# Functional Linear Models - PowerPoint PPT Presentation

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Functional Linear Models. Extend linear model ideas to FDA linear regression ANOVA. Outline. Chapter 9 Introduce functional linear model Fitting the model Assessing the fit Computational issues. Functional linear models. In formal term: Inner product representation: Matrix version:.

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Functional Linear Models

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## Functional Linear Models

Extend linear model ideas to FDA

linear regression

ANOVA

### Outline

Chapter 9

• Introduce functional linear model

• Fitting the model

• Assessing the fit

• Computational issues

### Functional linear models

• In formal term:

• Inner product representation:

• Matrix version:

### Fitting the model

• Extend the LS to the functional case.

Reinterpret the squared norm

To

### Assessing the fit

• Error sum of squares functions LMSSE

• Squared correlation functions RSQ

• F-ratio functions FRATIO

## Computational issues

Pointwise minimization

The goal is to estimate LMSSE()

Finding

Modeling with basis expansions1. Choosing a K-vector  of linearly independent functions2. Representing observed Y and estimatedparameter 3. The matrix system of linear equations

### Outline

Chapter 10

• Functional interpolation

• Regularization

• Conclusions for the data

## Functional interpolation

The model

Minimize LMSSE()

Perfectly fit without error at all

Use regularization to identify  uniquely

### Regularization methods

• By discretizing the function

• Using basis functions

a. re-expressing the model and data

b. smoothing by basis truncation

## Conclusions for the data

Higher precipitation is associated with higher temperatures in the last three months of the year and with lower temperatures in spring and early summer.