Stochastic Process, Extrapolation of a

the prediction of the value of a stochastic process at some future moment of time on the basis of the observed values of the process. The term “extrapolation” is also used in the more general sense of prediction on the basis of the observed values of some statistically related process in the past and future; an example of such a related process is the sum of the process being extrapolated and random noise distorting the observations.

In nearly all cases of interest, the predicted value of the process X(t) at the moment t = t₁ cannot be precisely determined from the available observational data. It is possible only to attempt to make the random error of the prediction Δ = X₁(t₁) - X(t₁) be on the average as small as possible; X₁(t₁) here is the predicted value of X(t₁). In the theory of extrapolation of stochastic processes the optimal, or best, prediction is usually considered to be the prediction for which the mathematical expectation of the square of Δ is a minimum. Such an optimal prediction coincides with the conditional mathematical expectation of the random variable X(t₁) given fixed values (known from the observations) of the variables on which the prediction is based.

An important place in the theory of extrapolation of stochastic processes is occupied by the theory of optimal linear extrapolation. This theory deals with the methods for finding from the observational data the linear function for which the mean square of its deviations from X(t₁) is smallest. In a number of cases of practical importance, such an optimal linear extrapolation coincides with the general optimal extrapolation.

The general theory of optimal linear extrapolation for stationary stochastic processes was developed by A. N. Kolmogorov and N. Wiener. Much progress has also been made in the theory of optimal (both linear and general nonlinear) extrapolation of processes that are consituents of Markov processes.