Table Of ContentData and Knowledge in a Changing World
Scientific Editorial Board
E. Fluck (Chairman) (Germany) J.-E. Dubois (France)
A. Kolaskar (India) P. Glaeser (France)
K.W. Reynard (U.K.) F.A. Kuznetsov (Russian Federation)
C.-H. Sun (China) G.H. Wood (USA)
M.A. Chinnery (USA)
Corresponding Members
D. Abir (Israel) P. Masclet (France)
M. Antoninetti (Italy) J.C. Menaut (France)
M. Attimonelli (Italy) B. Messabih (Algeria)
C. Bardinet (France) P. Mezey (Canada)
C. Bernard (France) F. Murtagh (Germany)
H. Bestougeff (France) G. Ostberg (Sweden)
Y. Bienvenu (France) W. Pillman (Austria)
F.A. Bisby (U.K.) J.R. Rodgers (Canada)
L. Blaine (USA) S.F. Rossouw (South Africa)
G. Blake (France) J.J. Royer (France)
J.P. Caliste (France) J. Rumble (USA)
M. Chastrette (France) S.K. Saxema (Sweden)
E.R. Cohen (USA) R. Sinding-Larsen (Norway)
M.-T. Cohen-Adad (France) R.J. Simpson (Australia)
A.-F. Cutting-Decelle (France) N. Swindells (U.K.)
M. Delamar (France) M. Tasumi (Japan)
J.-L. Delcroix (France) B.N. Taylor (USA)
R. Eckermann (Germany) J. Thiemann (USA)
N. Gershon (USA) A. Truyol (France)
E.R. Hilf (Germany) A. Tsugita (Japan)
S. Iwata (Japan) P. Uhlir (USA)
J.G. Kaufman (USA) F. Webster (USA)
H. Kehiaian (France) E.F. Westrum (USA)
A.D. Kozlov (Russia) G. Wood (Canada)
z.
H. Lalieu (Netherlands) Xu (China).
D. Lide (USA)
Springer
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J. o.
Dubois· A. Gvishiani
Dynamic Systems
and Dynamic Classification
Problems in Geophysical
Applications
With 76 Figures and 4 Colour Plates
Springer
CODATA Secretariat
Phyllis Glaeser, Exec. Director
51, Boulevard de Montmorency
F - 75016 Paris
Authors:
Prof. Dr. Jacques Octave Dubois Prof. Dr. Alexei Gvishiani
Institut de Physique Centre of Geophysical Data Studies
du Globe de Paris and Telematics Applications
B89, 4 place Jussieu Molodezhnaya Str., 3
75252 Paris Cedex 05 117 296 Moscow
France Russian Federation
e-mail: [email protected] e-mail: [email protected]
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ISBN 978-3-642-49953-1 ISBN 978-3-642-49951-7 (eBook)
DOl 10.1007/978-3-642-49951-7
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Die Deutsche Bibliothek - Cip-Einheitsaufnahme
Dubois, Jacques 0.:
Dynamic systems and dynamic classification problems in geophysical applications /
J. O. Dubois; A. Gvishiani. ICSU CODATA. -Berlin; Heidelberg; New York;
Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore;
Tokyo: Springer, 1998
(Data and knowledge in a changing world)
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Foreword
This book is the latest volume in the series entitled " Data and Knowledge
in a Changing World ", published by the Committee on Data for Science
and Technology (CODATA) of the International Council of Scientific Unions
(Icsu). This series was established to collect together, from many diverse
fields, the wealth of information pertaining t.o the intelligent exploitation of
data in the conduct of science and technology.
This volume is the first in a two-volume series that will discuss techniques for
the analysis of natural dynamic systems, and their applications to a variety
of geophysical problems. The present volume lays out the theoretical foun
dations for these techniques. The second volume will use these techniques
in applications to fields such as seismology, geodynamics, geoelectricity, ge
omagnetism, aeromagnetics, topography and bathymetry.
The book consists of two parts, which describe two complementary ap
proaches to the analysis of natural systems. The first, written by A. Gvishi
ani, deals with dynamic pattern recognition. It lays out the mathematical
VI Foreword
theory and the formalized algorithms that. forms the basis for the classifi
cation of vector objects and the use of this classification in the study of
dynamical systems, with particular emphasis on the prediction of system
behavior in space and time. It discusses the construction of classification
schemes, and the evaluation of their stability and reliability.
The second, written by J. Dubois, is concerned with various theoretical tools
that may be applied to the modeling of natural systems using large sets of
geophysical data. Dynamic systems are characterized in terms of attrac
tors and repellers, and methods for geometrical classification are described.
These are integrated together with elements of chaos theory into the con
cept of self-organized criticality, and this used to describe the evolution of
dynamic systems.
While most applications are to be discussed in the second volume, a few
illustrations are included in the present book, to demonstrate the need for the
analytical tools. These are drawn from the field of seismic hazard assessment,
and the use of classification schemes in the determination of earthquake
prone areas. These methods have been applied in a number of geographical
areas with considerable success.
The book is expected to be of interest to scientists, engineers, applied mathe
maticians and computer programmers who use artificial intelligence methods
in the analysis of large data sets. The authors are particularly concerned
with geophysical data sets, but the techniques described will have applica
tions in many other fields. It will be of particular interest to those concerned
with the prediction of different kinds of natural processes that develop in
space and time, and with the monitoring and testing of various prediction
algorithms.
Dr. Michael A. Chinnery
CODATA
Boulder, Colorado 80303-3526, U.S.A.
Table of Contents
Acknowledgement ...................................................................................................... XI
Introduction .................................................................................................................. 1
Chapter 1
Why the Techniques Developed in this Book are Important?
(A Few Examples of Applications) .............................................................................. 5
Part I
Foundations
Chapter 2
Basic Mathematical Facts ........................................................................................... 23
2.1 Elements of Set Theory ............................................................................. 23
2.1.1 Sets ............................................................................................................ 23
2.1.2 Functions ................................................................................................... 27
2.2 Elements of Linear Algebra ....................................................................... 29
2.3 Elements of Boolean Algebra .................................................................... 32
2.4 Metric spaces ............................................................................................. 37
Part II
Dynamic Pattern Recognition Problems and Control
over Classification Reliabity
Chapter 3
"Voting by a Set of Features" Algorithms .................................................................. 43
3.1 Basic Definitions ....................................................................................... 43
3.2 "Voting by Elementary Features" Algorithms ........................................... 45
3.3 Definition of Classes of Algorithms "Voting by a Set of Features"
(V SF) and "Voting by Elementary Features"(VEF). Normal Weights ...... 49
3.4 Class of Algorithms "CORA-i" ................................................................. 52
3.5 "Neighbours" as a VSF Algorithm ............................................................ 54
3.6 "Threshold" Classification Algorithms ..................................................... 56
3.7 "Bayes" as a VEF Algorithm ..................................................................... 58
VIII Table ofC ontents
Chapter 4
Dynamic and Limit Classification Problems .............................................................. 61
4.1 Formulation of the Dynamic Recognition Problem ................................... 61
4.2 Stability of Prediction in Dynamic Classification Problems ..................... 65
4.3 Stability of Prediction, Algorithm VEF-O .................................................. 66
4.4 Stability Conditions in the Case of the Algorithm VEF-l ......................... 69
4.5 Stability Zone for Prediction Obtained by the Algorithm VEF-2 .............. 73
4.6 Stability Zone for Prediction Obtained by a VSF Algorithm .................... 77
4.7 Stability of Prediction in Case of Normal Weights ................................... 80
Chapter 5
Dual Systems of Sets and Local Stability of Classification ....................................... 85
5.1 Dual Systems of Sets ................................................................................. 85
5.2 S-Theorems and S-Counter-examples ....................................................... 87
5.3 Local Stability and Stabilizing Sets ........................................................... 89
5.4 Local Stability and Cluster Analysis ......................................................... 93
Chapter 6
Investigation of Earthquake-prone Areas as a Limit Pattern
Recognition Problem .................................................................................................. 97
6.1 Formalizing the Recognition of Strongly
Eearthquake-prone Areas .......................................................................... 97
6.2 Correspondance Between Prediction and Classification
Problems .................................................................................................. 101
Chapter 7
Control Experiments for Evaluating Classification Reliability ................................ 105
7.1 Comparison of Quality Solutions for Real and Random
Learning Materials ................................................................................... 105
7.2 Estimating the Probability of the Classification ErrOL ............................ 110
7.3 Estimation of Parameters of the Classification Problem
using Estimation of Non-randomness and Reliability
Functions ................................................................................................. 113
7.4 Additional Arguments for Evaluating the Classification
Reliability ................................................................................................ 116
Table a/Contents IX
Part III
Dynamic Systems
Chapter 8
Basic Definitions and Facts ...................................................................................... 123
8.1 Measure and Dimension .......................................................................... 123
8.1.1 Hausdorff measure...................................................... ............................. 124
8.1.2 Hausdorff dimension ............................................................................... 125
8.2 Discrete Dynamic Systems ...................................................................... 126
8.3 Continuous Dynamic Systems ................................................................. 127
8.4 Representation and Study of Dynamic Systems ...................................... 128
8.4.1 Iterative scheme as a tool in dynamic system study ................................ 129
8.4.2 Bifurcation ............................................................................................... 140
8.5 Attraction and Repulsion Cycles ............................................................. 146
8.5.1 Areas contraction and consequences ....................................................... 147
8.6 Fractal Attractors, Basin of Attraction, Repellers .................................... 148
8.7 Lyapunov Exponents ............................................................................... 150
Chapter 9
Geometry of Attractors ............................................................................................. 155
9.1 Classical Examples of Strange Attractors ................................................ 155
9. 1.1 The Lorenz attractor ................................................................................ 155
9.1.2 The Henon attractor ................................................................................. 158
9.1.3 Some other classical attractors ................................................................. 158
9.2 Correlation Function Method .................................................................. 161
9.3 What Time Delay? ................................................................................... 164
9.3.1 The mutual information criterion ............................................................ 165
9.3.2 The minimum embedding dimension ...................................................... 165
9.3.3 The Lyapunov exponents computation (Sano and Sawada Method) ...... 167
9.3.4 Calculation of Lyapunov exponents (the method of Wolf et al.) ............ 170
9.4 First Return Mapping ............................................................................... 172
Chapter 10
Bifurcation, Cascades and Chaos ............................................................................. 173
10.1 Floquet Matrix, Hopf Bifurcation ............................................................ 173
10.1.1 Floquet matrix ......................................................................................... 174
10.1.2 Hopfbifurcation ...................................................................................... 176
10.1.3 Passing from torus? to torus 13. Ruelle-Takens theory ........................ 178
10.2 Curry and Yorke Model and Route to Chaos .......................................... 180
X Table a/Contents
10.3 Defmition of Chaos ................................................................................. 183
10.4 Control of Chaos ...................................................................................... 183
Chapter 11
Self Organisation ...................................................................................................... 189
11.1 The Evolving Sandpile Experiment ......................................................... 189
11.2 General Aspects of Self-Organized Criticality ........................................ 193
Chapter 12
Multifractals ............................................................................................................. 195
12.1 Generalized Dimensions of Fractals and Strange Attractors ................... 195
12.2 Defmitions and Properties ....................................................................... 198
12.2.1 A Simple Definition ................................................................................. 198
12.2.2 A Measure on the Middle Third Cantor Set ............................................ 199
12.2.3 Practical Methods of Constructing a Multifractal Spectrum ................... 199
Part IV
Convex Programming and Systems of Rigid Blocks
with Deformable Layers
Chapter 13
Systems of Rigid Blocks with Thin Deformable Layers (SRBTDL) ....................... 209
Chapter 14
System of Rigid and Deformable Blocks (SRDB) ................................................... 215
Part V
Bibliography ............................................................................................................ 225
Part VI
Index ........................................................................................................................ 243
Colour Plates ........................................................................................................... 251
