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Functional Linear ModelsPowerPoint Presentation

Functional Linear Models

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Functional Linear Models

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Functional Linear Models

Extend linear model ideas to FDA

linear regression

ANOVA

Chapter 9

- Introduce functional linear model
- Fitting the model
- Assessing the fit
- Computational issues

- In formal term:
- Inner product representation:
- Matrix version:

- Extend the LS to the functional case.
Reinterpret the squared norm

To

- Error sum of squares functions LMSSE
- Squared correlation functions RSQ
- F-ratio functions FRATIO

Computational issues

Pointwise minimization

The goal is to estimate LMSSE()

Minimizing the regularized RSS

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

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

- 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.