Modelling and Forecasting Financial Data Techniques of Nonlinear Dynamics

STUDIES IN COMPUTATIONAL FINANCE

Editor-in-Chief;

Apostolos-Paul Refenes, London Business School, UK

Editorial Board:

Y. Abu-Mostafa, CalTech, USA F. Diebold, University of Pennsylvania, USA A. Lo, MIT, USA J. Moody, Oregon Graduate Institute, USA M . Steiner, University of Augsburg, Germany H. White, UCSD, USA

S. Zenios, University of Pennsylvania, The Wharton School, USA

Volume I ADVANCES IN QUANTITATIVE ASSET MANAGEMENT edited by Christian L.Dunis Volume II MODELLING AND FORECASTING FINANCIAL DATA Techniques of Nonlinear Dynamics edited by A iol S. Sooft and Liangyue Cao

Modelling and Forecasting Financial Data

Techniques of Nonlinear Dynamics

edited by

Abdol S. Soofi University of Wisconsin-Platteville

and

Liangyue Cao University of Western Australia

W Springer Science+Business Media, LLC

ISBN 978-1-4613-5310-2 ISBN 978-1-4615-0931-8 (eBook) DOI 10.1007/978-1-4615-0931-8

Library of Congress Cataloging-in-Publication Data Modelling and forecasting financial data: techniques of nonlinear dynamics / edited by Abdol S. Soofi and Liangyue Cao.

p. cm.--(Studies in computational finance ; v.2) Includes bibliographical references and index. ISBN 978-1-4613-5310-2

1. Finance-Mathematical models. I. Soofi, Abdol S. II. Cao, Liangyue. III. Studies in computational finance; .2.

HG173.M6337 2002 332\01'5118-dc21 2001058519

Copyright ® 2002 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 2002 Softcover reprint of the hardcover 1 st edition 2002

A l l rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+ Business Media, L L C .

Printed on acid-free paper.

Contents

List of Figures

List of Tables

Preface

Contributing Authors

Introduction Abdol S. Sooft and Liangyue Gao

Part I EMBEDDING THEORY: TIME-DELAY PHASE SPACE RECON­STRUCTION AND DETECTION OF NONLINEAR DYNAMICS

1

vii

xv

xvii

xxi

1

Embedding Theory:Introduction and Applications to Time Series Analysis 11 F. Strozzi and J. M. Zaldivar

1.1 Introduction 11 1.2 Embedding Theories 14 1.3 Chaotic Time Series Analysis 18 1.4 Examples of Applications in Economics 32 1.5 Conclusions 37

2 Determining Minimum Embedding Dimension Liangyue Gao

2.1 Introduction 2.2 Major existing methods 2.3 False nearest neighbor method 2.4 Averaged false nearest neighbor method 2.5 Examples 2.6 Summary

3 Mutual Information and Relevant Variables for Predictions Bernd Pompe

3.1 Introduction 3.2 Theoretical Background 3.3 Mutual Information Analysis 3.4 Mutual Information Algorithm

43

43 44 45 47 49 59

61

61 64 69 72

VI

3.5 Examples 3.6 Conclusions Appendix A.1 The Best LMS Predictor A.2 A Property of MI A.3 A Property of GMI

MODELLING AND FORECASTING

78 88 89 89 89 90

Part II METHODS OF NONLINEAR MODELLING AND FORECASTING

4 State Space Local Linear Prediction 95 D. K ugiumtzis

4.1 Introduction 96 4.2 Local prediction 97 4.3 Implementation of Local Prediction Estimators on Time Series 104 4.4 Discussion 109

5 Local Polynomial Prediction and Volatility Estimation in Financial Time 115

Series Zhan-Qian Lu

5.1 Introduction 115 5.2 Local polynomial method 117 5.3 Technical setup for statistical theory 119 5.4 Prediction methods 123 5.5 Volatility estimation 126 5.6 Risk analysis of AOL stock 128 5.7 Concluding remarks 132

6 Kalman Filtering of Time Series Data David M. Walker

137

6.1 Introduction 6.2 Methods 6.3 Examples 6.4 Summary

7

137 138 147 156

Radial Basis Functions Networks 159 A. Braga, A. C. Carvalho, T. Ludermir, M. de Almeida, E. Lacerda

7.1 Introduction 160 7.2 Radial Functions 161 7.3 RBF Neural Networks 161 7.4 An example of using RBF for financial time-series forecasting 172 7.5 Discussions 173 7.6 Conclusions 175 7.7 Acknowledgements 176

8 Nonlinear Prediction of Time Series Using Wavelet Network Method 179 Liangyue Cao

8.1 Introduction 179 8.2 Nonlinear predictive model 180 8.3 Wavelet network 181 8.4 Examples 185 8.5 Discussion and conclusion 192

Contents Vll

Part III MODELLING AND PREDICTING MULTIVARIATE AND INPUT­OUTPUT TIME SERIES

9 Nonlinear Modelling and Prediction of Multivariate Financial Time 199

Series Liangyue Cao

9.1 Introduction 199 9.2 Embedding multivariate data 200 9.3 Prediction and relationship 202 9.4 Examples 203 9.5 Conclusions and discussions 209

10 Analysis of Economic Time Series Using NARMAX Polynomial Models 213 Luis Antonio Aguirre, Antonio Aguirre

10.1 Introduction 213 10.2 NARMAX Polynomial Models 216 10.3 Algorithms 220 10.4 Illustrative Results 223 10.5 Discussion 233

11 Modeling dynamical systems by Error Correction Neural Networks 237 Hans-Georg Zimmermann, Ralph Neuneier, Ralph Grothmann

11.1 Introduction 238 11.2 Modeling Dynamic Systems by Recurrent Neural Networks 239 11.3 Modeling Dynamic Systems by Error Correction 246 11.4 Variants-Invariants Separation 250 11.5 Optimal State Space Reconstruction for Forecasting 253 11.6 Yield Curve Forecasting by ECNN 260 11. 7 Conclusion 262

Part IV PROBLEMS IN MODELLING AND PREDICTION

12 Surrogate Data Test on Time Series D. K ugiumtzis

13

12.1 The Surrogate Data Test 12.2 Implementation of the Nonlinearity Test 12.3 Application to Financial Data 12.4 Discussion

267

269 273 276 277

Validation of Selected Global 283 Models

C. Letellier, O. Menard, L. A. Aguirre 13.1 Introduction 284 13.2 Bifurcation diagrams for model with parameter dependence 294 13.3 Synchronization 296 13.4 Conclusion 300

14 Testing Stationarity in Time Series Annette Witt, Jurgen Kurths

303

Vlll MODELLING AND FORECASTING

15

14.1 Introduction 14.2 Description of the tests 14.3 Applications 14.4 Summary and discussion

303 306 312 323

Analysis of Economic Delayed-Feedback Dynamics 327 Henning U. Voss, Jurgen Kurths

15.1 Introduction 328 15.2 Noise-like behavior induced by a Nerlove-Arrow model with time

delay 329 15.3 A nonparametric approach to analyze delayed-feedback dynamics 332 15.4 Analysis of Nerlove-Arrow models with time delay 336 15.5 Model improvement 337 15.6 Two delays and seasonal forcing 339 15.7 Analysis of the USA gross private domestic investment time series 341 15.8 The ACE algorithm 343 15.9 Summary and conclusion 345

16 Global Modeling and Differential Embedding J. Maquet, C. Letellier, and G. Gouesbet

351

17

16.1 Introduction 16.2 Global modeling techniques 16.3 Applications to Experimental Data 16.4 Discussion on applications 16.5 Conclusion

Estimation of Rules Underlying Fluctuating Data S. Siegert, R. Friedrich, Ch. Renner, J. Peinke

18

17.1 Introduction 17.2 Stochastic Processes 17.3 Dynamical Noise 17.4 Algorithm for Analysing Fluctuating Data Sets 17.5 Analysis Examples of Artificially Created Time Series 17.6 Scale Dependent Complex Systems 17.7 Financial Market 17.8 Turbulence 17.9 Conclusions

351 352 367 369 371

375

375 376 378 378 381 389 390 393 396

Nonlinear Noise Reduction 401 Rainer Hegger, Holger Kantz and Thomas Schreiber

18.1 Noise and its removal 402 18.2 Local projective noise reduction 403 18.3 Applications of noise reduction 407 18.4 Conclusion and outlook: Noise reduction for economic data 413

19 Optimal Model Size Jianming Ye

417

19.1 Introduction 19.2 Selection of Nested Models 19.3 Information Criteria: General Estimation Procedures 19.4 Applications and Implementation Issues

417 419 420 425

Contents IX

20 Influence of Measured Time Series in the Reconstruction of Nonlinear 429

Multivariable Dynamics C. Letellier, L. A. Aguirre

20.1 Introduction 429 20.2 Non equivalent observables 432 20.3 Discussions on applications 444 20.4 Conclusion 448

Part V APPLICATIONS IN ECONOMICS AND FINANCE

21 Nonlinear Forecasting of Noisy Financial Data Abdo18. 800fi, Liangyue Cao

22

21.1 Introduction 21.2 Methodology 21.3 Results 21.4 Conclusions

Canonical Variate Analysis and its Applications to Financial Data Berndt Pilgram, Peter Verhoeven, Alistair Mees, Michael McAleer

22.1 Non-linear Markov Modelling 22.2 Implementation of Forecasting 22.3 The GARCH(1,1}-t Model 22.4 Data Analysis 22.5 Empirical Results 22.6 Discussion

Index

455

455 457 459 462

467

470 473 474 475 476 479

483

List of Figures

1.1 Schematic representation of nonlinear time series anal-ysis using delay coordinate embedding 19

1.2 Space-Time Separation Plots 21 1.3 Estimated H, using the standard scaled window variance

method 24 1.4 Recurrence plots of Time series 31 1.5 Phase space of the Long Wave Model 35 1.6 False nearest neighours 36 1.7 Observed and predicted unfilled orders for capital 38 2.1 The values of E1 and E2 for the British pound/US dol-

lar time series, where "(1008 d.p.)" means that the E1 and the E2 curves were estimated using 1008 data points. 51

2.2 The percentages of false nearest neighbors for the British pound/US dollar time series. 51

2.3 The values of E1 and E2 for the Japanese yen/US dollar time series. 52

2.4 The percentages offalse nearest neighbors for the Japanese yen/US dollar time series. 52

2.5 The values of E1 and E2 for the Mackey-Glass time se-ries with only 200 data points used in the calculation, in comparison with the results obtained using 10000 data points. 54

2.6 The percentages of false nearest neighbors for the Mackey-Glass time series with 200 data points used in the cal-culation, in comparison with the percentages obtained using 10000 data points. 55

2.7 The values of E1 and E2 for the time series of total value of retail sales in China. 55

2.8 The percentages of false nearest neighbors for the time series of total value of retail sales in China. 56

2.9 The values of E1 and E2 for the time series of gross output value of industry in China. 57

2.10 The percentages of false nearest neighbors for the time series of gross output value of industry in China. 57

XlI MODELLING AND FORECASTING

2.11 The values of El and E2 for the US CPI time series. 58 2.12 The percentages of false nearest neighbors for the US

CPI time series. 58 3.1 Scheme of the different informations of our prediction

problem 63 3.2 Results of mutual information analysis of a I-dimensional

chaotic orbit 80 3.3 Results of mutual information analysis of a 3-dimensional

chaotic orbit 82 3.4 Rise of the information on the future with increasing

embedding-dimension 83 3.5 Daily US dollar exchange rates of five different curren-

cies 84 4.1 Singular values and filter factors for the Ikeda map 104 4.2 OLS and regularised prediction for the Henon map 106 4.3 The first differences of the monthly exchange rates CBP /USD 108 4.4 Prediction of the exchange rate data with OLS, RR and

PCR for a range of nearest neighbours and embedding dimensions 109

4.5 OLS and regularised prediction of the exchange rate data for selected number of nearest neighbours and em-bedding dimensions 110

5.1 Power-law relation in spread-volume of AOL stock. 130 5.2 AOL closing price return rate series. 131 5.3 Moving CARCH fits of AOL return series. 132 5.4 Comparison of local ARCH, CARCH, and loess fits. 133 6.1 Prediction and correction steps of Kalman filtering 142 6.2 Time series observations of a linear system 148 6.3 Kalman filter state estimates of a linear system 149 6.4 Reconstructed state space of Ikeda map 150 6.5 Parameter estimation of Ikeda map 152 6.6 Final predictions of the French currency exchange rate

using a radial basis model reconstructed with the Kalman filter 153

6.7 Predictions of a random walk model of the French cur-rency exchage rate 155

6.8 The predictions and innovations produced by the Kalman filter while estimating the parameters of a radial basis model to predict the French currency exchange rate 156

7.1 Schematic view of a one output RBF. 161 7.2 Format of some radial functions. 162 7.3 Distribution of centers on a regular grid. 163

List of Figures Xlll

7.4 Identifying clusters by k-means algorithm. 164 7.5 The effect of radial functions radius on generalization

and training. 169 7.6 Squared error surface E as a function of the weights. 171 7.7 Daily exchange rate between dollar and pound. 173 7.8 Dollar x Pound: I-step ahead prediction. 174 7.9 Dollar x Pound: 2-steps ahead prediction. 175 7.10 Dollar x Pound: 3-steps ahead prediction. 176 8.1 Prediction results on the time series generated from

chaotic Ikeda map. 186 8.2 Prediction results on the time series generated from

chaotic Ikeda map with additive noise. 189 8.3 Prediction results on the time series generated from

chaotic Ikeda map with a parameter varying randomly over time. 191

8.4 Prediction results on the time series of daily British Pound/US Dollar exchange rate. 193

9.1 The differenced-log time series of the Japanese yen/U.S. dollar exchange rate (the top one) and the money-income (the bottom one). 205

9.2 The differenced-log time series of the ten-year treasury constant maturity rate (the top one) and the three-month commercial paper rate (the bottom one). 207

10.1 Monthly price time series of calves and of finished steers. 224 10.2 Out-of-sample predictions obtained from identified models. 226 10.3 Detrended observed data and 6-month-ahead predictions. 227 10.4 Residuals of original price time series. 229 10.5 Frequency responses of linear and nonlinear models fit-

ted to the period Mar/54-Feb/66. 230 10.6 Frequency responses of linear and nonlinear models fit-

ted to the period Jun/70-May/82. 231 10.7 Static relations between calf and steer prices. 232 11.1 Identification of a dynamic system. 240 11.2 A time-delay recurrent neural network. 241 11.3 Finite unfolding in time. 242 11.4 Concept of overshooting. 244 11.5 Error Correction Neural Network. 248 11.6 Combining Overshooting and ECNN. 249 11.7 Combining Alternating Errors and ECNN. 250 11.8 Dynamics of a pendulum. 251 11.9 Variant-invariant separation of a dynamics. 11.10 Variant - invariant separation by neural networks.

251 252

xiv MODELLING AND FORECASTING

11.11 Combining Variance - Invariance Separation and Forecasting. 253 11.12 State space transformation. 254 11.13 Nonlinear coordinate transformation. 255 11.14 Unfolding in Space and time neural network (phase 1). 256

11.15 Unfolding in Space and time neural network (phase 2). 257 11.16 Unfolding in Space and time neural network using smooth-

ness penalty. 258 11.17 The unfolding of singularities. 259 11.18 Unfolding in Space and Time by Neural Networks. 260 11.19 Realized potential forecasting the German yield curve. 261

12.1 The statistics IqTUpl, IqBDSI and qLAM for the noisy Lorenz data 271

12.2 The statistics from the polynomial fits for the volatility exchange rate data 278

13.1 The two strips of the Rossler attractor. They define two regions whose topological properties are different. 286

13.2 Template of the Rossler attractor. A permutation be-tween the strips is required to meet the standard inser-tion convention. 287

13.3 First-return map to a Poincare section of the Rossler system: (a, b, c) =(0.398,2,4). 287

13.4 The linking number lk(1011, 1) = ~[-4l = -2 counted on a plane projection ofthe orbit couple (1011,1). Cross-ings are signed by inspection on the third coordinate. 288

13.5 Location of the folding in the xy-plane projection of the 3D attractor. The negative peak reveals a negative fold­ing located around e = 0.0 according to our definition ~e. ~9

13.6 Projection in the XY-plane of the attractor generated by the copper electro dissolution. 290

13.7 Template of the copper attractor. 291 13.8 Plane projection of an orbit couple. The linking number

lk(1011,10) is found to be equal to +3. 291 13.9 Model attractor for the copper electrodissolution gen-

erated by integrating the model with the modelling pa-rameters (295,14,52). 292

13.10 Limit cycle generated by the model for the copper elec­trodissolution with the modelling parameters (470,61,51). It is encoded by (100110). 292

13.11 Reconstructed state portrait starting from the experi-mental data. A first-return map exhibits an unusual shape. 293

List of Figures xv

13.12 Phase portrait generated by the autoregressive model. The locations of the foldings are quite similar to those observed on the experimental portrait. 293

13.13 The discrete model (b) is favourably compared to the Henon map (a) although the bifurcation diagrams present some slight departures; from (Aguirre & Mendes, 1996). 295

13.14 Validation by comparing the "bifurcation diagram" ver-sus the amplitude of the input with the diagram associ-ated with the original system. 297

13.15 Time evolution of the error e = X - y for different values of the coupling parameter). between the original Rossler system and the differential model. 298

13.16 Evolution of the minimum value of ). for synchronizing the model with the original Rossler system versus the difference oa on the bifurcation parameter a used for the model. 299

14.1 Examples of time series 305 14.2 Autocorrelation functions of filtered and unfiltered AR

processes 313 14.3 Autocorrelation function of fractional Brownian motion 314 14.4 Distributions of the test variable (logistic map) 316 14.5 Time series produced by the Kuramoto-Sivashinsky-equation 316 14.6 Time series of a standard-deviation normalised AR process 318 14.7 The L).14C-record 319 14.8 Financial time series 321 14.9 Mean standard deviation against window length for the

financial time series 322 15.1 Analysis of the Nerlove-Arrow Model with Time Delay 331 15.2 A Schematic View of Nonparametric Nonlinear Regression 334 15.3 Optimal Transformations for the Nerlove-Arrow Model 337 15.4 Optimal Transformations for an Inappropriate Model 339 15.5 Two-Delay Maximal Correlation 341 15.6 Optimal Transformations for the Two-Delay Model 342 15.7 Gross Domestic Investment and Related Series 344 16.1 The numerical search for the best model is performed

with the help of visual inspection of the model attractor. The modeling parameter (Nv, Np , N k ) are varied. 362

16.2 Comparison between the reconstructed phase portrait and the model attractor. Case of the x-variable of the Rossler system. (Nv, N p , N k ) = (100,10,35). 364

16.3 Comparison between the attractor reconstructed from the z-variable of the Rossler system and the attractor generated by the 4D model. (Nv, N p , N k ) = (150,14,35). 365

XVI MODELLING AND FORECASTING

16.4 Phase portraits of the recontructed Lorenz system and the differents models obtained without and with struc­ture selection.

16.5 The reconstructed attractor and the 3D model attrac­tor obtained from the current time series in the copper

367

electrodissolution experiments (Nv , Np , N k ) = (295,14,52). 368

16.6 XY-plane projections ofthe reconstructed and the model attractors for the Belousov-Zhabotinskii reaction. 369

16.7 Phase portraits of the noisy Duffing system and its model (A = 7.5). 370

17.1 Variable Xl, resp. X2 over time t. Extracts of the artificially created time series of system (17.20), (17.21). 382

17.2 State space xl - x2. Part of the artificially created tra-jectory of system (17.20), (17.21) in phase space. 382

17.3 State space x1-x2: Numerically determined vector field ofthe deterministic parts of system (17.20), (17.21), cal­culated according to the discussed algorithm. The tra­jectories, starting in the inner and outer region of the limit cycle, have been integrated along the vector field. 383

17.4 Variable Xl, resp. X2 over time t. Time series a) and b) are artificially created according to the dynamical sys­tem (17.22), (17.23). Time series c) has been calculated according to relation (17.18), using only the data of time series a). 385

17.5 State space xl - x2. An extract of the artificially cre-ated time series of system (17.22), (17.23) is shown as trajectory in phase space. 386

17.6 State space xl - x2. Vector field of the determinis-tic part of system (17.22), (17.23), presentation like fig. 17.3. 387

17.7 State space xl - x2. For comparison, the exact trajec­tories of system (17.22), (17.23) with the same starting points as in fig. 17.6 and the affiliated vector field are plotted. 388

17.8 Probability densities (pdf) p(q(t), Llt) ofthe price changes Q(Llt, t) = Y(t + Llt) - Y(t) for the time delays Llt = 5120,10240,20480,409608 (from bottom to top). 391

17.9 Contour plot ofthe conditional pdfp(ql, Lltllq2, Llt2) for Lltl = 36008 and Llt2 = 51208, the directly evaluated pdf (solid lines) is compared with the integrated pdf (dot-ted lines). 392

List of Figures XVll

17.10 The coefficient M(l) (q, i).it, i).t2 - i).t l ) as a function of the price increment q for i).tl = 5120s and i).t2 - Atl = 1500s (circles). The data are well reproduced by a linear fit (solid line); after (Friedrich et al., 2000a). 393

17.11 The coefficient M(2) (q, i).t l , i).t2 - i).t l ) presentation as in fig. 17.10. 394

17.12 The coefficient M(l)(q, I, I' -I) as a function of the ve-locity increment q for I = L/2 and I' - I = () (circles); after (Renner et al., 2000). 394

17.13 The coefficient M(2) (q, l, I' - I) as a function of the ve-locity increment q for I = L/2, I' -I = () (circles) and the fitting polynomial of degree two (solid line); after (Renner et al., 2000). 395

17.14 Comparison of the numerical solution of the Fokker-Planck equation (solid lines) for the pdfs p( q( x), l) with the pdfs obtained directly from the experimental data (bold symbols). The scales I are (from top to bottom): I = L, 0.6L, 0.35L, 0.2L and O.IL; after (Renner et al., 2000). 395

18.1 Schematic representation of the noise reduction method 406

18.2 The noise reduction applied to Henon data 408

18.3 Time series of a voice signal 409

18.4 Noise reduction applied to a speech signal 410

18.5 Nonlinear noise reduction applied to physiological data 411

18.6 Time series of a random sawtooth 412

18.7 Comparison of power spectra of a random sawtooth map 412

18.8 Time delay embedding of a $ US to Swiss francs ex-change rate 414

20.1 Diagram showing the relation between original and re-constructed spaces and functions. 431

20.2 The three attractors reconstructed from the dynamical variables of the Rossler system and the estimations of their embedding dimension. 433

20.3 Plane projection ofthe phase portrait reconstructed from the quantity s = y+z. Its embedding dimension is found of be equal to 4. 441

20.4 Plane projection of the nine state portraits induced by the different dynamical variables of the 9D Lorenz system. 443

20.5 The estimation of the embedding dimension is slightly affected by the choice of the observable. Nevertheless, curves suggest that the embedding dimension is equal to 4. 444

XVlll MODELLING AND FORECASTING

20.6 The estimation of the observability of the 9D Lorenz system for parameter values corresponding to a hyper-chaotic behavior. 445

20.7 Phase portrait of the Duffing system driven by a sinu-soidal constraint. A 4D model may then be obtained. (A = 0.05, B = 7.5). 447

20.8 Phase portrait of the Duffing system driven by a Gaus-sian random noise. (A = 0.05, B = 7.5). 448

List of Tables

1.1 Constants in Economic Long Wave Nolinear Model. 34 3.1 Results of the auto mutual information analysis of the

daily Canadian$ /US$ exchange rates returns 85 3.2 Results of the auto mutual information analysis of the

daily dollar exchange rates returns 86 3.3 Results of some cross mutual information analysis of the

daily dollar exchange rates returns 87 5.1 Comparison of GARCH and local ARCH models. 131 9.1 Results on the yen/U.S. dollar exchange rate time series. 206 9.2 Results on the U.S. interest rate time series. 208 14.1 Results of test B 322 16.1 Coefficients of the model obtained from the x-variable

of the Lorenz system. 366 18.1 Performance of the different filter techniques 413 19.1 Selection of Artificial Networks 427 21.1 Results of embedding dimension, number of neighbor-

hoods and RMSE for non-filtered and filtered data, re-spectively, where for SVD and LP methods, q = 10 was used. 465

21.2 Results of embedding dimension, number of neighbor-hoods and RMSE for non-filtered and filtered data, re-spectively, where for SVD and LP methods, q = 5 was used. 465

21.3 Tests of statistical significance of differences between the prediction errors with filtered data and with non-filtered data. 465

Preface

Recent developments in nonlinear sciences and information technology, in particular, developments in nonlinear dynamics and computer technology, have made detailed and quantitative assessments of complex and nonlinear dynamical systems such as economies and markets, which are often volatile and adaptive, possible. These complex systems evolve based on their internal dynamics, however, their evolutions may also be influenced by the external forces acting on the systems.

The development of nonlinear deterministic dynamics, especially the time­delay embedding theorems developed by Takens and later by Sauer et al. that allow to reconstruct dynamics of the underlying systems through only a scalar observed time series, plus rapid development of powerful computers in recent years which have made numerical implementations of techniques of nonlinear dynamics feasible, are instrumental in the studies of complex dynamical systems.

In this volume we have brought together a set of contributions which cover most up-to-dated methods developed recently in nonlinear dynamics, espe­cially in nonlinear deterministic time series analysis. The focus of the whole book is to present recent methodologies in nonlinear time series modelling and prediction. Although we have a large number of contributors to this book, we believe the chapters in the book are integrated and complemen­tary. Each chapter presents a particular method or methods for some typical applications of nonlinear time series modelling and prediction.

Many of the methods discussed in this book have emerged from physics, mathematics and signal processing. Accordingly, we are very honored to have a number of scientists specializing in the areas of nonlinear science as the contributors to this book.

When we invited these experts in nonlinear sciences to contribute to this book we had the following quotation from Alfred Marshall, the famed Cam­bridge economist, who called for contributions of 'trained scientific minds' of Cambridge University to attend to the problems of economics, in mind:

There is wanted wider and more scientific knowledge of facts: an organon stronger and more complete, more able to analyse and help in the solution of the economic problems of the age. To develop and apply the organon rightly

xxu MODELLING AND FORECASTING

is our most urgent need: and this requires all the faculties of a trained scientific mind. Eloquence and erudition have been lavishly spent in the service of Economics. They are good in their way; but what is most wanted now is the power of keeping the head cool and clear in tracing and analysing the combined action of many combined causes. Exceptional genius being left out of account, this power is rarely found save among those who have gone through a severe course of work in the more advanced sciences .... But may I not appeal to some of those who have not the taste or the time for the whole of the Moral Sciences, but who have the trained scientific minds which Economics is so urgently craving? May I not ask them to bring to bear some of their stored up force; to add a knowledge of the economic organon to their general training, and thus to take part in the great work of inquiring how far it is possible to remedy the economic evils of the present day?(Marshall, 1924}

This book could not have been completed without invaluable help and support from all contributors to this volume. We are very grateful to those contributors who reviewed others' contributions to this book in a very pro­fessional and timely manner. Specifically, we would like to mention Luis A. Aguirre, Andre Carlos P. L. F. Carvalho, Rainer Hegger, Christophe Letellier, John Lu, Berndt Pilgram, Bernd Pompe, Henning Voss, and Jose Manuel Zaldivar for reviewing the chapters. We specially thank Drs. Bernd Pompe, Luis Aguirre and Christophe Letellier for their great help: "beyond the call of duty" throughout the development of the book project.

Additionally, we would like to thank Allard Winterink, former acquisition editor of Kluwer Academic Publisher, Carolyn O'Neil, and Deborah Doherty for their assistance in different phases of the project development.

ABDOL S. SooFr AND LIANGYUE CAO

Abdol Sooft dedicates this book to the loving

memory of Rosteen S.Sooft (1975-1994), to

Rima Ellard, and to Shauheen S. Sooft.

Liangyue Cao dedicates this book to his wife, Hong Wu, and to his

son, Daniel.

Contributing Authors

Antonio Aguirre Department of Economics, Federal University of de Minas Gerais, Brazil.

Luis Anotnio Aguirre Department of Electrical Engineering, Federal University of de Minas Gerais, Brazil.

Marcelo Barros de Almeida, Department of Electronics, Federal University of Minas Gerais, Brazil.

Antonio de Padua Braga, Department of Electronics, Federal University of Minas Gerais, Brazil.

Liangyue Cao, Department of Mathematics, University of Western Australia.

Andre Carlos P.L.F. Carvalho, Department of Computing, University of Guelph, Canada.

Rudolph Friedrich, Institute for Theoretical Physics, University of Stuttgart, Germany.

Gerard Gousbet, CORIA UMR 6614, National Institute for Applied Sciences (INSA) of Rouen, France.

Ralph Grothmann, Siemens AG Corporation, Germany.

Rainer Hegger, Institute for Physical and Theoretical Chemistry

J. W. Goethe-University, Germany

XXVl MODELLING AND FORECASTING

Holger Kantz Max Planck Institute for the Physics of Complex Systems, Germany

Dimitris Kugiumtzis, Department of Mathematical and Physical Sciences, Aristotle University of Thessaloniki,

Greece

Jiirgen Kurths, Department of Physics, University of Potsdam, Germany.

Christophe Letellier, Department of Physics, CORIA UMR 6614 University of Rouen, France

Estefane Lacerda, Informatics Department,Federal University of Pernambuco, Brazil.

Zhan-Qian Lu, Statistical Engineering Div, ITL National Institute of Standards and Technology, USA

Teresa Bernarda Ludermir, Department of Electronics, Federal University of Minas Gerais, Brazil.

Jean Maquet, CORIA UMR 6614 National Institute for Applied Sciences (INSA) of Rouen, France

Michael McAleer, Department of Economics, University of Western Australia, Australia.

Alistair Mees, Department of Mathematics and Statistics, University of Western Australia, Australia.

Olivier. Menard Department of Physics, CORIA UMR 6614 University of Rouen, France

Ralph Neuneier, Siemens AG Corporation, Germany.

Joachim Peinke, Department of Physics, University of Oldenburg, Germany.

Contributing Authors xxvii

Berndt Pilgram Department of Mathematics and Statistics, University of Western Australia, Australia.

Bernd Pompe, Ernst-Montz-Arndt- Univesrity Greifswald, Institute of Physics, Germany.

Christoph Renner, Department of Physics, University of Oldenburg,Germany.

Thomas Schreiber, Max Planck Institute for the Physics of Complex Systems, Germany.

Silke Siegert, Institute for Theoretical Physics, University of Stuttgart, Germany.

Abdol S. Soofi, Department of Economics, University of Wisconsin-Platteville, USA.

Fernanda Strozzi Universita Carlo Cattaneo, Engineering Department, Italy.

Peter Verhoeven, School of Economics and Finance, Curtin University of Technology, Australia.

Henning U. Voss, Department of Physics, University of Freiburg, Germany.

David M. Walker, Centre for Applied Dynamics and Optimization, Department of Mathematics and Statistics,

University of Western Australia.

Annette Witt, Department of Physics, University of Potsdam, Germany.

Jianming Ye, Department of Information Technology, City University of New York, USA.

Jose-Manuel Zaldivar European Commission, Joint Research Centre, Environment Institute, Italy.

XXVlll

Hans-Georg Zimmermann, Siemens AG Corporation, Germany.

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