josefina lópez herrera advisors: françois cellier gabriela cembrano ioc - ua - iri time series...

Josefina López Herrera

Advisors:

François Cellier

Gabriela Cembrano

IOC - UA - IRI

Time Series PredictionUsing

Inductive Reasoning Techniques

Table of Contents

• Contributions principales. • Antecedents.• Time Series Analysis Techniques. • Fuzzy Inductive Reasoning (FIR) for Time Series Analysis.• Time Series Characteristics.• Conclusions and Future Research.

Contributions

• Evaluation of Prediction Error.

• Confidence Measures for Prediction in FIR.

• Dynamic Mask Allocation.

• Estimation of Horizon of Predictability.

• Applications:– Early Warning Using Smart Sensors.– Signal Predictive Control Using FIR

Antecedents

• George Klir at the State University of New York Uyttenhove 1978, Klir 1985

• François Cellier at the University of Arizona Cellier and Yandell 1987, D. Li and Cellier 1990,Cellier 1991,Cellier et al. 1996, Cellier et al. 1998

• Rafael Huber and Gabriela Cembrano at the IRI Institute (UPC-CSIC)

PhD. Dissertations UPC-UA

• Angela Nebot Castells (1994)

Qualitative Modeling and Simulation of Biomedical Systems using FIR

• Francisco Múgica (1995)

Diseño Sistemático de Controladores Difusos Usando Razonamiento Inductivo

• Alvaro de Albornoz Bueno (1996)

Inductive Reasoning and Reconstruction Analysis: Two Complementary Tools for Qualitative Fault Monitoring of Large-Scale Systems

LinearModelsLinearModels Non-Linear

ModelsNon-Linear

Models

FuzzyLogicFuzzyLogic

Pattern-Based Approaches

Pattern-Based Approaches

Time Series Analysis Techniques

Time Series Analysis Techniques

FIRFIR

• Stationarity will be assumed.

• Prefiltering of data may be necessary.

• Probabilistic Reasoning.

• Ljung 1999, Brockwell and David 1991, 1996 ,Box Jenkins 1994.

• Stochastic Time Series.

LinearModelsLinearModels

• Parametric Models, Learning Techniques• At least Quasi-stationary• Deterministic Elements• State Space Models (Casdagli and Eubank 1992)• Neural Networks (Weigend and Gershenfeld

1994)• Hybrid Models (Delgado 1998, Telecom 1994)

Non-LinearModels

Non-LinearModels

• Non-parametric Models, Synthesized Techniques • At least Quasi-stationary, Deterministic Elements• Fuzzy Neural Networks (Jang 1997)• FIR (López et al. 1996)• Mixed Models : Burr 1998, Takagi and Sugeno 1991

FuzzyLogicFuzzyLogic

• Fuzzification: Conversion to qualitative variables (Fuzzy Recoding)• Qualitative Modeling:Find the best qualitative relationship between inputs and outputs (Fuzzy Modeling)• Qualitative Simulation: Forecasting of future qualitative outputs (Fuzzy Simulation)• Defuzzification : Conversion to quantitative variables (Regeneration)

FIRFIR

Qualitative Modeling

Qualitative Simulation

31123

32212

11211

14321

oiiii

31123

32212

11211

14321

oiiii

Behavior Matrix

32121 yyyuu

Raw Data

Matrix

33?21

21?12

12?12

21?32

32?21

Optimal

Matrix

32131

21321

11311

1 ?

23

1

?1123

32121 yyyuu

Input Pattern

Distance Computation

Euclideandj

Output Forecast

Computation

fi=F(W*5-NN-out)

Forecast Value

5-Nearest Neighbors

Matched Input Pattern

Class

Side

Member

Time Series Forecasting

• In univariate time series, only a single variable has been observed, the future values of which are to be predicted on the basis of their own past.

1

1

1

1

1

1

u

t

tt

t7t

t14t

time 1

1

1

1

1

1

1

u

t

tt

t7t

t14t

time 1

• In this case, the mask candidate matrix has n-rows and one column. In order to decide the depth of the mask, the autocorrelation function is used.

Characteristics of Time Series

Natural BL Synthetic V

Stationary LV Non-stationary BTime invariant LV Time varying BLow dimensional LV Stochastic BClean BVL NoisyShort B Long LVDocumented BLV BlindLinear Non-linear BLVScalar BLV VectorSingle recording BLV Multiple

recordingsContinuous BLV DiscreteDormant Active BLV

Natural BL Synthetic V

Stationary LV Non-stationary BTime invariant LV Time varying BLow dimensional LV Stochastic BClean BVL NoisyShort B Long LVDocumented BLV BlindLinear Non-linear BLVScalar BLV VectorSingle recording BLV Multiple

recordingsContinuous BLV DiscreteDormant Active BLV

B-Barcelona water demand time series

V-Van-der-Pol oscillator time series

L- chaotic intensity pulsation of a single-mode far infrared NH3 laser beam

Weigend and Gershenfeld 1994

Water Demand Prediction

• Data Daily Demand in Barcelona. Jan 1985 - Nov 1986.

• The process is quasi-stationary, and its variance is roughly constant.


• The water demand on any given day is strongly correlated with the demand seven days earlier.

• Autocorrelation function of daily demand series.


• The result of prediction was:

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u

t

tt

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t14t

time 1

Prediction ErrorPrediction Error

)(0.25iiii dynstdmeantot errerrerrerr

))()(( terrterrmeanerriii simabsdyn ))()(( terrterrmeanerr

iii simabsdyn


)))),(ˆ(())),(((()))(ˆ())(((

tymeanabstymeanabsmaxtymeantymeanabs

erri

imeani

)))),(ˆ(())),((((

)))(ˆ())(((tymeanabstymeanabsmax

tymeantymeanabserr

i

imeani

)))),ˆ(())),(((()))(ˆ())(((

i

istd ystdabstystdabsmax

tystdtystdabserr

i

)))),ˆ(())),((((

)))(ˆ())(((i

istd ystdabstystdabsmax

tystdtystdabserr

i


))(ˆ),(( tytymaxy imax ))(ˆ),(( tytyminy imin

),()(

)(minmax

minnorm yymax

ytyty

),()(

)(minmax

minnorm yymax

ytyty

),(

)(ˆ)(minmax

mininorm yymax

ytyty

i

),()(ˆ)(

minmax

mininorm yymax

ytyty

i

))()(()( tytyabsterrii normnormabs ))()(()( tytyabsterr

ii normnormabs

)),(),((

))(),(()(

tytymax

tytymintsim

i

i

normnorm

normnormi

)),(),((

))(),(()(

tytymax

tytymintsim

i

i

normnorm

normnormi )(0.1 tsimerr isimi

)(0.1 tsimerr isimi

Qualitative Simulation with FIR

)4,9()3,8()2,7()1,6()5(

)4,8()3,7()2,6()1,5()4(

)4,7()3,6()2,5()1,4()3(

)4,6()3,5()2,4()1,3()2(

)4,5()3,4()2,3()1,2()(

)4,4()3,3()2,2()1,()(

)4,3()3,2()2,()1,()(

)4,2()3,()2,()1,()2(

)4,()3,()2,()1,2()3(

)4,()3,()2,2()1,3()4(

ttyttyttyttytty

ttyttyttyttytty

ttyttyttyttytty

ttyttyttyttytty

ttyttyttyttytty

ttyttyttyttyty

ttyttyttytytty

ttyttytyttytty

ttytyttyttytty

tyttyttyttytty

Y

)4,9()3,8()2,7()1,6()5(

)4,8()3,7()2,6()1,5()4(

)4,7()3,6()2,5()1,4()3(

)4,6()3,5()2,4()1,3()2(

)4,5()3,4()2,3()1,2()(

)4,4()3,3()2,2()1,()(

)4,3()3,2()2,()1,()(

)4,2()3,()2,()1,()2(

)4,()3,()2,()1,2()3(

)4,()3,()2,2()1,3()4(

ttyttyttyttytty

ttyttyttyttytty

ttyttyttyttytty

ttyttyttyttytty

ttyttyttyttytty

ttyttyttyttyty

ttyttyttytytty

ttyttytyttytty

ttytyttyttytty

tyttyttyttytty

Y

prediction for time ),( ktnty tnt using k steps

real data predicted data

Comparison of FIR with other Methodologies for the Barcelona

Water Demand Time Series

TMethodogy ARIMA NAR ANN FIR

New errorRelativeerror

7.88

4.21%

9.15

3.9%

8.74

4.1%

4.6

3.7%

TMethodogy ARIMA NAR ANN FIR

New errorRelativeerror

7.88

4.21%

9.15

3.9%

8.74

4.1%

4.6

3.7%

without intervention analysis

Box Jenkins *)

Quevedo et al. 19884.5 % relative error

ANN*)Griñó 1992 4.2% relative error

Box Jenkins *)

Quevedo et al. 19884.5 % relative error

ANN*)Griñó 1992 4.2% relative error

*) with intervention analysis

Related Investigation

Comparison of FIR with other Methodologies

ConfidenceMeasures

ConfidenceMeasures

CrispCrispFuzzyLogicFuzzyLogic

ProximityProximitySimilaritySimilarity

Sources of Uncertainty in Predictions

•Dispersion among neighbors in input space.•Uncertainty related to quantity of measurements.

•Dispersion among neighbors in output space.•Uncertainty related to quality of measurements.

Proximity Measure

• This measure is related to establishing the distance between the testing input state and the training input states of its five nearest neighbors in the experience data base and to establishing distance measures between the output states of the five nearest neighbors among themselves.

Similarity Measure

• (Dubois and Pradé 1980).

BA

BABAS

),(1BA

BABAS

),(1

A=B then S1(A,B) = 1.0A=B then S1(A,B) = 1.0

A disjoint B then S1(A,B) = 0.0A disjoint B then S1(A,B) = 0.0

• This measure is defined without the explicit use of a distance function, the similarity measure presented is based on intersection, union and cardinality.

Similarity Measure

• The similarity of the ith m-input of the jth nearest neighbor to the testing m-input based on intersection can be defined as follows:

j

ii

jiij

i q,qmax

q,qminsim

• The overall similarity of the jth neighbor is defined as the average similarity of all its m-inputs in the input space:

n

1i

ji

jin sim

n

1sim

where qi are normalized values in the range from 0 to 1.

Similarity Measure

• The similarity of the jth neighbor to the estimated testing m-output based on intersection can be defined as follows:

jout

joutj

out q,qmaxq,qmin

sim

5

1j

jout

jin

jrelsim simsimwconf

• A confidence value based on similarity measures can thus be defined :

FIR Confidence Measures for NH3 Time Series

• Deterministic process

• Similarity and Proximity

FIR Confidence Measures for Barcelona Time Series

• Stochastic Process with deterministic elements.

• The relationship between the prediction error and the confidence measures is less evident.

• The two are positively correlated.

Evaluation of Confidence Measures

• The similarity measure is more sensitive to the prediction error because the similarity measure preserves the qualitative difference between a new input state and its neighbors in the experience data base.

• The confidence measures are indicators of how well the series may be fitted by an autoregressive or deterministic model.

FIRMask #1

FIRMask #2

Mask Selector

FIRMask #n

Switch Selector

c1

c2

yn

y1

y2

cn

Best mask

y

Ts

yi predicted output using mask mi

ci estimated confidence

Dynamic Mask Allocation in Fuzzy Inductive Reasoning (DMAFIR)

Optimal and Suboptimal Mask for Barcelona Time Series

Dynamic Mask Allocation Applied to Barcelona Time Series

Prediction and Simulation

• FIR Predictions use different masks to predict future values n-steps into the future, avoiding the use of already predicted (contamined) data in the predictions.

• FIR Simulations use the optimal mask of the single step prediction recursively, minimizing the distance of extrapolation at the expense of recursively using already contamined data.

Qualitative Prediction

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1-step prediction

Mask candidate matrix Optimal Mask

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2-step prediction

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3-step prediction

Simulation and Prediction

• Without dynamic mask allocation for Barcelona time series.

• Comparison of FIR qualitative simulation and prediction with dynamic mask allocation for Barcelona time series.

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: 0)1( 2 xxxx

x1• By choosing the outputs of the two integrators as two state

variables:

x2• The following state-space model is obtained:

21 12

212 )1(

2y

2 Output Time Series

DMAFIR Algorithm to Predict Time Series with Multiple Regimes

9146.0))((~5.3

9085.0))((~5.2

9342.0))47(),((~5.1

Regime

ttyfy

ttyfy

ttyttyfy

QualityMask

* the input/output behaviors will be different because of the different training data used by the two models

Prediction Errors for Van-der-Pol Series

• FIR during the prediction looks for five good neighbors, it only encounters four that are truly pertinent.

Series =1.5 =2.5 =3.5

Models

=1.5 2.63 6.76 10.39

=2.5 2.96 0.97 4.65

=3.5 4.27 2.57 1.83

Series =1.5 =2.5 =3.5

Models

=1.5 2.63 6.76 10.39

=2.5 2.96 0.97 4.65

=3.5 4.27 2.57 1.83

• The values along the diagonal are

smallest and the values in the two remaining corners are largest.

One-day Predictions of the Van-der-Pol

Multiple Regimes Series.• A time series was 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 by 3.5 .The multiple regimes series consists of 553 samples.

Prediction Errors for Multiple Regimes Van-der-Pol Series

1195.1

9317.15.3

2978.25.2

8759.55.1

DMAFIR

errorModel

• 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-varying system exhibits an entire spectrum of different behavioral patterns. To demonstrate 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.

One-day Predictions of the Van-der-Pol Time-varying Series Using DMAFIR

with the Similarity Confidence Measure

2997.1

8791.15.3

4864.15.2

7431.55.1

DMAFIR

for

for

for

errorModel

• Predictions Errors for

Time-varying Van-der-Pol Series.

Predicting the Predictability Horizon

• The errors are likely to accumulate during iterative predictions of future values of a time series.

• It is thus of much interest to the user of such a tool to be able to assess the quality of predictions made not only locally, but as a function of time.

• When the predictions depend on previously predicted data points these are by themselves associated with a degree of uncertainty already.

• In the first step of a multiple-step prediction, the predicted value depends entirely on measurement data.

• The local error can be indirectly estimated using the proximity or similarity measure.

• Either measure can easily be extended to become an estimator of accumulated confidence

Water Demand of the City of Barcelona Multiple Step simulation using FIR

))14(),7(),((~

)( ttyttyttyfty ))14(),7(),((~

)( ttyttyttyfty 3

))14()7()(()()(

ttcttcttctctc aaa

la

3

))14()7()(()()(

ttcttcttctctc aaa

la

Conclusions• The prediction made by CIR (Causal Inductive Reasoning) were not significantly better.• The confidence measure of FIR are an indirect prediction error estimate.• A new formula to assess the error of predictions of a univariate time series, the FIR filters out

what it considers to be a noise.• FIR provides the model automatically, not requires a significant development effort as well as

knowledge about the nature of the process form wich the series was derived.• The confidence measures provide at least a statistical estimate for the quality of the prediction.• Several suboptimal mask are used to make, in parallel forecast of the same time series. Each of

the forecast is accompanied by an estimate of its quality. In each step, the one forecast is kept as the true forecast to be reported back to the user that shows the highest confidence value.

• A set of formulae has been devised to estimate the effects of data contamination on the accunulated confidence over multiple prediction steps.

• The FIR is a robust methodology, after López et al. 96 some UPC groups use FIR like Prediction Module in an Optimation Tool for Water Distribution Networks, Quevedo et al. 1999.

Publications• Cellier, F. And J. López (1995). Causal Inductive Reasoning. A new paradigm for data-

driven qualitative simulation of continuous-time dynamical systems. Systems Analysis Modelling Simulation 18(1), pp.26-43.

• Cellier F., J. López, A. Nebot, G. Cembrano (1996), Means for estimating the forecasting error in Fuzzy Inductive Reasoning, ESM´96:European Simulation Multiconference, Budapest, Hungary, June 2-6, pp.654-660.

• López J., G. Cembrano, F, Cellier (1996), Time series prediction using Fuzzy Inductive Reasoning, ESM´96:European Simulation Multiconference, Budapest, Hungary, June 2-6, pp.765-770.

• Cellier F., J. López, A. Nebot, G. Cembrano (1998), Confidence measures in Fuzzy Inductive Reasoning, International Journal of General Systems, in print.

• López J., F. Cellier (1999), Improving the Forecasting Capability of Fuzzy Inductive Reasoning by Means of Dynamic Mask Allocation, ESM´99:European Simulation Multiconference, in print.

• López J., F. Cellier, G. Cembrano, L. Ljung, (1999), Estimating the horizon of predictability in time series predictions using inductive modeling tools, International Journal of General Systems, submitted for publication.

Future Research

• Use of time-series predictors in the design of smart sensors with look-ahead capabilities. If a sensor with look-ahead capability can anticipate the crossing of a critical threshold, it may issue an early warning that might enable the plant operator to do something about the problem before it ever occurs. (Appendix A)

• The design of signal predictive controllers that make use of smart sensors of the class introduced in Appendix A, to improve the control performance of feedback control systems.

josefina lópez herrera advisors: françois cellier gabriela cembrano ioc - ua - iri time series...

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