Institute for Statics und Dynamics of Structures Fuzzy Time Series
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1 Institute for Statics und Dynamics of Structures Fuzzy Time Series Bernd Möller
2 1 Description of fuzzy time series Conclusions Folie 2 von slide422
3 1 Description of fuzzy time series Conclusions Folie 3 von slide42 3
4 outreach.ecology.uga.edu time series with fuzzy data forecasting? slide 4
5 Fuzzy time series slide 5
6 Fuzzy variables -discretization slide 6
7 Fuzzy variables -Discretization = discretization with increments -increments slide 7
8 Fuzzy variables -Discretization slide 8
9 Fuzzy variables for slide 9
10 Graphical description of fuzzy time series Plots slide 10
11 Graphical description of fuzzy time series Plots slide 11
12 Numerical description of fuzzy time series Fuzzy component model with: functional value of the fuzzy trend function functional value of the fuzzy cycle function realization of a fuzzy random noise process applicable by non-stationary fuzzy time series slide 12
13 Description of fuzzy time series by empirical parameters applicable by stationary and ergodic fuzzy time series empirical fuzzy mean value empirical -covariance function slide 13
14 1 Description of fuzzy time series Conclusions Folie 14slide von 14 42
15 Fuzzy time series realization of a fuzzy random process family of fuzzy random variables set of all fuzzy variables on space of the random elementary events realizations of a fuzzy random variable are fuzzy variables each realization of a fuzzy random process yields a sequence of fuzzy variables at discrete time points slide 15
16 Fuzzy random variables 8 realizations µ(x) µ(x) µ(x) µ(x) x x x x µ(x) µ(x) µ(x) µ(x) x x x x slide 16
17 Fuzzy random variables α-level sets are random sets interval bounds of the α-level sets are random variables slide 17
18 Fuzzy random variables -Discretization random -level sets bounds are random variables slide 18
19 Fuzzy random variables -Discretization -Representation of a fuzzy random variable correlated random variables slide 19
20 Fuzzy random process first and second order moments in -representation fuzzy expected value function -covariance function -variance for slide 20
21 Fuzzy-white-noise-process constant constant stationary and ergodic process for for increments are dependent increments are independent probability density function of a random increment slide 21
22 Fuzzy-ARMA-process Fuzzy-AR-process Fuzzy-MA-process with:, real valued [2n,2n] parameter matrices fuzzy random variable of a fuzzy-white-noise-process Forecasting presupposes the estimation of the parameter matrices. slide 22
23 Estimation of the parameter {A 1,,A p, B 1,,B q } = P 3 strategies for parameter estimation: Idea 1: Minimization of the differences between empirical parameters and model parameters applicable for stationary and ergodic fuzzy time series slide 23
24 Estimation of the parameter {A 1,,A p, B 1,,B q } = P Idea 2: Minimization of the average distance between measured fuzzy data and optimal 1-step-forecasting distance function HAUSDORF distance applicable for non-stationary fuzzy time series slide 24
25 Estimation of the parameter {A 1,,A p, B 1,,B q } = P Idea 3: Minimization of the square error between measured fuzzy data and optimal 1-step-forecasting incremental improvement of the parameter e.g.: applicable for non-stationary fuzzy time series slide 25
26 1 Description of fuzzy time series Conclusions Folie 26slide von 26 42
27 Forecasting strategies future values of a fuzzy time series are realizations of a fuzzy random forecast process Fuzzy random forecast process =family of conditional random variables conditional fuzzy random variable with realizations of the forecast process are future values of the fuzzy time series slide 27
28 Forecasting strategies 1. Optimal forecasting optimal forecasted value conditional fuzzy expected value of optimal 1-step-forecasting (Fuzzy-ARMA[p,q]-process) optimal h-step-forecasting (Fuzzy-ARMA[p,q]-process) with for for and for for slide 28
29 Forecasting strategies 2. Fuzzy forecast intervals A fuzzy forecast interval includes realizations with probability fuzzy forecast interval optimal forecasted value s future realizations are simulated by Monte Carlo Simulation slide 29
30 Forecasting strategies 3. Fuzzy random forecasting objective: estimation of the fuzzy random variable Monte Carlo simulation of fuzzy variables statistical evaluation of the simulated fuzzy variables e.g. fuzzy probability distribution function fuzzy expected value -covariance function slide 30
31 Forecasting of structural responses non-measurable parameters indirect forecasting of structural responses fuzzy stochastic structural analysis measurable parameters fuzzy time series of impact direct forecasting of impact slide 31
32 1 Description of fuzzy time series Conclusions Folie 32slide von 32 42
33 measured data of heavy goods vehicle traffic June 2002 to May 2003 slide 33
34 Measured data of heavy goods vehicle traffic May 2003 stationary and ergodic Fuzzy-ARMA[10,0]-process minimization of the differences between empirical and model characteristics optimal h-step-forecast slide 34
35 over 4 years Description of fuzzy time series Time series with measured settlements (over 4 years) slide 35
36 Measured data of an extensometer non-stationary Fuzzy-ARMA[10,3]-process minimization of square error control simulation by optimal 1-step-forecasting optimal h-step-forecasting fuzzy forecast intervals slide 36
37 Damage of a T-beam plate indirect forecasting slide 37
38 Damage of a T-beam plate Time series of live laods monthly goods in a storehouse non-stationary Fuzzy-ARMA[4,4]-process Fuzzy-ARMA[4,4]-process optimal h-step-forecasting fuzzy forecast intervals slide 38
39 Damage of a T-beam plate Fuzzy random forecasting Monte Carlo Simulation of 100 future realizations 8 realizations at time point slide 39
40 Damage of a T-beam plate indirect forecasting Fuzzy damage indicator not strengthened strengthened slide 40
41 Conclusions Time series with fuzzy data can be modeled as realizations of fuzzy random processes New -representation of fuzzy random variables enables the modeling of fuzzy time series as realizations of fuzzy random processes Fuzzy-ARMA-processes allow the simulation and forecasting of fuzzy time series slide 41
42 Thank you! slide 42
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