Generalized functional linear model

The Generalized Functional Linear Model(GFLM) is an extension of the Generalized linear model(GLM) that allows one to regress univariate responses of various types (continuous or discrete) on functional predictors, which are mostly random trajectories generated by a square-integrable stochastic processes. Similarly to GLM, a link function relates the expected value of the response variable to a linear predictor, which in case of GFLM is obtained by forming the scalar product of the random predictor function $X$ with a smooth parameter function $\beta$. Functional Linear Regression, Functional Poisson Regression and Functional Binomial Regression, with the important Functional Logistic Regression included, are special cases of GFLM. Applications of GFLM include classification and discrimination of stochastic processes and functional data.

Overview

A key aspect of GFLM is estimation and inference for the smooth parameter function $\beta$ which is usually obtained by dimension reduction of the infinite dimensional functional predictor. A common method is to expand the predictor function $X$ in an orthonormal basis of L² space, the Hilbert space of square integrable functions with the simultaneous expansion of the parameter function in the same basis. This is then applied to reduce the contribution of the of the parameter function $\beta$ in the linear predictor to a finite number of regression coefficients.Functional principal component analysis (FPCA) , that employs the Karhunen-Loève expansion is a common and parsimonious approach to accomplish this. However other orthogonal expansions, like Fourier expansions and B-spline expansions may also be employed for the dimension reduction step. Akaike Information Criteria(AIC) can be used for selecting the number of included components but minimization of cross-validation prediction errors is another criteria often used in classification applications. Once the dimension of the predictor process is reduced, the simplified linear predictor allows the usage of GLM and quasi-likelihood estimation techniques to obtain estimates of the finite dimensional regression coefficients which in turn provide an estimate of the parameter function $\beta$ in the GFLM.

Model Components

Linear Predictor

The predictor function $\textstyle X(t),t\in T$ typically are square integrable stochastic processes on a real interval $T$ and the unknown smooth parameter function $\beta (t),t\in T$ is assumed to be square integrable on $T$. Given a real measure $dw$ on $T$, the linear predictor is given by $\eta =\alpha +\int X(t)\beta (t)dw(t)$. Inclusion of the intercept $\alpha$ allows us to require $E(X(t))=0$ for all $t$ in $T$.