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arXiv1996

Statistical Inference, Occam's Razor and Statistical Mechanics on The Space of Probability Distributions

The task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions. Using the techniques of low-temperature expansions, we arrive at a systematic series for the Bayesian posterior probability of a model family that significantly extends known results in the literature. In particular, we arrive at a precise understanding of how Occam's Razor, the principle that simpler models should be preferred until the data justifies more complex models, is automatically embodied by probability theory. These results require a measure on the space of model parameters and we derive and discuss an interpretation of Jeffreys' prior distribution as a uniform prior over the distributions indexed by a family. Finally, we derive a theoretical index of the complexity of a parametric family relative to some true distribution that we call the {\it razor} of the model. The form of the razor immediately suggests several interesting questions in the theory of learning that can be studied using the techniques of statistical mechanics.

cond-mat↗
arXiv1996

Self-Organized Criticality and $1/f$ Noise in Traffic

Phantom traffic jams may emerge ``out of nowhere'' from small fluctuations rather than being triggered by large, exceptional events. We show how phantom jams arise in a model of single lane highway traffic, which mimics human driving behavior. Surprisingly, the optimal state of highest efficiency, with the largest throughput, is a critical state with traffic jams of all sizes. We demonstrate that open systems self-organize to the most efficient state. In the model we study, this critical state is a percolation transition for the phantom traffic jams. At criticality, the individual jams have a complicated fractal structure where cars follow an intermittent stop and go pattern. We analytically derive the form of the corresponding power spectrum to be $1/f^α$ with $α=1$ exactly. This theoretical prediction agrees with our numerical simulations and with observations of $1/f$ noise in real traffic.

cond-mat↗
arXiv1996

High dimensional properties of quenched noise growth models

We discuss the behavior of bounded slope quenched noise invasion models in high dimensions. We first observe that the roughness of such a steady state interface is generated by the combination of the roughness of the invasion process $χ_c$ and the roughness of the underlying interface dynamics. In high enough dimension we argue that $χ_c$ decreases to zero. This defines a critical dimension for the problem, over which it reduces to the correlated annealed dynamics, which we show to have the same roughness as the annealed equation at five dimensions. We argue that on the Cayley tree with one additional height coordinate the associated processes are fractal. The critical behavior is anomalous due to strong effects of rare events. Numerical simulations of the model on a Cayley tree and high dimensional lattices support those theoretical predictions.

cond-mat↗
arXiv1996

Continuum Theory with Memory for Avalanches in Self-Organized Criticality

The propagator for the activity in a broad class of self-organized critical models obeys an imaginary-time Schrödinger equation with a nonlocal, history-dependent potential representing memory. Consequently, the probability for an avalanche to spread beyond a distance $r$ in time $t$ has an anomalous tail $\exp{[-C\,x^{1/(D-1)}]}$ for $x=r^D/t \gg 1$ and $D>2$, indicative of glassy dynamics. The theory is verified for an exactly solvable model, where $D=4$ and $C=3/4$, and for the Bak-Sneppen model where it is tested numerically.

cond-mat↗
arXiv1998

Simple Model of Large Scale Organization in Evolution

A mathematical model of interacting species filling ecological niches left by the extinction of others is introduced. Species organize themselves into genera of all sizes. The size of a genus on average grows linearly with its age, confirming a general relation between Age and Area proposed by Willis. The ecology exhibits punctuated equilibrium. Analytic and numerical results show that the probability distribution of genera sizes, genera lifetimes, and extinction event sizes are the same power law $P(x) \sim 1/x^2$, consistent with paleontological data.

cond-mat↗
arXiv1997

Randomness and Apparent Fractality

We show that when the standard techniques for calculating fractal dimensions in empirical data (such as the box counting) are applied on uniformly random structures, apparent fractal behavior is observed in a range between physically relevant cutoffs. This range, spanning between one and two decades for densities of 0.1 and lower, is in good agreement with the typical range observed in experiments. The dimensions are not universal and depend on density. Our observations are applicable to spatial, temporal and spectral random structures, all with non-zero measure. Fat fractal analysis does not seem to add information over routine fractal analysis procedures. Most significantly, we find that this apparent fractal behavior is robust even to the presence of moderate correlations. We thus propose that apparent fractal behavior observed experimentally over a limited range in some systems, may often have its origin in underlying randomness.

cond-mat↗
arXiv1997

Competition in a Fitness Landscape

We present an extension of Eigen's model for quasi-species including the competition among individuals, proposed as the simplest mechanism for the formation of new species in a smooth fitness landscape. We are able to obtain analytically the critical threshold for species formation. The comparison with numerical simulations is very good.

cond-mat.stat-mech↗
arXiv1997

Scaling and correlation in financial data

The statistical properties of the increments x(t+T) - x(t) of a financial time series depend on the time resolution T on which the increments are considered. A non-parametric approach is used to study the scale dependence of the empirical distribution of the price increments x(t+T) - x(t) of S&P Index futures, for time scales T, ranging from a few minutes to a few days using high-frequency price data. We show that while the variance increases linearly with the timescale, the kurtosis exhibits anomalous scaling properties, indicating a departure from the iid hypothesis. Study of the dependence structure of the increments shows that although the autocorrelation function decays rapidly to zero in a few minutes, the correlation of their squares exhibits a slow power law decay with exponent 0.37, indicating persistence in the scale of fluctuations. We establish a link between the scaling behavior and the dependence structure of the increments : in particular, the anomalous scaling of kurtosis may be explained by "long memory" properties of the square of the increments.

cond-mat.stat-mech↗
arXiv1997

Self Organized Criticality in Digging Myopic Ant Model

We demonstrate the phenomenon of self organized criticality (SOC) in a simple random walk model described by a random walk of a myopic ant. The ant acts on the underlying lattice aiming at uniform digging of the surface but is unaffected by the underlying lattice. In 1-d, 2-d and 3-d we have explored this model and have obtained power laws in the time intervals between consecutive events of `digging'. Being a simple random walk, the power laws in space translate to power laws in time. We also study the finite size scaling of asymptotic scale invariant process as well as dynamic scaling in this system. This model differs qualitatively from the cascade models of SOC.

cond-mat↗
arXiv1997

Scaling in stock market data: stable laws and beyond

The concepts of scale invariance, self-similarity and scaling have been fruitfully applied to the study of price fluctuations in financial markets. After a brief review of the properties of stable Levy distributions and their applications to market data we indicate the shortcomings of such models and describe the truncated Levy flight as an alternative model for price movements. Furthermore, studying the dependence structure of the price increments shows that while their autocorrelation function decreases rapidly to zero, the correlation of their squares and absolute values shows a slow power law decay, indicating persistence in the scale of fluctuations, a property which can be related to the anomalous scaling of the kurtosis. In the last section we review, in the light of these empirical facts, recent attempts to draw analogies between scaling in financial markets and in turbulent flows.

cond-mat.stat-mech↗
arXiv1997

Introduction to the statistical theory of Darwinian evolution

These lectures contain a brief description of evolutionary models inspired by the statistical mechanics of disordered systems. After an introduction describing the Darwinian paradigm of evolving populations, the deterministic quasispecies equation is described, and the simplest fitness landscapes are discussed. The effect of finite population size is then considered, from the opposing points of view leading to stochastic escape and to adaptive walks. A synthesis is attempted. Finally the effects of coevolution are considered, and the promising models of large-scale inspired by the Bak-Sneppen models are described.

cond-mat.dis-nn↗
arXiv1997

Threshold Noise as a Source of Volatility in Random Synchronous Asymmetric Neural Networks

We study the diversity of complex spatio-temporal patterns of random synchronous asymmetric neural networks (RSANNs). Specifically, we investigate the impact of noisy thresholds on network performance and find that there is a narrow and interesting region of noise parameters where RSANNs display specific features of behavior desired for rapidly `thinking' systems: accessibility to a large set of distinct, complex patterns.

cond-mat.dis-nn↗
arXiv1998

Garden of Eden states in traffic models

We investigate the allowed configurations in the stationary state of the cellular automaton model for single-lane traffic. It is found that certain states in the configuration space can not be reached if one uses parallel dynamics. These so-called Garden of Eden (GoE) states do not exist for random-sequential dynamics and are responsible for the strong short-ranged correlations found in parallel dynamics. By eliminating the GoE states we obtain a simple and effective approximative description of the model. For $v_{max}=1$ the exact solution is recovered. For $v_{max}=2$ this elimination leads to much higher values of the flux compared to the mean-field result which are in good agreement with Monte Carlo simulations.

cond-mat.stat-mech↗
arXiv1998

Stochastic Dynamics in Game Theory

We introduce a simple stochastic dynamics for game theory. It assumes ``local'' rationality in the sense that any player climbs the gradient of his utility function in the presence of a stochastic force which represents deviation from rationality in the form of a ``heat bath''. We focus on particular games of a large number of players with a global interaction which is typical of economic systems. The stable states of this dynamics coincide with the Nash equilibria of game theory. We study the gaussian fluctuations around these equilibria and establish that fluctuations around competitive equilibria increase with the number of players. In other words, competitive equilibria are characterized by very broad and uneven distributions among players. We also develop a small noise expansion which allows to compute a ``free energy'' functional. In particular we discuss the problem of equilibrium selection when more than one equilibrium state is present.

cond-mat.stat-mech↗
arXiv1998

Modeling Market Mechanism with Evolutionary Games

This is an essay solicited by Europhysics News, published in its March/April 1998 issue with slight modifications. We outline some highlights of the econophysics models, especially the so-called Minority model of competition and evolution. Even without the usual math, this essay offers an analytical solution to the Minority model, revealing some key features of the solution.

cond-mat.soft↗
arXiv1998

Stochastic Lotka-Volterra Systems of Competing Auto-Catalytic Agents Lead Generically to Truncated Pareto Power Wealth Distribution, Truncated Levy Distribution of Market Returns, Clustered Volatility, Booms and Craches

We give a microscopic representation of the stock-market in which the microscopic agents are the individual traders and their capital. Their basic dynamics consists in the auto-catalysis of the individual capital and in the global competition/cooperation between the agents mediated by the total wealth invested in the stock (which we identify with the stock-index). We show that such systems lead generically to (truncated) Pareto power-law distribution of the individual wealth. This, in turn, leads to intermittent market (short time) returns parametrized by a (truncated) Levy distribution. We relate the truncation in the Levy distribution to the (truncation in the Pareto Power Law i.e. to the) fact that at each moment no trader can own more than the current total wealth invested in the stock. In the cases where the system is dominated by the largest traders, the dynamics looks similar to noisy low-dimensional chaos. By introducing traders memory and/or feedback between individual and collective wealth fluctuations (the later identified with the stock returns), one obtains clustered "volatility" as well as market booms and crashes. The basic feedback loop consists in: - computing the market price of the stock as the sum of the individual wealths invested in the stock by the traders and - determining the time variation of the individual trader wealth as his/her previous wealth multiplied by the stock return (i.e. the variation of the stock price).

cond-mat.dis-nn↗
arXiv1998

Electrodynamical model of quasi-efficient financial market

The modelling of financial markets presents a problem which is both theoretically challenging and practically important. The theoretical aspects concern the issue of market efficiency which may even have political implications \cite{Cuthbertson}, whilst the practical side of the problem has clear relevance to portfolio management \cite{Elton} and derivative pricing \cite{Hull}. Up till now all market models contain "smart money" traders and "noise" traders whose joint activity constitutes the market \cite{DeLong,Bak}. On a short time scale this traditional separation does not seem to be realistic, and is hardly acceptable since all high-frequency market participants are professional traders and cannot be separated into "smart" and "noisy". In this paper we present a "microscopic" model with homogenuous quasi-rational behaviour of traders, aiming to describe short time market behaviour. To construct the model we use an analogy between "screening" in quantum electrodynamics and an equilibration process in a market with temporal mispricing \cite{Ilinski,Dunbar}. As a result, we obtain the time-dependent distribution function of the returns which is in quantitative agreement with real market data and obeys the anomalous scaling relations recently reported for both high-frequency exchange rates \cite{Breymann}, S&P500 \cite{Stanley} and other stock market indices \cite{Bouchaud,Australia}.

cond-mat.stat-mech↗