-categorical
... skewVal <- skewness(df.cmplt.norm[,i])
... print(paste(i, skewVal, sep = ": "))
... if(abs(skewVal) > 0.5){
... skewedVars <- c(skewedVars, i)
... }
... }
... }
... }
[1] "Year: -0.0966073203178181"
[1] "State: 0"
[1] "LabrFrcPerct: 0.252455838759805"
[1] "Employed: -0.222298401708258"
[1] "LabrFrc: -0.210048778006162"
[1] "OutLabrFrc: -0.299617325738179"
</code></pre></div></div>
<p>As we can see now, the skewed variables are now normalized.</p>
<ul>
<li><strong>Correlation detection</strong></li>
</ul>
<p>We now checked for variables with high correlations to each other. Correlation measures the relationship between two variables. When two variables are so highly correlated that they explain each other (to the point that one can predict the variable with the other), then we have <em>collinearity</em> (or <em>multicollinearity</em>) problem. Therefore, its is important to treat collinearity problem. Let us now check, if our data has this problem or not.</p>
<p>Again, it is important to note that correlation works only for continuous variables. We can calculate the correlations by using the <code class="highlighter-rouge">cor()</code> as shown;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> correlations<- cor(df.cmplt.norm)
</code></pre></div></div>
<p>We then plotted the correlations shown in Fig-12. For this, we used the package <code class="highlighter-rouge">corrplot</code>;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(corrplot)
> corrplot(correlations, method = "number")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-corplot.png" alt="corplot" /></p>
<p>Fig-12: Correlation plot</p>
<p>As we can see from Fig-12, there are high correlations between variables, <code class="highlighter-rouge">Employed - LaborForce</code>; <code class="highlighter-rouge">Employed - OutsideLaborForce</code> and <code class="highlighter-rouge">LaborForce - OutsideLaborForce</code>.</p>
<ul>
<li><strong>Multicollinearity</strong></li>
</ul>
<p>Multicollinearity occurs because two (or more) variables are related or they measure the same thing. If one of the variables in your model doesn’t seem essential to your model, removing it may reduce multicollinearity. Examining the correlations between variables and taking into account the importance of the variables will help you make a decision about what variables to drop from the model.</p>
<p>There are several methods for dealing with multicollinearity. The simplest is to construct a correlation matrix and corresponding scatterplots. If the correlations between predictors approach 1, then multicollinearity might be a problem. In that case, one can make some educated guesses about which predictors to retain in the analysis.</p>
<p>Use, <em>Variance Inflation Factor (VIF)</em>. The VIF is a metric computed for every <em>X</em> variable that goes into a linear model. If the VIF of a variable is high, it means the information in that variable is already explained by the other <em>X</em> variables present in the given model, which means, more redundant is that variable. According to some references, if the VIF is too large(more than 5 or 10), we consider that the multicollinearity is existent. So, <strong>lower the VIF (<2) the better it is</strong>. VIF for a X var is calculated as; <img src="https://duttashi.github.io/images/vif.png" alt="vif0" /></p>
<p>where, Rsq is the Rsq term for the model with given X as response against all other Xs that went into the model as predictors.</p>
<p>Practically, if two of the X′s have high correlation, they will likely have high VIFs. Generally, VIF for an X variable should be less than 4 in order to be accepted as not causing multicollinearity. The cutoff is kept as low as 2, if you want to be strict about your X variables. Now, assume we want to predict <code class="highlighter-rouge">UnempRatePect</code> (unemployment rate percent) from rest of the predictors, so we regress it over others as given below in the equation; <code class="highlighter-rouge">> mod<- lm(Employed~., data=df.cmplt)</code>. We then calculate the VIF for this model by using the <code class="highlighter-rouge">vif()</code> method from the <code class="highlighter-rouge">DAAG</code> library, and find that the variables <code class="highlighter-rouge">Employed</code>, <code class="highlighter-rouge">LabrFrc</code>, <code class="highlighter-rouge">OutLabrFrc</code>, <code class="highlighter-rouge">State</code> are correlated.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> vfit<-vif(mod)
> sqrt(vif(mod)) > 2
Year State UnempRatePerct LabrFrcPerct LabrFrc OutLabrFrc
FALSE FALSE FALSE TRUE TRUE TRUE
</code></pre></div></div>
<ul>
<li><strong>Multicollinearity Treatment</strong></li>
</ul>
<p><strong>Principal Component Analysis (PCA): unsupervised data reduction method</strong></p>
<p>Principal Component Analysis (PCA) reduces the number of predictors to a smaller set of uncorrelated components. Remember, the PCA method can only be applied to continuous variables.</p>
<p>We aim to find the components which explain the maximum variance. This is because, we want to retain as much information as possible using these components. So, higher is the explained variance, higher will be the information contained in those components.</p>
<p>The base R function <code class="highlighter-rouge">princomp()</code> from the <code class="highlighter-rouge">stats package</code> is used to conduct the PCA test. By default, it centers the variable to have mean equals to zero. With parameter scale. = T, the variables (or the predictors) can be normalized to have standard deviation equals to 1. Since, we have already normalized the variables, we will not be using the scale option.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(stats) # for princomp()
> df.cmplt.norm.pca<- princomp(df.cmplt.norm, cor = TRUE)
> summary(df.cmplt.norm.pca)
Importance of components:
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7
Standard
deviation 1.7588184 1.2020620 1.0730485 0.8807122 0.73074799 0.02202556837 0.0063283251
Proportion
of Variance 0.4419203 0.2064219 0.1644904 0.1108077 0.07628466 0.00006930367 0.0000057211
Cumulative
Proportion 0.4419203 0.6483422 0.8128326 0.9236403 0.99992498 0.99999427890 1.0000000000
</code></pre></div></div>
<p>From the above summary, we can see that the <code class="highlighter-rouge">Comp.1</code> explains <code class="highlighter-rouge">44% variance</code>, <code class="highlighter-rouge">Comp.2</code> explains <code class="highlighter-rouge">20% variance</code> and so on. Also we can see that Comp.1 to Comp.5 have the highest standard deviation which indicates the number of components to retain (for further data analysis) as they explain maximum variance in the data.</p>
<ul>
<li><strong>Plotting the PCA (biplot) components</strong></li>
</ul>
<p>A PCA would not be complete without a bi-plot. In a biplot, the arrows point in the direction of increasing values for each original variable. The closeness of the arrows means that the variables are highly correlated. In Fig-13, notice the closeness of the arrows for variables, <code class="highlighter-rouge">OutLabrFrc</code>,<code class="highlighter-rouge">Employed</code> and <code class="highlighter-rouge">LabrFrc</code> indicates strong correlation. Again, notice the mild closeness of arrows for variable <code class="highlighter-rouge">LabrFrcPerct</code>,<code class="highlighter-rouge">State</code> and <code class="highlighter-rouge">UnempRatePerct</code> indicate mild correlation. Finally, notice the perpendicular distance between variables, <code class="highlighter-rouge">Year</code> and <code class="highlighter-rouge">OutLabrFrc</code> that indicates no correlation.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> # Plotting
> biplot(df.cmplt.norm.pca)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-biplot.png" alt="biplot" /></p>
<p>Fig-13: Biplot for PCA components</p>
<ul>
<li><strong>Determining the contribution (%) of each parameter in the calculated PCA</strong></li>
</ul>
<p>Now, the important question is how to determine the percentage of contribution (of each parameter) to each PC? simply put, how to know that <code class="highlighter-rouge">Comp.1</code> consist of say 35% of parameter1, 28% of parameter2 and so on.</p>
<p>The answer lies in computing the proportion of variance explained by each component, we simply divide the variance by sum of total variance. Thus we see that the first principal component <code class="highlighter-rouge">Comp.1</code> explains 44% of variance. The second component <code class="highlighter-rouge">Comp.2</code> explains 20% variance, the third component <code class="highlighter-rouge">Comp.3</code> explains 16% variance and so on.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> std_dev<- df.cmplt.norm.pca$sdev
> df.cmplt.norm.pca.var<- std_dev^2
> round(df.cmplt.norm.pca.var)
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7
3 1 1 1 1 0 0
#proportion of variance explained
prop_varex <- df.cmplt.norm.pca.var/sum(df.cmplt.norm.pca.var)
> round(prop_varex,3)
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7
0.442 0.206 0.164 0.111 0.076 0.000 0.000
</code></pre></div></div>
<p>Although, we have identified that <code class="highlighter-rouge">Comp.1</code> to <code class="highlighter-rouge">Comp.5</code> explain the maximum variance in the data but we use a scree plot for a visual identification too. A scree plot is used to access components or factors which explains the most of variability in the data. The cumulative scree plot in Fig-14, shows that 5 components explain about 99% of variance in the data. Therefore, in this case, we’ll select number of components as 05 [PC1 to PC5] and proceed to the modeling stage. For modeling, we’ll use these 05 components as predictor variables and follow the subsequent analysis.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> plot(cumsum(prop_varex), xlab = "Principal Component",
... ylab = "Cumulative Proportion of Variance Explained",
... type = "b")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-pca-screeplot.png" alt="screeplot" /></p>
<p>Fig-14: Cumulative Scree Plot for PCA</p>
<p>Now, we know that there are at least 5 components or variables in this dataset that exhibit maximum variance. Let us now see, what variables are these;</p>
<p>It is worth mentioning here that the principal components are located in the <code class="highlighter-rouge">loadings</code> component of the <code class="highlighter-rouge">princomp()</code> function. And if using the <code class="highlighter-rouge">prcomp</code> function, than the principal components are located in the <code class="highlighter-rouge">rotation</code> component.</p>
<p>Let’s now look at the first 5 PCA in first 5 rows</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> df.cmplt.norm.pca$loadings[1:5,1:5]
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
Year -0.15571810 0.59924346 -0.33488893 -0.17721252 0.68781319
State -0.01783084 -0.31630022 -0.66890937 -0.63612401 -0.21804005
UnempRatePerct 0.12931025 -0.60105660 0.34584708 -0.29005678 0.64662656
LabrFrcPerct -0.12043003 -0.40426976 -0.53298376 0.68221483 0.24047888
Employed -0.56396551 -0.08143999 0.07256198 -0.01140229 -0.02185788
</code></pre></div></div>
<p>We now demonstrate the relative contribution of each loading per column and express it as as a proportion of the column (loading) sum, taking care to use the absolute values to account for negative loading. See, this <a href="http://stackoverflow.com/questions/12760108/principal-components-analysis-how-to-get-the-contribution-of-each-paramete">SO solution</a></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> load <- with(df.cmplt.norm.pca, unclass(loadings))
> round(load,3)
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7
Year -0.156 0.599 -0.335 -0.177 0.688 0.004 0.000
State -0.018 -0.316 -0.669 -0.636 -0.218 -0.001 0.000
UnempRatePerct 0.129 -0.601 0.346 -0.290 0.647 -0.006 -0.010
LabrFrcPerct -0.120 -0.404 -0.533 0.682 0.240 0.121 -0.003
Employed -0.564 -0.081 0.073 -0.011 -0.022 -0.423 -0.700
LabrFrc -0.563 -0.091 0.077 -0.015 -0.012 -0.399 0.714
OutLabrFrc -0.556 -0.035 0.160 -0.118 -0.053 0.804 -0.014
</code></pre></div></div>
<p>And, this final step then yields the proportional contribution to the each principal component.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> aload <- abs(load) ## save absolute values
> round(sweep(aload, 2, colSums(aload), "/"),3)
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7
Year 0.074 0.282 0.153 0.092 0.366 0.002 0.000
State 0.008 0.149 0.305 0.330 0.116 0.001 0.000
UnempRatePerct 0.061 0.282 0.158 0.150 0.344 0.003 0.007
LabrFrcPerct 0.057 0.190 0.243 0.353 0.128 0.069 0.002
Employed 0.268 0.038 0.033 0.006 0.012 0.241 0.485
LabrFrc 0.267 0.043 0.035 0.008 0.006 0.227 0.495
OutLabrFrc 0.264 0.016 0.073 0.061 0.028 0.457 0.010
</code></pre></div></div>
<p>We already know that there are five components/variables with maximum variance in them. Now all that is left is to determine what are these variables. This can be determined easily from the above result. Remember, <code class="highlighter-rouge">Comp.1</code> shows variables with maximum variance, followed by <code class="highlighter-rouge">Comp.2</code> and so on. Now, in the column, <code class="highlighter-rouge">Comp.1</code> we keep only those variables that are greater than <code class="highlighter-rouge">0.05</code>. Therefore, the variables to keep are, <code class="highlighter-rouge">Year</code>, <code class="highlighter-rouge">UnempRatePerct</code>,<code class="highlighter-rouge">Employed</code>, <code class="highlighter-rouge">LabrFrc</code>, <code class="highlighter-rouge">LabrFrcPerct</code> and <code class="highlighter-rouge">OutLabrFrc</code>.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> vars_to_retain<- c("Year","Employed","UnempRatePerct","LabrFrc","LabrFrcPerct","OutLabrFrc")
> newdata<- df.cmplt.norm[,vars_to_retain]
> str(newdata)
'data.frame': 544 obs. of 6 variables:
$ Year : num 1982 1983 1984 1985 1986 ...
$ Employed : num 12.4 12.4 12.4 12.5 12.5 ...
$ UnempRatePerct: num 1.77 1.93 2.32 2.49 2.89 ...
$ LabrFrc : num 12.4 12.5 12.5 12.5 12.6 ...
$ LabrFrcPerct : num 6.02 6.04 6.03 6.04 6.05 ...
$ OutLabrFrc : num 11.5 11.5 11.6 11.6 11.6 ...
</code></pre></div></div>
<p>Note: We will be building the model on the normalized data stored in the variable, <code class="highlighter-rouge">df.cmplt.norm</code>.</p>
<h3 id="4-predictive-data-analytics">4. Predictive Data Analytics</h3>
<p>In this section, we will discuss various approaches applied to model building, predictive power and their trade-offs.</p>
<p><strong>A. Creating the train and test dataset</strong></p>
<p>We now divide the data into 75% training set and 25% testing set. We also created a root mean square evaluation function for model testing.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ratio = sample(1:nrow(newdata), size = 0.25*nrow(newdata))
> test.data = newdata[ratio,] #Test dataset 25% of total
> train.data = newdata[-ratio,] #Train dataset 75% of total
> dim(train.data)
[1] 408 4
> dim(test.data)
[1] 136 4
</code></pre></div></div>
<p><strong>B. Model Building - Evaluation Method</strong></p>
<p>We created a custom root mean square function that will evaluate the performance of our model.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Evaluation metric function
RMSE <- function(x,y)
{
a <- sqrt(sum((log(x)-log(y))^2)/length(y))
return(a)
}
</code></pre></div></div>
<p><strong>C. Model Building - Regression Analysis</strong></p>
<p>Regression is a supervised technique, a statistical process for estimating the relationship between a response variable and one or more predictors. Often the outcome variable is also called the response variable or the dependent variable and the and the risk factors and confounders are called the predictors, or explanatory or independent variables. In regression analysis, the dependent variable is denoted <code class="highlighter-rouge">y</code> and the independent variables are denoted by <code class="highlighter-rouge">x</code>.</p>
<p>The response variable for this study is continuous in nature therefore the choice of regression model is most appropriate.</p>
<p>Our multiple linear regression model for the response variable <code class="highlighter-rouge">Employed</code> reveals that the predictors, <code class="highlighter-rouge">UnempRatePerct</code> and <code class="highlighter-rouge">LabrFrc</code> are the most significant predictors such that if included in the model will enhance the predictive power of the response variable. The remaining predictors do not contribute to the regression model.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> linear.mod<- lm(Employed~., data = train.data)
> summary(linear.mod)
Call:
lm(formula = Employed ~ ., data = train.data)
Residuals:
Min 1Q Median 3Q Max
-0.060829 -0.002058 0.001863 0.004615 0.184889
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.38447474 0.34607122 -1.111 0.267
Year 0.00009844 0.00008295 1.187 0.236
UnempRatePerct -0.03869329 0.00119011 -32.512 <2e-16 ***
LabrFrc 0.97901237 0.01634419 59.900 <2e-16 ***
LabrFrcPerct 0.03468488 0.04784967 0.725 0.469
OutLabrFrc 0.02223528 0.01624485 1.369 0.172
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.01452 on 402 degrees of freedom
Multiple R-squared: 0.9999, Adjusted R-squared: 0.9999
F-statistic: 1.223e+06 on 5 and 402 DF, p-value: < 2.2e-16
</code></pre></div></div>
<p>The t value also known as the t-test which is positive for predictors, <code class="highlighter-rouge">Year</code>,<code class="highlighter-rouge">LabrFrc</code>,<code class="highlighter-rouge">LabrFrcPerct</code> and <code class="highlighter-rouge">OutLabrFrc</code> indicating that these predictors are associated with <code class="highlighter-rouge">Employed</code>. A larger t-value indicates that that it is less likely that the coefficient is not equal to zero purely by chance.</p>
<p>Again, as the p-value for variables, <code class="highlighter-rouge">UnempRatePerct</code> and <code class="highlighter-rouge">LabrFrc</code> is less than 0.05 they are both statistically significant in the multiple linear regression model for the response variable, <code class="highlighter-rouge">Employed</code> . The model’s, <code class="highlighter-rouge">p-value: < 2.2e-16</code> is also lower than the statistical significance level of <code class="highlighter-rouge">0.05</code>, this indicates that we can safely reject the null hypothesis that the value for the coefficient is zero (or in other words, the predictor variable has no explanatory relationship with the response variable).</p>
<p>We tested this model using the root mean square evaluation method. The RMSE is 0.003.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> RMSE0<- RMSE(predict, test.data$Employed)
> RMSE0<- round(RMSE0, digits = 3)
> RMSE0
[1] 0.003
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-residuals.png" alt="residuals" /></p>
<p>Fig-14: Residuals vs Fitted values for the response variable, “Employed”</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> actuals_preds <- data.frame(cbind(actuals=test.data$Employed, predicteds=predict)) # make actuals_predicteds dataframe.
> correlation_accuracy <- cor(actuals_preds)
> correlation_accuracy # 99%
actuals predicteds
actuals 1.0000000 0.9999386
predicteds 0.9999386 1.0000000
> min_max_accuracy <- mean (apply(actuals_preds, 1, min) / apply(actuals_preds, 1, max))
> min_max_accuracy
[1] 0.9988304
> mape <- mean(abs((actuals_preds$predicteds - actuals_preds$actuals))/actuals_preds$actuals)
> mape
[1] 0.001170885
</code></pre></div></div>
<p>The AIC and the BIC model diagnostics values are low too. <code class="highlighter-rouge">> AIC(linear.mod) [1] -2287.863</code> and <code class="highlighter-rouge">> BIC(linear.mod) [1] -2259.784</code>.</p>
<p><strong>D. Model Building - other supervised algorithms</strong></p>
<ul>
<li>Regression Tree method</li>
</ul>
<p>The regression tree method gives an accuracy of 0.037</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(rpart)
> model <- rpart(Employed ~., data = train.data, method = "anova")
> predict <- predict(model, test.data)
> RMSE1 <- RMSE(predict, test.data$Employed)
> RMSE1 <- round(RMSE1, digits = 3)
> RMSE1
[1] 0.037
</code></pre></div></div>
<ul>
<li>Random Forest method</li>
</ul>
<p>The random forest method gives an accuracy of 0.009. Look at the <code class="highlighter-rouge">IncNodePurity plot</code> in Fig-15. We see that important predictors are <code class="highlighter-rouge">Year</code>, <code class="highlighter-rouge">UnempRatePerct</code> ,<code class="highlighter-rouge">LabourFrcPerct</code></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(randomForest)
> model.forest <- randomForest(Employed ~., data = train.data, method = "anova",
... ntree = 300,
... mtry = 2, #mtry is sqrt(6)
... replace = F,
... nodesize = 1,
... importance = T)
> varImpPlot(model.forest)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Labor-vifplot.png" alt="vifplot" /></p>
<p>Fig-15: VIF plot</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> prediction <- predict(model.forest,test.data)
> RMSE3 <- sqrt(mean((log(prediction)-log(test.data$Employed))^2))
> round(RMSE3, digits = 3)
[1] 0.009
</code></pre></div></div>
<p><strong>D.1 Model Performance comparison</strong></p>
<p>As a rule of thumb, smaller the RMSE value better is the model. See this <a href="http://stats.stackexchange.com/questions/56302/what-are-good-rmse-values">SO post</a>. So its feasible to state that the multiple linear regression model yields the best predictive performance as it has the lowest RMSE value of <code class="highlighter-rouge">0.003</code>.</p>
<p>Multiple Linear Regression RMSE: 0.003</p>
<p>Random Forest RMSE: 0.009</p>
<p>Regression Tree RMSE: 0.037</p>
<h3 id="5-conclusion">5. Conclusion</h3>
<p>In this analytical study, we have explored three supervised learning models to predict the factors contributing to an accurate prediction of employed persons by state in Malaysia. Our multiple linear regression model for the response variable <code class="highlighter-rouge">Employed</code> reveals that the predictors, <code class="highlighter-rouge">UnempRatePerct</code>and <code class="highlighter-rouge">LabrFrc</code> are the most significant predictors such that if included in the model will enhance the predictive power of the response variable. The other predictors such as <code class="highlighter-rouge">Year</code>, <code class="highlighter-rouge">OutLabrFrc</code>, <code class="highlighter-rouge">LabrFrcPerct</code>does not contribute to the regression model. This model gives an <strong>accuracy of 99%</strong> on unseen data and has the lowest RMSE of <code class="highlighter-rouge">0.003</code> as compared to the other supervised learning methods. Again, its worthwhile to mention here the reason for such a high accuracy of the predictive model because we chose the correct model for the response variable and ensured to carry out a rigorous data preprocessing and modeling activities.</p>
<p>The complete code is listed on my Github repository in <a href="https://github.com/duttashi/learnr/blob/master/scripts/Full%20Case%20Studies/CaseStudy-MY-LaborForce.R">here</a></p>
<![CDATA[Predicting rubber plantation yield- A regression analysis approach]]>https://duttashi.github.io/blog/predicting-rubber-yield-case-study2017-02-09T00:00:00+00:002017-02-09T00:00:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<h3 id="introduction">Introduction</h3>
<p>Malaysia is the leading producer of natural rubber in the world. Being a leader in the production of natural rubber, Malaysia is contributing around 46% of total rubber production in the world. The rubber plantation was started in Malaysia in 1877.</p>
<p>The favorable rubber plantation climate requires a mean temperature of 27°C, never falling below 22°C. It also requires heavy rainfall above 200 cm. with no drought including deep rich soils with good drainage preferably brittle, well-oxidized and acidic in reaction. Sufficient supply of labour is an important factor for the collection and plantation of rubber over large holdings.</p>
<p>In Malaysia, rubber can grow anywhere, because of the suitability of climate and top soil; but most of the rubber estates are located in the western coastal plains of Malaysia. The plantation in coastal zone gets the benefit of nearest port for its export. Yet very low areas are avoided in order not to suffer from stagnation of water. The greatest production is in it’s Johor State of Southern Malaysia. Over here the rubber cultivation occupies about 4-2 million acres or about 66% of the total cultivated area in the nation.</p>
<p>This report consist of the following sections;</p>
<ol>
<li>
<p>Business/Research Question</p>
</li>
<li>
<p>Data Source</p>
</li>
<li>
<p>Making data management decisions</p>
</li>
</ol>
A. Exploratory Data Analysis (EDA)
* Data preprocessing (rename and round)
* Data preprocessing (joining the tables)
B. Data visualization
C. Data transformation
* Skewed variable treatment
D. Feature importance
<ol>
<li>Predictive Data Analytics</li>
</ol>
A. Creating the train and test dataset
B. Model Building - Evaluation Method
C. Model Building - Regression Analysis
D. Model Performance on various supervised algorithms
* Regression Tree method
* Random Forest method
D.1. Comparison of Predictive Model Performance
E. Model Diagnostics
* The p Value: Checking for statistical significance
* Check the AIC and BIC
* The R-Squared and Adjusted R-Squared
* How do you know if the model is best fit for the data?
* Residuals
F. Model Inference Summary
G. Calculate prediction accuracy and error rates
<ol>
<li>Conclusion</li>
</ol>
<p>References</p>
<h3 id="1-businessresearch-question">1. Business/Research Question</h3>
<p>Determine the factors which contribute to accurately predicting high rubber yield per kg based on historical rubber plantation data.</p>
<h3 id="2-data-source">2. Data Source</h3>
<p>The data comes from the Department of Statistics, Malaysia. This is an open data source portal and the data files can be accessed from their official <a href="http://www.dosm.gov.my/v1/index.php?r=column3/accordion&menu_id=aHhRYUpWS3B4VXlYaVBOeUF0WFpWUT09">website</a></p>
<h3 id="3-making-data-management-decisions">3. Making data management decisions</h3>
<p>Initially, the dataset consisted of six comma-separated files. Each file provided data (from year 1965 to year 2014) on factors like number of rubber estates in Malaysia, total planted area, production of natural rubber, tapped area, yield per hectare and total number of paid employees in the rubber estate.</p>
<p><strong>A. Exploratory Data Analysis (EDA)</strong></p>
<p>Each data file had the same dimension of 51 rows in 2 continuous variables. On knowing that each of the six-data file had the same dimensions, it confirmed our initial assumption that the actual dataset was divided into six separate files.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> df1<- read.csv("data/rubberestate/rubber-paidemployee.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
> df2<- read.csv("data/rubberestate/rubber-plantedarea.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
> df3<- read.csv("data/rubberestate/rubber-production.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
> df4<- read.csv("data/rubberestate/rubber-taparea.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
> df5<- read.csv("data/rubberestate/rubber-yield.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
> dim(df1)
[1] 51 2
> dim(df2)
[1] 51 2
> dim(df3)
[1] 51 2
> dim(df4)
[1] 51 2
> dim(df5)
[1] 51 2
</code></pre></div></div>
<p>Another peculiarity found was the column headings were too long for each of the data file. We decided to merge the six data files into a single dataset and rename the column names to short succinct names. For data analysis we are using the R programming language (Ihaka & Gentleman, 1996).</p>
<p>Besides, we also found that column value for number of employees was expressed in decimals! Now, there cannot be 2.5 employees so we decided to round all such values.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> names(df1)
[1] "Year"
[2] "Total.Number.of.Paid.Employee.During.the.Last.Pay.Period..Estate."
> names(df2) # additional space after column names. do formatting
[1] "Year" "Planted.Area..Estate....000..Hectare"
> names(df3)
[1] "Year" "Production..Estate....000..Tonne"
> names(df4)
[1] "Year" "Tapped.Area..Estate....000..Hectare"
> names(df5)
[1] "Year" "Yeild.per.Hectare..Estate...Kg."
> head(df1) # You cannot have employees in decimals. Round this variable
Year Total.Number.of.Paid.Employee.During.the.Last.Pay.Period..Estate.
1 1965 262.1
2 1966 258.4
3 1967 235.4
4 1968 209.8
5 1969 212.7
6 1970 205.4
</code></pre></div></div>
<p>So, we first decided to perform basic data management tasks that were identified above. For this we use the <code class="highlighter-rouge">rename</code> function in the <code class="highlighter-rouge">plyr</code> library (Wickham, 2015).</p>
<p>You will need to load this library in the R environment first before you can use the <code class="highlighter-rouge">rename</code> function.</p>
<ul>
<li>
<p>Data preprocessing (rename and round)</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> # Basic Data Management
> # Renaming the column name
> library(plyr)
> df1<- rename(df1, c("Total.Number.of.Paid.Employee.During.the.Last.Pay.Period..Estate." = "TotalPaidEmployee"))
> df2<-rename(df2, c("Planted.Area..Estate....000..Hectare" = "AreaPlantedHect"))
> df3<-rename(df3, c("Production..Estate....000..Tonne" = "ProduceTonne"))
> df4<-rename(df4, c("Tapped.Area..Estate....000..Hectare" = "TapAreaHect"))
> df5<-rename(df5, c("Yeild.per.Hectare..Estate...Kg." = "YieldperHectKg"))
> # Rounding the column value for TotalPaidEmployee because there can’t be example 2.5 employees
> df1$TotalPaidEmployee<- round(df1$TotalPaidEmployee)
</code></pre></div> </div>
</li>
<li>
<p>Data pre-processing (joining the tables)</p>
<p>We also notice that all the six data files have a common column which is, <code class="highlighter-rouge">Year</code>. So, we now join the files on this common column and save the resultant in a master data frame called, <code class="highlighter-rouge">df.master</code>. This process is known as the <code class="highlighter-rouge">inner join</code>.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code> > # Inner Join the data frames on common column
> df.m1<- merge(df1,df2, by="Year")
> df.m2<- merge(df3,df4, by="Year")
> df.m3<- merge(df.m2, df5, by="Year")
> df.master<- merge(df.m1, df.m3, by="Year")
</code></pre></div> </div>
<p>Now, that the dataset is ready for inspection, the first step would be to summarize it using the <code class="highlighter-rouge">summary</code> function call.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code> > summary(df.master)
Year TotalPaidEmployee AreaPlantedHect ProduceTonne TapAreaHect YieldperHectKg
Min. :1965 Min. : 10.00 Min. : 49.70 Min. : 53.00 Min. : 38.50 Min. : 937
1st Qu.:1977 1st Qu.: 16.75 1st Qu.: 87.47 1st Qu.: 88.62 1st Qu.: 64.62 1st Qu.:1304
Median :1990 Median :105.00 Median :354.85 Median :414.70 Median :307.05 Median :1381
Mean :1990 Mean :103.62 Mean :352.94 Mean :364.27 Mean :277.42 Mean :1347
3rd Qu.:2002 3rd Qu.:174.50 3rd Qu.:554.25 3rd Qu.:580.98 3rd Qu.:433.40 3rd Qu.:1420
Max. :2014 Max. :262.00 Max. :788.50 Max. :684.60 Max. :542.30 Max. :1525
NA's :1 NA's :1 NA's :1 NA's :1 NA's :1 NA's :1
</code></pre></div> </div>
</li>
</ul>
<p>We see that the minimum yield per hectare is 937 kg and the minimum area planted is 49.7 hectares. Besides, there is also one data point with missing value.</p>
<ul>
<li>
<p>Missing data treatment</p>
<p>We have applied the predictive mean modeling method for missing data imputation. This method is available in the <code class="highlighter-rouge">mice</code> (Buuren & Groothuis-Oudshoorn, 2011) library. You will need to load it in the R environment first.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code> >library(mice)
> tempData <- mice(df.master,m=5,maxit=50,meth='pmm',seed=1234)
> df.master<- mice::complete(tempData,1)
> colSums(is.na(df.master))
Year TotalPaidEmployee AreaPlantedHect ProduceTonne TapAreaHect YieldperHectKg
0 0 0 0 0 0
</code></pre></div> </div>
</li>
</ul>
<p>Now, the dataset is ready for visualization. This will help us in determining a research question. At this point it’s best to describe about our dataset. for this, we use the method <code class="highlighter-rouge">describe</code> from the <code class="highlighter-rouge">psych</code> library (Revelle, 2014). A basic example can be see <a href="http://www.statmethods.net/stats/descriptives.html">here</a></p>
<p><strong>B. Data visualization: visualizing data in pursuit of finding relationship between predictors</strong></p>
<p>We begin the data exploration by univariate data visualization. Here, we will be using the <code class="highlighter-rouge">%>%</code> or the pipe operator from the <code class="highlighter-rouge">magrittr</code> package (Bache & Wickham, 2014) and <code class="highlighter-rouge">select</code> statement from the <code class="highlighter-rouge">dplyr package</code> (Wickham & Francois, 2015) to visualize all the predictors excluding Year.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(magrittr)
> library(dplyr)
> # Method 1: selecting individual predictor name
> boxplot(df.master %>%
... select(AreaPlantedHect,YieldperHectKg,ProduceTonne,TapAreaHect,TotalPaidEmployee))
> # Method 2: Use the minus sign before the predictor you dont want to plot such that the remaining predictors are plotted
> boxplot(df.master %>%
... select(-Year),
... col = c("red","sienna","palevioletred1","royalblue2","brown"),
... ylab="Count", xlab="Predictors"
... )
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-boxplot1.png" alt="boxplot1" /></p>
<p>Fig-1: Boxplot</p>
<p>From Fig-1, it seems that there are some outlier’s for the <code class="highlighter-rouge">YieldperHectKg</code> predictor. We will come to it later, for now, we continue exploring the data.</p>
<p>Now, we use the line plots to determine relationships between continuous predictors.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ggplot(df.master)+ geom_line(aes(x=AreaPlantedHect, y=YieldperHectKg, color=”red”))
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-lineplot1.png" alt="lineplot1" /></p>
<p>Fig-2: Line Plot for predictors <code class="highlighter-rouge">AreaPlantedHect</code> and <code class="highlighter-rouge">YieldperHectKg</code></p>
<p>An interesting pattern is revealed in Fig-2. The <em>yield per hectare has a sharp decline (after 600 hectares) as plantation area increases</em>.</p>
<p>Lets’ explore the remaining predictors;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ggplot(df.master)+ geom_line(aes(x=AreaPlantedHect, y=ProduceTonne, color="red"))
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-lineplot2.png" alt="lineplot2" /></p>
<p>Fig-3: Line Plot for predictors <code class="highlighter-rouge">AreaPlantedHect</code> and <code class="highlighter-rouge">ProduceTonne</code></p>
<p>We see that produce increases with area but then it begins to decline after 600 hectares. There is a positive linear relationship between area planted and tap area as shown below in Fig-4.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ggplot(df.master)+ geom_line(aes(x=AreaPlantedHect, y=TapAreaHect, color="red"))
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-lineplot3.png" alt="lineplot3" /></p>
<p>Fig-4: Line Plot for predictors <code class="highlighter-rouge">AreaPlantedHect</code> and <code class="highlighter-rouge">TapAreaHect</code></p>
<p>Again, in Fig-5, we notice a positive linear relationship between area planted and paid employees but there is a sharp decline at 600 hectares persists.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ggplot(df.master)+ geom_line(aes(x=AreaPlantedHect, y=TotalPaidEmployee, color="red"))
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-lineplot4.png" alt="lineplot4" /></p>
<p>Fig-5: Line Plot for predictors <code class="highlighter-rouge">AreaPlantedHect</code> and <code class="highlighter-rouge">TotalPaidEmployee</code></p>
<p>The evidence of strong positive linear relationship between the predictors, <code class="highlighter-rouge">AreaPlantedHect</code>, <code class="highlighter-rouge">TapAreaHect</code>, <code class="highlighter-rouge">TotalPaidEmployee</code> and <code class="highlighter-rouge">ProduceTonne</code> cannot be overlooked. We, cross-check this phenomenon by deducing the correlation between them.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> cor(df.master$AreaPlantedHect, df.master$TapAreaHect) # very strong positive correlation
[1] 0.9930814
> cor(df.master$AreaPlantedHect, df.master$ProduceTonne) # very strong positive correlation
[1] 0.9434092
> cor(df.master$AreaPlantedHect, df.master$TotalPaidEmployee) # very strong positive correlation, as land size increases more labour is required
[1] 0.9951871
> cor(df.master$AreaPlantedHect, df.master$YieldperHectKg) # negative correlation, proving the point above that the yield per hectare decreases as plantation size increases
[1] -0.5466433
</code></pre></div></div>
<p>we now have ample evidence that the predictors, <code class="highlighter-rouge">TotalPaidEmployee</code>,<code class="highlighter-rouge">AreaPlantedHect</code>,<code class="highlighter-rouge">ProduceTonee</code> and <code class="highlighter-rouge">TapAreaHect</code> have a strong positive correlationship. Let’s visualize it.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> correlations<- cor(df.master)
> corrplot(correlations, method="number")
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-corrplot.png" alt="corrplot-1" /></p>
<p>Fig-6: Correlation Plot for predictors and response variables.</p>
<p>As seen in Fig-6 above, the predictors <code class="highlighter-rouge">Year</code> and <code class="highlighter-rouge">YieldPerHect</code> have low positive correlation with each other; <code class="highlighter-rouge">TotalPaidEmployee</code> and <code class="highlighter-rouge">YieldHect</code> have a semi-strong negative correlation; others like <code class="highlighter-rouge">AreaPlantedHect</code> and <code class="highlighter-rouge">YieldPerHect</code> have a strong negative correlation and <code class="highlighter-rouge">ProduceTonne</code> and <code class="highlighter-rouge">YieldperhectKg</code> have a low negative correlation with each other.</p>
<p>We can also create a scatter plot matrix (see Fig-7) to plot correlations among the continuous predictors by using the <code class="highlighter-rouge">pairs</code> function from the <code class="highlighter-rouge">ggplot2</code> library(Wickham, 2016)</p>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-corrplot-1.png" alt="corrplot-2" /></p>
<p>Fig-7: Scatter plot matrix for predictor and response variable correlation</p>
<p>We end this discussion by a simple question. Does the yield increase if the plantation area increases? Lets find this out in the following graph, see Fig-8.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library (RColorBrewer)
# We will select the first 4 colors in the Set1 palette
> cols<-brewer.pal(n=4,name="Set1")
# cols contain the names of four different colors
> plot(Training$AreaPlantedHect, Training$YieldperHectKg, pch=16,col=cols,
main=" Does high plantation area yield more rubber?",
xlab = "Area planted (in hectare)",
ylab = "Yield in Kg (per hectare)"
)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-plot1.png" alt="plot" /></p>
<p>Fig-7: Scatter plot matrix for predictor and response variable correlation</p>
<p><strong>C. Data transformation</strong></p>
<ul>
<li>Skewed variable treatment</li>
</ul>
<p>A variable is considered ‘highly skewed’ if its absolute value is greater than 1. A variable is considered ‘moderately skewed’ if its absolute value is greater than 0.5. let’s check if any of the predictors are skewed or not.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> for(i in names(df.master)){
... if(is.numeric(df.master[,i])){
... if(i != "YieldperHectKg"){
... # Enters this block if variable is non-categorical
... skewVal <- skewness(df.master[,i])
... print(paste(i, skewVal, sep = ": "))
... if(abs(skewVal) > 0.5){
... skewedVars <- c(skewedVars, i)
... }
... }
... }
... }
[1] "Year: 0.0380159253762087"
[1] "TotalPaidEmployee: 0.238560934226388"
[1] "AreaPlantedHect: 0.118115337328111"
[1] "ProduceTonne: -0.184114105316565"
[1] "TapAreaHect: -0.0526176590077839"
</code></pre></div></div>
<p>There are no skewed predictors.</p>
<p><strong>D. Feature importance</strong></p>
<p>Now, that we have statistically quantified the validity of the predictors, we proceed to determining the most relevant features. Such features when found will help in building a robust predictive model. We will use the <code class="highlighter-rouge">Boruta</code> package (Kursa & Rudnicki, 2010).</p>
<p>We are interested in predicting the variable Yield per hectare in kg (<code class="highlighter-rouge">YieldperHectKg</code>) therefore we will remove it from the feature selection process and perform the analysis on the remaining predictors.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(Boruta)
> set.seed(1234) # for code reproducibility
> response <- df.master$YieldperHectKg
> response <- df.master$YieldperHectKg
> bor.results <- Boruta(df.master,response,
... maxRuns=101,
... doTrace=0)
> cat("\n\nRelevant Attributes:\n")
Relevant Attributes:
> getSelectedAttributes(bor.results)
[1] "Year" "TotalPaidEmployee" "AreaPlantedHect" "ProduceTonne" "TapAreaHect"
[6] "YieldperHectKg"
> plot(bor.results)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-boruta.png" alt="plot" /></p>
<p>Fig-8: Feature importance plot</p>
<p>We see from Fig-8, that <code class="highlighter-rouge">Boruta</code> predicts all the features to be important for building a predictive model. Let us know proceed to building the predictive model.</p>
<h2 id="4-predictive-data-analytics">4. Predictive Data Analytics</h2>
<p>In this section, we will discuss various approaches in model building, predictive power and their trade-offs.</p>
<p><strong>A. Creating the train and test dataset</strong></p>
<p>Researchers and data practitioners have always emphasized on building a model that is intensively trained on a larger sample of the train data. Therefore, we will divide the dataset into 70% training data and 30% testing data.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ratio = sample(1:nrow(df.master), size = 0.25*nrow(df.master))
> Test = df.master[ratio,] #Test dataset 25% of total
> Training = df.master[-ratio,] #Train dataset 75% of total
> dim(Training)
[1] 39 6
> dim(Test)
[1] 12 6
</code></pre></div></div>
<p><strong>B. Model Building - Evaluation Method</strong></p>
<p>We created a custom root mean square function that will evaluate the performance of our model.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Evaluation metric function
RMSE <- function(x,y)
{
a <- sqrt(sum((log(x)-log(y))^2)/length(y))
return(a)
}
</code></pre></div></div>
<p><strong>C. Model Building - Regression Analysis</strong></p>
<p>Regression is a supervised technique, a statistical process for estimating the relationship between a response variable and one or more predictors. Often the outcome variable is also called the response variable or the dependent variable and the and the risk factors and confounders are called the predictors, or explanatory or independent variables. In regression analysis, the dependent variable is denoted <code class="highlighter-rouge">y</code> and the independent variables are denoted by <code class="highlighter-rouge">x</code>.</p>
<p>Regression analysis is a widely used technique which is useful for evaluating multiple independent variables. It serves to answer the question, “Which factors matter the most?”. Interested readers should see (Kleinbaum, Kupper and Muller, 2013) for more details on regression analysis and its many applications.</p>
<p>We then, created a multiple linear regression model for the response variable <code class="highlighter-rouge">YieldperHectKg</code> and the summary statistic showed that the predictors, <code class="highlighter-rouge">TapAreaHect</code>, <code class="highlighter-rouge">ProduceTonne</code> and <code class="highlighter-rouge">TotalPaidEmployee</code> are the most significant predictors such that if included in the model will enhance the predictive power of the response variable.
The other predictors like <code class="highlighter-rouge">Year</code> and <code class="highlighter-rouge">AreaPlantedHect</code> do not contribute to the regression model.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> linear.mod<- lm(YieldperHectKg~., data = Training)
> summary(linear.mod)
Call:
lm(formula = YieldperHectKg ~ ., data = Training)
Residuals:
Min 1Q Median 3Q Max
-73.203 -23.203 -1.562 13.087 108.326
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1839.5867 5221.4502 -0.352 0.72684
Year 1.6199 2.5965 0.624 0.53699
TotalPaidEmployee 2.1835 0.7680 2.843 0.00761 **
AreaPlantedHect -0.4247 0.4927 -0.862 0.39490
ProduceTonne 2.1643 0.2541 8.518 0.000000000764 ***
TapAreaHect -3.2198 0.9014 -3.572 0.00111 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 36.69 on 33 degrees of freedom
Multiple R-squared: 0.9244, Adjusted R-squared: 0.913
F-statistic: 80.74 on 5 and 33 DF, p-value: < 2.2e-16
</code></pre></div></div>
<p>The t value also known as the t-test which is positive for predictors, <code class="highlighter-rouge">Year</code>, <code class="highlighter-rouge">TotalPaidEmployee</code>, <code class="highlighter-rouge">AreaPlantedHect</code>,<code class="highlighter-rouge">ProduceTonne</code> and <code class="highlighter-rouge">TapAreaHect</code> indicating that these predictors are associated with <code class="highlighter-rouge">YieldperHectKg</code>. A larger t-value indicates that that it is less likely that the coefficient is not equal to zero purely by chance.</p>
<p>Again, as the p-value for <code class="highlighter-rouge">ProduceTonne</code>, <code class="highlighter-rouge">TapAreaHect</code> and <code class="highlighter-rouge">TotalPaidEmployee</code> is less than 0.05 they are both statistically significant in the multiple linear regression model for <code class="highlighter-rouge">YieldperHectKg</code> response variable. The model’s, <code class="highlighter-rouge">p-value: < 2.2e-16</code> is also lower than the statistical significance level of <code class="highlighter-rouge">0.05</code>, this indicates that we can safely reject the null hypothesis that the value for the coefficient is zero (or in other words, the predictor variable has no explanatory relationship with the response variable).</p>
<p>In Regression, the Null Hypothesis is that the coefficients associated with the variables is equal to zero. The alternate hypothesis is that the coefficients are not equal to zero (i.e. there exists a relationship
between the independent variable in question and the dependent variable).</p>
<p>We tested this model using the root mean square evaluation method.</p>
<p>Note, we did not remove the non-contributing predictors from the regression model and found the RMSE to be quite low of 0.045. This model has an F-statistic of 80.74 which is considerably high and better.</p>
<p>Next, we performed the model prediction on unseen data.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> predict<- predict(linear.mod, Test)
> RMSE0<- RMSE(predict, Test$YieldperHectKg)
> RMSE0
[1] 0.04533296
</code></pre></div></div>
<p><strong>D. Model Performance on various supervised algorithms</strong></p>
<p>We now test the model performance on some supervised algorithms to determine the model’s prediction accuracy.</p>
<ul>
<li>
<p><strong>Regression Tree method</strong></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code> > library(rpart)
> model <- rpart(YieldperHectKg ~., data = Training, method = "anova")
> predict <- predict(model, Test)
# RMSE
> RMSE1 <- RMSE(predict, Test$YieldperHectKg)
> RMSE1 <- round(RMSE1, digits = 3)
> RMSE1
> [1] 0.098
</code></pre></div> </div>
</li>
<li>
<p><strong>Random Forest method</strong></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code> > model.forest <- randomForest(YieldperHectKg ~., data = Training, method = "anova",
ntree = 300,
mtry = 2, #mtry is sqrt(6)
replace = F,
nodesize = 1,
importance = T)
> varImpPlot(model.forest) # Look at the IncNodePurity plot. From this plot we see that important vars are `TotalPaidEmployee`, `ProduceTonne` and `TapAreaHect`
> prediction <- predict(model.forest,Test)
> rmse <- sqrt(mean((log(prediction)-log(Test$YieldperHectKg))^2))
> round(rmse, digits = 3) # 0.049
</code></pre></div> </div>
</li>
</ul>
<p>The Variance Inflation Factor (VIF) plot shows the predictors, <code class="highlighter-rouge">TotalPaidEmployee</code>, <code class="highlighter-rouge">ProduceTonne</code> and <code class="highlighter-rouge">TapAreaHect</code> as most important.</p>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-VIFPlot.png" alt="plot" /></p>
<p>Fig-9: VIF plot</p>
<p><strong>D.1. Comparison of Predictive Model Performance</strong></p>
<p>So to predict the response variable, <code class="highlighter-rouge">YieldperHectKg</code> the best results were given by Regression Tree based model which gave an accuracy of <code class="highlighter-rouge">98%</code> as compared to others;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Linear Regression: 0.04533296
Regression Tree RMSE: 0.098
Random Forest RMSE: 0.049
</code></pre></div></div>
<p><strong>E. Model Diagnostics</strong></p>
<p><strong>i. The p Value: Checking for statistical significance</strong></p>
<p>It is extremely important for the model to be statistically significant before we can go ahead and use it to predict (or estimate) the dependent variable, otherwise, the confidence in predicted values from that model reduces and may be construed as an event of chance.</p>
<p>In this model <code class="highlighter-rouge">linear.mod</code> the p-Values of the predictors are well below the 0.05 threshold, so we can conclude our model is indeed statistically significant. This can visually be interpreted by the significance stars at the end of the row. The more the stars beside the variable’s p-Value, the more significant the variable is.</p>
<p><strong>ii. Check the AIC and BIC</strong></p>
<p>The Akaike’s Information Criterion AIC (Akaike, 1974) and the Bayesian Information Criterion BIC (Schwarz, 1978) are measures of the goodness of fit of an estimated statistical model and can also be used for model selection.</p>
<p>Both criteria depend on the maximized value of the likelihood function L for the
estimated model.</p>
<p>The AIC is defined as:
AIC = (−2) • ln (L) + 2 • k
where k is the number of model parameters and the BIC is defined as:
BIC = (−2) • ln(L) + k • ln(n)
where n is the sample size.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> AIC(linear.mod)
[1] 399.1521
> BIC(linear.mod)
[1] 410.797
</code></pre></div></div>
<p>For model comparison, the model with the lowest AIC and BIC score is preferred. Suppose, we had build another linear model with only two predictors, <code class="highlighter-rouge">ProduceTonne</code> and <code class="highlighter-rouge">TapAreaHect</code> given as;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> linear.mod1<- lm(YieldperHectKg~ProduceTonne+TapAreaHect, data = Training)
> AIC(linear.mod1)
[1] 402.8458
> BIC(linear.mod1)
[1] 409.5001
</code></pre></div></div>
<p>The <code class="highlighter-rouge">AIC</code> & <code class="highlighter-rouge">BIC</code> for <code class="highlighter-rouge">linear.mod</code> is <strong>lower</strong> than the <code class="highlighter-rouge">linear.mod1</code> therefore, <code class="highlighter-rouge">linear.mod</code> is a <strong>better model</strong> for predicting the response variable.</p>
<p><strong>iii. The R-Squared and Adjusted R-Squared</strong></p>
<p>The actual information in a data is the total variation it contains.
What R-Squared tells us is the proportion of variation in the dependent (response) variable that has been explained by this model.</p>
<p>Also, we do not necessarily have to discard a model based on a low R-Squared value. It’s a better practice to look at the AIC and prediction accuracy on validation sample when deciding on the efficacy of a model.</p>
<p>What about the adjusted R-Squared? As you add terms to your model, the R-Squared value of the new model will always be greater than that of its subset. This is because, since all the variables in the original model is also present, their contribution to explain the depend variable still remains in the super-set and
therefore, whatever new variable we add can only enhance (if not significantly) what was already explained.</p>
<p>Here is how, the adjusted R-Squared value comes to help. Adj R-Squared penalizes total value for the number of terms (read predictors) in your model.</p>
<p>Therefore, when comparing nested models, it is a good practice to look at adj-R-squared value over R-squared.</p>
<p>We also have an adjusted r-square value (we’re now looking at adjusted R-square as a more appropriate metric of variability as the adjusted R-squared increases only if the new term added ends up improving the model more than would be expected by chance). In this model, we arrived in a larger R-squared number of 0.94</p>
<p><strong>iv. How do you know if the model is best fit for your data?</strong></p>
<p>The most common metrics to look at while selecting the model are:</p>
<p>r-squared- Higher the better</p>
<p>Adj. r-squared- Higher the better</p>
<p>AIC- Lower the better</p>
<p>BIC- Lower the better</p>
<p>MAPE (Mean Absolute Percentage Error)- Lower the better</p>
<p>MSE (Mean Squared Error)- Lower the better</p>
<p>Min_Max Accuracy- Higher the better</p>
<p>RMSE- lower the better</p>
<p><strong>v. Residuals</strong></p>
<p>The difference between the observed value of the dependent variable (y) and the predicted value (ŷ) is called the residual (e). Each data point has one residual.</p>
<p>Residual = Observed value - Predicted value
<code class="highlighter-rouge">e = y - ŷ</code></p>
<p>Both the sum and the mean of the residuals are equal to zero. That is, <code class="highlighter-rouge">Σ e = 0</code> and <code class="highlighter-rouge">e = 0</code>.</p>
<p>A residual plot is a graph that shows the residuals on the vertical axis and the independent variable on the horizontal axis. If the points in a residual plot are randomly dispersed around the horizontal axis, a linear regression model is appropriate for the data; otherwise, a non-linear model is more appropriate.</p>
<p><img src="https://duttashi.github.io/images/casestudy-MY-Rubber-residualplot.png" alt="plot" /></p>
<p>Fig-10: Residual plot</p>
<p>From the residual plot in Fig-10, we see the points are randomly distributed, thus the choice of our multiple linear regression was appropriate in predicting the response variable.</p>
<p><strong>F. Model Inference Summary</strong></p>
<p>From the model diagnostics, we see that the model p value and predictor’s p value are less than the significance level, so we know we have a statistically significant model. Also, the R-Sq and Adj R-Sq are comparative to the
original model built on full data.</p>
<p><strong>G. Calculate prediction accuracy and error rates</strong></p>
<p>A simple correlation between the actuals and predicted values can be used as a form of accuracy measure.</p>
<p>A higher correlation accuracy implies that the actuals and predicted values have similar directional movement, i.e. when the actuals values increase the predicted also increase and vice-versa.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> predict<- predict(linear.mod, Test)
> actuals_preds <- data.frame(cbind(actuals=Test$YieldperHectKg, predicteds=predict)) # make actuals_predicteds dataframe.
> correlation_accuracy <- cor(actuals_preds)
> correlation_accuracy
actuals predicteds
actuals 1.0000000 0.9447834
predicteds 0.9447834 1.0000000
</code></pre></div></div>
<p>The prediction accuracy of the model <code class="highlighter-rouge">linear.mod</code> on unseen data is <strong>94%</strong></p>
<p>Now let’s calculate the Min Max accuracy and MAPE</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> min_max_accuracy <- mean (apply(actuals_preds, 1, min) / apply(actuals_preds, 1, max))
> min_max_accuracy
[1] 0.9721728
> mape <- mean(abs((actuals_preds$predicteds - actuals_preds$actuals))/actuals_preds$actuals)
> mape
[1] 0.02970934
</code></pre></div></div>
<p>Looks like we have a good model in here because the MAPE value is <strong>0.029</strong> which is quite low and min max accuracy of <strong>0.97</strong> which is quite high.</p>
<h3 id="5-conclusion">5. Conclusion</h3>
<p>In building a data powered case study, the primary component is the <em>research/business question</em>, that takes precedence above anything else. Experience has taught us that if one cannot think of a feasible research question then its best to perform exploratory data analysis first. This exploratory phase serves many purposes like it gives you a first hand account of the data at hand (<em>in terms of missing value, outliers, skewness, relationships etc</em>). During the exploratory phase, ensure to document and justify data management decisions so as to maintain <em>data accountability</em> and <em>data transparency</em>. This process subsequently leads in formulating the research question. Another approach could be to perform an extensive literature review, find the gap in existing literature, formulate the problem and then acquire the relevant dataset to answer the problem. Both approaches are correct but at the beginner level we would recommend the former approach because you will be more closer to <em>active action</em> rather than <em>passive thinking</em>.</p>
<p>Continuing further, in tree based models where the response or target variable can take a finite set of values are called, <em>classification tree’s</em>. In these tree structures, the <em>leaves</em> represent the <em>class labels</em> and the <em>branches</em> represent the <em>node</em> of features that lead to those class labels. On the contrary the decision trees where the response or target variable can take continuous value <em>(like price of a house)</em> are called <em>regression trees</em>. The term, <em>Classification and Regression Trees (CART)</em> is thus an umbrella term that combines both the procedures.</p>
<p>As we have seen so far, a rigorous model testing must be applied to build an efficient model. The predictors, <code class="highlighter-rouge">ProduceTonne</code> is most significant for prediction of the response variable, <code class="highlighter-rouge">YieldperHectKg</code> and is closely followed by other predictors, <code class="highlighter-rouge">TotalPaidEmployee</code> and <code class="highlighter-rouge">TapAreaHect</code>. We also see that <strong>Regression tree</strong> based approach give <strong>98% accuracy</strong> in predicting the response variable while Random Forest model (<strong>0.049%</strong>) does not even come close.</p>
<p>The reason we achieved such an high predictive accuracy for regression tree based model was because there was a strong positive linear relationship between the predictors and this works best for regression tree accuracy. In Fig-11, we show a graphical representation of which type of decision tree to use. The random forest algorithm would have served a response variable with finite set of values better. A simple and good introduction to understanding random forest is given <a href="http://blog.echen.me/2011/03/14/laymans-introduction-to-random-forests/">here.</a></p>
<p><img src="https://duttashi.github.io/images/which-type-of-decision-tree-to-use.png" alt="plot" /></p>
<p>Fig-11: Which Decision Tree method to use</p>
<p>Another, hat tip for beginners in data science is to look at the response variable in deciding which algorithm to use. In this case study, the response variable was continuous in nature with strong positive linear relationship among the predictors. Therefore, the choice of regression trees was ideal.</p>
<p>The complete code is listed on my Github repository in <a href="https://github.com/duttashi/LearningR/blob/master/scripts/Full%20Case%20Studies/CaseStudy-MY-RubberPlantation.R">here</a></p>
<h3 id="references">References</h3>
<p>Bache, S. M., & Wickham, H. (2014). Magrittr: A forward-pipe operator for R. R package version, 1(1).</p>
<p>Buuren, S., & Groothuis-Oudshoorn, K. (2011). mice: Multivariate imputation by chained equations in R. Journal of Statistical Software, 45(3).</p>
<p>Ihaka, R., & Gentleman, R. (1996). R: a language for data analysis and graphics. Journal of computational and graphical statistics, 5(3), 299-314.</p>
<p>Kursa, M. B., & Rudnicki, W. R. (2010). Feature selection with the Boruta package: Journal of Statistical Software.</p>
<p>Revelle, W. (2014). psych: Procedures for personality and psychological research. Northwestern University, Evanston. R package version, 1(1).</p>
<p>Wickham, H. (2015). plyr: Tools for splitting, applying and combining data. R package version 1.8. 1. R Found. Stat. Comput., Vienna.</p>
<p>Wickham, H. (2016). ggplot2: elegant graphics for data analysis: Springer.</p>
<p>Wickham, H., & Francois, R. (2015). dplyr: A grammar of data manipulation. R package version 0.4, 1, 20.</p>
<p>Kleinbaum, D., Kupper, L., Nizam, A., & Rosenberg, E. (2013). Applied regression analysis and other multivariable methods. Nelson Education.</p>
<![CDATA[Basic assumptions to be taken care of when building a predictive model]]>https://duttashi.github.io/blog/basic-assumptions-to-be-taken-care-of-when-building-a-predictive-model2017-01-18T00:00:00+00:002017-01-18T00:00:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>Before starting to build on a predictive model in R, the following assumptions should be taken care off;</p>
<p><strong>Assumption 1</strong>: <strong>The parameters of the linear regression model must be numeric and linear in nature</strong>.
If the parameters are non-numeric like categorical then use one-hot encoding (python) or dummy encoding (R) to convert them to numeric.</p>
<p><strong>Assumption 2</strong>: <strong>The mean of the residuals is Zero</strong>.
Check the mean of the residuals. If it zero (or very close), then this assumption is held true for that model. This is default unless you explicitly make amends, such as setting the intercept term to zero.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> set.seed(2)
> mod <- lm(dist ~ speed, data=cars)
> mean(mod$residuals)
[1] 8.65974e-17 Since the mean of residuals is approximately zero, this assumption holds true for this model.
</code></pre></div></div>
<p><strong>Assumption 3</strong>: <strong>Homoscedasticity of residuals or equal variance</strong>:
This assumption means that the variance around the regression line is the same for all values of the predictor variable (X).</p>
<p><em>How to check?</em></p>
<p>Once the regression model is built, set</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> par(mfrow=c(2, 2)) then, plot the model using
> plot(lm.mod) This produces four plots. The top-left and bottom-left plots shows how the residuals vary as the fitted values increase. First, I show an example where heteroscedasticity is present. To show this, I use the mtcars dataset from the base R dataset package.
> set.seed(2) # for example reproducibility
> par(mfrow=c(2,2)) # set 2 rows and 2 column plot layout
> mod_1 <- lm(mpg ~ disp, data=mtcars) # linear model
> plot(mod_1)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/lrassum-1.png" alt="image" /></p>
<p>Figure 1: An example of heteroscedasticity in mtcars dataset</p>
<p>From Figure 1, look at the first plot (top-left), as the fitted values along x increase, the residuals decrease and then increase. This pattern is indicated by the red line, which should be approximately flat if the disturbances are homoscedastic. The plot on the bottom left also checks and confirms this, and is more convenient as the disturbance term in Y axis is standardized. In this case, there is a definite pattern noticed. So, there is heteroscedasticity. Lets check this on a different model. Now, I will use the cars dataset from the base r dataset package in R.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> set.seed(2) # for example reproducibility
> par(mfrow=c(2,2)) # set 2 rows and 2 column plot layout
> mod <- lm(dist ~ speed, data=cars)
> plot(mod)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/lrassum-2.png" alt="image" /></p>
<p>Figure 2: An example of homoscedasticity in cars dataset</p>
<p>From Figure 2, looking at the first plot (top-left) the points appear random and the line looks pretty flat, with no increasing or decreasing trend. So, the condition of homoscedasticity can be accepted.</p>
<p><strong>Assumption 4</strong>: <strong>No autocorrelation of residuals</strong>
Autocorrelation is specially applicable for time series data. It is the correlation of a time series with lags of itself. When the residuals are autocorrelated, it means that the current value is dependent of the previous (historic) values and that there is a definite unexplained pattern in the Y variable that shows up in the disturbances.
So how do I check for autocorrelation?
There are several methods for it like the runs test for randomness <code class="highlighter-rouge">(R: lawstat::runs.test())</code>, <code class="highlighter-rouge">durbin-watson test (R: lmtest::dwtest())</code>, <code class="highlighter-rouge">acf plot</code> from the ggplot2 library. I will use the <code class="highlighter-rouge">acf plot()</code>.</p>
<p><em>Method : Visualise with acf plot from the base R package</em></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ?acf # check the help page the acf function
> data(cars) # using the cars dataset from base R
> acf(cars) # highly autocorrelated, see figure 3.
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/acf.png" alt="acf" /></p>
<p>Figure 3: Detecting Auto-Correlation in Predictors</p>
<p>The X axis corresponds to the lags of the residual, increasing in steps of 1. The very first line (to the left) shows the correlation of residual with itself (Lag0), therefore, it will always be equal to 1.
If the residuals were not autocorrelated, the correlation (Y-axis) from the immediate next line onwards will drop to a near zero value below the dashed blue line (significance level). Clearly, this is not the case here. So we can conclude that the residuals are autocorrelated.</p>
<p><strong>Remedial action to resolve Heteroscedasticity</strong></p>
<p>Add a variable named resid1 (can be any name for the variable) of residual as an X variable to the original model. This can be conveniently done using the slide function in DataCombine package. If, even after adding lag1 as an X variable, does not satisfy the assumption of autocorrelation of residuals, you might want to try adding lag2, or be creative in making meaningful derived explanatory variables or interaction terms. This is more like art than an algorithm. For more details, see <a href="https://stat.ethz.ch/R-manual/R-devel/library/stats/html/acf.html">here</a></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(DataCombine)
> set.seed(2) # for example reproducibility
> lmMod <- lm(dist ~ speed, data=cars)
> cars_data <- data.frame(cars, resid_mod1=lmMod$residuals)
> cars_data_1 <- slide(cars_data, Var="resid_mod1", NewVar = "lag1", slideBy = -1)
> cars_data_2 <- na.omit(cars_data_1)
> lmMod2 <- lm(dist ~ speed + lag1, data=cars_data_2)
> acf(lmMod2$residuals)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/acf1-2.png" alt="acf1" /></p>
<p>Figure 4: Homoscedasticity of residuals or equal variance</p>
<p><strong>Assumption 5</strong>: <strong>The residuals and the X variables must be uncorrelated</strong></p>
<p>How to check correlation among predictors, use the cor.test function</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> set.seed(2)
> mod.lm <- lm(dist ~ speed, data=cars)
> cor.test(cars$speed, mod.lm$residuals) # do correlation test
Pearson's product-moment correlation
data: cars$speed and mod.lm$residuals
t = 5.583e-16, df = 48, p-value = 1
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.2783477 0.2783477
sample estimates:
cor
8.058406e-17
</code></pre></div></div>
<p>Since p value is greater than zero, it is high, so the null hypothesis that the true correlation is Zero cannot be rejected. So the assumption holds true for this model.</p>
<p><strong>Assumption 6</strong>: <strong>The number of observations must be greater than the number of predictors or X variables</strong></p>
<p>This can be observed by looking at the data</p>
<p><strong>Assumption 7</strong>: <strong>The variability in predictors or X values is positive</strong>
What this infers to is that the variance in the predictors should not all be the same (or even nearly the same).</p>
<p>How to check this in R?</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> var(cars$dist)
[1] 664.0608
</code></pre></div></div>
<p>The variance in the X variable above is much larger than 0. So, this assumption is satisfied.</p>
<p><strong>Assumption 8</strong>: <strong>No perfect multicollinearity between the predictors</strong>
What this means is that there should not be a perfect linear relationship between the predictors or the explanatory variables.</p>
<p><em>How to check for multicollinearity?</em></p>
<p>Use Variance Inflation Factor (VIF). VIF is a metric computed for every X variable that goes into a linear model. If the VIF of a variable is high, it means the information in that variable is already explained by other X variables present in the given model, which means, more redundant is that variable. according to some references, if the VIF is too large(more than 5 or 10), we consider that the multicollinearity is existent. So, lower the VIF (less than 2) the better. VIF for a X var is calculated as:</p>
<p><img src="https://duttashi.github.io/images/vif.png" alt="vif" /></p>
<p>Figure 5: Variance Inflation Factor</p>
<p>where, <em>Rsq</em> is the Rsq term for the model with given X as response against all other Xs that went into the model as predictors.</p>
<p>Practically, if two of the X′s have high correlation, they will likely have high VIFs. Generally, VIF for an X variable should be less than 4 in order to be accepted as not causing multi-collinearity. The cutoff is kept as low as 2, if you want to be strict about your X variables.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> mod1 <- lm(mpg ~ ., data=mtcars)
> library(car) # load the car package which has the vif()
> vif(mod1)
cyl disp hp drat wt qsec vs am gear carb
15.373833 21.620241 9.832037 3.374620 15.164887 7.527958 4.965873 4.648487 5.357452 7.908747 From here, we can see that the VIF for data mtcars is high for all X’s variables or predictors indicating high multicollinearity.
</code></pre></div></div>
<p><em>How to remedy the issue of multicollinearity</em></p>
<p>In order to solve this problem, there are 2 main approaches. Firstly, we can use robust regression analysis instead of OLS(ordinary least squares), such as ridge regression, lasso regression and principal component regression. On the other hand, statistical learning regression is also a good method, like regression tree, bagging regression, random forest regression, neural network and SVR(support vector regression). In R language, the function <code class="highlighter-rouge">lm.ridge()</code> in package <a href="https://cran.r-project.org/web/packages/MASS/index.html">MASS</a> could implement ridge regression(linear model). The sample codes and output as follows</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(corrplot)
corrplot(cor(mtcars[, -1]))
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/corplot.png" alt="corrplot" /></p>
<p>Figure 6: Correlation Plot</p>
<p><strong>Assumption 9</strong>: <strong>The normality of the residuals</strong>
The residuals should be normally distributed. This can be visually checked by using the qqnorm() plot.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> par(mfrow=c(2,2))
> mod <- lm(dist ~ speed, data=cars)
> plot(mod)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/qqnorm-plot.png" alt="qqnormplot" /></p>
<p>Figure 7: The qqnorm plot to depict the residuals</p>
<p>The qqnorm() plot in top-right evaluates this assumption. If points lie exactly on the line, it is perfectly normal distribution. However, some deviation is to be expected, particularly near the ends (note the upper right), but the deviations should be small, even lesser that they are here.</p>
<p><strong>Check the aforementioned assumptions automatically</strong></p>
<p>The <code class="highlighter-rouge">> gvlma()</code> from the gvlma package offers to check for the important assumptions on a given linear model.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> install.packages("gvlma")
> library(gvlma)
> par(mfrow=c(2,2)) # draw 4 plots in same window
> mod <- lm(dist ~ speed, data=cars)
Call:
lm(formula = dist ~ speed, data = cars)
Coefficients:
(Intercept) speed
-17.579 3.932
ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
Level of Significance = 0.05
Call:
gvlma::gvlma(x = mod)
Value p-value Decision
Global Stat 15.801 0.003298 Assumptions NOT satisfied!
Skewness 6.528 0.010621 Assumptions NOT satisfied!
Kurtosis 1.661 0.197449 Assumptions acceptable.
Link Function 2.329 0.126998 Assumptions acceptable.
Heteroscedasticity 5.283 0.021530 Assumptions NOT satisfied!
> plot(mod)
</code></pre></div></div>
<p>Three of the assumptions are not satisfied. This is probably because we have only 50 data points in the data and having even 2 or 3 outliers can impact the quality of the model. So the immediate approach to address this is to remove those outliers and re-build the model. Take a look at the diagnostic plot below.</p>
<p><img src="https://duttashi.github.io/images/rplot1-1.png" alt="diagnosticplot" /></p>
<p>Figure 8: The diagnostic plot</p>
<p>As we can see in the above plot (figure 7), the data points: 23, 35 and 49 are marked as outliers. Lets remove them from the data and re-build the model.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> mod <- lm(dist ~ speed, data=cars[-c(23, 35, 49), ])
> gvlma::gvlma(mod)
Call:
lm(formula = dist ~ speed, data = cars[-c(23, 35, 49), ])
Coefficients:
(Intercept) speed
-15.137 3.608
ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
Level of Significance = 0.05
Call:
gvlma::gvlma(x = mod)
Value p-value Decision
Global Stat 7.5910 0.10776 Assumptions acceptable.
Skewness 0.8129 0.36725 Assumptions acceptable.
Kurtosis 0.2210 0.63831 Assumptions acceptable.
Link Function 3.2239 0.07257 Assumptions acceptable.
Heteroscedasticity 3.3332 0.06789 Assumptions acceptable.
</code></pre></div></div>
<p>Post removing the outliers we can see from the results that all our assumptions have been met in the new model.</p>
<p><img src="https://duttashi.github.io/images/rplot1-3.png" alt="normalisedmodelplot" /></p>
<p>Figure 9: Normalised variables plot</p>
<p>Note: For a good regression model, the red smoothed line should stay close to the mid-line and no point should have a large cook’s distance (i.e. should not have too much influence on the model.). On plotting the new model, the changes look minor, it is more closer to conforming with the assumptions.</p>
<p><strong>End thoughts</strong>
Given a dataset, its very important to first ensure that it fulfills the aforementioned assumptions before you begin with any sort or inferential or predictive modeling. Moreover, by taking care of these assumptions you are ensuring a robust model that will survive and yield high predictive values.</p>
<![CDATA[Data Transformations in R]]>https://duttashi.github.io/blog/data-transform2017-01-11T00:00:00+00:002017-01-11T00:00:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>A number of reasons can be attributed to when a predictive model crumples such as:</p>
<ul>
<li>
<p>Inadequate data pre-processing</p>
</li>
<li>
<p>Inadequate model validation</p>
</li>
<li>
<p>Unjustified extrapolation</p>
</li>
<li>
<p>Over-fitting</p>
</li>
</ul>
<p>(Kuhn, 2013)</p>
<p>Before we dive into data preprocessing, let me quickly define a few terms that I will be commonly using.</p>
<p><em>Predictor/Independent/Attributes/Descriptors</em> – are the different terms that are used as input for the prediction equation.</p>
<p><em>Response/Dependent/Target/Class/Outcome</em> – are the different terms that are referred to the outcome event that is to be predicted.</p>
<p>In this article, I am going to summarize some common data pre-processing approaches with examples in R</p>
<p>a. <strong>Centering and Scaling</strong></p>
<p>Variable centering is perhaps the most intuitive approach used in predictive modeling. To center a predictor variable, the average predictor value is subtracted from all the values. as a result of centering, the predictor has zero mean.
To scale the data, each predictor value is divided by its standard deviation (sd). This helps in coercing the predictor value to have a <code class="highlighter-rouge">sd</code> of one. Needless to mention, centering and scaling will work for continuous data. The drawback of this activity is loss of interpretability of the individual values.
An R example:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Load the default datasets
> library(datasets)
> data(mtcars)
> dim(mtcars)
32 11
> str(mtcars)
'data.frame': 32 obs. of 11 variables:
$ mpg : num 21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
$ cyl : num 6 6 4 6 8 6 8 4 4 6 ...
$ disp: num 160 160 108 258 360 ...
$ hp : num 110 110 93 110 175 105 245 62 95 123 ...
$ drat: num 3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
$ wt : num 2.62 2.88 2.32 3.21 3.44 ...
$ qsec: num 16.5 17 18.6 19.4 17 ...
$ vs : num 0 0 1 1 0 1 0 1 1 1 ...
$ am : num 1 1 1 0 0 0 0 0 0 0 ...
$ gear: num 4 4 4 3 3 3 3 4 4 4 ...
$ carb: num 4 4 1 1 2 1 4 2 2 4 ...
> cov(mtcars$disp, mtcars$cyl) # check for covariance
[1] 199.6603
> mtcars$disp.scl<-scale(mtcars$disp, center = TRUE, scale = TRUE)
> mtcars$cyl.scl<- scale(mtcars$cyl, center = TRUE, scale = TRUE)
> cov(mtcars$disp.scl, mtcars$cyl.scl) # check for covariance in scaled data
[,1]
[1,] 0.9020329
</code></pre></div></div>
<p>b. <strong>Resolving Skewness</strong></p>
<p>Skewness is a measure of shape. A common appraoch to check for skewness is to plot the predictor variable. As a rule, negative skewness indicates that the mean of the data values is less than the median, and the data distribution is left-skewed. Positive skewness would indicates that the mean of the data values is larger than the median, and the data distribution is right-skewed.</p>
<ul>
<li>If the skewness of the predictor variable is 0, the data is perfectly symmetrical,</li>
<li>If the skewness of the predictor variable is less than -1 or greater than +1, the data is highly skewed,</li>
<li>If the skewness of the predictor variable is between -1 and -1/2 or between +1 and +1/2 then the data is moderately skewed,</li>
<li>If the skewness of the predictor variable is -1/2 and +1/2, the data is approximately symmetric.</li>
</ul>
<p>I will use the function <code class="highlighter-rouge">skewness</code> from the <code class="highlighter-rouge">e1071 package</code> to compute the skewness coefficient</p>
<p>An R example:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(e1071)
> engine.displ<-skewness(mtcars$disp) > engine.displ
[1] 0.381657
</code></pre></div></div>
<p>So the variable displ is moderately positively skewed.</p>
<p>c. <strong>Resolving Outliers</strong></p>
<p>The outliers package provides a number of useful functions to systematically extract outliers. Some of these are convenient and come handy, especially the <code class="highlighter-rouge">outlier()</code> and <code class="highlighter-rouge">scores()</code> functions.</p>
<p><em>Outliers</em></p>
<p>The function <code class="highlighter-rouge">outliers()</code> gets the extreme most observation from the mean.
If you set the argument <code class="highlighter-rouge">opposite=TRUE</code>, it fetches from the other side.</p>
<p>An R example:</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> set.seed(4680) # for code reproducibility
> y<- rnorm(100) # create some dummy data > library(outliers) # load the library
> outlier(y)
[1] 3.581686
> dim(y)<-c(20,5) # convert it to a matrix > head(y,2)# Look at the first 2 rows of the data
[,1] [,2] [,3] [,4] [,5]
[1,] 0.5850232 1.7782596 2.051887 1.061939 -0.4421871
[2,] 0.5075315 -0.4786253 -1.885140 -0.582283 0.8159582
> outlier(y) # Now, check for outliers in the matrix
[1] -1.902847 -2.373839 3.581686 1.583868 1.877199
> outlier(y, opposite = TRUE)
[1] 1.229140 2.213041 -1.885140 -1.998539 -1.571196
</code></pre></div></div>
<p>There are two aspects the the <code class="highlighter-rouge">scores()</code> function.
Compute the normalised scores based on <code class="highlighter-rouge">z</code>, <code class="highlighter-rouge">t</code>, <code class="highlighter-rouge">chisq</code> etc.</p>
<p>Find out observations that lie beyond a given percentile based on a given score.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> set.seed(4680)
> x = rnorm(10)
> scores(x) # z-scores => (x-mean)/sd
[1] 0.9510577 0.8691908 0.6148924 -0.4336304 -1.6772781...
> scores(x, type="chisq") # chi-sq scores => (x - mean(x))^2/var(x)
[1] 0.90451084 0.75549262 0.37809269 0.18803531 2.81326197 . . .
> scores(x, type="t") # t scores
[1] 0.9454321 0.8562050 0.5923010 -0.4131696 -1.9073009
> scores(x, type="chisq", prob=0.9) # beyond 90th %ile based on chi-sq
[1] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
> scores(x, type="chisq", prob=0.95) # beyond 95th %ile
[1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
> scores(x, type="z", prob=0.95) # beyond 95th %ile based on z-scores
[1] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
> scores(x, type="t", prob=0.95) # beyond 95th %ile based on t-scores
[1] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
</code></pre></div></div>
<p>d. <strong>Outlier Treatment</strong></p>
<p>Once the outliers are identified, you may rectify it by using one of the following approaches.</p>
<ul>
<li>Imputation</li>
</ul>
<p>Imputation with mean / median / mode.</p>
<ul>
<li>Capping</li>
</ul>
<p>For missing values that lie outside the <code class="highlighter-rouge">1.5 * IQR limits</code>, we could cap it by replacing those observations outside the lower limit with the value of <code class="highlighter-rouge">5th%ile</code> and those that lie above the upper limit, with the value of <code class="highlighter-rouge">95th%ile</code>. For example, it can be done like this as shown;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> par(mfrow=c(1, 2)) # for side by side plotting
> x <- mtcars$mpg > plot(x)
> qnt <- quantile(x, probs=c(.25, .75), na.rm = T)
> caps <- quantile(x, probs=c(.05, .95), na.rm = T)
> H <- 1.5 * IQR(x, na.rm = T)
> x[x < (qnt[1] - H)] <- caps[1]
> x[x > (qnt[2] + H)] <- caps[2]
> plot(x)
</code></pre></div></div>
<p>e. Missing value treatment</p>
<ul>
<li>Impute Missing values with median or mode</li>
<li>Impute Missing values based on K-nearest neighbors</li>
</ul>
<p>Use the library <code class="highlighter-rouge">DMwR</code> or <code class="highlighter-rouge">mice</code> or <code class="highlighter-rouge">rpart</code>. If using <code class="highlighter-rouge">DMwR</code>, for every observation to be imputed, it identifies ‘k’ closest observations based on the euclidean distance and computes the weighted average (weighted based on distance) of these ‘k’ obs. The advantage is that you could impute all the missing values in all variables with one call to the function. It takes the whole data frame as the argument and you don’t even have to specify which variabe you want to impute. But be cautious not to include the response variable while imputing.</p>
<p>There are many other types of transformations like treating colinearity, dummy variable encoding, covariance treatment which I will cover in another post.</p>
<p><strong>Reference</strong></p>
<p>Kuhn, M., Johnson, K. (2013). Applied predictive modeling (pp. 389-400). New York: Springer.</p>
<![CDATA[Sold! How do home features add up to its price tag?]]>https://duttashi.github.io/blog/sold-how-do-home-features-add-up-to-its-price-tag2016-09-06T13:20:00+00:002016-09-06T13:20:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p><span style="color:#000000;">I begin with a new project. It is from the</span> <a href="https://www.kaggle.com/c/house-prices-advanced-regression-techniques" target="_blank">Kaggle</a> <span style="color:#000000;">playground wherein the objective is to build a regression model <em>(as the response variable or the outcome or dependent variable is continuous in nature) </em>from a given set of predictors or independent variables. </span></p>
<p><span style="color:#000000;">My motivation to work on this project are the following;</span></p>
<ul>
<li><span style="color:#000000;">Help me to learn and improve upon <em>feature engineering</em> and advanced regression algorithms like <em>random forests, gradient boosting with xgboost</em></span></li>
<li><span style="color:#000000;">Help me in articulating compelling data powered stories </span></li>
<li><span style="color:#000000;">Help me understand and build a complete end to end data powered solution</span></li>
</ul>
<h4><strong>The Dataset</strong></h4>
<p><span style="color:#000000;">From the Kaggle page,</span> “<em>The <a class="pdf-link" href="http://www.amstat.org/publications/jse/v19n3/decock.pdf" target="_blank">Ames Housing dataset</a> <span style="color:#000000;">was compiled by Dean De Cock for use in data science education. It’s an incredible alternative for data scientists looking for a modernized and expanded version of the often cited Boston Housing dataset.”</span></em></p>
<h4><strong>The Data Dictionary</strong></h4>
<p><span style="color:#000000;">The data dictionary can be accessed from</span> <a href="http://www.amstat.org/publications/jse/v19n3/decock/DataDocumentation.txt" target="_blank">here</a>.</p>
<h4><strong>Objective</strong></h4>
<p><span style="color:#000000;">With 79 explanatory variables describing (almost) every aspect of residential homes in Ames, Iowa, this competition challenges you to predict the final price of each home.</span></p>
<h4 class="page-name"><strong>Model Evaluation</strong></h4>
<p><span style="color:#000000;">Submissions are evaluated on</span> <a href="https://en.wikipedia.org/wiki/Root-mean-square_deviation" target="_blank">Root-Mean-Squared-Error (RMSE)</a> <span style="color:#000000;">between the logarithm of the predicted value and the logarithm of the observed sales price. (Taking logs means that errors in predicting expensive houses and cheap houses will affect the result equally.) In simple terms this means that the lower the RMSE value, greater is the accuracy of your prediction model.</span></p>
<h4><strong>About the dataset</strong></h4>
<p><span style="color:#000000;">The dataset is split into training and testing files where the training dataset has <em>81</em> variables in <em>1460</em> rows and the testing dataser has <em>80</em> variables in <em>1459</em> rows. These variables focus on the quantity and quality of many physical attributes of the real estate property. <em> </em></span></p>
<p><span style="color:#000000;">There are a large number of categorical variables (23 nominal, 23 ordinal) associated with this data set. They range from 2 to 28 classes with the smallest being STREET (gravel or paved) and the largest being NEIGHBORHOOD (areas within the Ames city limits). The nominal variables typically identify various types of dwellings, garages, materials, and environmental conditions while the ordinal variables typically rate various items within the property.</span></p>
<p><span style="color:#000000;">The 14 discrete variables typically quantify the number of items occurring within the house. Most are specifically focused on the number of kitchens, bedrooms, and bathrooms (full and half) located in the basement and above grade (ground) living areas of the home.</span></p>
<p><span style="color:#000000;">In general the 20 continuous variables relate to various area dimensions for each observation. In addition to the typical lot size and total dwelling square footage found on most common home listings, other more specific variables are quantified in the data set.</span></p>
<p><span style="color:#000000;">“<em>A strong analysis should include the interpretation of the various coefficients, statistics, and plots associated with their model and the verification of any necessary assumptions.”</em></span></p>
<p><span style="color:#000000;"><em><strong>An interesting feature of the dataset</strong></em> is that <span style="text-decoration:underline;">several of the predictors are labelled as NA when actually they are not missing values and correspond to actual data points</span>. This can be verified from the data dictionary where variable like Alley, Pool etc have NA value that correspond to <em>No Alley Access</em> and <em>No Pool </em>respectively.<em> </em>This <a href="http://stackoverflow.com/questions/19379081/how-to-replace-na-values-in-a-table-for-selected-columns-data-frame-data-tab" target="_blank">SO question</a> that was answered by the user ‘flodel’ solves this problem of recoding specific columns of a dataset. </span></p>
<p><span style="color:#000000;">A total of <i>357</i> missing values are present in training predictors (<em>LotFrontage-259, MasVnrType-8, MasVnrArea-8, Electrical-1, GarageYrBlt-81</em>) and <i>358</i> missing values in testing dataset predictors (<em>MSZoning-4,</em> <em>LotFrontage-227, Exterior1st-1, Exterior2nd-1, MasVnrType-16, MasVnArea-15, BsmtFinSF1-1, BsmtFinType2-1, BsmtFinSF2-1, BsmtUnfSF-1, TotalBsmtSF-1, BsmtFullBath-2, BsmtHalfBath-2, KitchenQual-1, Functional-2, GarageYrBlt-78, SaleType-1</em>).</span></p>
<h4><strong>Data Preprocessing</strong></h4>
<p><span style="color:#000000;">Some basic problems that need to be solved first namely, <em>data dimensionality reduction, missing value treatment, correlation, dummy coding. </em>A common question that most ask is that how to determine the relevant predictors in a high dimensional dataset as this. The approach that I will use for dimensionality reduction will be two fold, first I will check for zero variance predictors. </span></p>
<h4>(a) <em>Check for Near Zero Variance Predictors</em></h4>
<p><span style="color:#000000;">A predictor with zero variability does not contribute anything to the prediction model and can be removed. </span></p>
<p><span style="color:#000000;"><em><span style="text-decoration:underline;">Computing</span>: </em>This can easily be accomplished by using the <em>nearZeroVar() </em>method from the <em>caret package. </em>In training dataset, there are 21 near zero variance variables namely (</span><span style="color:#000000;"><em>‘Street’ ‘LandContour’ ‘Utilities’ ‘LandSlope’ ‘Condition2’ ‘RoofMatl’ ‘BsmtCond’ ‘BsmtFinType2’ ‘BsmtFinSF2’ ‘Heating’ ‘LowQualFinSF’ ‘KitchenAbvGr’ ‘Functional’ ‘GarageQual’ ‘GarageCond’ ‘EnclosedPorch’ ‘X3SsnPorch’ ‘ScreenPorch’ ‘PoolArea’ ‘MiscFeature’ ‘MiscVal’</em>) <em>and in the testing dataset there are 19 near zero variance predictors namely (‘Street’ ‘Utilities’ ‘LandSlope’ ‘Condition2’ ‘RoofMatl’ ‘BsmtCond’ ‘BsmtFinType2’ ‘Heating’ ‘LowQualFinSF’ ‘KitchenAbvGr’ ‘Functional’ ‘GarageCond’ ‘EnclosedPorch’ ‘X3SsnPorch’ ‘ScreenPorch’ ‘PoolArea’ ‘MiscVal’). </em>Post removal of these predictors from both the training and testing dataset, the data dimension is reduced to <em>60 predictors for train data </em>and <em>61 predictors </em>each. </span></p>
<p><strong>(b) <em>Missing data treatment</em></strong></p>
<p><span style="color:#000000;">There are two types of missing data;</span></p>
<p><span style="color:#000000;">(i) MCAR (Missing Completetly At Random) & (ii) MNAR (Missing Not At Random)</span></p>
<p><span style="color:#000000;">Usually, MCAR is the desirable scenario in case of missing data. For this analysis I will assume that MCAR is at play. Assuming data is MCAR, too much missing data can be a problem too. Usually a safe maximum threshold is 5% of the total for large datasets. If missing data for a certain feature or sample is more than 5% then you probably should leave that feature or sample out. We therefore check for features (columns) and samples (rows) where more than 5% of the data is missing using a simple function. Some good references are</span> <a href="http://stackoverflow.com/questions/4862178/remove-rows-with-nas-missing-values-in-data-frame" target="_blank">1</a> <span style="color:#000000;">and</span> <a href="http://stackoverflow.com/questions/4605206/drop-data-frame-columns-by-name" target="_blank">2</a>.</p>
<p><span style="color:#000000;"><span style="text-decoration:underline;"><em>Computing</em></span>: I have used the <em>VIM package </em>in R for missing data visualization. I set the threshold at 0.80, any predictors equal to or above this threshold need no imputation and should be removed. Post removal of the near zero variance predictors, I next check for high missing values and I find that there are no predictors with high missing values in either the train or test data. </span></p>
<p><span style="color:#000080;">Important Note:</span> <span style="color:#000000;">As per this <a href="https://www.r-bloggers.com/imputing-missing-data-with-r-mice-package/" target="_blank">r-blogger’s post</a>, it is not advisable to use mean imputation for continuous predictors because it can affect the variance in the data. Also, one should avoid using the mode imputation for categorical variables so I use the <em>mice library</em> for missing valueimputation for the continuous variables. </span></p>
<p><em><strong>(c) Correlation treatment</strong></em></p>
<p><span style="color:#000000;"><span class="Apple-style-span">Correlation refers to a technique used to measure the relationship between two or more variables.</span><span class="Apple-style-span">When two objects are correlated, it means that they vary together.</span><span class="Apple-style-span">Positive correlation means that high scores on one are associated with high scores on the other, and that low scores on one are associated with low scores on the other. Negative correlation, on the other hand, means that high scores on the first thing are associated with low scores on the second. Negative correlation also means that low scores on the first are associated with high scores on the second.</span></span></p>
<p style="font-weight:300;"><span style="color:#000000;">Pearson <em>r</em> is a statistic that is commonly used to calculate bivariate correlations. Or better said, its checks for linear relations. </span></p>
<p style="font-weight:300;"><span style="color:#000000;">For an Example Pearson <em>r</em> = -0.80, <em>p</em> < .01. What does this mean?</span></p>
<p style="font-weight:300;"><span style="color:#000000;">To interpret correlations, four pieces of information are necessary.</span>
<span style="color:#000000;"><b><strong>1. <em>The numerical value of the correlation coefficient.</em></strong></b>Correlation coefficients can vary numerically between 0.0 and 1.0. The closer the correlation is to 1.0, the stronger the relationship between the two variables. A correlation of 0.0 indicates the absence of a relationship. If the correlation coefficient is –0.80, which indicates the presence of a strong relationship.</span></p>
<p style="font-weight:300;"><span style="color:#000000;"><b style="line-height:1.7;"><strong><em>2. The sign of the correlation coefficient</em>.</strong></b>A positive correlation coefficient means that as variable 1 increases, variable 2 increases, and conversely, as variable 1 decreases, variable 2 decreases. In other words, the variables move in the same direction when there is a positive correlation. A negative correlation means that as variable 1 increases, variable 2 decreases and vice versa. In other words, the variables move in opposite directions when there is a negative correlation. The negative sign indicates that as class size increases, mean reading scores decrease.</span></p>
<p style="font-weight:300;"><span style="color:#000000;"><b style="line-height:1.7;"><strong><em>3. The statistical significance of the correlation. </em></strong></b><span style="line-height:1.7;">A statistically significant correlation is indicated by a probability value of less than 0.05. This means that the probability of obtaining such a correlation coefficient by chance is less than five times out of 100, so the result indicates the presence of a relationship.</span></span></p>
<p><span style="color:#000000;">In any data anlysis activity, the analyst should always check for highly correlated variables and remove them from the dataset because correlated predictors do not quantify </span></p>
<p><span style="color:#000000;"><strong>4. <i>The effect size of the correlation.</i></strong>For correlations, the effect size is called the coefficient of determination and is defined as <i>r</i><sup>2</sup>. The coefficient of determination can vary from 0 to 1.00 and indicates that the proportion of variation in the scores can be predicted from the relationship between the two variables.</span></p>
<p><span style="color:#000000;">A correlation can only indicate the presence or absence of a relationship, not the nature of the relationship. <i><strong>Correlation is not causation</strong>.</i></span></p>
<p><span style="color:#000000;">How Problematic is Multicollinearity?</span></p>
<p><span style="color:#000000;">Moderate multicollinearity may not be problematic. However, severe multicollinearity is a problem because it can increase the variance of the coefficient estimates and make the estimates very sensitive to minor changes in the model. The result is that the coefficient estimates are unstable and difficult to interpret. Multicollinearity saps the statistical power of the analysis, can cause the coefficients to switch signs, and makes it more difficult to specify the correct model. According to Tabachnick & Fidell (1996) the independent variables with a bivariate correlation more than .70 should not be included in multiple regression analysis.</span></p>
<p><span style="text-decoration:underline;color:#000000;"><em><strong>Computing</strong></em></span></p>
<p><span style="color:#000000;">To detect highly correlated predictors in the data, I used the <em>findCorrelation()</em> method of the caret library and I find that there are four predictors in the training dataset with more than 80% correlation and these are “YearRemodAdd”,”OverallCond”,”BsmtQual”,”Foundation” which I then remove from the train data thereby reducing the data dimension to 56. I follow the similar activity for the test data and I find that there are two predictors with more than 80% correlation and these are “Foundation” “LotShape” which I then remove from the test data.
The final data dimensions are 1460 rows in 56 columns in train data and 1460 rows in 59 columns in the test data.
</span></p>
<p><span style="color:#000000;">The R code used in this post can be can be accessed on my <a style="color:#000000;" href="https://github.com/duttashi/House-Price-Prediction/blob/master/scripts/data_preproc.R" target="_blank">github</a> account and my Kaggle notebook can be viewed <a href="https://www.kaggle.com/ashishdutt/house-prices-advanced-regression-techniques/ahoy-all-relevant-guests-on-board-let-s-sail" target="_blank">here</a>. </span></p>
<p><span style="color:#000000;">In the next post, I will discuss on the issue of outlier detection, skewness resolution and data visualization.</span></p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<![CDATA[Learning from data science competitions- baby steps]]>https://duttashi.github.io/blog/learning-from-data-science-competitions-xgboost-algorithm2016-08-24T08:17:00+00:002016-08-24T08:17:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>Off lately a considerable number of winner machine learning enthusiasts have used <a href="https://github.com/dmlc/xgboost" target="_blank">XGBoost</a> as their predictive analytics solution. This algorithm has taken a preceedence over the traditional tree based algorithms like Random Forests and Neural Networks.</p>
<p>The acronym <strong>Xgboost </strong>stands for e<strong>X</strong>treme G<strong>radient </strong><strong>B</strong>oosting package. The creators of this algorithm presented its <a href="https://www.kaggle.com/tqchen/otto-group-product-classification-challenge/understanding-xgboost-model-on-otto-data" target="_blank">implementation</a> by winning the Kaggle Otto Group competition. Another interesting tutorial is listed <a href="https://www.r-bloggers.com/an-introduction-to-xgboost-r-package/">here</a> and the complete documentation can be seen <a href="http://xgboost.readthedocs.io/en/latest/R-package/xgboostPresentation.html">here</a>. This page lists a comprehensive list of <a href="https://github.com/dmlc/xgboost/tree/master/demo#tutorials" target="_blank">awesome tutorials</a> on it and this one shows <a href="http://fr.slideshare.net/MichaelBENESTY/feature-importance-analysis-with-xgboost-in-tax-audit" target="_blank">feature importance</a> It is a classification algorithm and the reasons of its superior efficiency are,</p>
<ul>
<li>It's written in C++</li>
<li>It can be multithreaded on a single machine</li>
<li>It preprocesses the data before the training algorithm.</li>
</ul>
<p>Unlike its previous tree based predecessors it takes care of many of the inherent problems associated with tree based classification. For example, “By setting the parameter <code>early_stopping</code>,<code>xgboost</code> will terminate the training process if the performance is getting worse in the iteration.” [1]</p>
<p>As with all machine learning algorithms, xgboost works on numerical data. If categorical data is there then use one-hot encoding from the R caret package to transform the categorical data (factors) to numerical dummy variables that can be used by the algorithm. Here is a good <a href="http://stackoverflow.com/questions/24142576/one-hot-encoding-in-r-categorical-to-dummy-variables">SO discussion</a> on one-hot encoding in R. This <a href="https://www.quora.com/What-is-one-hot-encoding-and-when-is-it-used-in-data-science" target="_blank">Quora thread</a> discusses the question on “<em>when should you be using one-hot encoding in data science?”.</em></p>
<p>Okay, enough of background information. Now let’s see some action.</p>
<p><strong>Problem Description</strong></p>
<p>The objective is to predict whether a donor has donated blood in March 2007. To this effect, the dataset for this study is derived from <a href="https://www.drivendata.org/competitions/2/page/7/" target="_blank">DrivenData</a> which incidentally is also hosting a practice data science competition on the same.</p>
<p><strong>Problem Type: Classification</strong>.</p>
<p>And how did I figure this out? Well, one has to read the problem description carefully as well as the submission format. In this case, the submission format categorically states that the response variable to be either 1 or 0 which is proof enough that this is a classification problem.</p>
<p><strong>Choice of predictive algorithm</strong></p>
<p>Boy, that really let my head spinning for some time. You see I was torn between the traditionalist approach and the quickie (get it out there) approach. First, I thought let me learn and explore what story is the data trying to tell me (<em>traditionalist approach) </em>but then I gave up on this idea because of my past experiences. Once I venture this path, I get stuck somewhere or keep digging in a quest to perfect my solution and time slips away. So this time, I said to myself, “<em>Enough! let me try the quickie approach that is get it (read the solution) out of the lab as quickly as possible. And I can later continue to improve the solution”</em>. So following this intuition and a very much required <em>self-morale boost</em> (<em>that is what happens to you when you are out in the laboratory all by yourself</em>) I decided to choose XGBoost as the preliminary predictive classification algorithm. Being neck deep into clustering algorithms (<em>which is my research area) </em>and if truth be told I never really had a penchant for supervised algorithms (<em>once again a gut feeling that they were too easy because you already know the outcome. Dammn! I was so wrong)</em></p>
<p><strong>Choice of tool</strong></p>
<p>For sometime now, I had been juggling between the choice of being a pythonist or an R user, <em>“To be or not to be, that is the question”. </em> The worldwide web has some great resources on this discussion and you can take your pick. In my case, I decided to chose and stick with R because of two reasons, primarily its a statistical programming language and two predictive analytics or machine learning has its roots in statistics.</p>
<p><strong>The</strong> <strong>Strategy</strong></p>
<p>“Visualize it, <em>Clean it, Smoothe it, Publish it”. </em></p>
<p>After reading the data in R, my first step was to plot as many meaningful graphs as possible to detect a trend or a relationship. I started with line plots but before I get into that, a brief about the dataset. The dataset was pre-divided into training and testing data. The training data had 576 observations in 6 continuous variables of which the last variable was the response. Similarly, the test data had 200 observations in 5 continuous variables.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Read in the data
train.data<- read.csv("data//blood_donation_train.csv", sep = ",", header=TRUE)
test.data<-read.csv("data//blood_donation_test.csv", sep = ",", header=TRUE)
# Check the data structure
> str(train.data)
'data.frame': 576 obs. of 6 variables:
$ ID : int 619 664 441 160 358 335 47 164 736 436 ...
$ Months.since.Last.Donation : int 2 0 1 2 1 4 2 1 5 0 ...
$ Number.of.Donations : int 50 13 16 20 24 4 7 12 46 3 ...
$ Total.Volume.Donated..c.c..: int 12500 3250 4000 5000 6000 1000 1750 3000 11500 750 ...
$ Months.since.First.Donation: int 98 28 35 45 77 4 14 35 98 4 ...
$ Made.Donation.in.March.2007: int 1 1 1 1 0 0 1 0 1 0 ...
> str(test.data)
'data.frame': 200 obs. of 5 variables:
$ ID : int 659 276 263 303 83 500 530 244 249 728 ...
$ Months.since.Last.Donation : int 2 21 4 11 4 3 4 14 23 14 ...
$ Number.of.Donations : int 12 7 1 11 12 21 2 1 2 4 ...
$ Total.Volume.Donated..c.c..: int 3000 1750 250 2750 3000 5250 500 250 500 1000 ...
$ Months.since.First.Donation: int 52 38 4 38 34 42 4 14 87 64 ...
</code></pre></div></div>
<p><b> a. Data Visualization</b></p>
<p>I first started with the base R graphics library, you know commands like <em>hist() or plot()</em> but honestly speaking the visualization was draconian, awful. You see it did not appeal to me at all and thus my grey cells slumbered. Then, I chose the ggplot2 library. Now, that was something. The visualizations were very appealing inducing the grey mater to become active.</p>
<p><em><strong>Learning note</strong>: So far, I have not done any data massaging activity like centering or scaling. Why? The reason is one will find patterns in the raw data and not in a centered or scaled data.</em></p>
<p>Off the numerous graphs I plotted, I finally settled on the ones that displayed some proof of variablity. I wanted to see if there was a cohort of people who were donating more blood than normal. I was interested in this hypothesis because there are some cool folks out there (pun intended) for whom blood donation is a business. Anyway, if you look at the line plot 1 that explores my perceived hypothesis, you will notice a distinct cluster of people who donated between 100 cc to 5000 cc in approx 35 months range.</p>
<p><img src="https://duttashi.github.io/images/rplot-2-1.png" alt="image" /></p>
<p>Line plot 1: Distribution of total blood volume donated in year 2007-2010</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>highDonation2<- subset(train.data, Total.Volume.Donated..c.c..>=100 & Total.Volume.Donated..c.c..<=5000 & Months.since.Last.Donation<=35)
p5<- ggplot() +geom_line(aes(x=Total.Volume.Donated..c.c.., y=Months.since.Last.Donation, colour=Total.Volume.Donated..c.c..),size=1 ,data=highDonation2, stat = "identity")
p5 # Visualize it
highDonation2.3<- subset(train.data, Total.Volume.Donated..c.c..>800 & Total.Volume.Donated..c.c..<=5000 & Months.since.Last.Donation<=35)
str(highDonation2.3)
p6.3<- ggplot() +geom_line(aes(x=Total.Volume.Donated..c.c.., y=Months.since.Last.Donation, colour=Total.Volume.Donated..c.c..),size=1 ,data=highDonation2.3, stat = "identity")
p6.3 # Visualize it
highDonation2.4<- subset(train.data, Total.Volume.Donated..c.c..>2000 & Total.Volume.Donated..c.c..<=5000 & Months.since.Last.Donation<=6)
p6.2<- ggplot() +geom_line(aes(x=Total.Volume.Donated..c.c.., y=Months.since.Last.Donation, colour=Total.Volume.Donated..c.c..),size=1 ,data=highDonation2.4, stat = "identity")
p6.2 # Visualize it
</code></pre></div></div>
<p>I then took a subset of these people and I noticed that total observations was 562 which is just 14 observations less than the original dataset. Hmm.. maybe I should narrow my range down a bit more. so then I narrowed the range between 1000 cc to 5000 cc of blood donated in the 1 year and I find there are 76 people and when I further narrow it down to between 2000-5000 cc of blood donation in 6 months, there are 55 people out of 576 as shown in line plot 2.</p>
<p><img src="https://duttashi.github.io/images/rplot-2-2.png" alt="image" /></p>
<p>Line plot 2: Distribution of total blood volume (in cc) donated in 06 months of 2007</p>
<p>If you look closely at the line plot 2, you will notice a distinct spike between 4 and 6 months. (<em>Ohh baby, things are getting soo hot and spicy now, I can feel the mounting tension). </em>Let’s plot it. And lo behold there are 37 good folks who have donated approx 2000 cc to 5000 cc in the months of May and June, 2007.</p>
<p><img src="https://duttashi.github.io/images/rplot-2-3.png" alt="image" /></p>
<p>Line plot 3: Distribution of total blood volume (in cc) donated in May & June of 2007</p>
<p>I finally take this exploration one step further wherein I search for a pattern or a group of people who had made more than 20 blood donations in six months of year 2007. And they are 08 such good guys who were hyperactive in blood donation. This I show in line plot 4.</p>
<p><img src="https://duttashi.github.io/images/rplot2-4.png" alt="image" /></p>
<p>Line plot 4: Distribution of high blood donors in six months of year 2007</p>
<p>This post is getting too long now. I thin it will not be easier to read and digest it. So I will stop here and continue it in another post.</p>
<p><strong>Key Takeaway Learning Points</strong></p>
<p>A few important points that have helped me a lot.</p>
<ol>
<li>A picture is worth a thousand words. Believe in the power of visualizations</li>
<li>Always, begin the data exploration with a hypothesis or question and then dive into the data to prove it. You will find something if not anything.</li>
<li>Read and regurgiate on the research question, read material related to it to ensure that the data at hand is enough to answer your questions.</li>
<li>If you are a novice, don't you dare make assumptions or develop any preconceived notions about knowledge nuggets (<em>for example, my initial aversion towards supervised learning as noted above) </em>that you have not explored.</li>
<li>Get your fundamentals strong in statistics, linear algebra and probability for these are the base of data science.</li>
<li>Practice programming your learnings and it will be best if create an end to end project. Needless to mention, the more you read, the more you write and the more you code, you will get better in your craft.And stick to one programming tool.</li>
<li>Subscribe to data science blogs like R-bloggers, kaggle, driven data etc. Create a blog which will serve as your live portfolio.</li>
<li>I think to master the art of story telling with data takes time and a hell lot of reading and analysis.</li>
</ol>
<p>In the next part of this post, I will elaborate and discuss on my strategy that i undertook to submit my initial entry for predicting blood donor, competition hosted at driven data.</p>
<p>References</p>
<p>“An Introduction To Xgboost R Package”. R-bloggers.com. N.p., 2016. Web. 23 Aug. 2016.</p>
<![CDATA[Data Splitting]]>https://duttashi.github.io/blog/data-splitting2016-08-08T13:41:00+00:002016-08-08T13:41:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>A few common steps in data model building are;</p>
<ul>
<li>Pre-processing the predictor data (predictor - independent variable's)</li>
<li>Estimating the model parameters</li>
<li>Selecting the predictors for the model</li>
<li>Evaluating the model performance</li>
<li>Fine tuning the class prediction rules</li>
</ul>
<p>“One of the first decisions to make when modeling is to decide which samples will be used to evaluate performance. Ideally, the model should be evaluated on samples that were not used to build or fine-tune the model, so that they provide an unbiased sense of model effectiveness. When a large amount of data is at hand, a set of samples can be set aside to evaluate the final model. The “training” data set is the general term for the samples used to create the model, while the “test” or “validation” data set is used to qualify performance.” (Kuhn, 2013)</p>
<p>In most cases, the training and test samples are desired to be as homogenous as possible. Random sampling methods can be used to create similar data sets.
Let’s take an example. I will be using R programming language and will use two datasets from the UCI Machine Learning repository.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># clear the workspace
> rm(list=ls())
# ensure the process is reproducible
> set.seed(2)
</code></pre></div></div>
<p>The first dataset is the Wisconsin Breast Cancer Database
Description: Predict whether a cancer is malignant or benign from biopsy details.
Type: Binary Classification
Dimensions: 699 instances, 11 attributes
Inputs: Integer (Nominal)
Output: Categorical, 2 class labels
UCI Machine Learning Repository: <a href="https://archive.ics.uci.edu/ml/datasets/Breast+Cancer+Wisconsin+(Original)" target="_blank">Description</a>
Published accuracy results: <a href="http://www.is.umk.pl/projects/datasets.html#Wisconsin" target="_blank">Summary</a></p>
<p><span style="color:#000080;"><strong>Splitting based on Response/Outcome/Dependent variable</strong></span></p>
<p>Let’s say, I want to take a sample of 70% of my data, I will do it like</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> BreastCancer[sample(nrow(BreastCancer), 524),] # 70% sample size
> table(smpl$Class)
benign malignant
345 179
</code></pre></div></div>
<p style="text-align:left;">And when I plot it is shown in figure 1 below;</p>
<p><img src="https://duttashi.github.io/images/data-split-1.png" alt="image" /></p>
<p>Figure 1: Plot of categorical class variable</p>
<p style="text-align:left;">However, if you want to give different probabilities of being selected for the elements, lets say, elements that cancer type is benign has probability 0.25, while those whose cancer type is malignant has prob 0.75, you should do like</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> prb <- ifelse(BreastCancer$Class =="benign",0.25, 0.75)
> smpl<- BreastCancer[sample(nrow(BreastCancer), 524, prob = prb),]
> table(smpl$Class)
benign malignant
299 225
</code></pre></div></div>
<p style="text-align:left;">And when I plot it is like shown in figure 2,</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> plot(smpl$Class)
</code></pre></div></div>
<p style="text-align:left;">
![image](https://duttashi.github.io/images/data-split-1-1.png)
Figure 2: Plot of categorical class variable with probability based sample split
If the outcome or the response variable is categorical then split the data using stratified random sampling that applies random sampling within subgroups (such as the classes). In this way, there is a higher likelihood that the outcome distributions will match. The function <em>createDataPartition </em>of the caret package can be used to create balanced splits of the data or random stratified split. I show it using an example in R as given;
> library(caret)
> train.rows<- createDataPartition(y= BreastCancer$Class, p=0.7, list = FALSE)
> train.data<- BreastCancer[train.rows,] # 70% data goes in here
> table(train.data$Class)
benign malignant
321 169
And the plot shown in figure 3
![image](https://duttashi.github.io/images/data-split-2-train.png)
Figure 3: Plot of categorical class variable from train sample data
Similarly, I do for the test sample data as given
> test.data<- BreastCancer[-train.rows,] # 30% data goes in here
> table(test.data$Class)
benign malignant
137 72
> plot(test.data$Class)
And I show the plot in figure 4,
![image](https://duttashi.github.io/images/data-split-2-test.png)
Figure 4: Plot of categorical class variable from test sample data
<span style="color:#000080;"><strong>Splitting based on Predictor/Input/Independent variables</strong></span>
So far we have seen the data splitting was based on the outcome or the response variable. However, the data can be split on the predictor variables too. This is achieved by <em>maximum dissimilarity sampling </em> as proposed by Willet (1999) and Clark (1997). This is particularly useful for unsupervised learning where there are no response variables. There are many methods in R to calculate dissimilarity. caret uses the proxy package. See the manual for that package for a list of available measures. Also, there are many ways to calculate which sample is “most dissimilar”. The argument obj can be used to specify any function that returns a scalar measure. caret includes two functions, minDiss and sumDiss, that can be used to maximize the minimum and total dissimilarities, respectfully.
References
Kuhn, M.,& Johnson, K. (2013). Applied predictive modeling (pp. 389-400). New York: Springer.
Willett, P. (1999), "Dissimilarity-Based Algorithms for Selecting Structurally Diverse Sets of Compounds," Journal of Computational Biology, 6, 447-457.
</p>
<![CDATA[Big or small-lets save them all- Visualizing Data]]>https://duttashi.github.io/blog/big-or-small-lets-save-them-all-visualizing-data2016-01-23T00:00:00+00:002016-01-23T00:00:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>I am revisiting the research question once again, “Can alcohol consumption increase the risk of breast cancer in working class women? and the variables to explore are;</p>
<ol>
<li>‘alcconsumption’- average alcohol consumption, adult (15+) per capita consumption in litres pure alcohol</li>
<li>‘breastcancerper100TH’- Number of new cases of breast cancer in 100,000 female residents during the certain year</li>
<li>‘femaleemployrate’- Percentage of female population, age above 15, that has been employed during the given year</li>
</ol>
<p>From the research question, the <span style="text-decoration:underline;">dependent variable</span> or the response or the outcome variable <span style="text-decoration:underline;">is breast cancer per 100<sup>th</sup> women</span> and the <span style="text-decoration:underline;">independent variables</span> are <span style="text-decoration:underline;">alcohol consumption and female employ rate</span></p>
<p>Let us now look at the measures of center and spread of the aforementioned variables. This will help us to better understand our quantitative variables. In python, to measure the mean, median, mode, minimum and maximum value, standard deviation and percentiles of a quantitative variable can be computed using the describe() function as shown below</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>#using the describe function to get the standard deviation and other descriptive statistics of our variables
desc1=data['breastcancerper100th'].describe()
desc2=data['femaleemployrate'].describe()
desc3=data['alcconsumption'].describe()
print "\nBreast Cancer per 100th person\n", desc1
print "\nfemale employ rate\n", desc2
print "\nAlcohol consumption in litres\n", desc3
</code></pre></div></div>
<p>and the result will be</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Breast Cancer per 100th person
count 173.000000
mean 37.402890
std 22.697901
min 3.900000
25% 20.600000
50% 30.000000
75% 50.300000
max 101.100000
</code></pre></div></div>
<p>So, on an average there are 37 women per 100th in whom breast cancer is reported with a standard deviation of +- 22.</p>
<p>Similarly, I next find the mean and standard deviation of the variable, ‘femalemployrate’</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>female employ rate
count 178.000000
mean 47.549438
std 14.625743
min 11.300000
25% 38.725000
50% 47.549999
75% 55.875000
max 83.300003
</code></pre></div></div>
<p>I can say that on an average there are 47% women employed in a given year with a deviation of +-15.</p>
<p>Finally, I find the mean and deviation of the variable, ‘alcconsumption’ given as</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Alcohol consumption in litres
count 187.000000
mean 6.689412
std 4.899617
min 0.030000
25% 2.625000
50% 5.920000
75% 9.925000
max 23.010000
</code></pre></div></div>
<p>This can be interpreted as among adults (15+) the average alcohol consumption in liters per capita income is 7 liters (rounding off) with a standard deviation of +-5 (rounding off).</p>
<p>Therefore the inference will be that in 47% <em>(+-15)</em> employed women in a given year the average alcohol consumption is 7 liters (+-5) per capita and the number of breast cancer cases reported on an average are 37 (+-22) per 100th female residents.</p>
<p>Another, alternative method of finding descriptive statistic for your variables is to use the describe() on the dataframe which in this case is called ‘data’ as given</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>data.describe()
</code></pre></div></div>
<p>I now provide the univariate data analysis of the individual variables</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Now plotting the univariate quantitative variables using the distribution plot
sub5=sub4.copy()
sns.distplot(sub5['alcconsumption'].dropna(),kde=True)
plt.xlabel('Alcohol consumption in litres')
plt.title('Breast cancer in working class women')
plt.show()
'''Note: Although there is no need to use the show() method for ipython notebook as %matplotlib inline does the trick but I am adding it here because matplotlib inline does not work for an IDE like Pycharm and for that i need to use plt.show'''
</code></pre></div></div>
<p>And the barchart is</p>
<p><img class=" size-full wp-image-1227 aligncenter" src="https://edumine.files.wordpress.com/2016/01/fd1.png" alt="fd1" width="521" height="380" /></p>
<p style="text-align:center;">Bar Chart 1: Alcohol consumption in liters</p>
<p>Notice, we have two peaks in bar chart 1. So it is a bimodal distribution which means that there are two distinct groups of data. The two groups are evident from the bar chart 1, where the first group (or the first peak) is centered at 5 liters of alcohol consumption and the second group (or the second peak) is centered at 35 liters of alcohol consumption</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>sns.distplot(sub5['breastcancerper100th'].dropna(),kde=True)
plt.xlabel('Breast cancer per 100th women')
plt.title('Breast cancer in working class women')
plt.show()
</code></pre></div></div>
<p>And the barchart is</p>
<p><img class=" size-full wp-image-1233 aligncenter" src="https://edumine.files.wordpress.com/2016/01/fd2.png" alt="fd2" width="501" height="365" /></p>
<p style="text-align:center;">Bar Chart 2: Breast cancer per 100th women</p>
<p style="text-align:left;">Similarly, in bar chart 2, there are two peaks so it is a bimodal distribution where the first group is centered at 35 cases of new breast cancer reported and the second group is centered at 86 cases of new breast cancer reported.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>sns.distplot(sub5['femaleemployrate'].dropna(),kde=True)
plt.xlabel('Female employee rate')
plt.title('Breast cancer in working class women')
plt.show()
</code></pre></div></div>
<p>And the bar chart is</p>
<p><img class=" size-full wp-image-1237 aligncenter" src="https://edumine.files.wordpress.com/2016/01/fd3.png" alt="fd3" width="495" height="365" /></p>
<p style="text-align:center;">Bar Chart 3: Female Employed Rate above 15+ (in %age) in a given year</p>
<p style="text-align:left;">In bar chart 3 we see a unimodal distribution where there is one group with maximum number of 42.</p>
<p>Now that we have seen the individual variable visually, I will now come back to the research question to see if there is any relationship between the research questions. Recall, for this study the various hypotheses were;</p>
<p>H<sub>0 </sub>(Null Hypothesis) = Breast cancer is not caused by alcohol consumption</p>
<p>H<sub>1 </sub>(Alternative Hypothesis) = Alcohol consumption causes breast cancer</p>
<p>H<sub>2 </sub>(Alternative Hypothesis) = Female employee are susceptible to increased risk of breast cancer.</p>
<p>So, let’s check if there is any relationship between the breast cancer and alcohol consumption.</p>
<p>Please note here that since all the variables of this study are quantitative in nature so I will be using the scatter plot to visualize them.</p>
<p>Note that a histogram is not a bar chart. Histograms are used to show distributions of variables while bar charts are used to compare variables. Histograms plot quantitative data with ranges of the data grouped into bins or intervals while bar charts plot categorical data. For Dell Statistica, you can take a look <a href="http://documents.software.dell.com/Statistics/Textbook/Graphical-Analytic-Techniques">here</a> for the graphical data visualization and in Python it can be done using matplotlib library as shown <a href="https://plot.ly/matplotlib/bar-charts/">here</a> and a good SO question <a href="http://stackoverflow.com/questions/11617719/how-to-plot-a-very-simple-bar-chart-python-matplotlib-using-input-txt-file">here</a></p>
<ul>
<li>When visualizing a categorical to categorical relationship we use a Bar Chart.</li>
<li>When visualizing a categorical to quantitative relationship we use a Bar Chart.</li>
<li>When visualizing a quantitative to quantitative relationship we use a Scatter Plot.</li>
</ul>
<p>Also, please note that it is very important to bear in mind when plotting association between two variables, the <span style="text-decoration:underline;">independent or the explanatory variable is ‘X’ plotted on the x-axis</span> and the <span style="text-decoration:underline;">dependent or the response variable is ‘Y’ plotted on the y-axis</span></p>
<p><img class=" wp-image-1350 aligncenter" src="https://edumine.files.wordpress.com/2016/01/ind_dep_graph.png" alt="ind_dep_graph" width="268" height="265" /></p>
<p>So to check if the relationship exist or not, I code it in python as follows</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># using scatter plot the visulaize quantitative variable.
# if categorical variable then use histogram
scat1= sns.regplot(x='alcconsumption', y='breastcancerper100th', data=data)
plt.xlabel('Alcohol consumption in liters')
plt.ylabel('Breast cancer per 100th person')
plt.title('Scatterplot for the Association between Alcohol Consumption and Breast Cancer 100th person')
</code></pre></div></div>
<p>And the corresponding scatter plot is <img class=" size-full wp-image-1251 aligncenter" src="https://edumine.files.wordpress.com/2016/01/sct1.png" alt="sct1" width="527" height="365" /></p>
<p style="text-align:center;">Scatter Plot 1</p>
<p>From the scatter plot 1, its evident that we have a positive relationship between the two variables. And this proves the alternative hypothesis (H<sub>1</sub>) that higher alcohol consumption by women has increased chances of breast cancer in them. Thus we can safely reject the null hypothesis that alcohol consumption does not cause breast cancer in women. Also, the points on the scatter plot are densely scattered around the linear line therefore the strength of the relationship is strong. This means that we have a statistically significant and strong positive relationship between higher alcohol consumption causing increased number of breast cancer patients in women.</p>
<p>Now, let us check if the other alternative hypothesis (H<sub>2</sub>), “Female employee are susceptible to increased risk of breast cancer” is true or not. To verify this claim, I code it as</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>scat2= sns.regplot(x='femaleemployrate', y='breastcancerper100th', data=data)
plt.xlabel('Female Employ Rate')
plt.ylabel('Breast cancer per 100th person')
plt.title('Scatterplot for the Association between Female Employ Rate and Breast Cancer per 100th Rate')
</code></pre></div></div>
<p>And the scatter plot is <img class=" size-full wp-image-1263 aligncenter" src="https://edumine.files.wordpress.com/2016/01/sct2.png" alt="sct2" width="529" height="365" /></p>
<p style="text-align:center;">Scatter Plot 2</p>
<p>From scatter plot 2, we can see that there is a negative relationship between the two variables. That means as the number of female employment count increases the number of breast cancer patients in employed women decreases. Also the strength of this relationship is weak as the number of points are sparsely located on the linear line. So, I will say that although the relationship is statistically significant but it is weak thus its safe to conclude that female employment rate does not necessarily contribute to breast cancer in women.</p>
<p>I now come to the conclusion of this analytical series. After performing descriptive and exploratory data analysis on the gapminder dataset using python as a programming tool, I have been successful in determining that higher alcohol consumption by women increases the chance of breast cancer in them. I have also been successful in determining that breast cancer occurrence in employed females has a weak correlation. Perhaps, there are other factors that could prove this.</p>
<p>Finally, to conclude this exploratory data analysis series of posts has been very fruitful and immensely captivating to me. In the next post, I will discuss on the statistical relationships between the variables and testing the hypotheses in the context of Analysis of Variance (when you have one quantitative variable and one categorical variable). And since the dataset that I chose does not have any categorical variable, I will also show how to categorize a quantitative variable.</p>
<p>The complete python code is listed on my GitHub account <a href="https://github.com/duttashi/Data-Analysis-Visualization/blob/master/gapminder_data_analysis.py" target="_blank">here</a></p>
<p>Cheers!</p>
<![CDATA[Big or small-lets save them all- Making Data Management Decisions]]>https://duttashi.github.io/blog/big-or-small-lets-save-them-all-making-data-management-decisions2016-01-15T00:00:00+00:002016-01-15T00:00:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>So far I have discussed the data-set, the research question, introduced the variables to analyze and performed some exploratory data analysis in which I showed how to get a brief overview of the data using python. Continuing further, I have now reached a stage wherein I must ‘dive into’ the data-set and make some strategic data management decisions. This stage cannot be taken lightly because it lays the foundation of the entire project. A misjudgment here can spell doom to the entire data analysis cycle.</p>
<p>The first step is to see, if the data is complete or not? By completeness, I mean to check the rows and the columns of the data-set for any missing values or junk values. (<em>Do note, here I have asked two questions. In this post I will answer the first question only. In another post i will answer the second question</em>); a) How to deal with missing values and b) How to deal with junk values.</p>
<p>To answer the first question, I use the following code to get the sum of missing values by rows thereafter I use the is.null().sum() as given to display the column count of the missing values.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Create a copy of the original dataset as sub4
sub4=data
print "Missing data rows count: &amp;quot;,sum([True for idx,row in data.iterrows() if any(row.isnull())]) I would see that there are 48 rows of missing data as shown
Missing data rows count: 48 Now how about I want to see the columns that have missing data. For that I use the isnull().sum() function as given
print sub4.isnull().sum() This line of code will give me the column-vise missing data count as shown
country 0
breastcancerper100th 40
femaleemployrate 35
alcconsumption 26
dtype: int64
</code></pre></div></div>
<p>So now, how to deal with this missing data? There are some excellent papers written that have addressed this issue. For interested reader, I refer to two such examples <a href="http://www.unt.edu/rss/class/mike/6810/articles/roth.pdf">here</a> and <a href="http://myweb.brooklyn.liu.edu/cortiz/PDF%20Files/Best%20practices%20for%20missing%20data%20management.pdf">here</a>.
<span style="text-decoration:underline;"><strong>Dealing with Missing Values</strong></span>
So what do I do with a data set that has 3 continuous variables which off-course as always is dirty (<em><strong>brief melodrama now: </strong>hands high in air and I shout “Don’t you have any mercy on me! When will you give me that perfect data set. God laughs and tells his accountant pointing at ‘me’..”look at that earthly fool..while all fellows at his age ask for wine, women and fun he wants me to give him “clean data” which even I don’t have”</em>). So how do I mop it clean? Do i remove the missing values? “Nah” that would be apocalyptic in data science ..hmmm..so what do I do? How about I code all the missing values as Zero. <strong>NO! Not to underestimate the Zero. </strong>So what do I do?</p>
<p>One solution is to impute the missing continuous variables with the mean of the neighboring values in the variable. Note: to impute the missing categorical values, one can try imputing the mode (highest occurring frequency value). Yeah..that should do the trick. So I code it as given;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>sub4.fillna(sub4['breastcancerper100th'].mean(), inplace=True)
sub4.fillna(sub4['femaleemployrate'].mean(), inplace=True)
sub4.fillna(sub4['alcconsumption'].mean(), inplace=True)
</code></pre></div></div>
<p>So here, I have used the fillna() method of pandas library. You can see here the <a href="http://pandas.pydata.org/pandas-docs/stable/missing_data.html">documentation</a> . Now I show the output before missing value imputation as</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Missing data rows count: 48
country 0
breastcancerper100th 40
femaleemployrate 35
alcconsumption 26
dtype: int64
</code></pre></div></div>
<p>and the output after the missing values were imputed using the fillna() function as</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>country 0 breastcancerper100th 0 femaleemployrate 0 alcconsumption 0 dtype: int64
</code></pre></div></div>
<p>Continuing further, I now categorize the quantitative variables based on customized splits using the cut function and why I am doing this because it will help me later to view a nice elegant frequency distribution.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># categorize quantitative variable based on customized splits using the cut function
sub4['alco']=pd.qcut(sub4.alcconsumption,6,labels=["0","1-4","5-9","10-14","15-19","20-24"])
sub4['brst']=pd.qcut(sub4.breastcancerper100th,5,labels=["1-20","21-40","41-60","61-80","81-90"])
sub4['emply']=pd.qcut(sub4.femaleemployrate,4,labels=["30-39","40-59","60-79","80-90"])
</code></pre></div></div>
<p><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;"><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;">Now, that I that I have split the continuous variables, I will now show there frequency distributions so as to understand my data better.</span></span></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>fd1=sub4['alco'].value_counts(sort=False,dropna=False)
fd2=sub4['brst'].value_counts(sort=False,dropna=False)
fd3=sub4['emply'].value_counts(sort=False,dropna=False)
</code></pre></div></div>
<p><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;"><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;">I will now print the frequency distribution for alcohol consumption as given</span></span></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Alcohol Consumption
0 36
1-4 35
5-9 36
10-14 35
15-19 35
20-24 36
dtype: int64
</code></pre></div></div>
<p><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;"><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;">then, the frequency distribution for breast cancer per 100th women as </span></span></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Breast Cancer per 100th
1-20 43
21-40 43
41-60 65
61-80 19
81-90 43
dtype: int64
</code></pre></div></div>
<p><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;"><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;"><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;">and finally the female employee rate as </span></span></span></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Female Employee Rate
30-39 73
40-59 34
60-79 53
80-90 53
dtype: int64 <span style="font-family:Consolas, Monaco, monospace;line-height:1.7;"><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;"><span style="font-family:Consolas, Monaco, monospace;line-height:1.7;">Now, this looks better. So if I have to summarize it the frequency distribution for alcohol consumption per liters among adults (age 15+). I will say that there are 36 women who drink no alcohol at all (and still they are breast cancer victims...hmmm ..nice find..will explore it further later). The count of women who drink between 5-9 liters and 20-24 liters of pure alcohol is similar! Then there are about 73% of women who have been employed in a certain year and roughly about 43 new breast cancer cases are reported per 100th female residents. </span>
</code></pre></div></div>
<p>Stay tuned, next time I will provide a visual interpretation of these findings and more.</p>
<p>Cheers!</p>
<![CDATA[Big or small-lets save them all-Exploratory Data Analysis]]>https://duttashi.github.io/blog/big-or-small-lets-save-them-all-exploratory-data-analysis2016-01-09T00:00:00+00:002016-01-09T00:00:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>In my previous post, I had discussed at length on the research question, the dataset, the variables and the various research hypothesis.
For the sake of brevity, I will restate the research question and the variables of study.</p>
<p>Research question: Can alcohol consumption increase the risk of breast cancer in working class women.
Variables to explore are:</p>
<ol>
<li>‘alcconsumption’- average alcohol consumption, adult (15+) per capita consumption in litres pure alcohol</li>
<li>‘breastcancerper100TH’- Number of new cases of breast cancer in 100,000 female residents during the certain year</li>
<li>‘femaleemployrate’- Percentage of female population, age above 15, that has been employed during the given year</li>
</ol>
<p>In this post, I present to the readers an exploratory data analysis of the gapminder dataset.</p>
<p>Although, for this course we are provided with the relevant dataset, however if you are not taking this course and are interested in the source of the data, then you can get it from <a href="http://www.gapminder.org/data/">here</a>. In the List of indicators search box type “breast cancer, new cases per 100,000 women” to download the dataset.</p>
<p>I will be using python for Exploratory Data Analysis (EDA). I begin by importing the libraries pandas and numpy as</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Importing the libraries
import pandas as pd
import numpy as np
</code></pre></div></div>
<p>I have already downloaded the dataset which is .csv (comma seperated value format) and will now load/read it in a variable called datausing pandas library as given</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Reading the data where low_memory=False increases the program efficiency
data = pd.read_csv('data/train.csv', low_memory=False)
</code></pre></div></div>
<p>To get a quick look at the number of rows and columns and the coulmn headers, you can do the following;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>print (len(data)) # shows the number of rows, here 213 rows
print (len(data.columns))# shows the number of cols, here 4 columns# Print the column headers/headings
names=data.columns.values
print names
</code></pre></div></div>
<p>You will see the output as</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>213
4
213
['country' 'breastcancerper100th' 'femaleemployrate' 'alcconsumption']
</code></pre></div></div>
<p>Now, to see the frequency distribution of these four variables I use the <strong>value_counts() </strong>function to generate the frequency counts of the breast cancer dependence variables. Note, if you want to see the data with the missing values then choose the flag <strong>dropna=False</strong> as shown. For this dataset, majority of variable values have a frequency of 1.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>print "\nAlcohol Consumption\nFrequency Distribution (in %)"
c1=data['alcconsumption'].value_counts(sort=False,dropna=False)
print c1
print "\nBreast Cancer per 100th"
c2=data['breastcancerper100th'].value_counts(sort=False)
print c2
print "\nFemale Employee Rate"
c3=data['femaleemployrate'].value_counts(sort=False)
print c3
</code></pre></div></div>
<p>The output will be <code class="highlighter-rouge">Alcohol Consumption 5.25 1 9.75 1 0.50 1 9.50 1 9.60 1</code></p>
<p>In the above output, values 5.25,9.75,0.50,5.05 are the alcohol consumption in litres and the value 0.004695 is the percentage count of the value. The flag sort=False means that values will not be sorted according to their frequencies. Similarly, I show the frequency distribution for the other two variables</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Breast Cancer per 100th
23.5 2
70.5 1
31.5 1
62.5 1
19.5 6
</code></pre></div></div>
<p>and</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Female Employee Rate
45.900002 2
55.500000 1
35.500000 1
40.500000 1
45.500000 1
</code></pre></div></div>
<p>I now subset the data to explore my research question in a bid to see if it requires any improvement or not. I want to see which countries are prone to greater risk of breast cancer among female employee where the average alcohol intake is 10L;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># Creating a subset of the data
sub1=data[(data['femaleemployrate']>40) & (data['alcconsumption']>=20)& (data['breastcancerper100th']<50)]
# creating a copy of the subset. This copy will be used for subsequent analysis
sub2=sub1.copy()
</code></pre></div></div>
<p>and the result is;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>country breastcancerper100th femaleemployrate alcconsumption
9 Australia 83.2 54.599998 10.21
32 Canada 84.3 58.900002 10.20
50 Denmark 88.7 58.099998 12.02
63 Finland 84.7 53.400002 13.10
90 Ireland 74.9 51.000000 14.92
185 Switzerland 81.7 57.000000 11.41
202 United Kingdom 87.2 53.099998 13.24
</code></pre></div></div>
<p>Interestingly, countries with stable economies like Australia, Canada, Denmark, Finland, Ireland, Switzerland & UK top the list of high breast cancer risk among working women class. These countries are liberal to women rights. Now, this can be an interesting question that will be explored later.</p>
<p>How about countries with very low female employee rates- how much is there contribution to alcohol consumption and breast cancer risk? <em>(I set the threshold for high employee rate as greater than 40% and threshold for high alcohol consumption to be greater than 20 liters and breast cancer risk at less than 50%). And the winner is,</em> <strong>Moldova </strong>a landlocked country in Eastern Europe. Here we can see that Moldova contributes to approximately 50% of new breast cancer cases reported per 100,000th female residents with a per capita alcohol consumption of 23%. So with a low female employee rate of 43% (as compared to the threshold of 40%) this country does have a significant amount of new breast cancer cases reported because of high alcohol consumption by the relatively less number of adult female residents. ((on a side note: “Heaven’s! Moldavian working class women drink a lot :-) ))</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>print "\nContries where Female Employee Rate is greater than 40 &" \
" Alcohol Consumption is greater than 20L & new breast cancer cases reported are less than 50\n"
print sub2
print "\nContries where Female Employee Rate is greater than 50 &" \
" Alcohol Consumption is greater than 10L & new breast cancer cases reported are greater than 70\n"
sub3=data[(data['alcconsumption']>10)&(data['breastcancerper100th']>70)&(data['femaleemployrate']>50)]
print sub3
</code></pre></div></div>
<p>the result is</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Contries where Female Employee Rate is greater than 40 & Alcohol Consumption is greater than 20L & new breast cancer cases reported are less than 50
country incomeperperson alcconsumption armedforcesrate \
126 Moldova 595.874534521728 23.01 .5415062
breastcancerper100th co2emissions femaleemployrate hivrate \
126 49.6 149904333.333333 43.599998 .4
internetuserate lifeexpectancy oilperperson polityscore \
126 40.122234699607 69.317 8
relectricperperson suicideper100th employrate urbanrate
126 304.940114846777 15.53849 44.2999992370606 41.76
</code></pre></div></div>
<p>The complete python code is listed on my <a href="https://github.com/duttashi/Data-Analysis-Visualization/blob/master/gapminder%20data%20analysis.ipynb" target="_blank">github account</a></p>
<p>This series will be continued….</p>
<![CDATA[Batch Geo-coding in R]]>https://duttashi.github.io/blog/batch-geo-coding-in-r2015-07-05T02:34:00+00:002015-07-05T02:34:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>Geocoding (sometimes called forward <b>geocoding</b>) is the process of enriching a description of a location, most typically a postal address or place name, with geographic coordinates from spatial reference data such as building polygons, land parcels, street addresses, postal codes (e.g. ZIP codes, CEDEX) and so on.</p>
<p>Google API for Geo-coding restricts coordinates lookup to 2500 per IP address per day. So if you have more than this limit of addresses then searching for an alternative solution is cumbersome.</p>
<p>The task at hand was to determine the coordinates of a huge number of addresses to the tune of over 10,000.</p>
<p>The question was how to achieve this in R?</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library(RgoogleMaps)
> DF <- with(caseLoc, data.frame(caseLoc, t(sapply(caseLoc$caseLocation, getGeoCode))))
#caseLoc is the address file and caseLocation is the column header
</code></pre></div></div>
<![CDATA[To read multiple files from a directory and save to a data frame]]>https://duttashi.github.io/blog/to-read-multiple-files-from-a-directory-and-save-to-a-data-frame2015-06-23T16:16:00+00:002015-06-23T16:16:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>There are various solution to this questions like these but I will attempt to answer the problems that I encountered with there working solution that either I found or created by my own.</p>
<p>Question 1: My initial problem was how to read multiple .CSV files and store them into a single data frame.</p>
<p>Solution: Use a lapply() function and rbind(). One of the working R code I found <a href="http://stackoverflow.com/questions/23190280/issue-in-loading-multiple-csv-files-into-single-dataframe-in-r-using-rbind">here</a> provided by Hadley.</p>
<p>The code is;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code># The following code reads multiple csv files into a single data frame
load_data <- function(path)
{
files <- dir(path, pattern = '\\*.csv', full.names = TRUE)
tables <- lapply(files, read.csv)
do.call(rbind, tables)
}
</code></pre></div></div>
<p>And then use the function like</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> load_data("D://User//Temp")
</code></pre></div></div>
<![CDATA[Installing Apache Spark on Windows 7 environment]]>https://duttashi.github.io/blog/how-to-install-apache-spark-on-a-windows-7-environment2015-06-11T18:10:00+00:002015-06-11T18:10:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>Apache Spark is a lightening fast cluster computing engine conducive for big data processing. In order to learn how to work on it currently there is a MOOC conducted by UC Berkley <a href="https://courses.edx.org/courses/BerkeleyX/CS100.1x/1T2015/info">here</a>. However, they are using a pre-configured VM setup specific for the MOOC and for the lab exercises. But I wanted to get a taste of this technology on my personal computer. I invested two days searching the internet trying to find out how to install and configure it on a windows based environment. And finally, I was able to come up with the following brief steps that lead me to a working instantiation of Apache Spark.</p>
<p>To install Spark on a windows based environment the following prerequisites should be fulfilled first.</p>
<p><strong>Requirement 1:</strong></p>
<ul>
<li>If you are a Python user then Install Python 2.6+ or above otherwise this step is not required. If you are not a python user then you also do not need to setup the python path as the environment variable</li>
<li>Download a pre-built Spark binary for Hadoop. I chose Spark release 1.2.1, package type Pre-built for Hadoop 2.3 or later from <a href="https://spark.apache.org/downloads.html">here</a>.</li>
<li>Once downloaded I unzipped the *.tar file by using WinRar to the D drive. (You can unzip it to any drive on your computer)</li>
<li>The benefit of using a pre-built binary is that you will not have to go through the trouble of building the spark binaries from scratch.</li>
<li>Download and install Scala version 2.10.4 from <a href="http://www.scala-lang.org/download/" target="_blank">here</a> only if you are a Scala user otherwise this step is not required. If you are not a scala user then you also do not need to setup the scala path as the environment variable</li>
<li>Download and install <a href="http://en.osdn.jp/projects/win-hadoop/downloads/62852/hadoop-winutils-2.6.0.zip/" target="_blank">winutils.exe</a> and place it in any location in the D drive. Actually, the official release of Hadoop 2.6 does not include the required binaries (like winutils.exe) which are required to run Hadoop. Remember, Spark is a engine built over Hadoop.</li>
</ul>
<p><strong>Setting up the PATH variable in Windows environment :</strong></p>
<p>This is the most important step. If the Path variable is not properly setup, you will not be able to start the spark shell. Now how to access the path variable?</p>
<ul>
<li>Right click on Computer- Left click on Properties</li>
<li>Click on Advanced System Settings</li>
<li>Under Start up & Recovery, Click on the button labelled as "Environment Variable"</li>
<li>You will see the window divided into two parts, the upper part will read User variables for username and the lower part will read System variables. We will create two new system variables, So click on "New" button under System variable</li>
<li>Set the variable name as <code> JAVA_HOME </code></li>
(in case JAVA is not installed on your computer then follow these steps). Next set the variable value as the <code>JDK PATH</code>. In my case it is <code> C:\Program Files\Java\jdk1.7.0_79\</code>
(please type the path without the single quote)
<li>Similarly, create a new system variable and name it as</li>
<code>PYTHON_PATH</code>
Set the variable value as the Python Path on your computer. In my case it is <code> C:\Python27\ </code>
(please type the path without the single quote)
<li>Create a new system variable and name it as</li>
<code>HADOOP_HOME</code>
Set the variable value as <code>C:\winutils</code>
(Note: There is no need to install Hadoop. The spark shell only requires the Hadoop path which in this case holds the value to winutils that will let us compile the spark program on a windows environment.
<li>Create a new system variable and name it as </li>
<code>SPARK_HOME </code>
Assign the variable value as the path to your Spark binary location. In my case it is in <code>C:\SPARK\BIN</code>
<span style="text-decoration:underline;"><strong>NOTE:</strong></span><strong> Apache Maven installation is an optional step. </strong>I am mentioning it here because I want to install SparkR a R version of Spark.
<ul>
<li>Download Apache Maven 3.1.1 from <a href="https://maven.apache.org/download.cgi" target="_blank">here</a> </li>
<li>Choose Maven 3.1.1. (binary zip) and unpack it using WinZip or WinRAR. Create a new system variable and name it as</li> <code> MAVEN_HOME and M2_HOME</code>
</ul>
<strong> </strong>Assign the both these variables the value as the path to your Maven binary location. In my case it is in <code>D:\APACHE-MAVEN-3.1.1\BIN</code> so I have <code> MAVEN_HOME=D:\APACHE-MAVEN-3.1.1\BIN </code>
and <code> M2_HOME=D:\APACHE-MAVEN-3.1.1\BIN </code>
Now, all you have to do is append these four system variables namely JAVA_HOME, PYTHON_PATH, HADOOP_HOME & SPARK_HOME to your Path variable.
This can be done as follows <code> %JAVA_HOME%\BIN; %PYTHON_PATH%; %HADOOP_HOME%; %SPARK_HOME%; %M2_HOME%\BIN %MAVEN_HOME%\BIN </code> (Note: Do not forget to end each entry with a semi-colon)
Click on Ok to close the Environment variable window and then similarly on System properties window.
<strong>How to start Spark on windows</strong>
To run spark on windows environment
<ol>
<li>Open up the command prompt terminal</li>
<li>Change directory to the location where the spark directory is. For example in my case its present in the D directory</li>
<li>Navigate into the bin directory like cd bin</li>
<li>Run the command spark-shell and you should see the spark logo with the scala prompt</li>
</ol>
![image](https://duttashi.github.io/images/spark-shell1.png)
![image](https://duttashi.github.io/images/spark-shell2.png)
<ol>
<li>Open up the web browser and type localhost:4040 in the address bar and you shall see the Spark shell application UI<a href="https://edumine.files.wordpress.com/2015/06/spark-shell2.png">
</a></li>
</ol>
![image](https://duttashi.github.io/images/spark-ui.png)
<ol>
<li>To quit Spark, at the command prompt type <code> exit</code> </li>
</ol>
That is all to install and run a standalone spark cluster on a windows based environment.
Cheers!
</ul>
<![CDATA[Gini index to compute inequality or impurity in the data]]>https://duttashi.github.io/blog/gini-index-to-compute-inequality-or-impurity-in-the-data2015-05-18T14:25:00+00:002015-05-18T14:25:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>“Gini index measures the extent to which the distribution of income or consumption expenditure among individuals or households within an economy deviates from a perfectly equal distribution” [1]. A Lorenz curve plots the cumulative percentages of total income received against the cumulative number of recipients, starting with the poorest individual or household. The Gini index measures the area between the Lorenz curve and a hypothetical line of absolute equality, expressed as a percentage of the maximum area under the line. Thus a Gini index of 0 represents perfect equality, while an index of 100 implies perfect inequality.</p>
<p>Simply put Gini index measures the impurity of data D. Some refer to it as inequality. </p>
<p><img src="https://duttashi.github.io/images/gini_formula.jpg" alt="image" /></p>
<p>Now let me show the practical example for the same in R. The package that we use here is <a href="http://cran.r-project.org/web/packages/ineq/ineq.pdf">ineq</a> (To see the help file use ??help (ineq)) otherwise you can create your own Gini function as shown <a href="http://r.789695.n4.nabble.com/Function-Gini-or-Ineq-td2525852.html">here</a> like</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>gini if (!is.numeric(x))
{
warning("'x' is not numeric; returning NA")
return(as.numeric(NA))
}
if (any(na.ind if (!na.rm)
stop("'x' contain NAs")
else
x
}
n mu N ox dd dd / (mu * N)
}
</code></pre></div></div>
<p>Either way, whether you use the ineq package or the aforementioned function it will give you the same result. Now, let us measure the inequality of the river Nile data. In R it can be done as follows;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> data (Nile)
> install.packages(“ineq”)
> library(ineq)
> data(Nile)
> ineq(Nile, type=”Gini”)
[1] 0.1031993
</code></pre></div></div>
<p>So there is 10% inequality in the Nile data. And if you want to plot this inequality you can do so by using the Lorenzo curve as given</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> plot(Lc(Nile),col="darkred",lwd=2)
</code></pre></div></div>
<p>This will generate the given plot where the dark red line represents the inequality in the Nile data. It is worthwhile to note that Gini index computation works well for univariate categorical data but it is biased towards multi-valued attributes and has difficulty when the number of classes is large.</p>
<p><img src="https://duttashi.github.io/images/rplot01.png" alt="image" /></p>
<p>Reference</p>
<p>The World Bank. 2014. GINI index (World Bank estimate). [ONLINE] Available at: http://data.worldbank.org/indicator/SI.POV.GINI. [Accessed 18 May 15].</p>
<![CDATA[Assessing Clustering Tendency in R]]>https://duttashi.github.io/blog/assessing-clustering-tendency-in-r2015-05-13T14:51:00+00:002015-05-13T14:51:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>While searching for a R package that applied ‘Hopkin statistic’ (mentioned in chapter 10, example 10.9 page 484 of the book) that determines if a given non-random or non-uniform dataset has the possibility of cluster’s present in it or not, I accidentally discovered this <a href="http://cedric.cnam.fr/fichiers/art_2554.pdf" target="_blank">R package</a> for finding the best number of clusters. The <em>NbClust</em> package provides most of the popular indices for cluster validation. It also proposes to the user the best clustering scheme from the different results obtained by varying all combinations of number of clusters,distance measures, and clustering methods.</p>
<p>Besides the package, I would also recommend to any interested learner to read this paper <a href="http://web.itu.edu.tr/sgunduz/courses/verimaden/paper/validity_survey.pdf" target="_blank">part 1</a> and <a href="http://www.sigmod.org/publications/sigmod-record/0209/a1.partii_clvalidity1.pdf" target="_blank">part II</a> by (Halkidi, Batistakis & Vazirgiannis, 2001) who have provided a comprehensive discussion on cluster validation techniques.</p>
<p>Clustering algorithms impose a classification on a dataset even if there are no clusters present for example k-means. To avoid this, clustering tendency assessment is used. On a given dataset it will determines if the dataset D has a non-random or a non-uniform distribution of data structure that will lead to meaningful clusters. To determine this “cluster tendency” a measure called Hopkins statistic can be used.</p>
<p>Anyway, in clustering one of major problem a researcher/analyst face is how to determine an optimal number of clusters in a dataset and how to validate the clustered results.</p>
<p>So searching for the Hopkin statistic package in R I discovered NBClust package. I provide below an example of how it can be used in R to determine an initial number of clusters that a given dataset can have.</p>
<ol>
<li>If you do not have the package installed, you can do so by typing the following command in R console as</li>
</ol>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> install.packages("NbClust")
</code></pre></div></div>
<p>You can then load it in memory as;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library("NbClust")
</code></pre></div></div>
<p>To see a list of default datasets in R type</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> data()
# and you will see example datasets
# For this example, I will work with iris and mtcars dataset [/sourcecode] To load the dataset in R type at the console
> data(iris)
# To see the dataset column headings and datatype
>str(iris)
>head(iris)
</code></pre></div></div>
<p>Remove the column ‘Species’ from this dataset because if you dont you will get an error when executing the NbClust method</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>>iris$Species=NULL
# Now apply the NbClust method as given but first set the seed function to any value so that your result is reproducible.
>set.seed(26)
clusterNo=NbClust(iris,distance="euclidean",
min.nc=2,max.nc=10,method="complete",index="all")
# where distance function is euclidean distance, min.nc is minimum number of clusters, max.nc is the maximum number of clusters, method can be single, complete, ward, average etc and index=all means test for all 30 indices with the given parameters.
</code></pre></div></div>
<p>For package documentation see <a href="https://cran.r-project.org/web/packages/NbClust/NbClust.pdf">here</a></p>
<h4 id="you-will-then-get-a-brief-summary-of-the-results-as-shown-below">You will then get a brief summary of the results as shown below</h4>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>*******************************************************
Among all indices:
* 2 proposed 2 as the best number of clusters
* 13 proposed 3 as the best number of clusters
* 5 proposed 4 as the best number of clusters
* 1 proposed 6 as the best number of clusters
* 2 proposed 10 as the best number of clusters
***** Conclusion *****
*According to the majority rule,
the best number of clusters is 3
*******************************************************
</code></pre></div></div>
<p>Now, lets take another example in which I will change the dataset to quakes which provides the location of earthquakes in Fiji. Since the above code is the same with only the dataset as a change, I will not provide the comments.</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> data(quakes)
> str(quakes)
> clusterNo=NbClust(quakes,distance="euclidean",min.nc=2,max.nc=10,
method="complete",index="all")
*******************************************************
Among all indices:
* 6 proposed 2 as the best number of clusters
* 3 proposed 3 as the best number of clusters
* 3 proposed 4 as the best number of clusters
* 2 proposed 5 as the best number of clusters
* 7 proposed 7 as the best number of clusters
* 3 proposed 8 as the best number of clusters
***** Conclusion *****
* According to the majority rule,
the best number of clusters is 7
********************************************************
</code></pre></div></div>
<p>Thus you can see that the best number of clusters for the iris dataset is 3 and that for the quakes dataset is 7. Now that you know this, its time to smile because its is proved now that these two datasets are non-uniformly distributed which is a requirement of clustering.</p>
<p>In a similar fashion you can play with the various arguments in the NbClust function. In my opinion, once you get an idea that your dataset actually has clusters in it then trust me you will feel very happy. Because, as I have already stated in the first paragraph, clustering algorithms will partition/divide your dataset into clusters because that is what they are supposed to do. Its analogous to using a knife. The property of a knife is to cut and it will cut any object that is given to it. In the same way are the clustering algorithms. But the quintessential question that matters is “does the dataset has any inherent clusters in it or not? And if yes, then how do you figure it out?” This question is what I have attempted to answer in this post</p>
<p>Cheers.</p>
<![CDATA[Packages for data mining algorithms in R and Python]]>https://duttashi.github.io/blog/packages-for-data-mining-algorithms-in-r-and-python2015-05-11T17:47:00+00:002015-05-11T17:47:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>Although there is abundance of such data both in print and electronic format but it is mostly either buried deep in voluminous books or in a long threaded conversation?
I think it will be appropriate to “cluster” all such useful packages as used in two popular data mining languages R and Python in a single thread.</p>
<ol>
<li>Hierarchical Clustering Methods</li>
<li>For hierarchical clustering methods use the cluster package in R. An example implementation is posted on this<a href="https://class.coursera.org/clusteranalysis-001/forum/thread?thread_id=481">thread</a> In the same package you can find methods for <a href="http://cran.r-project.org/web/packages/clues/index.html">clues</a>, clara, clarans, Diana, ClustOfVar algorithms</li>
<li>BIRCH methods- The<a href="http://cran.r-project.org/web/packages/birch/index.html">R package has been removed</a> from the CRAN repository. You can either use the <a href="http://cran.r-project.org/src/contrib/Archive/birch">earlier versions</a> found here or else you can modify the code. For Python, you can use <a href="http://scikit-learn.org/dev/modules/generated/sklearn.cluster.Birch.html">sckit-learn</a></li>
<li>Agglomerative Clustering- the r function is agnes found in the cluster package</li>
<li>Expectation-Maximization algorithm- the r package is<a href="http://cran.r-project.org/web/packages/EMCluster/index.html">EMCluster</a></li>
<li><a href="http://cran.r-project.org/web/packages/kmodR/index.html">K-modesR</a>for classical k-means, <a href="http://cran.r-project.org/web/packages/kernlab/index.html">kernlab</a>, <a href="http://cran.r-project.org/web/packages/flexclust/index.html">Flexclust</a></li>
<li><a href="http://cran.r-project.org/web/packages/clValid/vignettes/clValid.pdf">Clustering and Cluster validation in R</a>Package - <a href="http://cran.r-project.org/web/packages/fpc/">fpc</a>, <a href="http://cran.r-project.org/web/packages/RANN/index.html">RANN for k-nearest neighbors</a></li>
<li>For clustering mixed-type dataset, the R package is<a href="http://cran.r-project.org/web/packages/clue/index.html">Cluster Ensembles</a></li>
<li>In Python- Text processing tasks can be handled by<a href="http://www.nltk.org/">Natural Language Toolkit (NLP)</a> is a mature, well-documented package for NLP, <a href="http://textblob.readthedocs.org/en/dev/">TextBlob</a> is a simpler alternative, <a href="http://honnibal.github.io/spaCy/">spaCy</a> is a brand new alternative focused on performance. The R package for text processing is <a href="http://cran.r-project.org/web/packages/tm/index.html">tm</a> package</li>
<li><a href="http://cran.r-project.org/web/views/">CRAN Task View</a>- contains a list of packages that can be used for finding groups in data and modeling unobserved cross-sectional heterogeneity. This is one place where you can find both the function name and its description.
Is data cleaning your objective? So if your focus is on data cleaning also known as data munging then python is more powerful in my experience because its backed by <a href="https://docs.python.org/2/library/re.html">regular expression</a></li>
</ol>
<p>Is data exploration your objective? The <a href="http://pandas.pydata.org/">pandas</a> package in Python is very powerful and extremely flexible but its equally challenging to learn too. Similarly, the <a href="http://cran.r-project.org/web/packages/dplyr/index.html">dplyr</a> package in R can be used for the same.</p>
<p>Is data visualization your objective? If so then in R, <a href="http://ggplot2.org/">ggplot2</a> is an excellent package for data visualization. Similarly, you can use <a href="http://ggplot.yhathq.com/">ggplot for python</a> for graphics</p>
<p>And finally, like the CRAN-R project is a single repository for R packages the <a href="https://store.continuum.io/cshop/anaconda/">Anaconda distribution for Python</a> has a similar package management system</p>
<![CDATA[How to create a dissimilarity matrix for mixed type dataset]]>https://duttashi.github.io/blog/how-to-create-a-dissimilarity-matrix-for-mixed-type-dataset2015-05-07T09:33:00+00:002015-05-07T09:33:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>Today, I will discuss on how to create a dissimilarity matrix for mixed type dataset. A quick recap of what a dissimilarity matrix and mixed type dataset is should be good enough to grab your attention.</p>
<p><img src="https://duttashi.github.io/images/matrix.gif" alt="image" /></p>
<p>Let me begin the discussion with the following question,</p>
<p>Question: What is Similarity and Dissimilarity measure?</p>
<p>Answer: Similarity measure is a numerical measure on how similar are two objects. It often falls between 0 (meaning no similarity) and 1 (meaning completely similar).</p>
<p>Dissimilarity measure is a numerical measure on how different or dissimilar are two objects. Its range is inverse of similarity. 0 (meaning completely similar) and infinity (meaning no similarity). There is an excellent tutorial on this that can be found <a href="https://onlinecourses.science.psu.edu/stat857/node/28">here</a>.</p>
<p>I will work with the mixed data set from the UCI Machine Learning repository that can be found <a title="Link: https://archive.ics.uci.edu/ml/datasets/Contraceptive+Method+Choice" href="https://archive.ics.uci.edu/ml/datasets/Contraceptive+Method+Choice">here</a>. The dataset has 1473 attributes in 10 objects. Please note that this dataset is already standardised and scaled. However, real world dataset when given to you for analysis may or may not be scaled. Its then you will have to do all the dirty work.</p>
<p>Step 1: I load the dataset in R and name the dataframe as cmc</p>
<p>Step 2: I now create a dissimilarity matrix by using the distance function of the cluster package as (Note: if package cluster is not loaded then you can load it as;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> library("cluster")
> cmcTrain.dis=dist(cmc, method="euclidean")
</code></pre></div></div>
<div>but this step does not create the matrix. Now I use the as.matrix() to coerce this to matrix as</div>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>cmcTrain.matx=as.matrix(cmcTrain.dis)
</code></pre></div></div>
<p>Step 4: Now, I can apply any clustering algorithm to this dissimilarity matrix. for example, if I want to apply agglomerative hierarchical clustering</div></p>
<p>I will do it as;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> cmcTrain.hc=agnes(cmcTrain.matx,diss=TRUE,metric="euclidean",stand="TRUE",method="average")
</code></pre></div></div>
<p>Notice parameter diss=”TRUE” because cmcTrain.matx is a dissimilarity matrix</pre></p>
<p>I will now plot it as</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> plot(cmcTrain.pamx)
</code></pre></div></div>
<p>Similarly, I can apply the Partition Around Medoids (PAM) clustering algorithm as</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> cmcTrain.pamx=pam(cmcTrain.matx, 2,diss="TRUE",metric="euclidean")
</code></pre></div></div>
<p>I will now plot it as
> plot(cmcTrain.pamx)</p>
<p>And the resultant plot is;</p>
<p><img src="https://duttashi.github.io/images/diss_matrix.png" alt="image" /> </p>
<p>So you see the important is data pre-processing followed by data scaling and standardisation and after that you can apply whatever clustering or classification algorithm to it and interpret the results accordingly.</p>
<![CDATA[Hierarchical Clustering Methods implementation in R- A Case Study]]>https://duttashi.github.io/blog/hierarchical-clustering-methods-implementation-in-r-a-case-study2015-05-05T14:13:00+00:002015-05-05T14:13:00+00:00Ashish Dutthttps://duttashi.github.ioashishdutt@yahoo.com.myblog<p>Today I will present the implementation of agglomerative hierarchical clustering in R. You can do a similar implementation in the language of your choice.</p>
<p>I will use the iris dataset here for explanation purpose. If you don’t have it in your R version you can download it from <a href="https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data">here</a></p>
<p>If you don’t have it in your R version you can download it from here. copy and save it to a text file called “iris.txt”. Please note, that there are no column headings to this dataset as of now. Now there are two ways to add the column headings.</p>
<p>Method 1: Open a new excel sheet and name the first row as sepalLength, sepalWidth, petalLength, petalWidth and after that you copy the dataset and paste it in excel sheet</p>
<p>Method 2: if you want to do it in R, you can use the function like <em> colnames() </em></p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> colnames(iris)=c("sepalLength","sepalWidth","petalLeangth","petalWidth","Species") Now if you view the dataset like
> View(iris) you would see that it has the column names rather than the default V1, V2, V3, V4,V5. On the R console if you type
> summary(iris) you will see that it has 149 values in 5 variables of which the first four are numeric and the last variable Species is categorical.
</code></pre></div></div>
<p>Now, you can install the cluster package using</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> install.packages(“cluster”) and after that load it using the library function like
>library(cluster)
</code></pre></div></div>
<p>I would recommend you to take a look at the cluster package so as to know how to implement the various clustering algorithms in it. To check that use</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> ??cluster
</code></pre></div></div>
<p>I will now begin with the Agglomerative Clustering algorithm implementation in R first using the cluster package.</p>
<p>It is advisable to draw a random sample of data first otherwise the cluster dendogram will be messy because there are more than 100 values in 5 variables. To do this you can do like</p>
<p>my.dataframe[sample(nrow(dataset), size=), ] so for our example, the command will be</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> iris.sample=iris[sample(nrow(iris),size=40),]
</code></pre></div></div>
<p>So now if you view the data like</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> View(iris)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/random.png" alt="image" /></p>
<p>you will see that it’s randomly sampled. Notice the column row.names it lists random row numbers. This proves that the data is randomly sampled.</p>
<p>Now to apply agglomerative hierarchical clustering I will use the agnes function of the cluster package.</p>
<p>Let me first briefly describe the agnes function parameters here;</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>agnes(x, diss = inherits(x, “dist”), metric = “euclidean”, stand = FALSE, method = “average”, par.method,keep.diss = n < 100, keep.data = !diss, trace.lev = 0)
</code></pre></div></div>
<p>where x= data frame or data matrix or the dissimilarity function.</p>
<p>In case of data matrix, all variables must be numeric. Missing Values (NA) are allowed. diss= logical flag if TRUE (which is the default value) then x is considered a dissimilarity matrix, if set FALSE then x is a matrix of observations by variable metric= euclidean distance or manhattan distance, stand= logical flag if TRUE (default value) then measurement in x are standardised before calculating the dissimilarity function. If x is already a dissimilarity matrix, then this argument will be ignored method= defines the clustering method to be used.</p>
<p>There are 6 types, “single”, “complete”, “average”,”ward”,”weighted”,”flexible” Now to apply the agglomerative hierarchical clustering, I will use the agnes function of the cluster package</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> iris.hc=agnes(iris.sample, diss=FALSE, metric="euclidean", stand="TRUE", method="average")
</code></pre></div></div>
<p>Note: I have kept the diss=FALSE because iris.sample is a dataframe and not a dissimilarity matrix. I have chosen euclidean distance metric. And the clustering method is single link. You can try others.</p>
<p>Now to plot this dendogram use the plot function like</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> plot(iris.hc)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/irisplot1.png" alt="image" /></p>
<p>You will notice that the dendogram has row numbers of the random sample which is visually not very interpretive. At least for me this dendogram is not visually interpretive. So I will use the labels command in the plot function to show the names like</p>
<div class="highlighter-rouge"><div class="highlight"><pre class="highlight"><code>> plot(iris.hc, labels=iris.sample$Species)
</code></pre></div></div>
<p><img src="https://duttashi.github.io/images/rplot.png" alt="image" /></p>
<p>Clearly there are three clusters as shown in the plot.</p>
<p>Hope this helps.</p>