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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.
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Hans-Georg Zimmermann, Siemens AG Corporation, Germany.
MODELLING AND FORECASTING