Functional Linear Models Extend linear model ideas to

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 1. The goal is to estimate LMSSE( ) 2. Minimizing the regularized RSS 3. Finding

• Modeling with basis expansions 1. Choosing a K-vector of linearly independent functions 2. Representing observed Y and estimated parameter 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 1. By discretizing the function 2. 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.