1. Josefina López Herrera Institut d’Informàtica i Robòtica Industrial Universitat Politècnica de Catalunya Edifici Nexus Gran Capità 2-4 Barcelona 08034, Spain jlopez@iri.upc.es Improving the Forecasting Capability of Fuzzy Inductive Reasoning by Means of Dynamic Mask Allocation François E. Cellier Electrical & Computer Engineering Dept. University of Arizona P.O.Box 210104 Tucson, AZ 85721-0104 U.S.A Cellier@ECE.Arizona.Edu

2. Table of Contents Introduction. Dynamic Mask Allocation. DMAFIR and QDMAFIR. Multiple Regimes. Variable Structure Systems. Conclusions.

3. Qualitative Simulation Using FIR Qualitative FIR Model Inputs Confidence in Prediction Predicted Output

4. Dynamic Mask Allocation in Fuzzy Inductive Reasoning (DMAFIR) FIR Mask #1 FIR Mask #2 Mask Selector FIR Mask #n Switch Selector c1c1 c2c2 ynyn y1y1 y2y2 cncn Best mask y TsTs y i predicted output using mask m i c i estimated confidence

5. Quality-adjusted Dynamic Mask Allocation (QDMAFIR) Q i is the mask quality of the selected mask m i

6. Optimal and Suboptimal Mask for Barcelona Time Series

7. 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.

8. Qualitative Simulation with FIR prediction for time using k steps real data predicted data

10. Prediction Error

12. 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

13. 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.

14. 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

15. 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.

16. One-day Predictions of the Van-der-Pol Series Using FIR With The model can not predict the peaks of the time series with FIR can only predict behaviors that it has seen before. Model

17. 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.

18. 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.

19. 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.

20. 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.

21. One-day Prediction of the Van-der-Pol Time-Varying Series

22. 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.

23. 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)