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 with a smooth parameter function . 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 which is usually obtained by dimension reduction of the infinite dimensional functional predictor. A common method is to expand the predictor function 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 representation is then combined with a truncation step to reduce the contribution of the parameter function 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. Other orthogonal expansions, like Fourier expansions and B-spline expansions may also be employed for the dimension reduction step. The Akaike information criterion (AIC) can be used for selecting the number of included components. Minimization of cross-validation prediction errors is another criterion often used in classification applications. Once the dimension of the predictor process has been reduced, the simplified linear predictor allows to use 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 in the GFLM.
Model components
Linear predictor
The predictor functions , typically are square integrable stochastic processes on a real interval and the unknown smooth parameter function , is assumed to be square integrable on . Given a real measure on , the linear predictor is given by where is the centered predictor process and is a scalar that serves as an intercept.
Response variable and variance function
The outcome is typically a real valued random variable which may be either continuous or discrete. Often the conditional distribution of given the predictor process is specified within the exponential family. However it is also sufficient to consider the functional quasi-likelihood set up, where instead of the distribution of the response one specifies the conditional variance function, , as a function of the conditional mean, .
Link function
The link function is a smooth invertible function, that relates the conditional mean of the response with the linear predictor . The relationship is given by .
Formulation
In order to implement the necessary dimension reduction, the centered predictor process and the parameter function are expanded as,
where is an orthonormal basis of the function space such that where if and otherwise.
The random variables are given by and the coefficients as for .
and and denoting , so .
From the orthonormality of the basis functions , it follows immediately that .
The key step is then approximating by for a suitably chosen truncation point .
FPCA gives the most parsimonious approximation of the linear predictor for a given number of basis functions as the eigenfunction basis explains more of the variation than any other set of basis functions.
For a differentiable link function with bounded first derivative, the approximation error of the -truncated model i.e. the linear predictor truncated to the summation of the first components, is a constant multiple of .
A heuristic motivation for the truncation strategy derives from the fact that which is a consequence of the Cauchy–Schwarz inequality and by noting that the right hand side of the last inequality converges to 0 as since both and are finite.
For the special case of the eigenfunction basis, the sequence corresponds to the sequence of the eigenvalues of the covariance kernel .
