IE582 Fall 2021 Assignment-4 Due Date- 10/23/2021 11:59 PM Instructions • Please complete this assignment in the same group formed for the assignment2. • There are three question sets in this assignment. ◦

Question Set 1: Simple Linear Regression Calculation ◦

Question Set 2: Multiple Linear Regression Matrix Derivation ◦

Question Set 3: Forward, Backward and Stepwise Selection Linear Regression

• Question sets 1 and 2 are for calculator-based calculations and derivations. • You are only required to answer interpretative questions based on the Appendix Output section. (Question Sets 3) Question Set 1: Simple Linear Regression Calculation The table 1 dataset provides ht (Height in cms) and wt (Weight in kgs) for , 18-year-old girls. This dataset is obtained from alr4 package. Based on the dataset, please answer the following questions by assuming ht to be the independent variable.

## [1] "Question Set 1 Dataset"

## ht wt

## 1 169.6 71.2

## 2 166.8 58.2

## 3 157.1 56.0

## 4 181.1 64.5

## 5 158.4 53.0

## 6 165.6 52.4

## 7 166.7 56.8

## 8 156.5 49.2

## 9 168.1 55.6

## 10 165.3 77.8

• Compute estimates of the slope and the intercept for the simple linear regression.

• Find the residuals or errors associated with the fitted model. Calculate associated mean and standard deviation(sample).

• Estimate for the fitted line. What would be the correlation between two variables?

• Predict the weight range for a group having the height range in 140 cm to 190 cm.

Question Set 2: Multiple Linear Regression Matrix n = 10 β1 β0 R2 Page 1 of 35IE582_F2021_Assignment_4.utf8 10/19/2021file:///C:/Users/zhong/OneDrive/Desktop/IE582_F2021_Assignment_4-1.html Derivation Consider the following equation for observation of dependent variable using as an indepedent variable. Where

• Based on the provided set of equations, first state the necessary assumptions to convert the same into Multiple Linear Regression framework

• Derive the appropriate solution for and using Matrix Calculus.

• Verify your solution with the simple linear regression equations. Please provide sufficient reasoning for your steps Question Set 3: Forward, Backward and Stepwise Selection Linear Regression Introduction: The assignment is based on the bike sharing dataset listed on UCI Machine Learning Repository. The dataset is provided at two levels:

• Day Level: Question Set 3 is based on the same.

Useful links:

• Data Dictionary File: http://personal.psu.edu/mzm6664/Readme.txt (http://personal.psu.edu/mzm6664/Readme.txt) ith Y X = + + + ⋯ + + +Yi β0 β1Xi1 β2Xi2 βp−1Xi(p−1) βpXip ei i = 1, 2, . . . ,n j = 0, 1, 2, . . . , p =  indepedent variable value for   observation Xij jth ith =  error assocaited with the prediction of   observation ei ith X e Page 2 of 35IE582_F2021_Assignment_4.utf8 10/19/2021file:///C:/Users/zhong/OneDrive/Desktop/IE582_F2021_Assignment_4-1.html • Original Data Source Link: https://archive.ics.uci.edu/ml/datasets/Bike+Sharing+Dataset# (https://archive.ics.uci.edu/ml/datasets/Bike+Sharing+Dataset#) • Processed day level data: http://personal.psu.edu/mul490/cnt_day.csv (http://personal.psu.edu/mul490/cnt_day.csv)

This question is based on Day Level dataset. An approach is presented to develop a linear regression model using combination of Forward, Backward and Stepwise selection methods with VIF based insights. Three iterations have been utilized to decide the final model. Please answer the following

questions:

• Please verify if all the three methods- Forward, Backward and Stepwise selection result to the same model at the end of iteration one. Report the number of steps taken by each of the methods to reach to the final model of the first iteration.

• By interpreting results, please provide reasoning for the second and third stage of iteration. • Based on the beta values of the final model(at the end of third iteration), identify factors that are negatively related to count of renting bikes on a given day.

• Please comments on residuals and standardized residuals graphs with respect to multiple linear regression assumptions.

• Post results on the given approach, please list possible two actions that can be taken to improve the model performance. (You are only required to list actions.) Appendix Output: The R Output for questions

3

## [1] "Forward selection using AIC" Page 3 of 35IE582_F2021_Assignment_4.utf8 10/19/2021file:///C:/Users/zhong/OneDrive/Desktop/IE582_F2021_Assignment_4-1.html

## Start: AIC=11066.88

## cnt ~ 1

## ## Df Sum of Sq RSS AIC

## + atemp 1 1091003307 1648532085 10698

## + dteday 1 1083287662 1656247730 10701

## + temp 1 1078688585 1660846807 10703

## + yr 1 879828893 1859706499 10786

## + season 1 451797359 2287738033 10937

## + weathersit 1 242288753 2497246639 11001

## + mnth 1 214744463 2524790929 11009

## + windspeed 1 150705556 2588829836 11028

## + hum 1 27757373 2711778019 11061

## + holiday 1 12797494 2726737898 11066

## + weekday 1 12461089 2727074303 11066

## + workingday 1 10246038 2729289354 11066

##2739535392 11067