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In mathematics, the Wiener series, or Wiener G-functional expansion, originates from the 1958 book of
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher ...
. It is an orthogonal expansion for nonlinear functionals closely related to the
Volterra 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 ...
and having the same relation to it as an orthogonal Hermite polynomial expansion has to a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
. For this reason it is also known as the Wiener–Hermite expansion. The analogue of the coefficients are referred to as Wiener kernels. The terms of the series are orthogonal (uncorrelated) with respect to a statistical input of
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines ...
. This property allows the terms to be identified in applications by the ''Lee–Schetzen method''. The Wiener series is important in nonlinear system identification. In this context, the series approximates the functional relation of the output to the entire history of system input at any time. The Wiener series has been applied mostly to the identification of biological systems, especially in
neuroscience Neuroscience is the science, scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a Multidisciplinary approach, multidisciplinary science that combines physiology, an ...
. The name Wiener series is almost exclusively used in
system theory Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its stru ...
. In the mathematical literature it occurs as the Itô expansion (1951) which has a different form but is entirely equivalent to it. The Wiener series should not be confused with the
Wiener filter In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant ( LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and ...
, which is another algorithm developed by Norbert Wiener used in signal processing.


Wiener G-functional expressions

Given a system with an input/output pair (x(t),y(t)) where the input is white noise with zero mean value and power A, we can write the output of the system as sum of a series of Wiener G-functionals y(n) = \sum_p (G_p x)(n) In the following the expressions of the G-functionals up to the fifth order will be given: : (G_0 x)(n) = k_0 = E\left\; : (G_1 x)(n) = \sum_^ k_1 (\tau _1 )x(n - \tau _1 ); : (G_2 x)(n) = \sum_^ k_2 (\tau _1 ,\tau _2 )x(n - \tau _1 )x(n - \tau _2) - A\sum_^ k_2 (\tau _1 ,\tau _1 ); : (G_3 x)(n) = \sum_^ k_3 (\tau _1 ,\tau _2 ,\tau _3 ) x(n - \tau _1 )x(n - \tau _2)x(n - \tau _3) - 3A \sum_^ \sum_^k_3 (\tau _1 ,\tau _2 ,\tau _2 ) x(n - \tau _1 ); : \begin (G_4 x)(n) = & \sum_^ k_4 (\tau_1 ,\tau_2 ,\tau_3 ,\tau_4 ) x(n - \tau_1 )x(n - \tau_2 )x(n - \tau_3 )x(n - \tau_4 ) + \\ pt& - 6A \sum_^ \sum_^ k_4 (\tau_1, \tau_2, \tau_3 ,\tau_3) x(n - \tau_1 )x(n - \tau_2) + 3A^2 \sum_^ k_4 (\tau_1 ,\tau_1 ,\tau_2 ,\tau_2 ) ; \end : \begin (G_5 x)(n) = & \sum_^ k_5 (\tau_1, \tau_2, \tau_3, \tau_4, \tau_5 ) x(n - \tau_1 )x(n - \tau_2 )x(n - \tau_3 )x(n - \tau_4 )x(n - \tau_5 ) + \\ pt& - 10A\sum_^ \sum_^ k_5 (\tau_1, \tau _2 ,\tau_3, \tau_4, \tau_4 ) x(n - \tau_1 )x(n - \tau_2 )x(n - \tau_3 ) \\ pt& + 15A^2 \sum_^ \sum_^ k_5 (\tau_1, \tau_2, \tau_2 ,\tau_3 ,\tau_3 ) x(n - \tau_1 ). \end


See also

*
Volterra 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 ...
*
System identification The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. System identification also includes the optimal design of experiments for efficiently generating informative dat ...
*
Spike-triggered average The spike-triggered averaging (STA) is a tool for characterizing the response properties of a neuron using the spikes emitted in response to a time-varying stimulus. The STA provides an estimate of a neuron's linear receptive field. It is a useful ...


References

* * * Itô K "A multiple Wiener integral" J. Math. Soc. Jpn. 3 1951 157–169 * * * * {{cite journal , doi=10.1162/neco.2006.18.12.3097 , title=A unifying view of Wiener and Volterra theory and polynomial kernel regression , last=Franz , first=M , author2=Schölkopf, B. , journal=
Neural Computation Neural computation is the information processing performed by networks of neurons. Neural computation is affiliated with the philosophical tradition known as Computational theory of mind, also referred to as computationalism, which advances the t ...
, volume=18 , pages=3097–3118 , year=2006 , issue=12 * L.A. Zadeh On the representation of nonlinear operators. IRE Westcon Conv. Record pt.2 1957 105-113. Mathematical series Functional analysis