Modeling Stationary Data by a Class of Generalized Ornstein-Uhlenbeck Processes: The Gaussian Case

We analyze in this work the effect of the iterated application of the linear operator that maps a Wiener process onto an OrnsteinUhlenbeck process. The processes obtained after p iterations are called Ornstein-Uhlenbeck processes of order p (denoted OU(p)). Technically our composition of operators is easy to manipulate and its parameters can be computed efficiently because, as we show, in most cases the result of the iteration is a linear combination of the same operators, and exceptionally it involves simple generalizations of them. This provides a straightforward computation of covariances. We also give a state space model representation of OU(p) and from this setup show that the discrete process resulting from sampling the linear combination of Ornstein-Uhlenbeck processes, at equally spaced periods of time, is a parsimonious ARMA process. Experiments on real data show that the empirical autocorrelation for large lags can be fairly modeled with OU(p) processes with approximately half the number of parameters than ARMA processes.