Improving the Forecasting
Capability of Fuzzy Inductive
Reasoning by Means of
Dynamic Mask Allocation
Josefina López Herrera
Institut d’Informàtica i Robòtica
Industrial
Universitat Politècnica de Catalunya
Edifici Nexus
Gran Capità 2-4
Barcelona 08034, Spain
[email protected]
François E. Cellier
Electrical & Computer Engineering Dept.
University of Arizona
P.O.Box 210104
Tucson, AZ 85721-0104
U.S.A
[email protected]
Table of Contents
•
•
•
•
•
•
Introduction.
Dynamic Mask Allocation.
DMAFIR and QDMAFIR.
Multiple Regimes.
Variable Structure Systems.
Conclusions.
Qualitative Simulation Using FIR
Inputs
Qualitative
FIR
Model
Predicted Output
Confidence in
Prediction
Dynamic Mask Allocation in Fuzzy
Inductive Reasoning (DMAFIR)
FIR
Mask #1
Ts
FIR
Mask #2
c1
y1
c2
y2
Mask Selector
Best mask
Switch Selector
FIR
Mask #n
cn
yn
y
yi predicted output using mask
mi
ci estimated confidence
Quality-adjusted Dynamic Mask
Allocation (QDMAFIR)
Qi
Qrel 
Qopt
Qdyn (t)  Q rel i (t)  conf sim (t)
Qi is the mask quality of the selected mask mi
Optimal and Suboptimal Mask
for Barcelona Time Series
Dynamic Mask Allocation Applied to
Barcelona Time Series
• Comparison of FIR
and DMAFIR for
Barcelona time series.
• Comparison of FIR
and QDMAFIR for
Barcelona time series.
Qualitative Simulation with FIR



 y (t  4t )
 y (t  3t )

 y (t  2t )
 y (t  t )

 y (t )
Y 
 y (t  t )
 y (t  2t )

 y (t  3t )
 y (t  4t )

 y (t  5t )







y (t  3t ,1)
y (t  2t ,2)
y (t  t ,3)
y ( t ,4 )
y (t  2t ,1)
y (t  t ,1)
y (t  t ,2)
y ( t ,2 )
y (t ,3)
y (t  t ,3)
y (t  t ,4)
y (t  2t ,4)
y (t ,1)
y (t  t ,2)
y (t  2t ,3)
y (t  3t ,4)
y (t  t ,1)
y (t  2t ,2)
y (t  3t ,3)
y (t  4t ,4)
y (t  2t ,1)
y (t  3t ,1)
y (t  3t ,2)
y (t  4t ,2)
y (t  4t ,3)
y (t  5t ,3)
y (t  5t ,4)
y (t  6t ,4)
y (t  4t ,1)
y (t  5t ,2)
y (t  6t ,3)
y (t  7t ,4)
y (t  5t ,1)
y (t  6t ,2)
y (t  7t ,3)
y (t  8t ,4)
y (t  6t ,1)

y (t  7t ,2)

y (t  8t ,3)

y (t  9t ,4)


















real data
predicted data
 y (t  nt , k ) prediction for time t  nt using k steps
Prediction Error
Error
Prediction
errtoti  25.0  ( errmeani  errstdi  errdyni )
errdyni  mean(errabsi (t )  errsimi (t ))
Prediction Error
errmeani
err
std
i
abs(mean( y (t ))  mean( yˆ i (t )))

max(abs(mean( y (t ))), abs(mean( yˆ i (t ))),  )
abs ( std ( y ( t ))  std ( yˆ i ( t )))

max ( abs ( std ( y ( t ))), abs ( std ( yˆ i ))),  )
Prediction Error
ymax  max( y (t ), yˆi (t ))
y (t )  ymin
ynorm (t ) 
max ( ymax  ymin ,  )
sim i ( t ) 
ymin  min( y (t ), yˆi (t ))
ynormi (t ) 
yˆ i (t )  ymin
max ( ymax  ymin ,  )
min ( y norm ( t ), y norm i ( t ))
max ( y norm ( t ), y norm i ( t ),  )
errabsi (t )  abs ( ynorm (t )  ynormi (t ))
errsimi  1.0  simi (t )
DMAFIR Algorithm to Predict Time
Series with Multiple Regimes
• The behavioral patterns change between segments.
• Van-der-Pol oscillator series is introduced. This oscillator
is described by the following second-order differential
equation:
x    (1  x 2 )  x  x  0
• By choosing the outputs of the two integrators as two state
variables:
 x
1
2  x
• The following state-space model is obtained:
2    (1  12 )   2  1
1   2
2
Output Time Series
y  2
DMAFIR Algorithm to Predict Time
Series with Multiple Regimes
• To start the experiment, three different models were
identified using three different values of   1.5   2.5   3.5
• The first 80 data points of each time series were discarded,
as they represent the transitory period. The next 800 data
points were used to learn the behavior of each series and
the subsequent 200 data points were used as testing data.
• With a sampling rate of 0.05, 200 data points correspond
aprox. to one oscillation period. Four limit cycles were
used for training the model, and one limit cycle was used
for testing.
DMAFIR Algorithm to Predict Time
Series with Multiple Regimes
Regime
  1.5
  2.5
  3.5
Mask
~
y  f ( y (t  t ), y (t  47t ))
~
y  f ( y (t  t )) 
~

y  f ( y (t  t ))
Quality
0.9342
0.9085
0.9146
* the input/output behaviors will be different because of the
different training data used by the two models
Van-der-Pol Series Using FIR
• Only with Optimal Mask.
• Compares the real value with
their predictions.
• Because of the completely
deterministic nature of this
time series, the predictions
should be perfect. They are
not perfect due to data
deprivation. Since 800 data
points were used for training,
the experience data base
contains only four cycles.
One-day Predictions of the Van-der-Pol
Series Using FIR With   1.5
Model
• The model can not predict the peaks of the time series with
  2.5,   3.5
• FIR can only predict behaviors that it has seen before.
Prediction Errors for Van-der-Pol Series
Series
  1.5   2.5
  3.5
Model (   1.5) 2.6292
Model (   2.5) 2.9645
6.7597 10.3922
0.9747 4.6463
Model (   3.5)
2.5744
4.2691
1.8272
• The values along the diagonal are smallest and
the values in the two remaining corners are largest.
• FIR during the prediction looks for five good neighbors, it only
encounters four that are truly pertinent.
One-day Predictions of the Van-der-Pol
Multiple Regimes Series.
• A time series be
constructed in which
the variable 
assumes a value of
1.5 during one
segment, followed by
a value of 2.5 during
the second time
segment, followed 3.5
The multiple regimes
series consists of 553
samples.
Predictions Errors for Multiple Regimes
Van-der-Pol Series
Model
error
  1 .5
  2 .5
  3 .5
5.8759
2.2978
1.9317
DMAFIR 1.1195
• The model obtained for 
= 1.5 cannot predict the
higher peaks of the second
and third time segment
very well.
• The DMAFIR error demostrates that this new
technique can indeed be successfully applied to
the problem of predicting time series that operate
in multiple regimes.
Variable Structure System Prediction
with DMAFIR
• A time-varing system exhibits an entire spectrum of
different behavioral patterns. To demostrate DMAFIR’s
ability of dealing with time-varying systems, the Van-der-Pol
oscillator is used. A series was generated, in which 
changes its value continuously in the range from 1.0 to 3.5.
The time series contains 953 records sampled using a
sampling interval of 0.05. The time series contains 953
records sampled using a sampling interval of 0.05.
One-day Prediction of the Van-der-Pol
Time-Varying Series
One-day Predictions of the Van-der-Pol
Time-Varying Series Using DMAFIR
with the Similarity Confidence Measure
Model
error
for  1.5
5.7431
for  2.5
1.4864
for  3.5
1.8791
DMAFIR
1.2997
• Predictions Errors for
Time-varying Van-der-Pol
Series.
Conclusions
• FIRs confidence measure is exploited to
dynamically select the one of a set of
models that best predicts the behavior of the
output of the given time
• The algorithm is shown to improve the
quality of the forecasts made:
– single regime (Barcelona)
– multiple regimes (Van der Pol)
– time-varying systems (Van der Pol)