Functional regression is a version of regression analysis when responses or covariates include functional data.
A linear model with scalar response
denotes the inner product in Euclidean space,
is a random error with mean zero and finite variance.
FLMs can be divided into two types based on the responses.
in model (1) by a centered functional covariate
For implementation, regularization is needed and can be done through truncation,
[1] In addition, a reproducing kernel Hilbert space (RKHS) approach can also be used to estimate
is usually assumed to be a random process with mean zero and finite variance.
, is an extension of multivariate linear regression with the inner product in Euclidean space replaced by that in
An estimating equation motivated by multivariate linear regression is
,[9] it is reasonable to consider a historical functional linear model, where the current value of
Adding multiple functional covariates, model (4) can be extended to where for
is a centered functional covariate with domain
as a constant function yields a special case of model (5)
which is a FLM with functional responses and scalar covariates.
[11][12][13] Adding multiple functional covariates, model (6) can also be extended to
are multiple functional covariates with domain
[3] Functional polynomial models are an extension of the FLMs with scalar responses, analogous to extending linear regression to polynomial regression.
is a random error with mean zero and finite variance.
By analogy to FLMs with scalar responses, estimation of functional polynomial models can be obtained through expanding both the centered covariate
[14] A functional multiple index model is given by
yields a functional single index model.
, this model is problematic due to curse of dimensionality.
and relatively small sample sizes, the estimator given by this model often has large variance.
-component functional multiple index model can be expressed as
Estimation methods for functional single and multiple index models are available.
One form of FAMs is obtained by replacing the linear function of
[3][17] Another form of FAMs consists of a sequence of time-additive models:
[3][18] A direct extension of FLMs with scalar responses shown in model (2) is to add a link function to create a generalized functional linear model (GFLM) by analogy to extending linear regression to generalized linear regression (GLM), of which the three components are: