Computational Intelligence in Time Series Forecasting

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Part I Introduction 1 Computational Intelligence: An Introduction................................................3 1.1 Introduction..............................................................................................3 1.2 Soft Computing.........................................................................................3 1.3 Probabilistic Reasoning............................................................................4 1.4 Evolutionary Computation........................................................................6 1.5 Computational Intelligence.......................................................................8 1.6 Hybrid Computational Technology..........................................................9 1.7 Application Areas...................................................................................10 1.8 Applications in Industry.........................................................................11 References..............................................................................................12 2 Traditional Problem Definition.....................................................................17 2.1 Introduction to Time Series Analysis.....................................................17 2.2 Traditional Problem Definition...............................................................18 2.2.1 Characteristic Features..............................................................18 2.2.1.1 Stationarity ..................................................................18 2.2.1.2 Linearity ......................................................................20 2.2.1.3 Trend............................................................................20 2.2.1.4 Seasonality...................................................................21 2.2.1.5 Estimation and Elimination of Trend and Seasonality...................................................................21 2.3 Classification of Time Series..................................................................22 2.3.1 Linear Time Series....................................................................23 2.3.2 Nonlinear Time Series...............................................................23 2.3.3 Univariate Time Series..............................................................23 2.3.4 Multivariate Time Series...........................................................24 2.3.5 Chaotic Time Series..................................................................24 2.4 Time Series Analysis..............................................................................25 2.4.1 Objectives of Analysis..............................................................25 2.4.2 Time Series Modelling..............................................................26 2.4.3 Time Series Models...................................................................26 2.5 Regressive Models..................................................................................27 2.5.1 Auto regression Model ..............................................................27 2.5.2 Moving-average Model ............................................................28 2.5.3 ARMA Model...........................................................................28 2.5.4 ARIMA Model..........................................................................29 2.5.5 CARMAX Model......................................................................32 2.5.6 Multivariate Time Series Model................................................33 2.5.7 Linear Time Series Models.......................................................35 2.5.8 Nonlinear Time Series Models..................................................35 2.5.9 Chaotic Time Series Models.....................................................36 2.6 Time-domain Models..............................................................................37 2.6.1 Transfer-function Models..........................................................37 2.6.2 State-space Models....................................................................38 2.7 Frequency-domain Models.....................................................................39 2.8 Model Building.......................................................................................42 2.8.1 Model Identification..................................................................43 2.8.2 Model Estimation......................................................................45 2.8.3 Model Validation and Diagnostic Check..................................48 2.9 Forecasting Methods...............................................................................49 2.9.1 Some Forecasting Issues...........................................................50 2.9.2 Forecasting Using Trend Analysis............................................51 2.9.3 Forecasting Using Regression Approaches...............................51 2.9.4 Forecasting Using the Box-Jenkins Method..............................53 2.9.4.1 Forecasting Using an Autoregressive Model AR(p)....53 2.9.4.2 Forecasting Using a Moving-average Model MA(q)...54 2.9.4.3 Forecasting Using an ARMA Model...........................54 2.9.4.4 Forecasting Using an ARIMA Model..........................56 2.9.4.5 Forecasting Using an CARIMAX Model....................57 2.9.5 Forecasting Using Smoothing...................................................57 2.9.5.1 Forecasting Using a Simple Moving Average.............57 2.9.5.2 Forecasting Using Exponential Smoothing .................58 2.9.5.3 Forecasting Using Adaptive Smoothing......................62 2.9.5.4 Combined Forecast......................................................64 2.10 Application Examples.............................................................................66 2.10.1 Forecasting Nonstationary Processes........................................66 2.10.2 Quality Prediction of Crude Oil................................................67 2.10.3 Production Monitoring and Failure Diagnosis..........................68 2.10.4 Tool Wear Monitoring..............................................................68 2.10.5 Minimum Variance Control......................................................69 2.10.6 General Predictive Control........................................................71 References..............................................................................................74 Selected Reading....................................................................................74 Part II Basic Intelligent Computational Technologies 3 Neural Networks Approach...........................................................................79 3.1 Introduction............................................................................................79 3.2 Basic Network Architecture....................................................................80 3.3 Networks Used for Forecasting..............................................................84 3.3.1 Multilayer Perceptron Networks...............................................84 3.3.2 Radial Basis Function Networks...............................................85 3.3.3 Recurrent Networks ..................................................................87 3.3.4 Counter Propagation Networks.................................................92 3.3.5 Probabilistic Neural Networks..................................................94 3.4 Network Training Methods.....................................................................95 3.4.1 Accelerated Backpropagation Algorithm..................................99 3.5 Forecasting Methodology.....................................................................103 3.5.1 Data Preparation for Forecasting.............................................104 3.5.2 Determination of Network Architecture..................................106 3.5.3 Network Training Strategy......................................................112 3.5.4 Training, Stopping and Evaluation..........................................116 3.6 Forecasting Using Neural Networks.....................................................129 3.6.1 Neural Networks versus Traditional Forecasting....................129 3.6.2 Combining Neural Networks and Traditional Approaches.....131 3.6.3 Nonlinear Combination of Forecasts Using Neural Networks 132 3.6.4 Forecasting of Multivariate Time Series.................................136 References............................................................................................137 Selected Reading..................................................................................142 4 Fuzzy Logic Approach .................................................................................143 4.1 Introduction..........................................................................................143 4.2 Fuzzy Sets and Membership Functions................................................144 4.3 Fuzzy Logic Systems ...........................................................................146 4.3.1 Mamdani Type of Fuzzy Logic Systems.................................148 4.3.2 Takagi-Sugeno Type of Fuzzy Logic Systems........................148 4.3.3 Relational Fuzzy Logic System of Pedrycz.............................149 4.4 Inferencing the Fuzzy Logic System....................................................150 4.4.1 Inferencing a Mamdani-type Fuzzy Model.............................150 4.4.2 Inferencing a Takagi-Sugeno-type Fuzzy Model....................153 4.4.3 Inferencing a (Pedrycz) Relational Fuzzy Model....................154 4.5 Automated Generation of Fuzzy Rule Base..........................................157 4.5.1 The Rules Generation Algorithm............................................157 4.5.2 Modifications Proposed for Automated Rules Generation......162 4.5.3 Estimation of Takagi-Sugeno Rules’ Consequent Parameters...............................................................................166 4.6 Forecasting Time Series Using the Fuzzy Logic Approach..................169 4.6.1 Forecasting Chaotic Time Series: An Example.......................169 4.7 Rules Generation by Clustering............................................................173 4.7.1 Fuzzy Clustering Algorithms for Rule Generation..................173 4.7.1.1 Elements of Clustering Theory .................................174 4.7.1.2 Hard Partition............................................................175 4.7.1.3 Fuzzy Partition...........................................................177 4.7.2 Fuzzy c-means Clustering.......................................................178 4.7.2.1 Fuzzy c-means Algorithm..........................................179 4.7.2.1.1 Parameters of Fuzzy c-means Algorithm....180 4.7.3 Gustafson-Kessel Algorithm...................................................183 4.7.3.1 Gustafson-Kessel Clustering Algorithm....................184 4.7.3.1.1 Parameters of Gustafson-Kessel Algorithm....................................................185 4.7.3.1.2 Interpretation of Cluster Covariance Matrix.........................................................185 4.7.4 Identification of Antecedent Parameters by Fuzzy Clustering................................................................................185 4.7.5 Modelling of a Nonlinear Plant...............................................187 4.8 Fuzzy Model as Nonlinear Forecasts Combiner...................................190 4.9 Concluding Remarks............................................................................193 References............................................................................................193 5 Evolutionary Computation..........................................................................195 5.1 Introduction..........................................................................................195 5.1.1 The Mechanisms of Evolution................................................196 5.1.2 Evolutionary Algorithms.........................................................196 5.2 Genetic Algorithms...............................................................................197 5.2.1 Genetic Operators....................................................................198 5.2.1.1 Selection....................................................................199 5.2.1.2 Reproduction.............................................................199 5.2.1.3 Mutation ....................................................................199 5.2.1.4 Crossover...................................................................201 5.2.2 Auxiliary Genetic Operators...................................................201 5.2.2.1 Fitness Windowing or Scaling...................................201 5.2.3 Real-coded Genetic Algorithms..............................................203 5.2.3.1 Real Genetic Operators..............................................204 5.2.3.1.1 Selection Function......................................204 5.2.3.1.2 Crossover Operators for Real-coded Genetic Algorithms.....................................205 5.2.3.1.3 Mutation Operators.....................................205 5.2.4 Forecasting Examples.............................................................206 5.3 Genetic Programming...........................................................................209 5.3.1 Initialization............................................................................210 5.3.2 Execution of Algorithm...........................................................211 5.3.3 Fitness Measure.......................................................................211 5.3.4 Improved Genetic Versions.....................................................211 5.3.5 Applications............................................................................212 5.4 Evolutionary Strategies.........................................................................212 5.4.1 Applications to Real-world Problems ....................................213 5.5 Evolutionary Programming ..................................................................214 5.5.1 Evolutionary Programming Mechanism ................................215 5.6 Differential Evolution ..........................................................................215 5.6.1 First Variant of Differential Evolution (DE1).........................216 5.6.2 Second Variant of Differential Evolution (DE2).....................218 References............................................................................................218 Part III Hybrid Computational Technologies 6 Neuro-fuzzy Approach.................................................................................223 6.1 Motivation for Technology Merging....................................................223 6.2 Neuro-fuzzy Modelling ........................................................................224 6.2.1 Fuzzy Neurons........................................................................227 6.2.1.1 AND Fuzzy Neuron...................................................228 6.2.1.2 OR Fuzzy Neuron......................................................229 6.3 Neuro-fuzzy System Selection for Forecasting....................................230 6.4 Takagi-Sugeno-type Neuro-fuzzy Network..........................................232 6.4.1 Neural Network Representation of Fuzzy Logic Systems.......233 6.4.2 Training Algorithm for Neuro-fuzzy Network........................234 6.4.2.1 Backpropagation Training of Takagi-Sugeno-type Neuro-fuzzy Network................................................234 6.4.2.2 Improved Backpropagation Training Algorithm.......238 6.4.2.3 Levenberg-Marquardt Training Algorithm................239 6.4.2.3.1 Computation of Jacobian Matrix ...............241 6.4.2.4 Adaptive Learning Rate and Oscillation Control ......246 6.5 Comparison of Radial Basis Function Network and Neuro-fuzzy Network ..........................................................................247 6.6 Comparison of Neural Network and Neuro-fuzzy Network Training..248 6.7 Modelling and Identification of Nonlinear Dynamics .........................249 6.7.1 Short-term Forecasting of Electrical load ...............................249 6.7.2 Prediction of Chaotic Time Series...........................................253 6.7.3 Modelling and Prediction of Wang Data.................................258 6.8 Other Engineering Application Examples............................................264 6.8.1 Application of Neuro-fuzzy Modelling to Materials Property Prediction .................................................265 6.8.1.1 Property Prediction for C-Mn Steels ..........................266 6.8.1.2 Property Prediction for C-Mn-Nb Steels ....................266 6.8.2 Correction of Pyrometer Reading ...........................................266 6.8.3 Application for Tool Wear Monitoring ..................................268 6.9 Concluding Remarks............................................................................270 References............................................................................................271 7 Transparent Fuzzy/Neuro-fuzzy Modelling ..............................................275 7.1 Introduction .........................................................................................275 7.2 Model Transparency and Compactness................................................276 7.3 Fuzzy Modelling with Enhanced Transparency....................................277 7.3.1 Redundancy in Numerical Data-driven Modelling .................277 7.3.2 Compact and Transparent Modelling Scheme ........................279 7.4 Similarity Between Fuzzy Sets.............................................................281 7.4.1 Similarity Measure..................................................................282 7.4.2 Similarity-based Rule Base Simplification .............................282 7.5 Simplification of Rule Base..................................................................285 7.5.1 Merging Similar Fuzzy Sets....................................................287 7.5.2 Removing Irrelevant Fuzzy Sets.............................................289 7.5.3 Removing Redundant Inputs...................................................290 7.5.4 Merging Rules ........................................................................290 7.6 Rule Base Simplification Algorithms ..................................................291 7.6.1 Iterative Merging.....................................................................292 7.6.2 Similarity Relations.................................................................294 7.7 Model Competitive Issues: Accuracy versus Complexity....................296 7.8 Application Examples...........................................................................299 7.9 Concluding Remarks............................................................................302 References............................................................................................302 8 Evolving Neural and Fuzzy Systems...........................................................305 8.1 Introduction..........................................................................................305 8.1.1 Evolving Neural Networks......................................................305 8.1.1.1 Evolving Connection Weights...................................306 8.1.1.2 Evolving the Network Architecture...........................309 8.1.1.3 Evolving the Pure Network Architecture...................310 8.1.1.4 Evolving Complete Network.....................................311 8.1.1.5 Evolving the Activation Function..............................312 8.1.1.6 Application Examples................................................313 8.1.2 Evolving Fuzzy Logic Systems...............................................313 References............................................................................................317 9 Adaptive Genetic Algorithms.......................................................................321 9.1 Introduction..........................................................................................321 9.2 Genetic Algorithm Parameters to Be Adapted......................................322 9.3 Probabilistic Control of Genetic Algorithm Parameters.......................323 9.4 Adaptation of Population Size..............................................................327 9.5 Fuzzy-logic-controlled Genetic Algorithms.........................................329 9.6 Concluding Remarks............................................................................330 References............................................................................................330 Part IV Recent Developments 10 State of the Art and Development Trends..................................................335 10.1 Introduction..........................................................................................335 10.2 Support Vector Machines.....................................................................337 10.2.1 Data-dependent Representation...............................................342 10.2.2 Machine Implementation.........................................................343 10.2.3 Applications............................................................................344 10.3 Wavelet Networks................................................................................345 10.3.1 Wavelet Theory.......................................................................345 10.3.2 Wavelet Neural Networks.......................................................346 10.3.3 Applications............................................................................349 10.4 Fractally Configured Neural Networks.................................................350 10.5 Fuzzy Clustering...................................................................................352 10.5.1 Fuzzy Clustering Using Kohonen Networks...........................353 10.5.2 Entropy-based Fuzzy Clustering.............................................355 10.5.2.1 Entropy Measure for Cluster Estimation...................356 10.5.2.1 The Entropy Measure ..................................356 10.5.2.2 Fuzzy Clustering Based on Entropy Measure............358 10.5.2.3 Fuzzy Model Identification Using Entropy-based Fuzzy Clustering................................359 References............................................................................................360 Index....................................................................................................................363

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