Functional Linear Models

1. Functional Linear Models Extend linear model ideas to FDA linear regression ANOVA

2. Outline Chapter 9 • Introduce functional linear model • Fitting the model • Assessing the fit • Computational issues

5. Functional linear models • In formal term: • Inner product representation: • Matrix version:

6. Fitting the model • Extend the LS to the functional case. Reinterpret the squared norm To

9. Assessing the fit • Error sum of squares functions LMSSE • Squared correlation functions RSQ • F-ratio functions FRATIO

11. Computational issues Pointwise minimization The goal is to estimate LMSSE() Minimizing the regularized RSS Finding

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

13. Outline Chapter 10 • Functional interpolation • Regularization • Conclusions for the data

14. Functional interpolation The model Minimize LMSSE() Perfectly fit without error at all Use regularization to identify  uniquely

16. Regularization methods • By discretizing the function • Using basis functions a. re-expressing the model and data b. smoothing by basis truncation

19. 3.Regularization with roughness penalties cross-validation score

24. 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.