6.2 4 practice modeling fitting linear models to data answers

Data Science and Big Data Analytics

Chapter 6: Advanced Analytical Theory and Methods: Regression

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Chapter Sections

6.1 Linear Regression

6.2 Logical Regression

6.3 Reasons to Choose and Cautions

6.4 Additional Regression Models

Summary

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6 Regression

Regression analysis attempts to explain the influence that input (independent) variables have on the outcome (dependent) variable

Questions regression might answer

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

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

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6.1 Linear Regression

Models the relationship between several input variables and a continuous outcome variable

Assumption is that the relationship is linear

Various transformations can be used to achieve a linear relationship

Linear regression models are probabilistic

Involves randomness and uncertainty

Not deterministic like Ohm’s Law (V=IR)

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6.1.1 Use Cases

Real estate example

Predict residential home prices

Possible inputs – living area, #bathrooms, #bedrooms, lot size, property taxes

Demand forecasting example

Restaurant predicts quantity of food needed

Possible inputs – weather, day of week, etc.

Medical example

Analyze effect of proposed radiation treatment

Possible inputs – radiation treatment duration, freq

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6.1.2 Model Description

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6.1.2 Model Description Example

Predict person’s annual income as a function of age and education

Ordinary Least Squares (OLS) is a common technique to estimate the parameters

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6.1.2 Model Description Example

OLS

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6.1.2 Model Description Example

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6.1.2 Model Description With Normally Distributed Errors

Making additional assumptions on the error term provides further capabilities

It is common to assume the error term is a normally distributed random variable

Mean zero and constant variance

That is

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With this assumption, the expected value is

And the variance is

6.1.2 Model Description With Normally Distributed Errors

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Normality assumption with one input variable

E.g., for x=8, E(y)~20 but varies 15-25

6.1.2 Model Description With Normally Distributed Errors

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6.1.2 Model Description Example in R

Be sure to get publisher's R downloads: http://www.wiley.com/WileyCDA/WileyTitle/productCd-111887613X.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)

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Scatterplot

Examine bottom line

income~age: strong + trend

income~educ: slight + trend

income~gender: no trend

6.1.2 Model Description Example in R

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Quantify the linear relationship trends

> results <- lm(Income~Age+Education+Gender,income_input)

> summary(results)

Intercept: income of $7263 for newborn female

Age coef: ~1, year age increase -> $1k income incr

Educ coef: ~1.76, year educ + -> $1.76k 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

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Examine residuals – uncertainty or sampling error

Small p-values indicate statistically significant results

Age and Education highly significant, p<2e-16

Gender p=0.13 large, not significant at 90% confid. level

Therefore, drop variable gender from linear model

> results2 <- lm(Income~Age+Education,income_input)

> summary(results) # results about same as before

Residual standard error: residual standard deviation

R-squared (R2): variation of data explained by model

Here ~64% (R2 = 1 means model explains data perfectly)

F-statistic: tests entire model – here p value is small

6.1.2 Model Description Example in R

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6.1.2 Model Description Categorical Variables

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

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6.1.2 Model Description Confidence Intervals on the Parameters

Once an acceptable linear model is developed, it is often useful to draw some inferences

R provides confidence intervals using confint() function

> confint(results2, level = .95)

For example, Education coefficient was 1.76, and now the corresponding 95% confidence interval is (1.53. 1.99)

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6.1.2 Model Description Confidence Interval on Expected Outcome

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(results2,new_pt,level=.95,

interval=“confidence”)

> conf_int_pt

Expected income = $68699, conf interval ($67831,$69567)

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6.1.2 Model Description Prediction Interval on a Particular Outcome

The predict() function in R also provides upper/lower bounds on a particular outcome, prediction intervals

> pred_int_pt <- predict(results2,new_pt,level=.95,

interval=“prediction”)

> pred_int_pt

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

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6.1.3 Diagnostics Evaluating the Linearity Assumption

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

Class of dataset Groceries is transactions, containing 3 slots

transactionInfo # data frame with vectors having length of transactions

itemInfo # data frame storing item labels

data # binary evidence matrix of labels in transactions

> Groceries@itemInfo[1:10,]

> apply(Groceries@data[,10:20],2,function(r) paste(Groceries@itemInfo[r,"labels"],collapse=", "))

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6.1.3 Diagnostics Evaluating the Linearity Assumption

Income as a quadratic function of Age

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6.1.3 Diagnostics Evaluating the Residuals

The error terms was assumed to be normally distributed with zero mean and constant variance

> with(results2,{plot(fitted.values,residuals,ylim=c(-40,40)) })

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6.1.3 Diagnostics Evaluating the Residuals

Next four figs don’t fit zero mean, const variance assumption

Nonlnear trend in residuals

Residuals not centered on zero

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6.1.3 Diagnostics Evaluating the Residuals

Variance not

constant

Residuals not centered on zero

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6.1.3 Diagnostics Evaluating the Normality Assumption

The normality assumption still has to be validate

> hist(results2$residuals)

Residuals centered on zero and appear normally distributed

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6.1.3 Diagnostics Evaluating the Normality Assumption

Another option is to examine a Q-Q plot comparing observed data against quantiles (Q) of assumed dist

> qqnorm(results2$residuals)

> qqline(results2$residuals)

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6.1.3 Diagnostics Evaluating the Normality Assumption

Normally distributed residuals

Non-normally distributed residuals

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6.1.3 Diagnostics N-Fold Cross-Validation

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

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

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6.1.3 Diagnostics Other Diagnostic Considerations

The model might be improved by including additional input variables

However, the adjusted R2 applies a penalty as the number of parameters increases

Residual plots should be examined for outliers

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

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6.2 Logistic Regression Introduction

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 focuses on two-valued outcomes like true/false, pass/fail, or yes/no

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6.2.1 Logistic Regression Use Cases

Medical

Probability of a patient’s successful response to a specific medical treatment – input could include age, weight, etc.

Finance

Probability an applicant defaults on a loan

Marketing

Probability a wireless customer switches carriers (churns)

Engineering

Probability a mechanical part malfunctions or fails

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6.2.2 Logistic Regression Model Description

Logical regression is based on the logistic function

As y -> infinity, f(y)->1; and as y->-infinity, f(y)->0

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6.2.2 Logistic Regression Model Description

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.

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6.2.2 Logistic Regression Model Description: customer churn example

A wireless telecom company estimates probability of a customer churning (switching companies)

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

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6.2.2 Logistic Regression Model Description: customer churn example

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6.2.3 Diagnostics Model Description: customer churn example

> head(churn_input) # Churned = 1 if cust churned

> sum(churn_input$Churned) # 1743/8000 churned

Use the Generalized Linear Model function glm()

> Churn_logistic1<-glm(Churned~Age+Married+Cust_years+Churned_contacts,data=churn_input,family=binomial(link=“logit”))

> summary(Churn_logistic1) # Age + Churned_contacts best

> Churn_logistic3<-glm(Churned~Age+Churned_contacts,data=churn_input,family=binomial(link=“logit”))

> summary(Churn_logistic3) # Age + Churned_contacts

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6.2.3 Diagnostics Deviance and the Pseudo-R2

In logistic regression, deviance = -2logL

where L is the maximized value of the likelihood function used to obtain the parameter estimates

Two deviance values are provided

Null deviance = deviance based on only the y-intercept term

Residual deviance = deviance based on all parameters

Pseudo-R2 measures how well fitted model explains the data

Value near 1 indicates a good fit over the null model

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6.2.3 Diagnostics Receiver Operating Characteristic (ROC) Curve

Logistic regression is often used to classify

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 nC, we have

True Positive: predict C, when actually C

True Negative: predict nC, when actually nC

False Positive: predict C, when actually nC

False Negative: predict nC, when actually C

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6.2.3 Diagnostics Receiver Operating Characteristic (ROC) Curve

The Receiver Operating Characteristic (ROC) curve

Plots TPR against FPR

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6.2.3 Diagnostics Receiver Operating Characteristic (ROC) Curve

> library(ROCR)

> Pred = predict(Churn_logistic3, type=“response”)

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6.2.3 Diagnostics Receiver Operating Characteristic (ROC) Curve

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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.

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6.3 Reasons to Choose and Cautions

Linear regression – outcome variable continuous

Logistic regression – outcome variable categorical

Both models assume a linear additive function of the inputs variables

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”

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6.4 Additional Regression Models

Multicollinearity is the condition when several input variables are highly correlated

This can lead to inappropriately large coefficients

To mitigate this problem

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 more-than-two-state categorical outcome variable

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