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

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()

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


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


3.Regularization with roughness penalties cross-validation score


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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