Functional Linear Models

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# Functional Linear Models - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Functional Linear Models' - freira

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Presentation Transcript

### 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.