Josefina L

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Improving the Forecasting Capability of Fuzzy
Inductive Reasoning by Means of Dynamic Mask
Allocation

• Josefina López Herrera
• Institut dInformàtica i Robòtica Industrial
• Universitat Politècnica de Catalunya
• Edifici Nexus
• Gran Capità 2-4
• Barcelona 08034, Spain
• jlopez_at_iri.upc.es

• François E. Cellier
• Electrical Computer Engineering Dept.
• University of Arizona
• P.O.Box 210104
• Tucson, AZ 85721-0104
• U.S.A
• Cellier_at_ECE.Arizona.Edu

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Table of Contents

• Introduction.
• Dynamic Mask Allocation.
• DMAFIR and QDMAFIR.
• Multiple Regimes.
• Variable Structure Systems.
• Conclusions.

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Qualitative Simulation Using FIR
Qualitative FIR Model
Predicted Output
Inputs
Confidence in Prediction
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Dynamic Mask Allocation in Fuzzy Inductive
Reasoning (DMAFIR)
c1
FIR Mask 1
y1
Mask Selector
c2
y2
FIR Mask 2
Ts
Best mask
Switch Selector
y
cn
yi predicted output using mask mi ci estimated
confidence
FIR Mask n
yn
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Quality-adjusted Dynamic Mask Allocation (QDMAFIR)
Qi is the mask quality of the selected mask mi
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Optimal and Suboptimal Mask for Barcelona Time
Series
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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.

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Qualitative Simulation with FIR
real data
predicted data
using k steps
prediction for time
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Prediction Error
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Prediction Error
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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

• By choosing the outputs of the two integrators as
  two state variables

• The following state-space model is obtained

Output Time Series
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DMAFIR Algorithm to Predict Time Series with
Multiple Regimes

• To start the experiment, three different models
  were identified using three different values of

• 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.

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DMAFIR Algorithm to Predict Time Series with
Multiple Regimes
the input/output behaviors will be different
because of the different training data used by
the two models
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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.

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One-day Predictions of the Van-der-Pol Series
Using FIR With
Model

• The model can not predict the peaks of the time
  series with

• FIR can only predict behaviors that it has seen
  before.

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Prediction Errors for Van-der-Pol Series

• 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.

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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.
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Predictions Errors for Multiple Regimes
Van-der-Pol Series

• 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.

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Variable Structure System Prediction with DMAFIR

• A time-varing system exhibits an entire spectrum
  of different behavioral patterns. To demostrate
  DMAFIRs 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.

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One-day Prediction of the Van-der-Pol
Time-Varying Series
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One-day Predictions of the Van-der-Pol
Time-Varying Series Using DMAFIR with the
Similarity Confidence Measure

• Predictions Errors for Time-varying Van-der-Pol
  Series.

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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)