Wiener series

In mathematics, the Wiener series, developed by Norbert Wiener in the 1940s, is an orthogonal expansion for nonlinear functionals that is closely related to the Volterra series. The Wiener series is a polynomial expansion that is analogous to the Taylor series expansion of a function. The coefficients for the nth order term in the series (which are weights on nth-order monomials) are referred to as Wiener kernels.

The Wiener series plays an important role in the theory of nonlinear system identification. In this context, the series is usually used to approximate the functional that describes the mapping from the entire history of system input to the system output at a single time instant. Wiener kernels have been applied to the study of biological systems, especially in neuroscience.

The Wiener series differs from the Volterra series in that its terms are orthogonal with respect to a given stimulus or input distribution, which gives them better theoretical convergence properties in certain cases.

Wiener kernels should not be confused with the Wiener filter, which is an unrelated concept.

See also

• Volterra series
• System identification
• Spike-triggered average

References

• Measurement of the Wiener kernels of a non-linear system by cross-correlation. Lee and Schetzen. International Journal of Control, First Series, 2:237–254 (1965).
• White-noise analysis of a neuron chain: an application of the Wiener theory. Marmarelis, P.Z. & Naka, K. Science 175:1276–1278 (1972)
• The Volterra and Wiener Theories of Nonlinear Systems, Martin Schetzen (1980)
• Wiener Analysis of Nonlinear Feedback. Marmarelis, P.Z. Sensory Systems Annals of Biomedical Engineering 19:345–382 (1991)
• A unifying view of Wiener and Volterra theory and polynomial kernel regression. Franz, M. & Schölkopf, B. Neural computation 18:3097–3118. (2006)