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Statistical Methods for Fuzzy Data by Reinhard Viertl

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26.1 Linear approximation and prediction

A fuzzy stochastic process (Xt*)tT 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*, tT, from the p quantities Xt − 1*,…, Xt − p*, p > 0 with t > p. The approximation is assumed to be

(26.1) Numbered Display Equation

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

(26.2) Numbered Display Equation

In the case of real valued weakly stationary time series (Xt)tT, i.e. constant expectation Xt = μ for all tT and covariance Cov (Xt, Xs) depending only on the distance ts for all t, sT for Xt = 0 and Cov (Xt, Xs) = γts the coefficients αi, i = ...

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