26.1 Linear approximation and prediction
A fuzzy stochastic process (Xt*)t∈T is a family of fuzzy random quantities Xt*, where T is an index set, usually a subset of the set of real numbers . The first approach is looking for the ‘best’ linear approximation of the fuzzy random quantity Xt*, t ∈ T, from the p quantities Xt − 1*,…, Xt − p*, p > 0 with t > p. The approximation is assumed to be
with real numbers αi. The quality of the approximation is determined by the expectation of the squared distance between approximated value and observed value. This means for the coefficients αi looking for those values fulfilling
In the case of real valued weakly stationary time series (Xt)t ∈ T, i.e. constant expectation Xt = μ for all t ∈ T and covariance Cov (Xt, Xs) depending only on the distance t − s for all t, s ∈ T for Xt = 0 and Cov (Xt, Xs) = γt − s the coefficients αi, i = ...