TITLE: Penalized regularization for functional linear models

SPEAKER: Dr. Tim Randolph

ABSTRACT:

In a 'functional linear model' the aim is to estimate the linear relationship between a scaler response, y, and a random L2 function, x. This formally fits into the classical regression model, y = X*b + e, but the estimation of b is inherently ill-posed. A common method used to address this problem regularizes the solution by penalizing L*b, where L is a linear operator. This constrains b (which is nominally any L2 function) by bounding the norm of L*b. Special cases include Marquardt's generalized regression, Hoerl's ridge regression (L=I) and

Tikhonov-Philips' regularization (L=D, a differential operator). This talk considers (linear) penalized estimates in general and their expression via generalized singular vectors of the pair (X,L). This provides perspective on the functional structure of an estimate and on criteria for choosing the penalty operator. Corresponding explicit expressions for bias and variance provide insight regarding the trade-off between the properties of L versus X.