Volterra series

Model for approximating non linear effects, similar to a Taylor series
The Volterra series is a model for non linear behavior similar to the Taylor series. It differs from the Taylor series in its ability to capture "memory" effects. The Taylor series can be used for approximating the response of a nonlinear system to a given input if the output of the system depends strictly on the input at that particular time. In the Volterra series, the output of the nonlinear system depends on the input to the system at all other times. This provides the ability to capture the "memory" effect of devices like capacitors and inductors. It has been applied in the fields of medicine (biomedical engineering) and biology, especially neuroscience. It is also used in electrical engineering to model intermodulation distortion in many devices, including power amplifiers and frequency mixers. Its main advantage lies in its generalizability: it can represent a wide range of systems. Thus, it is sometimes considered a non parametric model. In mathematics, a Volterra series denotes a functional expansion of a dynamic, nonlinear, time invariant functional. The Volterra series are frequently used in system identification. The Volterra series, which is used to prove the Volterra theorem, is an infinite sum of multidimensional convolutional integrals.

The Volterra series is a modernized version of the theory of analytic functionals from the Italian mathematician Vito Volterra, in his work dating from 1887. Norbert Wiener became interested in this theory in the 1920s due to his contact with Volterra's student Paul Lévy. Wiener applied his theory of Brownian motion for the integration of Volterra analytic functionals. The use of the Volterra series for system analysis originated from a restricted 1942 wartime report of Wiener's, who was then a professor of mathematics at MIT. He used the series to make an approximate analysis of the effect of radar noise in a nonlinear receiver circuit. The report became public after the war. As a general method of analysis of nonlinear systems, the Volterra series came into use after about 1957 as the result of a series of reports, at first privately circulated, from MIT and elsewhere. The name itself, Volterra series, came into use a few years later.

Estimating the Volterra coefficients individually is complicated, since the basis functionals of the Volterra series are correlated. This leads to the problem of simultaneously solving a set of integral equations for the coefficients. Hence, estimation of Volterra coefficients is generally performed by estimating the coefficients of an orthogonalized series, e. g. the Wiener series, and then recomputing the coefficients of the original Volterra series. The Volterra series main appeal over the orthogonalized series lies in its intuitive, canonical structure, i. e. all interactions of the input have one fixed degree. The orthogonalized basis functionals will generally be quite complicated. An important aspect, with respect to which the following methods differ, is whether the orthogonalization of the basis functionals is to be performed over the idealized specification of the input signal (e. g. gaussian, white noise) or over the actual realization of the input (i. e. the pseudo random, bounded, almost white version of gaussian white noise, or any other stimulus). The latter methods, despite their lack of mathematical elegance, have been shown to be more flexible (as arbitrary inputs can be easily accommodated) and precise (due to the effect that the idealized version of the input signal is not always realizable).

Linear regression is a standard tool from linear analysis. Hence, one of its main advantages is the widespread existence of standard tools for solving linear regressions efficiently. It has some educational value, since it highlights the basic property of Volterra series: linear combination of non linear basis functionals. For estimation, the order of the original should be known, since the Volterra basis functionals are not orthogonal, and thus estimation cannot be performed incrementally.

This method was invented by Franz and Schölkopf and is based on statistical learning theory. Consequently, this approach is also based on minimizing the empirical error (often called empirical risk minimization). Franz and Schölkopf proposed that the kernel method could essentially replace the Volterra series representation, although noting that the latter is more intuitive.

This method was developed by van Hemmen and coworkers and utilizes Dirac delta functions to sample the Volterra coefficients.