fuzzy time series
DESCRIPTION
A brief explanation with examples on fuzzy time seriesTRANSCRIPT
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 1/42
Institute for Statics und Dynamics of Structures
Fuzzy Time Series
Bernd Möller
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 2/42
Institute for Static und Dynamics of Structures
slide 2
4 Examples
5 Conclusions
1 Description of fuzzy time series
2 Modelling of fuzzy time series
3 Forecasting of fuzzy time series
Folie 2 von 42
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 3/42
Institute for Static und Dynamics of Structures
slide 3
4 Examples
5 Conclusions
1 Description of fuzzy time series
2 Modelling of fuzzy time series
3 Forecasting of fuzzy time series
Folie 3 von 42
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 4/42
Institute for Static und Dynamics of Structures
slide 4
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
www.vbz.ch outreach.ecology.uga.edu
time series with fuzzy data forecasting
?
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 5/42
Institute for Static und Dynamics of Structures
slide 5
Fuzzy time series
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 6/42
Institute for Static und Dynamics of Structures
slide 6
-discretization
Fuzzy variables
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 7/42
Institute for Static und Dynamics of Structures
slide 7
-Discretization = discretization with increments
-increments
Fuzzy variables
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 8/42
Institute for Static und Dynamics of Structures
slide 8
-Discretization
Fuzzy variables
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 9/42
Institute for Static und Dynamics of Structures
slide 9
for
Fuzzy variables
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 10/42
Institute for Static und Dynamics of Structures
slide 10
PlotsGraphical description of fuzzy time series
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 11/42
Institute for Static und Dynamics of Structures
slide 11
PlotsGraphical description of fuzzy time series
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 12/42
Institute for Static und Dynamics of Structures
slide 12
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
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
Numerical description of fuzzy time series
applicable by non-stationary fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 13/42
Institute for Static und Dynamics of Structures
slide 13
empirical fuzzy mean value
empirical -covariance function
Description of fuzzy time series by empirical parameters
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
applicable by stationary and ergodic fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 14/42
Institute for Static und Dynamics of Structures
slide 14
4 Examples
5 Conclusions
1 Description of fuzzy time series
2 Modelling of fuzzy time series
3 Forecasting of fuzzy time series
Folie 14 von 42
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 15/42
Institute for Static und Dynamics of Structures
slide 15
realizations of a fuzzy random variable are fuzzy variables
… space of the random elementary events
… set of all fuzzy variables on
realization of a fuzzy random process
family of fuzzy random variables
Fuzzy time series
Modelling of fuzzy time series
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
each realization of a fuzzy random process yieldsa sequence of fuzzy variables at discrete time points
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 16/42
Institute for Static und Dynamics of Structures
slide 16
(x)
x
(x)
x
(x)
x
(x)
x
(x)
x
(x)
x
(x)
x
(x)
x
8 realizationsFuzzy random variables
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 17/42
Institute for Static und Dynamics of Structures
slide 17
-level sets are random sets
interval bounds of the -level sets are random variables
1.
2.
Fuzzy random variables
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 18/42
Institute for Static und Dynamics of Structures
slide 18
Fuzzy random variables
● random -level sets● bounds are random variables
-Discretization
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 19/42
Institute for Static und Dynamics of Structures
slide 19
Fuzzy random variables
-Representation
-Discretization
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
correlated
randomvariables
of a fuzzy random variable
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 20/42
Institute for Static und Dynamics of Structures
slide 20
fuzzy expected value function
-covariance function -variance
Fuzzy random process
first and second order moments in -representation
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
for
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 21/42
Institute for Static und Dynamics of Structures
slide 21
constant
constant
for
for
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
Fuzzy-white-noise-process
probability density function of a random increment
stationary and
ergodic process
increments are dependentincrements are independent
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 22/42
Institute for Static und Dynamics of Structures
slide 22
… real valued [2n,2n] parameter matrices
… fuzzy random variable of a fuzzy-white-noise-process
Fuzzy-AR-process Fuzzy-MA-process
Fuzzy-ARMA-process
with: ,
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
Forecasting presupposes the estimation
of the parameter matrices.
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 23/42
Institute for Static und Dynamics of Structures
slide 23
Estimation of the parameter {A1,…,Ap, B1,…,Bq} = P
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
applicable for stationary and ergodic fuzzy time series
Minimization of the differences betweenempirical parameters and model parameters
Idea 1:
3 strategies for parameter estimation:
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 24/42
Institute for Static und Dynamics of Structures
slide 24
Idea 2: Minimization of the average distance betweenmeasured fuzzy data and optimal 1-step-forecasting
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
applicable for non-stationary fuzzy time series
Estimation of the parameter {A1,…,Ap, B1,…,Bq} = P
distance function
HAUSDORF distance
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 25/42
Institute for Static und Dynamics of Structures
slide 25
Minimization of the square error between
measured fuzzy data and optimal 1-step-forecastingIdea 3:
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
Estimation of the parameter {A1,…,Ap, B1,…,Bq} = P
incremental improvement of the parameter
e.g.:
applicable for non-stationary fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 26/42
Institute for Static und Dynamics of Structures
slide 26
4 Examples
5 Conclusions
1 Description of fuzzy time series
2 Modelling of fuzzy time series
3 Forecasting of fuzzy time series
Folie 26 von 42
f f
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 27/42
Institute for Static und Dynamics of Structures
slide 27
Forecasting strategies
Fuzzy random forecast process=family of conditional random variables
with
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
future values of a fuzzy time series arerealizations of a fuzzy random forecast process
conditional fuzzy random variable
realizations of the forecast process arefuture values of the fuzzy time series
D i ti f f ti i
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 28/42
Institute for Static und Dynamics of Structures
slide 28
Forecasting strategies
1. Optimal forecasting
optimal 1-step-forecasting (Fuzzy-ARMA[p,q]-process)
optimal h-step-forecasting (Fuzzy-ARMA[p,q]-process)
for
for
for
for
with and
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
optimal forecasted value conditional fuzzy expected value of
Description of fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 29/42
Institute for Static und Dynamics of Structures
slide 29
2. Fuzzy forecast intervals
A fuzzy forecast interval includes realizationswith probability
Forecasting strategies
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
optimalforecasted
value
fuzzyforecast
interval
s future realizations are simulated by Monte Carlo Simulation
Description of fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 30/42
Institute for Static und Dynamics of Structures
slide 30
3. Fuzzy random forecasting
Monte Carlo simulation of fuzzy variables
statistical evaluation of the simulated fuzzy variables
e.g. fuzzy probability distribution function
fuzzy expected value
-covariance function
●
●
●
Forecasting strategies
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
objective: estimation of the fuzzy random variable
Description of fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 31/42
Institute for Static und Dynamics of Structures
slide 31
Forecasting of structural responses
fuzzy time series of impact
direct forecasting of impact
fuzzy stochastic structural analysis
indirect forecasting of structural responses
Description of fuzzy time seriesModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
measurable
parameters
non-measurableparameters
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 32/42
Institute for Static und Dynamics of Structures
slide 32
4 Examples
5 Conclusions
1 Description of fuzzy time series
2 Modelling of fuzzy time series
3 Forecasting of fuzzy time series
Folie 32 von 42
Description of fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 33/42
Institute for Static und Dynamics of Structures
slide 33
measured data of heavy goods vehicle traffic
p yModelling of fuzzy time series
Forecasting of fuzzy time seriesExamples
June 2002 to May 2003
Description of fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 34/42
Institute for Static und Dynamics of Structures
slide 34
optimal h-step-forecast
stationary and ergodic Fuzzy-ARMA[10,0]-process
minimization of the differences betweenempirical and model characteristics
Measured data of heavy goods vehicle traffic
Modelling of fuzzy time seriesForecasting of fuzzy time series
Examples
May 2003
Description of fuzzy time seriesd ll f f
over 4 years
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 35/42
Institute for Static und Dynamics of Structures
slide 35
Time series with measured settlements
Modelling of fuzzy time seriesForecasting of fuzzy time series
Examples
y
(over 4 years)
Description of fuzzy time seriesM d lli f f ti i
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 36/42
Institute for Static und Dynamics of Structures
slide 36
control simulation byoptimal 1-step-forecasting
optimal h-step-forecasting
fuzzy forecast intervals
non-stationary Fuzzy-ARMA[10,3]-process
minimization of square error
Measured data of an extensometer
Modelling of fuzzy time seriesForecasting of fuzzy time series
Examples
Description of fuzzy time seriesModelling of fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 37/42
Institute for Static und Dynamics of Structures
slide 37
Modelling of fuzzy time seriesForecasting of fuzzy time series
Examples
Damage of a T-beam plate indirect forecasting
Description of fuzzy time seriesModelling of fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 38/42
Institute for Static und Dynamics of Structures
slide 38
Modelling of fuzzy time seriesForecasting of fuzzy time series
Examples
Fuzzy-ARMA[4,4]-process
optimal h-step-forecasting
fuzzy forecast intervals
Damage of a T-beam plate
Time series of live laods – monthly goods – in a storehouse
non-stationary Fuzzy-ARMA[4,4]-process
Description of fuzzy time seriesModelling of fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 39/42
Institute for Static und Dynamics of Structures
slide 39
Modelling of fuzzy time seriesForecasting of fuzzy time series
Examples
Monte Carlo Simulation of 100 future realizations8 realizations at time point
Damage of a T-beam plate Fuzzy random forecasting
Description of fuzzy time seriesModelling of fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 40/42
Institute for Static und Dynamics of Structures
slide 40
Modelling of fuzzy time seriesForecasting of fuzzy time series
Examples
Fuzzy damage indicator
Damage of a T-beam plate
not strengthenedstrengthened
indirect forecasting
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 41/42
Institute for Static und Dynamics of Structures
slide 41
Time series with fuzzy data can be modeledas realizations of fuzzy random processes
Conclusions
New -representation of fuzzy random variables
enables the modeling of fuzzy time series asrealizations of fuzzy random processes
Fuzzy-ARMA-processes allow the simulationand forecasting of fuzzy time series
7/21/2019 Fuzzy Time Series
http://slidepdf.com/reader/full/fuzzy-time-series 42/42
Institute for Static und Dynamics of Structures
slide 42
Thank you!