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Abstract

Classical Volterra and Wiener theory of nonlinear systems
does not address the problem of noisy measurements in system
identification. This issue is treated in the present part of the
report. We first show how to incorporate the implicit estimation
technique for Volterra and Wiener series described in Part I into
the framework of regularised estimation without giving up the
orthogonality properties of the Wiener operators. We then proceed to
a more general treatment of polynomial estimators (Volterra and
Wiener models are two special cases) in the context of Gaussian
processes. The implicit estimation technique from Part I can be
interpreted as Gaussian process regression using a polynomial
covariance function. Polynomial covariance functions, however, have
some unfavorable properties which make them inferior to other, more
localised covariance functions in terms of generalisation error. We
propose to remedy this problem by approximating a covariance
function with more favorable properties at a finite set of input
points. Our experiments show that this additional degree of freedom
can lead to improved performance in polynomial regression.