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(e) Show that MSE for the OLS is 0. What is the relationship between y and span(X). (As a sanity check, sum of residuals should be 0.)

(e) Show that MSE for the OLS is 0. What is the relationship between y and span(X). (As a sanity check, sum of residuals should be 0.)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Please answer question (e)

 
A Special Case of Linear Regression 4. In this question, we fit two models: y = 0 + 0x₁ y = 0 + 0x₁+02x2 using L2 loss. The superscript S is to denote a Simple Linear Regression (SLR) and O is used to denote a Ordinary Least Square (OLS) with two features, respectively. The data are given below: Y -1 3 4 bias 1 1 1 1 -2 1 X2 -1 0 1 1 (a)Find and using the formulas derived in lecture 10 ( = rand = y). Specify which x you are using and show all steps. You may find it helpful to keep intermediate steps in the square root (they cancel out nicely at the end!).
Transcribed Image Text:A Special Case of Linear Regression 4. In this question, we fit two models: y = 0 + 0x₁ y = 0 + 0x₁+02x2 using L2 loss. The superscript S is to denote a Simple Linear Regression (SLR) and O is used to denote a Ordinary Least Square (OLS) with two features, respectively. The data are given below: Y -1 3 4 bias 1 1 1 1 -2 1 X2 -1 0 1 1 (a)Find and using the formulas derived in lecture 10 ( = rand = y). Specify which x you are using and show all steps. You may find it helpful to keep intermediate steps in the square root (they cancel out nicely at the end!).
[6] using the formula derived in lecture 12: ÔS = (X¹X)-¹X¹y. Explicitly write out the matrix X for this problem and show all steps. How does it -1 [1/a 0 to your answer to part a)? (Hint: [8] = [¹/ 19%] (b)Findo s compare (c) Find the MSE for the SLR model above. (As a sanity check, sum of residuals should be 0.) 0 (d) Finde o= using the formula derived in lecture 12: 60 = (XX)-¹XTy Explicitly write out the matrix X for this problem and show all steps. (Hint: The intercept and coefficient of ₁ for OLS are the same as SLR in this special example. Check remark at the end of the question to see why this is the case.) (e) Show that MSE for the OLS is 0. What is the relationship between y and span(X). (As a sanity check, sum of residuals should be 0.)
Transcribed Image Text:[6] using the formula derived in lecture 12: ÔS = (X¹X)-¹X¹y. Explicitly write out the matrix X for this problem and show all steps. How does it -1 [1/a 0 to your answer to part a)? (Hint: [8] = [¹/ 19%] (b)Findo s compare (c) Find the MSE for the SLR model above. (As a sanity check, sum of residuals should be 0.) 0 (d) Finde o= using the formula derived in lecture 12: 60 = (XX)-¹XTy Explicitly write out the matrix X for this problem and show all steps. (Hint: The intercept and coefficient of ₁ for OLS are the same as SLR in this special example. Check remark at the end of the question to see why this is the case.) (e) Show that MSE for the OLS is 0. What is the relationship between y and span(X). (As a sanity check, sum of residuals should be 0.)
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