Lus Amaral

Anomalous fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics

We discuss examples of complex systems composed of many interacting subsystems. We focus on those systems displaying nontrivial long-range correlations. These include the one-dimensional sequence of base pairs in DNA, the sequence of flight times of the large seabird Wandering Albatross, and the annual fluctuations in the growth rate of business firms. We review formal analogies in the models that describe the observed long-range correlations, and conclude by discussing the possibility that behavior of large numbers of humans (as measured, e.g., by economic indices) might conform to analogs of the scaling laws that have proved useful in describing systems composed of large numbers of inanimate objects.

Scaling and correlation in financial time series

We discuss the results of three recent phenomenological studies focussed on understanding the distinctive statistical properties of financial time series – (i) The probability distribution of stock price fluctuations: Stock price fluctuations occur in all magnitudes, in analogy to earthquakes – from tiny fluctuations to very drastic events, such as the crash of 19 October 1987, sometimes referred to as “Black Monday”. The distribution of price fluctuations decays with a power-law tail well outside the Levy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. In addition, this distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. (ii) Correlations in financial time series: While price fluctuations themselves have rapidly decaying correlations, the magnitude of fluctuations measured by either the absolute value or the square of the price fluctuations has correlations that decay as a power-law, persisting for several months. (iii) Volatility and trading activity: We quantify the relation between trading activity – measured by the number of transactions NΔt – and the price change GΔt for a given stock, over a time interval [t,t+Δt]. We find that NΔt displays long-range power-law correlations in time, which leads to the interpretation that the long-range correlations previously found for |GΔt| are connected to those of NΔt.

Application of statistical physics to heartbeat diagnosis

We present several recent studies based on statistical physics concepts that can be used as diagnostic tools for heart failure. We describe the scaling exponent characterizing the long-range correlations in heartbeat time series as well as the multifractal features recently discovered in heartbeat rhythm. It is found that both features, the long-range correlations and the multifractility, are weaker in cases of heart failure.

Scale invariance and universality: organizing principles in complex systems

This paper is a brief summary of a talk that was designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena – scale invariance and universality – can be useful in guiding research on a broad class of complex phenomena. We shall see that while scale invariance has been tested for many years, universality is relatively more rarely discussed. In particular, we shall develop a heuristic argument that serves to make more plausible the universality hypothesis in both thermal critical phenomena and percolation phenomena, and suggest that this argument could be developed into a possible coherent approach to understanding the ubiquity of scale invariance and universality in a wide range of complex systems.

A random matrix theory approach to financial cross-correlations

It is common knowledge that any two firms in the economy are correlated. Even firms belonging to different sectors of an industry may be correlated because of “indirect” correlations. How can we analyze and understand these correlations? This article reviews recent results regarding cross-correlations between stocks. Specifically, we use methods of random matrix theory (RMT), which originated from the need to understand the interactions between the constituent elements of complex interacting systems, to analyze the cross-correlation matrix C of returns. We analyze 30-min returns of the largest 1000 US stocks for the 2-year period 1994–1995. We find that the statistics of approximately 20 of the largest eigenvalues (2%) show deviations from the predictions of RMT. To test that the rest of the eigenvalues are genuinely random, we test for universal properties such as eigenvalue spacings and eigenvalue correlations, and demonstrate that C shares universal properties with the Gaussian orthogonal ensemble of random matrices. The statistics of the eigenvectors of C confirm the deviations of the largest few eigenvalues from the RMT prediction. We also find that these deviating eigenvectors are stable in time. In addition, we quantify the number of firms that participate significantly to an eigenvector using the concept of inverse participation ratio, borrowed from localization theory.

Scale invariance and universality of economic fluctuations

In recent years, physicists have begun to apply concepts and methods of statistical physics to study economic problems, and the neologism “econophysics” is increasingly used to refer to this work. Much recent work is focused on understanding the statistical properties of time series. One reason for this interest is that economic systems are examples of complex interacting systems for which a huge amount of data exist, and it is possible that economic time series viewed from a different perspective might yield new results. This manuscript is a brief summary of a talk that was designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena – scale invariance and universality – can be useful in guiding research on economics. We shall see that while scale invariance has been tested for many years, universality is relatively less frequently discussed. This article reviews the results of two recent studies – (i) The probability distribution of stock price fluctuations: Stock price fluctuations occur in all magnitudes, in analogy to earthquakes – from tiny fluctuations to drastic events, such as market crashes. The distribution of price fluctuations decays with a power-law tail well outside the Levy stable regime and describes fluctuations that differ in size by as much as eight orders of magnitude. (ii) Quantifying business firm fluctuations: We analyze the Computstat database comprising all publicly traded United States manufacturing companies within the years 1974–1993. We find that the distributions of growth rates is different for different bins of firm size, with a width that varies inversely with a power of firm size. Similar variation is found for other complex organizations, including country size, university research budget size, and size of species of bird populations.

Similarities and differences between physics and economics

In this opening talk, we discuss some of the similarities between work being done by economists, and by physicists seeking to contribute to economics. We also mention some of the differences in the approaches taken, and justify these different approaches by developing the argument that by approaching the same problem from different points of view new results might emerge. In particular, we review some recent results, for example the finding that there are two new universal scaling models in economics: (i) the fluctuation of price changes of any stock market is characterized by a PDF which is a simple power law with exponent 4 that extends over 102 standard deviations (a factor of 108 on the y-axis); (ii) for a wide range of economic organizations, the histogram that shows how size of organization is inversely correlated to fluctuations in size with an exponent ≈1/6. Neither of these two new laws has a firm theoretical foundation. We also discuss results that are reminiscent of phase transitions in spin systems, where the divergent behavior of the response function at the critical point (zero magnetic field) leads to large fluctuations.

Scaling and universality in animate and inanimate systems

We illustrate the general principle that in biophysics, econophysics and possibly even city growth, the conceptual framework provided by scaling and universality may be of use in making sense of complex statistical data. Specifically, we discuss recent work on DNA sequences, heartbeat intervals, avalanche-like lung inflation, urban growth, and company growth. Although our main focus is on data, we also discuss statistical mechanical models.

Collective behavior of stock price movements—a random matrix theory approach

We review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory (RMT), developed in physics to describe the energy levels of complex systems. RMT makes predictions for “universal” properties that do not depend on the interactions between the elements comprising the system; deviations from RMT provide clues regarding system-specific properties. We compare the statistics of the cross-correlation matrix C—whose elements Cij are the correlation coefficients of price fluctuations of stock i and j—against a random matrix having the same symmetry properties. It is found that RMT methods can distinguish random and non-random parts of C. The non-random part of C which deviates from RMT results, provides information regarding genuine collective behavior among stocks.

Quantifying fluctuations in economic systems by adapting methods of statistical physics

The emerging subfield of econophysics explores the degree to which certain concepts and methods from statistical physics can be appropriately modified and adapted to provide new insights into questions that have been the focus of interest in the economics community. Here we give a brief overview of two examples of research topics that are receiving recent attention. A first topic is the characterization of the dynamics of stock price fluctuations. For example, we investigate the relation between trading activity – measured by the number of transactions NΔt – and the price change GΔt for a given stock, over a time interval [t,t+Δt]. We relate the time-dependent standard deviation of price fluctuations – volatility – to two microscopic quantities: the number of transactions NΔt in Δt and the variance WΔt2 of the price changes for all transactions in Δt. Our work indicates that while the pronounced tails in the distribution of price fluctuations arise from WΔt, the long-range correlations found in ∣GΔt∣ are largely due to NΔt. We also investigate the relation between price fluctuations and the number of shares QΔt traded in Δt. We find that the distribution of QΔt is consistent with a stable Levy distribution, suggesting a Levy scaling relationship between QΔt and NΔt, which would provide one explanation for volume-volatility co-movement. A second topic concerns cross-correlations between the price fluctuations of different stocks. We adapt a conceptual framework, random matrix theory (RMT), first used in physics to interpret statistical properties of nuclear energy spectra. RMT makes predictions for the statistical properties of matrices that are universal, that is, do not depend on the interactions between the elements comprising the system. In physics systems, deviations from the predictions of RMT provide clues regarding the mechanisms controlling the dynamics of a given system, so this framework can be of potential value if applied to economic systems. We discuss a systematic comparison between the statistics of the cross-correlation matrix C – whose elements Cij are the correlation-coefficients between the returns of stock i and j – and that of a random matrix having the same symmetry properties. Our work suggests that RMT can be used to distinguish random and non-random parts of C; the non-random part of C, which deviates from RMT results provides information regarding genuine cross-correlations between stocks.