26
Stochastic methods in fuzzy time series analysis
In Chapter fuzzy time series are analyzed without assumptions on a stochastic model. The only stochastic element was the error component 
t* for moving averages.
In the present chapter the fuzzy observations are considered as realizations of a stochastic process whose elements are fuzzy stochastic quantities. The goal is to use dependencies of the observations for predictions of future values.
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 ...
