Exploratory Data Analysis Using Fisher Information

✍ Scribed by Roy Frieden, Robert A. Gatenby

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 375
• Edition: 1st Edition.
• Category: Library

✦ Synopsis

B.R. Frieden and R.A. Gatenby (Eds), Exploratory Data Analysis using Fisher Information (Springer, London 2007)
For some years now, Roy Frieden has been exploring the consequences of studying physical phenomena on the basis of Fisher information and extreme physical information (EPI). From the very beginning, the results were spectacular. From the slenderest beginnings, many of the fundamental equations of physics emerged from these EPI principles: the Klein-Gordon and Dirac equations of quantum mechanics as well as the Schrödinger equation; Newton's second law; Maxwell's equations; many of the equations of general relativity; and this does not exhaust the list. These ideas, gradually developed in a series of publications in very respectable and severely refereed scientific journals, were brought together in Physics from Fisher Information (1998) and its successor, Science from Fisher Information (2004).
It was clear from that work that the approach should not be limited to physics but the extent to which it has shown itself fruitful, charted in Frieden's latest book, is a revelation. This is not a monograph but a collection of essays, edited by Frieden and R.A. Gatenby, a life scientist, on a very wide range of topics, all of which are shown to benefit from the use of EPI. The book begins with an introduction by Frieden, in which the reader is told what Fisher information is and how to use it, employing the EPI approach. Eight chapters follow, contributed by the editors and 11 other authors, on financial economics (Frieden, R.J. Hawkins and and J.L. d'Anna); tissue growth and cancer (by the editors); statistical mechanics and thermal physics' - not very different from what I was taught to call thermodynamics (A. and A.R. Plastino); astrobiology (by Frieden and B.H. Soffer), which is described as a unification of biology and astrophysics; encryption (R.C. Venkatesan); the management of sustainable environmental systems (A.L. Mayer, C.W. Pawlowski, B.D. Fath and H. Cabezas); ecology (by the editors); and to conclude,Sociohistory: an information theory of social change' (M.I. Yolles and Frieden).
This makes for a very adventurous book, all of which makes fascinating reading though some chapters are more readable than others and occasionally, the authors seem unnecessarily on the defensive, as though they expect readers to have a red pencil at the ready. The list of chapters already gives a good idea of the diversity of the contents and even within individual chapters, the coverage is often surprising; thus Chapter 7 (Environmental systems) ends with a section on `Sociopolitical data', in which "state failure", the risk of a "catatastrophic collapse of a nation's governing body" is examined and illustrated with a histogram showing the stability of five countries, Sweden, France, Argentina, Sierra Leone and Haiti. The Fisher index based on eight criteria is very high (indicating great stability) for Sweden, low for Argentina, Sierra Leone and Haiti and only marginally better for France (in the years between 1961 and 1995)! The concluding chapter (Sociohistory) is the most difficult for readers from the exact sciences, unaccustomed to Kant's notion of the noumenon, the Hegelian doctrine of the dialectic and the autonomous holon, though the authors have tried hard to render the vocabulary of the sociologist palatable.
The very different nature of the topics examined makes it less easy to appreciate the remarkable role of EPI than in the earlier books, addressed to physicists in language with which they were familiar, however revolutionary the theory presented. I imagine that readers of this latest offering will peruse only the chapter that deals with their own particular interest. I therefore wish to emphasize that the truly original feature of this book - apart from EPI itself - is precisely its broad coverage; its demonstration that such a simple principle, easily grasped, is capable of yielding valuable results in such a wide range of fields of enquiry. I found Frieden's earlier books immensely original and intellectually thrilling and this one adds yet more weight to that opinion.
P.W. Hawkes
(M.A., Ph.D., Sc.D., Cambridge; emeritus Director of Research, CNRS)

✦ Table of Contents

Cover......Page 1
Copyright, Contents......Page 2
Title Page
......Page 3
Copyright Page
......Page 4
Table of Contents
......Page 5
Contributor Details......Page 7
1 Introduction to Fisher Information: Its Origin, Uses, and Predictions......Page 14
1.1.1. Partial Derivatives......Page 16
1.1.2.2. Solution......Page 17
1.1.2.4. Building in Constraints......Page 18
1.1.3. Dirac Delta Function......Page 19
1.1.4. Unitary Transformation......Page 20
1.2.1.1. Comparisonwith ShannonInformation......Page 21
1.2.1.3. UnbiasedEstimators......Page 22
1.2.1.4. Use of Schwarz Inequality......Page 23
1.2.1.6. Fisher Coordinates......Page 24
1.2.1.8. Examples of Tests for Efficiency......Page 25
1.2.2.1. Multiple Parameters and Data......Page 26
1.2.2.2. Shift-InvariantCases......Page 27
1.2.2.4. No Shift-Invariance, Discrete Data......Page 28
1.2.2.6. Tensor Form of I......Page 29
1.2.2.8. Complex Amplitude 1/J-Form of I......Page 30
1.2.2.10. I in Curved Space......Page 31
1.2.2.11. I for Gluon Four-Position Measurement......Page 32
1.3.1. EPI Zero Principle......Page 34
1.3.3. Fisher I -Theorem......Page 35
1.4. Extreme Physical Information (EPI)......Page 36
1.4.1. Relation to Anthropic Principle......Page 37
1.4.3. Data Information is Generic......Page 38
1.4.4. Underpinnings: A "Participatory" Universe......Page 39
1.4.5. A "Cooperative" Universe and Its Implications......Page 40
1.4.7. Drawbacks of Classical "Action Principle"......Page 44
1.5.2. Exhaustivity Property ofEPI......Page 45
1.5.3. Inexact, Classical Scenarios : Type (B) Deduction......Page 46
1.6. Information Game......Page 47
1.6.2. Saddlepoint Property......Page 48
1.6.5. Peirce Graphs......Page 49
1.6.6. Game Corollary......Page 50
1.7. Predictions of the EPI Approach......Page 51
2.1.1. Summary......Page 55
2.1.3. Variational Approaches to the Determination of PriceValuation Fluctuation......Page 56
2.1.4. Trade as a Measurement in EPI Process......Page 57
2.1.5. Intrinsic vs Actual Data Values......Page 58
2.1.6. Incorporating Data Values into EPI......Page 59
2.1.7. Information J and the "Technical" Approach to Valuation......Page 60
2.1.9. SWE Solutions......Page 61
2.2.2. Background......Page 63
2.2.3. PDFin the Term Structure ofInterest Rates, and Yield Curve Construction......Page 64
2.2.4. PDF for a Perpetual Annuity......Page 68
2.2.5. Yield Curve Dynamics......Page 70
2.2.6. Relation to Nelson Siegel Approach and Dynamical Fokker-Planck Solution......Page 73
2.2.7. A Measure of Volatility......Page 74
2.2.9. Aoki Theory......Page 76
2.3.1. Summary......Page 77
2.3.2. Background......Page 78
2.3.3. Information I......Page 79
2.3.5. Phase Space......Page 80
2.3.6. Optimized Information and q-Theory......Page 82
2.3.7. Other Optimized Strategies......Page 83
2.3.8. Investment Parameters......Page 84
2.3.10. Market Efficiency......Page 85
3.1.1. Summary......Page 87
3.1.2. Introduction......Page 88
3.1.3. Ideal Requirements of Biological Information......Page 89
3.1.5. Kullback-Leibler Information......Page 91
3.1.6. Principle ofExtreme K-L Information......Page 92
3.1.9. Biological Interplay ofSystem and Reference Probabilities......Page 93
3.1.11. Shannon Information Types......Page 94
3.1.13. Some Problems with Biological Uses of Shannon Information......Page 96
3.1.14.1. Bioinformatics and Network Analysis......Page 97
3.1.14.2. Hub Dynamics......Page 99
3.1.15. Information and Cellular Fitness......Page 100
3.1.16. Cellular Information Utilization......Page 101
3.1.17. Potential Controversies-Is All Cellular Information Stored in the Genome and Transmitted by Proteins?......Page 102
3.1.18. Multicellular Information Dynamics......Page 104
3.1.19. Information and Disease......Page 105
3.2.1. Summary......Page 107
3.2.2. Introduction......Page 108
3.2.3. Bound and Free Intracellular Information......Page 109
3.2.4. Information Dynamics Before and After Reproduction,With and Without Mutation......Page 110
3.2.5. Limits ofInformation Degradation in Carcinogenesis......Page 112
3.2.5 .2. Angiogenesis......Page 113
3.2.6. The Evolving Microenvironment and Resulting Mutation Rate......Page 114
3.2.7. Application of EPI to Tumor Growth......Page 115
3.2.7.2. Brief Review of EPI......Page 116
3.2.8. Fisher Variable, Measurements......Page 117
3.2.9.2. Self-Consistency Approach, General Considerations......Page 118
3.2.11. Determining the Power by Minimizing the Information......Page 119
3.2.12. Experimental Verification......Page 120
3.2.13.1. Efficiency K......Page 121
3.2.13.2. Fibonacci Constant......Page 122
3.2.13.3 . Uncertainty in Onset Time......Page 123
3.2.14. Alternative Growth Model Using Monte Carlo Techniques......Page 124
3.2.15. Conclusions......Page 125
3.3.1. General Solution......Page 128
3.3.3. Error in the Estimated Duration of the Cancer......Page 130
4 Information and Thermal Physics......Page 132
4.1.2. Introduction......Page 133
4.1.3. A Scientific Theory's Structure......Page 134
4.1.4. Fisher-Related Activity in Contemporary Physics......Page 135
4.2.1. Summary......Page 136
4.2.2. Classical Statistical Mechanics a la Fisher......Page 137
4.2.4. General Quadratic Hamiltonians......Page 140
4.2.5. Free Particle......Page 141
4.2.7. A Nonlinear Problem: Paramagnetic System......Page 142
4.2.8. Conclusions......Page 143
4.3.3.1. Macroscopic Thermodynamics......Page 144
4.3 .3.2. Legendre Structure......Page 145
4.3.5. Axioms ofInformation Theory......Page 146
4.4.2. Jaynes' Reformulation......Page 147
4.4.3. Legendre Structure in Jaynes' Formulation......Page 148
4.4.5. Legendre Structure Preserved by a ChangeofMeasure Is --> IT......Page 149
4.4.6. Still More General Measures......Page 150
4.5.1. Summary......Page 151
4.5.3. Minimizing FIM Leads to a Schrodinger-Like Equation......Page 152
4.5.5. Shannon's S vs Fisher's I......Page 154
4.5.6. Discussion......Page 155
4.6.2. Steps ofthe Approach......Page 157
4.7.2. Establishing the Connection......Page 158
4.7.3.1. Boltzmann Equation in the Relaxation Approximation......Page 159
4.7.3.2. Generalities on Viscosity......Page 160
4.7.4. Comparison with the Grad Treatment......Page 164
4.7.5.1. Ground State......Page 165
4.7.5 .2. Admixture of Excited State s......Page 166
5.1.1. Summary......Page 168
5.1.2. Introduction......Page 169
5.1.3.1. Unitless Physical Constants......Page 171
5.1.3.2. Dirac Hypothesis for the Cosmological Constants......Page 172
5.1.5. Biological Attributes......Page 173
5.1.6. Corresponding Attributes ofBiology and Cosmology......Page 176
5.1.7. Discussion......Page 180
5.1.8. Concluding Remarks......Page 184
5.2.2. Introduction......Page 185
5.2.4. Schrodinger Equation......Page 186
5.2.5. Force-Free Medium......Page 187
5.2.6. Case ofa Complex Potential......Page 188
5.2.8. Growth Equation......Page 189
5.2.9. How Could Such a System Be Realized?......Page 190
5.2.10. Current Evidence for Nanolife......Page 191
6.1.1. Fisher Information and Extreme Physical Information......Page 194
6.1.2. Fisher Game......Page 195
6.1.4. Time-Independent Schrodinger Equation......Page 196
6.1.5. Real Probability Amplitudes and Lagrangians......Page 197
6.1.8. Solutions of the TISE and TISLE......Page 198
6.1.9. Information Theory and Securing Covert Information......Page 199
6.1.10. Rationale for Encrypting Covert Information in a Statistical Distribution......Page 201
6.1.12. Ill-Conditioned Nature of Encryption Problem......Page 202
6.1.14. Objectives to be Accomplished......Page 203
6.2.1. Operators and the Dirae Notation......Page 204
6.2.2. Eigenstructure of the Constraint Operator......Page 205
6.2.4. Generic Strategy for Encryption and Decryption......Page 206
6.3. Inference of the Host Distribution Using Fisher Information......Page 207
6.3.2. Amplitudes and Pseudo-Potentials Satisfying MFI and MaxEnt......Page 208
6.3.3. Modified Game Corollary......Page 209
6.3.4. Fisher Game vs MaxEnt......Page 212
6.4. Implementation of Encryption and Decryption Procedures......Page 216
6.4.1. Encryption Process......Page 219
6.4.2. Decryption......Page 223
6.4.3. Summary of the Encryption-Decryption Strategy......Page 224
6.4.4. Security Against Malicious Attacks......Page 225
6.5. Extension of the Encryption Strategy......Page 227
6.6. Summary of Concepts......Page 228
7.1. Summary......Page 230
7.2. Introduction......Page 231
7.3.1. Definition......Page 232
7.3.2. Shift-Invariant Cases......Page 233
7.3.3. Phase States s of Dynamic Systems......Page 234
7.4. Probability Law on State Variable s......Page 235
7.5. Evaluating the Information......Page 237
7.6. Dynamic Order......Page 238
7.7. Dynamic Regimes and Fisher Information......Page 239
7.8. Evaluation of Fisher Information......Page 240
7.9.1. Two-Species Model System......Page 242
7.9.2. Multispecies Model: Species and Trophic Levels......Page 246
7.9.3. Ecosystems with Pseudo-Economies: Agriculture and Industry......Page 248
7.10. Applications to Real Systems......Page 250
7.10.1. North Pacific Ocean......Page 251
7.10.2. Global Climate......Page 253
7.10.3. Sociopolitical Data......Page 254
7.11. Summary......Page 256
8.1.1. Summary......Page 258
8.1.3. A Biological Uncertainty Principle......Page 259
8.1.4. Ramification of the Uncertainty Principle......Page 260
8.1.5. Necessary Condition for Cataclysm......Page 261
8.1.6. Scenarios of Cataclysm from Fossil Record......Page 262
8.1.8. Resulting Changes in Population Occurrence Rates......Page 263
8.1.9. Getting the Information......Page 264
8.1.11. Final Decision Rule, Conditions of Use......Page 265
8.1.12. Ideally Breeding Rabbits......Page 267
8.1.13. Homo Sapiens......Page 268
8.2.2. Background......Page 269
8.2.3. Cramer-Rao Inequality and Efficient Estimation......Page 271
8.2.5. Objectives......Page 272
8.2.6. How Can the Efficiency Condition be Satisfied?......Page 273
8.2.7. Power-Law Solution......Page 274
8.2.9. Unbiasedness Condition......Page 275
8.2.10. Asymptotic Power b = 1+E, with E Small......Page 276
8.2.11. Discussion......Page 278
8.2.12. Experimental Evidence for a l/x Law......Page 279
8.3.1. Summary......Page 280
8.3.1.2. Biological Allometric Laws......Page 281
8.3.1.3. On Models for Biological Allometry......Page 282
8.3.2. PriorKnowledge Assumed......Page 283
8.3.3.1. Measurement, System Function......Page 284
8.3.4. Data Information I......Page 285
8.3.5.2. Fourier Analysis......Page 286
8.3.7. Synopsis of the Approach......Page 288
8.3.8. Primary Variation ofthe System Function Leads to a Family of Power Laws......Page 289
8.3.9. Variation of the Attribute Parameters Gives Powers a - an = n/4......Page 290
8.3.10. Secondary Extremization Through Choice of h(x)......Page 291
8.3.10.2. Resulting variational principle in Base Function h(x)......Page 292
8.3.10.4 . Result k(x ) = 0, Giving Base Function hex ) Proportional to x......Page 293
8.3.11. Final Allometric Laws......Page 294
8.3.13. Discussion......Page 295
9.1. Summary......Page 298
9.1.1. Philosophical Background......Page 299
9.1.2. Boundary Considerations......Page 300
9.1.3. Sociohistory: Historical Aspects......Page 301
9.1.5. Extreme Phenomenal Information......Page 302
9.1.7. fin-Yang Nature of Sociohistory......Page 303
9.1.9. Hegelian Doctrine of the Dialectic......Page 304
9.2. Social Cybernetics......Page 305
9.2.1. SVS Theory......Page 306
9.2.3. Global Noumenon......Page 308
9.3.1. The Ontological Basis for SVS Theory......Page 309
9.3.3. System Informations I, J......Page 314
9.3.4. Information Channel......Page 315
9.3.5. Information I......Page 316
9.3.7. EPI Zero Condition......Page 317
9.3.9. EPI Extremum Principle......Page 318
9.3.10. Knowledge Game......Page 319
9.4.1. Cultural Driving Forces......Page 320
9.4.2. Sensate and Ideational Aspects......Page 321
9.5.2. Dispersed Agents......Page 322
9.5.3. Ideational vs Sensate Dispersed Agents......Page 323
9.5.4. The Dynamics of Viable Holons......Page 324
9.5.5. Emergent States and EPI......Page 327
9.5.6. Sensate and Ideational Aspects ofthe Informations......Page 329
9.5.7. Role of Efficiency Constant K......Page 330
9.5.8. Coefficients ofInformation......Page 331
9.5.9. Role ofK in Defining States ofSociety......Page 332
9.5.10. Emerging Balance Between Cultural Dispositions......Page 333
9.6.1. Time Evolution of the Informations......Page 334
9.6.3. A Quantification Using Information Parameter K......Page 336
9.8. An Illustrative Application of EPI to Sociocultural Dynamics......Page 338
9.8.1. General Problem of Population Growth and Motion......Page 339
9.8.2. Population Growth and Depletion Coefficients......Page 340
9.8.3. EPI Solution......Page 341
9.9.1. Sensate vs Ideational Mindsets......Page 343
9.9.2. Cultural Instability......Page 344
9.9.3. Quantitative Growth Effects......Page 345
9.9.4. Manifestations in Political Power and Dominance......Page 346
9.10. Overview......Page 347
References......Page 349
Index......Page 369

✦ Subjects

Физика;Матметоды и моделирование в физике;