Question 4: Consider a data set with the response vector Y = (y₁,..., yn) and the data matrix 1 211 12 1 In In2 2 We model the relationship between X and Y using the linear regression model: y = 00 + 0₁x₁1 + 0₂₁2 + €¡, i = 1,...,n, where ~ N(0,0²). Let the parameter vector be denoted as = (0o,01,02). We wish to minimize the sum of squared residuals: SSE = Σi-1(Yi - (00+0₁₁+0₂¡2))². Let the fitted value be denoted as ŷ¡ = 0 + 0₁Xi1 + 0₂X12, and let the fitted value vector be denoted as Ŷ. a) Show that SSE=1(yi -ŷi)². b) Show that SSE = (Y – Ŷ)'(Y – Ŷ). c) Show that Ý = X0. d) Simplify the derivative equation OSSE 30 e) Find the solution of which solves the equation in part d. = = 0.
Question 4: Consider a data set with the response vector Y = (y₁,..., yn) and the data matrix 1 211 12 1 In In2 2 We model the relationship between X and Y using the linear regression model: y = 00 + 0₁x₁1 + 0₂₁2 + €¡, i = 1,...,n, where ~ N(0,0²). Let the parameter vector be denoted as = (0o,01,02). We wish to minimize the sum of squared residuals: SSE = Σi-1(Yi - (00+0₁₁+0₂¡2))². Let the fitted value be denoted as ŷ¡ = 0 + 0₁Xi1 + 0₂X12, and let the fitted value vector be denoted as Ŷ. a) Show that SSE=1(yi -ŷi)². b) Show that SSE = (Y – Ŷ)'(Y – Ŷ). c) Show that Ý = X0. d) Simplify the derivative equation OSSE 30 e) Find the solution of which solves the equation in part d. = = 0.
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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Transcribed Image Text:Question 4: Consider a data set with the response vector Y = (y₁,..., yn) and the data
matrix
-
1
X12
(EB)
1
Inl
=
We model the relationship between X and Y using the linear regression model: y =
00+0₁₁+0₂x₁2 + €₁, i 1,...,n, where ~ N(0,02). Let the parameter vector be
denoted as (00, 01,02). We wish to minimize the sum of squared residuals: SSE =
Σ(yi (00+0₁₁+0₂2))². Let the fitted value be denoted as ĝi = 0 + 0₁x₁1 + 0₂x₁2,
and let the fitted value vector be denoted as Ŷ.
=
2
a) Show that SSE = Σ-1(yi - ŷi)².
b) Show that SSE = (Y – Ŷ)¹(Y – Ý).
c) Show that Ŷ = XO.
d) Simplify the derivative equation
asse
20
e) Find the solution of which solves the equation in part d.
: 0.
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