By Bernd Möller
This publication bargains with uncertainty forecasting according to a fuzzy time sequence procedure, together with fuzzy random approaches and synthetic neural networks. A attention of information and dimension uncertainty complements forecasting in a variety of purposes, really within the fields of engineering, environmental technology and civil engineering.
Uncertain info are defined through a brand new incremental fuzzy illustration which allows an entire and exact estimation of uncertainty.
The e-book is aimed toward engineers in addition to pros operating in comparable fields. Descriptive, modeling and forecasting tools bearing on fuzzy time sequence are brought and defined intimately. Emphasis is put on forecasting by using fuzzy random techniques, similar to fuzzy ARMA strategies and fuzzy white-noise tactics, in addition to forecasting in response to man made neural networks.
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Additional info for Uncertainty Forecasting in Engineering
Is represented For the numerical computation the fuzzy random variable X by n α-level sets. , Xαn r , respectively, are thus regarded as the coordinates of the 2n-dimensional coordinate system of the interval bounds. 34. The fuzzy probability distribution function form II FX˜ ♣x ˜q ˜ which is discretized by n ➙ 2 random αof the fuzzy random variable X, level sets rXαi l ; Xαi r s, is defined as the 2n-dimensional probability distribution function of the random variables Xαi l and Xαi r according to Eq.
10) Eq. 112) holds. , 2n ˜ ✏ x appears with a probability of P ♣X ˜j q ✏ Each of the ten realizations 1 10 . Selected functional values of the associated fuzzy probability distribution function form II lr FX˜ ♣x ˜q (evaluated according to Eq. 3. 3. 25 are obtained (analogous to random variables X classical probability theory) by partial differentiation of the fuzzy probability distribution function form II lr FX˜ ♣x ˜q according to Eq. 113). 113) 2n The fuzzy probability distribution function form II lr FX˜ ♣x ˜q of a continuous ˜ may thus be obtained by inversion of the differentifuzzy random variable X ation given by Eq.
15 together with the empirical fuzzy probability distribution function form I according to Eq. 96). This depiction illustrates the interrelation between the fuzzy probability distribution function form II according to Eq. 104) and the fuzzy probability distribution function form I according to Eq. 96). The marginal distributions of the multidimensional probability distribution function according to Eq. 104) correspond to the left and right boundary functions of the fuzzy probability distribution function form I.