Linear and non-linear filters for linear, but not gaussian processes†

Abstract

Suppose {Xt} is a zero mean second order stationary process satisfying the difference equation Xt= θ Xt–1, where {ut} is a sequence of i.i.d. random variables. Further, we assume that we observe the process yt = Xt + vt, where {Vt} is also a sequence of i.i.d. random variables. The object is to obtain a recursive filter for Xt given {Yt}. If the process {ut} and {vt} are gaussian, then the recursive filter is linear, and this is the well known Kalman-Bucy filter. Otherwise, the optimal filters are non-linear. In this paper, we define two non-linear filters and compare their performance with respect to the linear filter.