Contents
Abstract ix
Acknowledgments xi
Author’s contribution xiii
Published papers . . . xiii
Submitted papers . . . xiii
List of Figures xvi List of Tables xviii Nomenclature xix Foreword 1 Introduction 3 1 Theory and methods 7 1.1 Introduction . . . 8
1.2 Inverse approach . . . 8
1.3 Data . . . 9
1.4 Maximum entropy principle . . . 10
1.5 Relation of maxent models to other approaches. . . 12
1.6 Entropy. . . 13
1.7 Variational methods . . . 15
1.8 Most probable state and fluctuations . . . 18
1.9 Testing the order of maxent models . . . 19
1.10 Equilibrium . . . 20
Contents
2.5 Distribution of the pairwise influences . . . 52
2.6 Conclusion . . . 56
3 Market structure explained by pairwise interactions 57 3.1 Introduction . . . 59
3.2 The model . . . 59
3.3 Order-disorder transition . . . 60
3.4 Dynamics of interactions . . . 62
3.5 Link to the graph-theoretic approach . . . 63
3.6 Conclusion . . . 68
4 A statistical perspective on criticality in financial markets 71 4.1 Introduction . . . 73
4.2 Criticality. . . 74
4.3 Why criticality is important . . . 75
4.4 Practical recipe . . . 76
4.5 Signatures of criticality . . . 77
4.6 Sampling indices and stock exchanges . . . 78
4.7 Results . . . 80
4.8 Link to maximum entropy models . . . 87
4.9 Discussion . . . 89
5 Predicting trend reversals using market instantaneous state 93 5.1 Introduction . . . 95
5.2 Collective states . . . 96
5.3 Results . . . 98
5.4 Noise and comparison to artificial networks. . . 103
5.5 Simultaneous trend reversals . . . 105
5.6 Conclusion . . . 105
Appendices 109 5.A Cleaning the data . . . 109
5.B Regularized pseudo-maximum likelihood . . . 109
5.C Confusion matrix . . . 109
5.D Dichotomized Gaussian model . . . 110
6 General conclusion 111 6.1 Introduction . . . 112
6.2 The Brock-Durlauf model . . . 112
6.3 Conclusion . . . 119
6.4 Perspectives . . . 119
Bibliography 121
List of Figures
0.1 Cumulative distribution of the log-returns. . . 3
0.2 Thought-line . . . 5
0.3 Tag cloud. . . 6
1.1 Direct and inverse approaches. . . 8
1.2 Utility function as log-likelihood . . . 9
1.3 Correlations induced by common influences. . . 11
1.4 Markov networks . . . 13
1.5 Entropy of a coin toss . . . 14
1.6 Projection and KLD . . . 16
1.7 Mutual information . . . 19
1.8 Statistical dependencies . . . 20
1.9 Equilibrium approximations . . . 23
1.10 Idealized city . . . 25
1.11 Monte Carlo estimation of the consensus . . . 25
1.12 Asymptotic and equilibrium solutions . . . 26
1.13 Entropy-utility relation . . . 29
1.14 Power-law . . . 30
1.15 Kolmogorov-Smirnov statistics and max-lik estimator . . . 31
1.16 Assets tree . . . 32
1.17 Length of a assets tree through time . . . 32
1.18 Entropy of independent signs . . . 34
1.19 Rate function for a Gaussian sample mean. . . 35
1.20 LDT, LLN and CLT . . . 37
1.21 Laplace approximation . . . 39
2.1 Indices eigen-mode . . . 45
2.2 Multi-information distribution . . . 46
2.3 Multi-information vs the number of stocks . . . 47
List of Figures
3.6 Determinant of the influence matrix . . . 64
3.7 Length of the Dow Jones assets tree. . . 64
3.8 The Dow Jones assets tree . . . 65
3.9 Clusters of the Dow Jones . . . 66
3.10 Clusters of the DAX. . . 66
3.11 Matrix maps . . . 67
3.12 Degree distribution . . . 67
3.13 Clusters of the SP100 . . . 68
3.14 Diagonal influences (large version) . . . 69
3.15 Entropy during crises (large version) . . . 69
4.1 Distributions of the net consensus for different interaction strengths. . . 74
4.2 The variances of the orientation and of the utility as a function ofT . . . 75
4.3 The mean orientation as a function of the scaling parameter . . . 75
4.4 Multi-information of an idealized city . . . 76
4.5 Schematic illustration of the pdf rescaling . . . 78
4.6 Statistical significance of data sets. . . 80
4.7 Statistical significance of the Dow Jones data set . . . 81
4.8 Variance of the log-likelihood . . . 81
4.9 The critical scaling parameter vs correlations . . . 82
4.10 Value of the critical scaling parameter (indices) . . . 82
4.11 Value of the critical scaling parameter (Dow Jones) . . . 82
4.12 Value of the critical scaling parameter (different time-windows) . . . 83
4.13 KLD between the critical and the scaled distributions . . . 84
4.14 Frequencies of correlation coefficients . . . 84
4.15 Testing the power-law hypothesis. . . 85
4.16 Empirical pdf of the MLE estimator . . . 86
4.17 Shannon entropy vs the opposite of the log-likelihood . . . 86
4.18 Linearity of the entropy (simulation) . . . 87
4.19 Evolution of the critical scaling parameter . . . 87
4.20 Order-disorder transition? . . . 89
4.21 The variances of the overlap parameter and of the log-likelihood . . . 90
4.22 Critical value of the scaling parameter (GARCH and MCMC) . . . 90
4.23 The relative size of clusters. . . 92
5.1 Cross-correlogram . . . 97
5.2 Predicted series . . . 98
5.3 ROC curves for European indices . . . 99
5.4 Accuracy . . . 99
5.5 Mean ROC curves for the Dow Jones (daily). . . 100
5.6 Mean ROC curves for the Dow Jones (min) . . . 101
5.7 Accuracy vs system size . . . 101
5.8 Testing the dependence on the testing block length . . . 102
5.9 Accuracy as a function of length of the testing block . . . 102
5.10 Testing the dependence on the distance of the testing block . . . 102
5.11 Accuracy vs the distance between the learning and testing blocks . . . 103
5.12 Accuracy pmf . . . 103
5.13 Schematic representation of noise level estimation in parameters inference . . . 104
5.14 The distributions of simultaneous trend reversals . . . 106
5.15 Comparison of empirical and theoretical PMF of simultaneous reversals . . . 106
5.16 Confusion matrix . . . 110
6.1 BD partition function . . . 115
6.2 Mean consensus pdf for different values ofβ . . . 116
6.3 φ(m)for heterogenous social networks. . . 116
6.4 The evolution of the mean consensus . . . 118
6.5 Thought-line, step∞ . . . 119
List of Tables
1.1 Multi-information criterion . . . 21
1.2 Correspondence with the LDT. . . 33
2.1 Noise quantification in influences estimation . . . 53
4.1 Statistical test of power-law hypothesis . . . 85
5.1 Quantification of the noise in influences inference . . . 104
5.2 Quantification of the reconstruction error . . . 105
5.3 Comparison of artificial accuracy and AUC to real accuracy and AUC . . . 105
5.4 Confusion matrix, a short example . . . 110
