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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.

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Statistical analysis
Statistical Analysis

Professor Lynne Stokes

Department of Statistical Science

Lecture 6QF

Matrix Solutions to Normal Equations


Direct kronecker products
Direct (Kronecker) Products

(Left) Direct Product

Some Properties

assuming all operations are valid


Solving the normal equations
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
Properties of X’X

  • Symmetric

  • Same rank as X

  • Has an inverse (X’X)-1 iff X has full column rank


Solving the normal equations1
Solving the Normal Equations

Residuals

Least Squares

Solution: Solve the Normal Equations


Solving the normal equations2
Solving the Normal Equations

Normal Equations

Problem: X’X is Singular, has no inverse

Show


Generalized inverse g or
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


Solving the normal equations3

Theorem: For any , X’X X’ = X’

(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
A Solution to the Normal Equations X’

One Generalized Inverse

Verification


A solution to the normal equations1
A Solution to the Normal Equations X’

Corresponds to the solution to the normal equations

with the constraint m = 0 imposed


Assignment
Assignment X’

  • 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


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