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Neural Networks as Nonlinear Adaptive Filters Perspective Neural networks, in particular recurrent neural networks, are cast into the framework of nonlinear adaptive filters. In this context, the relation between recurrent neural networks and polynomial filters is first established. Learning strategies and algorithms are then developed for neural adaptive system identifiers and predictors. Finally, issues concerning the choice of a neural architecture with respect to the bias and variance of the prediction performance are discussed | Recurrent Neural Networks for Prediction Authored by Danilo P. Mandic Jonathon A. Chambers Copyright 2001 John Wiley Sons Ltd ISBNs 0-471-49517-4 Hardback 0-470-84535-X Electronic 6 Neural Networks as Nonlinear Adaptive Filters Perspective Neural networks in particular recurrent neural networks are cast into the framework of nonlinear adaptive filters. In this context the relation between recurrent neural networks and polynomial filters is first established. Learning strategies and algorithms are then developed for neural adaptive system identifiers and predictors. Finally issues concerning the choice of a neural architecture with respect to the bias and variance of the prediction performance are discussed. Introduction Representation of nonlinear systems in terms of NARMA NARMAX models has been discussed at length in the work of Billings and others Billings 1980 Chen and Billings 1989 Connor 1994 Nerrand et al. 1994 . Some cognitive aspects of neural nonlinear filters are provided in Maass and Sontag 2000 . Pearson 1995 in his article on nonlinear input-output modelling shows that block oriented nonlinear models are a subset of the class of Volterra models. So for instance the Hammerstein model which consists of a static nonlinearity f applied at the output of a linear dynamical system described by its -domain transfer function H z can be represented1 by the Volterra series. In the previous chapter we have shown that neural networks be they feedforward or recurrent cannot generate time delays of an order higher than the dimension of the input to the network. Another important feature is the capability to generate subharmonics in the spectrum of the output of a nonlinear neural filter Pearson 1995 . The key property for generating subharmonics in nonlinear systems is recursion hence recurrent neural networks are necessary for their generation. Notice that as 1 Under the condition that the function f is analytic and that the Volterra series can be thought of