/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q20E Exercises 19 and 20 involve a de... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Q20E (page 331)

Exercises 19 and 20 involve a design matrix \(X\) with two or more columns and a least-squares solution \(\hat \beta \) of \({\bf{y}} = X\beta \). Consider the following numbers.

(i) \({\left\| {X\hat \beta } \right\|^2}\)鈥攖he sum of the squares of the 鈥渞egression term.鈥� Denote this number by \(SS\left( R \right)\).

(ii) \({\left\| {{\bf{y}} - X\hat \beta } \right\|^2}\)鈥攖he sum of the squares for error term. Denote this number by \(SS\left( E \right)\).

(iii) \({\left\| {\bf{y}} \right\|^2}\)鈥攖he 鈥渢otal鈥� sum of the squares of the -values. Denote this number by \(SS\left( T \right)\).

Every statistics text that discusses regression and the linear model \(y = X\beta + \in \) introduces these numbers, though terminology and notation vary somewhat. To simplify matters, assume that the mean of the -values is zero. In this case, \(SS\left( T \right)\) is proportional to what is called the variance of the set of \(y\)-values.

20. Show that \({\left\| {X\hat \beta } \right\|^2} = {\hat \beta ^T}{X^T}{\bf{y}}\). (Hint: Rewrite the left side and use the fact that \(\hat \beta \) satisfies the normal equations.) This formula for is used in statistics. From this and from Exercise 19, obtain the standard formula for \(SS\left( E \right)\):

\(SS\left( E \right) = {y^T}y - \hat \beta {X^T}y\)

Short Answer

Expert verified

It is verified that \({\left\| {X\hat \beta } \right\|^2} = {\hat \beta ^T}{X^T}{\bf{y}}\) and the standard formula for \(SS\left( E \right)\) is \(SS\left( E \right) = {\left\| {\bf{y}} \right\|^2} - {\hat \beta ^T}{X^T}{\bf{y}}\).

Step by step solution

01

Write the property

  1. \({\left\| {\bf{v}} \right\|^2} = {{\bf{v}}^T}{\bf{v}}\)
  2. \({\left( {AB} \right)^T} = {B^T}{A^T}\)
02

Solve \({\left\| {X\hat \beta } \right\|^2}\)

Solve \({\left\| {X\hat \beta } \right\|^2}\) by using the properties \({\left\| {\bf{v}} \right\|^2} &= {{\bf{v}}^T}{\bf{v}}\) and \({\left( {AB} \right)^T} &= {B^T}{A^T}\).

\(\begin{aligned}{\left\| {X\hat \beta } \right\|^2} = {\left( {X\hat \beta } \right)^T}\left( {X\hat \beta } \right)\\ = {{\hat \beta }^T}{X^T}X\hat \beta \end{aligned}\)

Use \({\bf{y}} = X\beta \) in the obtained expression.

\({\left\| {X\hat \beta } \right\|^2} = {\hat \beta ^T}{X^T}{\bf{y}}\)

Hence proved.

03

Find the value of \(SS\left( E \right)\)

From exercise (19),\(SS\left( T \right) = SS\left( R \right) + SS\left( E \right)\), which implies \(SS\left( E \right) = SS\left( T \right) - SS\left( R \right)\). It is given that \(SS\left( T \right) = {\left\| {\bf{y}} \right\|^2}\) and \(SS\left( R \right) = {\left\| {X\hat \beta } \right\|^2}\).

Substitute the required expressions into \(SS\left( T \right) = SS\left( R \right) + SS\left( E \right)\) and simplify by using \({\left\| {X\hat \beta } \right\|^2} = {\hat \beta ^T}{X^T}{\bf{y}}\).

\(\begin{aligned}SS\left( E \right) = {\left\| {\bf{y}} \right\|^2} - {\left\| {X\hat \beta } \right\|^2}\\ = {\left\| {\bf{y}} \right\|^2} - {{\hat \beta }^T}{X^T}{\bf{y}}\end{aligned}\)

Hence, the standard formula for \(SS\left( E \right)\) is \(SS\left( E \right) = {\left\| {\bf{y}} \right\|^2} - {\hat \beta ^T}{X^T}{\bf{y}}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises 11 and 12, find the closest point to\[{\bf{y}}\]in the subspace\[W\]spanned by\[{{\bf{v}}_1}\], and\[{{\bf{v}}_2}\].

11.\[y = \left[ {\begin{aligned}3\\1\\5\\1\end{aligned}} \right]\],\[{{\bf{v}}_1} = \left[ {\begin{aligned}3\\1\\{ - 1}\\1\end{aligned}} \right]\],\[{{\bf{v}}_2} = \left[ {\begin{aligned}1\\{ - 1}\\1\\{ - 1}\end{aligned}} \right]\]

Let \({\mathbb{R}^{\bf{2}}}\) have the inner product of Example 1. Show that the Cauchy-Schwarz inequality holds for \({\bf{x}} = \left( {{\bf{3}}, - {\bf{2}}} \right)\) and \({\bf{y}} = \left( { - {\bf{2}},{\bf{1}}} \right)\). (Suggestion: Study \({\left| {\left\langle {{\bf{x}},{\bf{y}}} \right\rangle } \right|^{\bf{2}}}\).)

In Exercises 13 and 14, find the best approximation to\[{\bf{z}}\]by vectors of the form\[{c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2}\].

13.\[z = \left[ {\begin{aligned}3\\{ - 7}\\2\\3\end{aligned}} \right]\],\[{{\bf{v}}_1} = \left[ {\begin{aligned}2\\{ - 1}\\{ - 3}\\1\end{aligned}} \right]\],\[{{\bf{v}}_2} = \left[ {\begin{aligned}1\\1\\0\\{ - 1}\end{aligned}} \right]\]

To measure the take-off performance of an airplane, the horizontal position of the plane was measured every second, from \(t = 0\) to \(t = 12\). The positions (in feet) were: 0, 8.8, 29.9, 62.0, 104.7, 159.1, 222.0, 294.5, 380.4, 471.1, 571.7, 686.8, 809.2.

a. Find the least-squares cubic curve \(y = {\beta _0} + {\beta _1}t + {\beta _2}{t^2} + {\beta _3}{t^3}\) for these data.

b. Use the result of part (a) to estimate the velocity of the plane when \(t = 4.5\) seconds.

In Exercises 1-6, the given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.

  1. \(\left( {\begin{aligned}{{}{}}3\\0\\{ - 1}\end{aligned}} \right),\left( {\begin{aligned}{{}{}}8\\5\\{ - 6}\end{aligned}} \right)\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.