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:.

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Functional Linear Models

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Functional linear models

Functional Linear Models

Extend linear model ideas to FDA

linear regression

ANOVA


Outline

Outline

Chapter 9

  • Introduce functional linear model

  • Fitting the model

  • Assessing the fit

  • Computational issues


Functional linear models1

Functional linear models

  • In formal term:

  • Inner product representation:

  • Matrix version:


Fitting the model

Fitting the model

  • Extend the LS to the functional case.

    Reinterpret the squared norm

    To


Assessing the fit

Assessing the fit

  • Error sum of squares functions LMSSE

  • Squared correlation functions RSQ

  • F-ratio functions FRATIO


Computational issues

Computational issues

Pointwise minimization

The goal is to estimate LMSSE()

Minimizing the regularized RSS

Finding


Functional linear models

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


Outline1

Outline

Chapter 10

  • Functional interpolation

  • Regularization

  • Conclusions for the data


Functional interpolation

Functional interpolation

The model

Minimize LMSSE()

Perfectly fit without error at all

Use regularization to identify  uniquely


Regularization methods

Regularization methods

  • By discretizing the function

  • Using basis functions

    a. re-expressing the model and data

    b. smoothing by basis truncation


3 regularization with roughness penalties cross validation score

3.Regularization with roughness penalties cross-validation score


Conclusions for the data

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.


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