Estimation and Prediction of Nonlinear Functionals of Gaussian Processes.

Finite Dimensional Nonlinear Estimation in Continuous and Discrete Time.

“…Using variationa l expansions similar to those of Gilbert [ 13], it is straightforward to show tha t the kernels of the Volterra series ( 8 ) are also separable in the sense of [2J, (l2]; this is basically due to the fact that the kernels arise from the variational eq uations as products of pulse responses of linear systems . Brockett (11) has also shown that a cont inuous t ime finit e Volterra series has a bilinear realization if and only if it has separable kernels.…”

“…It is useful first to contrast these problems with the estimation and prediction problems considered by Ruang and Cambanis ( 8 ) . There the problem is that of estimating a nonlinear functional y of a Gaussian process {x(t), ttS }, given observations of ( x(t), tcS}, where S is a subset of S. In our problem the objective is…”

“…Although the elegant formulas of Huang and Cambanis cannot be applied here , the approach of utilizing the homogeneous chaos , or , equivalently , the Cameron-Martin orthogonal series decomposition of a Gaussian process (4]- [8], will prove to be quite useful in unifying and simplifying our results.…”

“…In both cases, the linear and nonlinear innovations are equal. Hence the estimation problem (l) - (3) or (4) - (6) Gaussian processes is the subject of (4) - [ 8], and the applica tion of these results to our recursive estimation problem is presented in the next section , where a new proof of Theorem 1 is presented and the corresponding proof of Theorem 2 is outlined .…”

“…We will not require all of the isomorphisms presented in [ 7 ] ; only the following decomposition in terms of Hermite polynomials will be utilized here [8]. Let x ( t ) , t t S , be any zero-mean second order Gaussian process defined on ( ,B ,P); for our purposes S will …”