Data Science and Big Data Analytics Chap 6

Data Science and Big Data Analytics Chap 6: Adv Analytical Theory and Methods: Regression Charles Tappert Seidenberg School of CSIS, Pace University

Chapter Sections n n n 6. 1 Linear Regression 6. 2 Logical Regression 6. 3 Reasons to Choose and Cautions 6. 4 Additional Regression Models Summary

6 Regression n n Regression analysis attempts to explain the influence that input (independent) variables have on the outcome (dependent) variable Questions regression might answer n n n What is a person’s expected income? What is probability an applicant will default on a loan? Regression can find the input variables having the greatest statistical influence on the outcome n n Then, can try to produce better values of input variables E. g. – if 10 -year-old reading level predicts students’ later success, then try to improve early age reading levels

6. 1 Linear Regression n Models the relationship between several input variables and a continuous outcome variable n n n Assumption is that the relationship is linear Various transformations can be used to achieve a linear relationship Linear regression models are probabilistic n n Involves randomness and uncertainty Not deterministic like Ohm’s Law (V=IR)

6. 1. 1 Use Cases n Real estate example n Predict residential home prices n n Demand forecasting example n Restaurant predicts quantity of food needed n n Possible inputs – living area, #bathrooms, #bedrooms, lot size, property taxes Possible inputs – weather, day of week, etc. Medical example n Analyze effect of proposed radiation treatment n Possible inputs – radiation treatment duration, freq

6. 1. 2 Model Description

6. 1. 2 Model Description Example n n Predict person’s annual income as a function of age and education Ordinary Least Squares (OLS) is a common technique to estimate the parameters

6. 1. 2 Model Description Example OLS

6. 1. 2 Model Description Example

6. 1. 2 Model Description With Normally Distributed Errors n n Making additional assumptions on the error term provides further capabilities It is common to assume the error term is a normally distributed random variable n n Mean zero and constant variance That is

6. 1. 2 Model Description With Normally Distributed Errors n With this assumption, the expected value is n And the variance is

6. 1. 2 Model Description With Normally Distributed Errors n Normality assumption with one input variable n E. g. , for x=8, E(y)~20 but varies 15 -25

6. 1. 2 Model Description Example in R Be sure to get publisher's R downloads: http: //www. wiley. com/Wiley. CDA/Wiley. Title/product. Cd-111887613 X. html > income_input = as. data. frame(read. csv(“c: /data/income. csv”)) > income_input[1: 10, ] > summary(income_input) > library(lattice) > splom(~income_input[c(2: 5)], groups=NULL, data=income_input, axis. line. tck=0, axis. text. alpha=0)

6. 1. 2 Model Description Example in R Scatterplot Examine bottom line income~age: strong + trend income~educ: slight + trend income~gender: no trend

6. 1. 2 Model Description Example in R

6. 1. 2 Model Description Example in R n Quantify the linear relationship trends > results <- lm(Income~Age+Education+Gender, income_input) > summary(results) n n n Intercept: income of $7263 for newborn female Age coef: ~1, year age increase -> $1 k income incr Educ coef: ~1. 76, year educ + -> $1. 76 k income + Gender coef: ~-0. 93, male income decreases $930 Residuals – assumed to be normally distributed – vary from -37 to +37 (more information coming)

6. 1. 2 Model Description Example in R n n Examine residuals – uncertainty or sampling error Small p-values indicate statistically significant results n n n Age and Education highly significant, p<2 e-16 Gender p=0. 13 large, not significant at 90% confid. level Therefore, drop variable gender from linear model > results 2 <- lm(Income~Age+Education, income_input) > summary(results) # results about same as before n n Residual standard error: residual standard deviation R-squared (R 2): variation of data explained by model n n Here ~64% (R 2 = 1 means model explains data perfectly) F-statistic: tests entire model – here p value is small

6. 1. 2 Model Description Categorical Variables n n n In the example in R, Gender is a binary variable Variables like Gender are categorical variables in contrast to numeric variables where numeric differences are meaningful The book section discusses how income by state could be implemented

6. 1. 2 Model Description Confidence Intervals on the Parameters n Once an acceptable linear model is developed, it is often useful to draw some inferences n n R provides confidence intervals using confint() function > confint(results 2, level =. 95) For example, the Education coefficient was 1. 76, and now the corresponding 95% confidence interval is (1. 53, 1. 99) which provides the amount of uncertainty in the estimate

6. 1. 2 Model Description Confidence Interval on Expected Outcome n n In the income example, the regression line provides the expected income for a given Age and Education Using the predict() function in R, a confidence interval on the expected outcome can be obtained > Age <- 41 > Education <- 12 > new_pt <- data. frame(Age, Education) > conf_int_pt <- predict(results 2, new_pt, level=. 95, interval=“confidence”) > conf_int_pt n Expected income = $68699, conf interval ($67831, $69567)

6. 1. 2 Model Description Prediction Interval on a Particular Outcome n The predict() function in R also provides upper/lower bounds on a particular outcome, prediction intervals > pred_int_pt <- predict(results 2, new_pt, level=. 95, interval=“prediction”) > pred_int_pt n n Expected income = $68699, pred interval ($44988, $92409) This is a much wider interval because the confidence interval applies to the expected outcome that falls on the regression line, but the prediction interval applies to an outcome that may appear anywhere within the normal distribution

> 6. 1. 3 Diagnostics Evaluating the Linearity Assumption n n A major assumption in linear regression modeling is that the relationship between the input and output variables is linear The most fundamental way to evaluate this is to plot the outcome variable against each income variable In the following figure a linear model would not apply In such cases, a transformation might allow a linear model to apply

> 6. 1. 3 Diagnostics Evaluating the Linearity Assumption n Income as a quadratic function of Age

> 6. 1. 3 Diagnostics Evaluating the Residuals n The error terms was assumed to be normally distributed with zero mean and constant variance > with(results 2, {plot(fitted. values, residuals, ylim=c(40, 40)) })

> 6. 1. 3 Diagnostics Evaluating the Residuals n Next four figs don’t fit zero mean, const variance assumption Nonlnear trend in residuals Residuals not centered on zero

> 6. 1. 3 Diagnostics Evaluating the Residuals not centered on zero Variance not constant

> 6. 1. 3 Diagnostics Evaluating the Normality Assumption n The normality assumption still has to be validated > hist(results 2$residuals) Residuals centered on zero and appear normally distributed

> 6. 1. 3 Diagnostics Evaluating the Normality Assumption n Another option is to examine a Q-Q plot comparing observed data against quantiles (Q) of assumed dist > qqnorm(results 2$residuals) > qqline(results 2$residuals)

> 6. 1. 3 Diagnostics Evaluating the Normality Assumption Normally distributed residuals Non-normally distributed residuals

> 6. 1. 3 Diagnostics N-Fold Cross-Validation n n To prevent overfitting, a common practice splits the dataset into training and test sets, develops the model on the training set and evaluates it on the test set If the quantity of the dataset is insufficient for this, an N -fold cross-validation technique can be used n n n Dataset randomly split into N dataset of equal size Model trained on N-1 of the sets, tested on remaining one Process repeated N times Average the N model errors over the N folds Note: if N = size of dataset, this is leave-one-out procedure

> 6. 1. 3 Diagnostics Other Diagnostic Considerations n The model might be improved by including additional input variables n n Residual plots should be examined for outliers n n n However, the adjusted R 2 applies a penalty as the number of parameters increases Points markedly different from the majority of points They result from bad data, data processing errors, or actual rare occurrences Finally, the magnitude and signs of the estimated parameters should be examined to see if they make sense

> 6. 2 Logistic Regression Introduction n n In linear regression modeling, the outcome variable is continuous – e. g. , income ~ age and education In logistic regression, the outcome variable is categorical, and this chapter section focuses on twovalued outcomes like true/false, pass/fail, or yes/no

> 6. 2. 1 Logistic Regression Use Cases n Medical n n Finance n n Probability an applicant defaults on a loan Marketing n n Probability of a patient’s successful response to a specific medical treatment – input could include age, weight, etc. Probability a wireless customer switches carriers (churns) Engineering n Probability a mechanical part malfunctions or fails

> 6. 2. 2 Logistic Regression Model Description n Logical regression is based on the logistic function n As y -> infinity, f(y)->1; and as y->-infinity, f(y)->0

> 6. 2. 2 Logistic Regression Model Description n With the range of f(y) as (0, 1), the logistic function models the probability of an outcome occurring In contrast to linear regression, the values of y are not directly observed; only the values of f(y) in terms of success or failure are observed. Called log odds ratio, or logit of p. Maximum Likelihood Estimation (MLE) is used to estimate model parameters. MLR is beyond the scope of this book. Note p =f(y)

> 6. 2. 2 Logistic Regression Model Description: customer churn example n A wireless telecom company estimates probability of a customer churning (switching companies) n n Variables collected for each customer: age (years), married (y/n), duration as customer (years), churned contacts (count), churned (true/false) After analyzing the data and fitting a logical regression model, age and churned contacts were selected as the best predictor variables

> 6. 2. 2 Logistic Regression Model Description: customer churn example

> 6. 2. 3 Diagnostics Model Description: customer churn example > head(churn_input) # Churned = 1 if cust churned > sum(churn_input$Churned) # 1743/8000 churned n Use the Generalized Linear Model function glm() > Churn_logistic 1<glm(Churned~Age+Married+Cust_years+Churned_contacts, data =churn_input, family=binomial(link=“logit”)) > summary(Churn_logistic 1) # Age + Churned_contacts best > Churn_logistic 3<glm(Churned~Age+Churned_contacts, data=churn_input, family =binomial(link=“logit”)) > summary(Churn_logistic 3) # Age + Churned_contacts

> 6. 2. 3 Diagnostics Deviance and the Pseudo-R 2 n In logistic regression, deviance = -2 log. L n where L is the maximized value of the likelihood function used to obtain the parameter estimates Two deviance values are provided n n Null deviance = deviance based on only the y-intercept term Residual deviance = deviance based on all parameters Pseudo-R 2 measures how well fitted model explains the data n Value near 1 indicates a good fit over the null model

> 6. 2. 3 Diagnostics Receiver Operating Characteristic (ROC) Curve n Logistic regression is often used to classify n n n In the Churn example, a customer can be classified as Churn if the model predicts high probability of churning Although 0. 5 is often used as the probability threshold, other values can be used based on desired error tradeoff For two classes, C and n. C, we have n n True Positive: predict C, when actually C True Negative: predict n. C, when actually n. C False Positive: predict C, when actually n. C False Negative: predict n. C, when actually C

> 6. 2. 3 Diagnostics Receiver Operating Characteristic (ROC) Curve n The Receiver Operating Characteristic (ROC) curve n Plots TPR against FPR

> 6. 2. 3 Diagnostics Receiver Operating Characteristic (ROC) Curve > library(ROCR) > Pred = predict(Churn_logistic 3, type=“response”)

> 6. 2. 3 Diagnostics Receiver Operating Characteristic (ROC) Curve

> 6. 2. 3 Diagnostics Histogram of the Probabilities It is interesting to visualize the counts of the customers who churned and who didn’t churn against the estimated churn probability.

> 6. 3 Reasons to Choose and Cautions n n n Linear regression – outcome variable continuous Logistic regression – outcome variable categorical Both models assume a linear additive function of the inputs variables n n n If this is not true, the models perform poorly In linear regression, the further assumption of normally distributed error terms is important for many statistical inferences Although a set of input variables may be a good predictor of an output variable, “correlation does not imply causation”

> 6. 4 Additional Regression Models n Multicollinearity is the condition when several input variables are highly correlated n n To mitigate this problem n n n This can lead to inappropriately large coefficients Ridge regression applies a penalty based on the size of the coefficients Lasso regression applies a penalty proportional to the sum of the absolute values of the coefficients Multinomial logistic regression – used for a morethan-two-state categorical outcome variable