arXiv subjects

D. Sornette

Publications and source records attributed to D. Sornette.

At least 109 records · Page 6Linked to original sources

arXiv2007

Multifractal Omori Law for Earthquake Triggering: New Tests on the California, Japan and Worldwide Catalogs

The Multifractal Stress-Activated (MSA) model is a statistical model of triggered seismicity based on mechanical and thermodynamic principles. It predicts that, above a triggering magnitude cut-off $M_0$, the exponent $p$ of the Omori law for the seismic decay of aftershocks is a linear increasing function $p(M) =a M+b$ of the main shock magnitude $M$. We previously reported empirical support for this prediction, using the Southern California SCEC catalog. Here, we confirm this law using an updated, longer version of the same catalog, as well as new methods to estimate $p$. One of this methods is the newly defined Scaling Function Analysis, adapted from the wavelet transform. This method is able to measure a singularity ($p$-value), erasing the possible regular part of a time series. The Scaling Function Analysis also proves particularly efficient to reveal the coexistence of several types of relaxation laws (typical Omori sequences and short-lived swarms sequences) which can be mixed within the same catalog. The same methods are used on data from the worlwide Harvard CMT and show results compatible with those of Southern California. For the Japanese JMA catalog, we still observe a linear dependence of $p$ on $M$, yet with a smaller slope. The scaling function analysis shows however that results for this catalog may be biased by numerous swarm sequences, despite our efforts to remove them before the analysis.

physics.geo-ph↗

arXiv2007

Numerical investigations of discrete scale invariance in fractals and multifractal measures

Fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems. Here, we focus on lattice multifractals which exhibit complex exponents associated with observable log-periodicity. We perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments. A novel numerical approach is proposed to extract the log-frequencies. In the non-lattice case, there is no visible log-periodicity, {\em{i.e.}}, no preferred scaling ratio since the set of complex exponents spread irregularly within the complex plane. A non-lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non-lattice one. An algorithm for the construction of the lattice sequence is proposed explicitly.

cond-mat.stat-mech↗

arXiv2007

"Illusion of control" in Minority and Parrondo Games

Human beings like to believe they are in control of their destiny. This ubiquitous trait seems to increase motivation and persistence, and is probably evolutionarily adaptive. But how good really is our ability to control? How successful is our track record in these areas? There is little understanding of when and under what circumstances we may over-estimate or even lose our ability to control and optimize outcomes, especially when they are the result of aggregations of individual optimization processes. Here, we demonstrate analytically using the theory of Markov Chains and by numerical simulations in two classes of games, the Minority game and the Parrondo Games, that agents who optimize their strategy based on past information actually perform worse than non-optimizing agents. In other words, low-entropy (more informative) strategies under-perform high-entropy (or random) strategies. This provides a precise definition of the "illusion of control" in set-ups a priori defined to emphasize the importance of optimization.

physics.soc-ph↗

arXiv2007

Properties of a simple bilinear stochastic model: estimation and predictability

We analyze the properties of arguably the simplest bilinear stochastic multiplicative process, proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise. By construction, it has no linear predictability (zero two-point correlation) but a certain nonlinear predictability (non-zero three-point correlation). It can thus be considered as a paradigm for testing the existence of a possible nonlinear predictbility in a given time series. We present a rather exhaustive study of the process, including its ability to produce fat-tailed distribution from Gaussian innovations, the unstable characteristics of the inversion of the key nonlinear parameters and of the two initial conditions necessary for the implementation of a prediction scheme and an analysis of the associated super-exponential sensitivity of the inversion of the innovations in the presence of a large impluse. Our study emphasizes the conditions under which a degree of predictability can be achieved and describes a number of different attempts, which overall illuminates the properties of the process. In conclusion, notwithstanding its remarkable simplicity, the bilinear stochastic process exhibits remarkably rich and complex behavior, which makes it a serious candidate for the modeling of financial times series and of other complex systems.

physics.data-an↗

arXiv2007

Illusion of Control in a Brownian Game

Both single-player Parrondo games (SPPG) and multi-player Parrondo games (MPPG) display the Parrondo Effect (PE) wherein two or more individually fair (or Llosing) games yield a net winning outcome if alternated periodically or randomly. (There is a more formal, less restrictive definition of the PE.) We illustrate that, when subject to an elementary optimization rule, the PG displays degraded rather than enhanced returns. Optimization provides only the illusion of control, when low-entropy strategies (i.e. which use more information) under-perform random strategies (with maximal entropy). This illusion is unfortuntately widespread in many human attempts to manage or predict complex systems. For the PG, the illusion is especially striking in that the optimization rule reverses an already paradoxical-seeming positive gain - the Parrondo effect proper - and turns it negative. While this phenomenon has been previously demonstrated using somewhat artificial conditions in the MPPG (L. Dinios and J.M.R. Parrondo. Europhysics Letters 63, 319 (2003); J.M.R. Parrondo et al. Advances in Condensed Matter and Statistical Mechanics, eds. E. Korutcheva and R. Cuerno, Nova Science Publishers, 2003), we demonstrate it in the natural setting of a history-dependent SPPG.

physics.soc-ph↗

arXiv2007

New Approach to the Characterization of Mmax and of the Tail of the Distribution of Earthquake Magnitudes

We develop a new method for the statistical esitmation of the tail of the distribution of earthquake sizes recorded in the Worldwide Harvard catalog of seismic moments converted to mW-magnitudes (1977-2004 and 1977-2006). We show that using the set of maximum magnitudes (the set of T-maxima) in windows of duration T days provides a significant improvement over existing methods, in particular (i) by minimizing the negative impact of time-clustering of foreshock / main shock /aftershock sequences in the estimation of the tail of the magnitude distribution, and (ii) by providing via a simulation method reliable estimates of the biases in the Moment estimation procedure (which turns out to be more efficient than the Maximum Likelihood estimation). Using a simulation method, we have determined the optimal window size of the T-maxima to be T=500 days. We have estimated the following quantiles of the distribution of T-maxima of earthquake magnitudes for the whole period 1977-2006: Q_{0.16}(Mmax)=9.3, Q_{0.5}(Mmax)=9.7 and Q_{0.84}(Mmax)=10.3. Finally, we suggest two more stable statistical characterristics of the tail of the distribution of earthquake magnitudes: the quantile QT(q) of a high probability level q for the T-maxima, and the probability of exceedence for a high threshold magnitude. We obtained the following sample estimates for the global Harvard catalog: QT(q=0.98)=8.6 +- 0.2 and a probability for the T-maxima to exceed magnitude 8 equal to 0.13-0.20. The comparison between our estimates for the two periods 1977-2004 and 1977-2006, where the later period includes the great Sumatra earthquake, 24.12.2004, mW=9.0, confirms the instability of the estimation of the parameter Mmax and the stability of the two other estimates.

physics.geo-ph↗

arXiv2007

A two-Factor Asset Pricing Model and the Fat Tail Distribution of Firm Sizes

In the standard equilibrium and/or arbitrage pricing framework, the value of any asset is uniquely specified from the belief that only the systematic risks need to be remunerated by the market. Here, we show that, even for arbitrary large economies when the distribution of the capitalization of firms is sufficiently heavy-tailed as is the case of real economies, there may exist a new source of significant systematic risk, which has been totally neglected up to now but must be priced by the market. This new source of risk can readily explain several asset pricing anomalies on the sole basis of the internal-consistency of the market model. For this, we derive a theoretical two-factor model for asset pricing which has empirically a similar explanatory power as the Fama-French three-factor model. In addition to the usual market risk, our model accounts for a diversification risk, proxied by the equally-weighted portfolio, and which results from an ``internal consistency factor'' appearing for arbitrary large economies, as a consequence of the concentration of the market portfolio when the distribution of the capitalization of firms is sufficiently heavy-tailed as in real economies. Our model rationalizes the superior performance of the Fama and French three-factor model in explaining the cross section of stock returns: the size factor constitutes an alternative proxy of the diversification factor while the book-to-market effect is related to the increasing sensitivity of value stocks to this factor.

physics.soc-ph↗

arXiv2007

Nonlinear Dynamical Model of Regime Switching Between Conventions and Business Cycles

We introduce and study a non-equilibrium continuous-time dynamical model of the price of a single asset traded by a population of heterogeneous interacting agents in the presence of uncertainty and regulatory constraints. The model takes into account (i) the price formation delay between decision and investment by the second-order nature of the dynamical equations, (ii) the linear and nonlinear mean-reversal or their contrarian in the form of speculative price trading, (iii) market friction, (iv) uncertainty in the fundamental value which controls the amplitude of mispricing, (v) nonlinear speculative momentum effects and (vi) market regulations that may limit large mispricing drifts. We find markets with coexisting equilibrium, conventions and business cycles, which depend on (a) the relative strength of value-investing versus momentum-investing, (b) the level of uncertainty on the fundamental value and (c) the degree of market regulation. The stochastic dynamics is characterized by nonlinear geometric random walk-like processes with spontaneous regime shifts between different conventions or business cycles. This model provides a natural dynamical framework to model regime shifts between different market phases that may result from the interplay between the effects (i-vi).

nlin.AO↗

arXiv2006

Algorithm for Model Validation: Theory and Applications

Validation is often defined as the process of determining the degree to which a model is an accurate representation of the real world from the perspective of its intended uses. Validation is crucial as industries and governments depend increasingly on predictions by computer models to justify their decisions. We propose to formulate the validation of a given model as an iterative construction process that mimics the often implicit process occurring in the minds of scientists. We offer a formal representation of the progressive build-up of trust in the model. We thus replace static claims on the impossibility of validating a given model by a dynamic process of constructive approximation. This approach is better adapted to the fuzzy, coarse-grained nature of validation. Our procedure factors in the degree of redundancy versus novelty of the experiments used for validation as well as the degree to which the model predicts the observations. We illustrate the new methodology first with the maturation of Quantum Mechanics as the arguably best established physics theory and then with several concrete examples drawn from some of our primary scientific interests: a cellular automaton model for earthquakes, a multifractal random walk model for financial time series, an anomalous diffusion model for solar radiation transport in the cloudy atmosphere, and a computational fluid dynamics code for the Richtmyer-Meshkov instability.

physics.data-an↗

arXiv2006

Self-Consistent Asset Pricing Models

We discuss the foundations of factor or regression models in the light of the self-consistency condition that the market portfolio (and more generally the risk factors) is (are) constituted of the assets whose returns it is (they are) supposed to explain. As already reported in several articles, self-consistency implies correlations between the return disturbances. As a consequence, the alpha's and beta's of the factor model are unobservable. Self-consistency leads to renormalized beta's with zero effective alpha's, which are observable with standard OLS regressions. Analytical derivations and numerical simulations show that, for arbitrary choices of the proxy which are different from the true market portfolio, a modified linear regression holds with a non-zero value $α_i$ at the origin between an asset $i$'s return and the proxy's return. Self-consistency also introduces ``orthogonality'' and ``normality'' conditions linking the beta's, alpha's (as well as the residuals) and the weights of the proxy portfolio. Two diagnostics based on these orthogonality and normality conditions are implemented on a basket of 323 assets which have been components of the S&P500 in the period from Jan. 1990 to Feb. 2005. These two diagnostics show interesting departures from dynamical self-consistency starting about 2 years before the end of the Internet bubble. Finally, the factor decomposition with the self-consistency condition derives a risk-factor decomposition in the multi-factor case which is identical to the principal components analysis (PCA), thus providing a direct link between model-driven and data-driven constructions of risk factors.

physics.soc-ph↗

arXiv2006

Comment on "Analysis of the Spatial Distribution between Successive Earthquakes" by Davidsen and Paczuski

By analyzing a southern California earthquake catalog, Davidsen and Paczuski [Phys. Rev. Lett. 94, 048501 (2005)] claim to have found evidence contradicting the theory of aftershock zone scaling in favor of scale-free statistics. We present four elements showing that Davidsen and Paczuski's results may be insensitive to the existence of physical length scales associated with aftershock zones or mainshock rupture lengths, so that their claim is unsubstantiated. (i) Their exponent smaller than 1 for a pdf implies that the power law statistics they report is at best an intermediate asymptotic; (ii) their power law is not robust to the removal of 6 months of data around Landers earthquake within a period of 17 years; (iii) the same analysis for Japan and northern California shows no evidence of robust power laws; (iv) a statistical model of earthquake triggering that explicitely obeys aftershock zone scaling can reproduce the observed histogram of Davidsen and Paczuski, demonstrating that their statistic may not be sensitive to the presence of characteristic scales associated with earthquake triggering.

physics.geo-ph↗

arXiv2006

Theory of Earthquake Recurrence Times

The statistics of recurrence times in broad areas have been reported to obey universal scaling laws, both for single homogeneous regions (Corral, 2003) and when averaged over multiple regions (Bak et al.,2002). These unified scaling laws are characterized by intermediate power law asymptotics. On the other hand, Molchan (2005) has presented a mathematical proof that, if such a universal law exists, it is necessarily an exponential, in obvious contradiction with the data. First, we generalize Molchan's argument to show that an approximate unified law can be found which is compatible with the empirical observations when incorporating the impact of the Omori law of earthquake triggering. We then develop the full theory of the statistics of inter-event times in the framework of the ETAS model of triggered seismicity and show that the empirical observations can be fully explained. Our theoretical expression fits well the empirical statistics over the whole range of recurrence times, accounting for different regimes by using only the physics of triggering quantified by Omori's law. The description of the statistics of recurrence times over multiple regions requires an additional subtle statistical derivation that maps the fractal geometry of earthquake epicenters onto the distribution of the average seismic rates in multiple regions. This yields a prediction in excellent agreement with the empirical data for reasonable values of the fractal dimension $d \approx 1.8$, the average clustering ratio $n \approx 0.9$, and the productivity exponent $α\approx 0.9$ times the $b$-value of the Gutenberg-Richter law.

physics.geo-ph↗

arXiv2006

Electro-Magnetic Earthquake Bursts and Critical Rupture of Peroxy Bond Networks in Rocks

We propose a mechanism for the low frequency electromagnetic emissions and other electromagnetic phenomena which have been associated with earthquakes. The mechanism combines the critical earthquake concept and the concept of crust acting as a charging electric battery under increasing stress. The electric charges are released by activation of dormant charge carriers in the oxygen anion sublattice, called peroxy bonds or positive hole pairs (PHP), where a PHP represents an $O_3X/^{OO}\backslash YO_3$ with $X,Y = Si^{4+}, Al^{3+}...$, i.e. an $O^-$ in a matrix of $O^{2-}$ of silicates. We propose that PHP are activated by plastic deformations during the slow cooperative build-up of stress and the increasingly correlated damage culminating in a large ``critical'' earthquake. Recent laboratory experiments indeed show that stressed rocks form electric batteries which can release their charge when a conducting path closes the equivalent electric circuit. We conjecture that the intermittent and erratic occurrences of EM signals are a consequence of the progressive build-up of the battery charges in the Earth crust and their erratic release when crack networks are percolating throughout the stressed rock volumes, providing a conductive pathway for the battery currents to discharge. EM signals are thus expected close to the rupture, either slightly before or after, that is, when percolation is most favored.

physics.geo-ph↗

arXiv2006

"Universal" Distribution of Inter-Earthquake Times Explained

We propose a simple theory for the ``universal'' scaling law previously reported for the distributions of waiting times between earthquakes. It is based on a largely used benchmark model of seismicity, which just assumes no difference in the physics of foreshocks, mainshocks and aftershocks. Our theoretical calculations provide good fits to the data and show that universality is only approximate. We conclude that the distributions of inter-event times do not reveal more information than what is already known from the Gutenberg-Richter and the Omori power laws. Our results reinforces the view that triggering of earthquakes by other earthquakes is a key physical mechanism to understand seismicity.

physics.geo-ph↗

arXiv2006

Generic Multifractality in Exponentials of Long Memory Processes

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent $ϕ+1/2$, where $ϕ>0$. This generalizes previous studies performed only with $ϕ=0$ (with a truncation at an integral scale), by showing that multifractality holds over a remarkably large range of dimensionless scales for $ϕ>0$. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation $ϕ$ from 1/2 and of another parameter $σ^2$ embodying information on the short-range amplitude of the memory kernel, the ultra-violet cut-off (``viscous'') scale and the variance of the white-noise innovations. In these processes, both a viscous scale and an integral scale naturally appear, bracketing the ``inertial'' scaling regime. We exhibit a surprisingly good collapse of the multifractal spectra $ζ(q)$ on a universal scaling function, which enables us to derive high-order multifractal exponents from the small-order values and also obtain a given multifractal spectrum $ζ(q)$ by different combinations of $ϕ$ and $σ^2$.

cond-mat.stat-mech↗

arXiv2005

Celebrating the Physics in Geophysics

As 2005, the International Year of Physics, comes to an end, two physicists working primarily in geophysical research reflect on how geophysics is not an applied physics. Although geophysics has certainly benefited from progress in physics and sometimes emulated the reductionist program of mainstream physics, it has also educated the physics community about some of the generic behaviors of strongly nonlinear systems. Dramatic examples are the insights we have gained into the ``emergent'' phenomena of chaos, cascading instabilities, turbulence, self-organization, fractal structure, power-law variability, anomalous scaling, threshold dynamics, creep, fracture, and so on. In all of these examples, relatively simple models have been able to explain the recurring features of apparently very complex signals and fields. It appears that the future of the intricate relation between physics and geophysics will be as exciting as its past has been characterized by a mutual fascination. Physics departments in our universities should capitalize on this trend to attract and retain young talent motivated to address problems that really matter for the future of the planet. A pressing topic with huge impact on populations and that is challenging enough for both physics and geophysics communities to work together like never before is the understanding and prediction of extreme events.

physics.geo-ph↗

arXiv2005

Optimal Prediction of Time-to-Failure from Information Revealed by Damage

We present a general prediction scheme of failure times based on updating continuously with time the probability for failure of the global system, conditioned on the information revealed on the pre-existing idiosyncratic realization of the system by the damage that has occurred until the present time. Its implementation on a simple prototype system of interacting elements with unknown random lifetimes undergoing irreversible damage until a global rupture occurs shows that the most probable predicted failure time (mode) may evolve non-monotonically with time as information is incorporated in the prediction scheme. In addition, both the mode, its standard deviation and, in fact, the full distribution of predicted failure times exhibit sensitive dependence on the realization of the system, similarly to ``chaos'' in spinglasses, providing a multi-dimensional dynamical explanation for the broad distribution of failure times observed in many empirical situations.

cond-mat.mtrl-sci↗

arXiv2005

The Donation-Payment Gift Card Concept: how to give twice with one card

Standard economic theory, starting with Adam Smith's invisible hand, holds that those who trade for their own selfish motives of maximizing their private preferences may contribute more to the public wealth than those who claim altruistic motives. Under restrictive conditions, this has been shown to result from a self-organizing mechanism acting at the global system level, which ensures that optimal allocation of resources derives from competition within the rule of law. In a nutshell, according to this view, being competitively selfish adds value for all. But to what degree can doing good add competitive value to products and services? We propose the creation of the Donation-Payment card as a solution to the legal and moral issues plaguing the emerging anonymous, direct-to-consumer, stored-value prepaid (SVP) card market. Our proposal to redirect millions of dollars whose status is in conflict not only improves society and empowers people, but also represents the best free-market solution to the SVP-Cards industry. The Donation-Payment card kills two birds with one stone by marrying business interests with societal benefits. In addition, think of the state of mind of a grand-parent giving a SVP-card to a grand-child, who also feels like "killing two birds with one stone" or should we say more aptly feels like "giving twice with one card".

physics.soc-ph↗