Statistical Analysis

1 / 11

# Statistical Analysis - PowerPoint PPT Presentation

Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 6QF Matrix Solutions to Normal Equations. Direct (Kronecker) Products. (Left) Direct Product. Some Properties. assuming all operations are valid. Solving the Normal Equations.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Statistical Analysis' - elizabeth-stout

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Statistical Analysis

Professor Lynne Stokes

Department of Statistical Science

Lecture 6QF

Matrix Solutions to Normal Equations

Direct (Kronecker) Products

(Left) Direct Product

Some Properties

assuming all operations are valid

Solving the Normal Equations

Single-Factor, Balanced Experiment

yij = m + ai + eij i = 1, ..., a; j = 1, ..., r

Matrix Formulation

y = Xb + e

y = (y11 y12 ... y1r ... ya1 ya2 ... yar)’

Show

Properties of X’X
• Symmetric
• Same rank as X
• Has an inverse (X’X)-1 iff X has full column rank
Solving the Normal Equations

Residuals

Least Squares

Solution: Solve the Normal Equations

Solving the Normal Equations

Normal Equations

Problem: X’X is Singular, has no inverse

Show

Generalized Inverse: G or

A

Definition

AGA = A

not unique

Moore-Penrose Generalized Inverse

• AGA = A
• GAG = G
• AG is symmetric
• GA is symmetric

unique

Some Properties

if A has full row rank, G = A’(AA’)-1

if A has full column rank, G = (A’A)–1A’

Common Notation

A

(X’X)

(X’X)

Solving the Normal Equations

Normal Equations

Solutions

• Every solution to the normal equations corresponds to a generalized inverse of X’X
• Every generalized inverse of X’X solves the normal equations
A Solution to the Normal Equations

One Generalized Inverse

Verification

A Solution to the Normal Equations

Corresponds to the solution to the normal equations

with the constraint m = 0 imposed

Assignment
• Find another generalized inverse for X’X in a one-factor balanced experiment
• Verify that it is a generalized inverse
• Solve the normal equations using the generalized inverse
• Determine what constraint on the model parameters correspond to the generalized inverse