System Identification Problems and the Method of Moments

Abstract

Let X(t) and W(t) be vectors of dimension N > 0. We are concerned with the problem of computing an N x N matrix A such that [see pdf for notation](1.1) where X'(t) is the rate of change of X(t) with respect to time t. Such problems frequently arise in the biosciences although it is not always immediately evident that they may be posed in the form of Eq. (1.1). In applications the data for X(t) and W(t) is obtained from experiments and the collection of such data is not always performed over equal time intervals nor is it always the case that data is obtained for the scalar functions Xi(t) and Wi(t), which form the components of X(t) and W(t) respectively, at the same instances of time. Also, certain entries in the matrix Aare known a priori or relationships between entries must be satisfied. The object of this paper is to introduce a method by which the system identification problem, i.e. the problem discussed above, may be analyzed. In particular, we consider the moments [see pdf for notation](1.2) of the data; or in the event that such integrals do not exist, we compute the moments of X(t)e-pt where p > 0 is sufficiently large. The method of moments may be developed from the principle that the matrix A should be determined in such a manner so as to achieve the best fit between the moments of X(t) as computed from the data and the moments of X(t) as computed from the differential equation (1.1). This is the point of view taken in [1].