Rosario N. Mantegna

Zipf plots and the size distribution of firms

We use a Zipf plot to demonstrate that the upper tail of the size distribution of firms is too thin relative to the log normal rather than too fat, as had previously been believed.

Correlation approach to identify coding regions in DNA sequences

Recently, it was observed that noncoding regions of DNA sequences possess long-range power-law correlations, whereas coding regions typically display only short-range correlations. We develop an algorithm based on this finding that enables investigators to perform a statistical analysis on long DNA sequences to locate possible coding regions. The algorithm is particularly successful in predicting the location of lengthy coding regions. For example, for the complete genome of yeast chromosome III (315,344 nucleotides), at least 82% of the predictions correspond to putative coding regions; the algorithm correctly identified all coding regions larger than 3000 nucleotides, 92% of coding regions between 2000 and 3000 nucleotides long, and 79% of coding regions between 1000 and 2000 nucleotides. The predictive ability of this new algorithm supports the claim that there is a fundamental difference in the correlation property between coding and noncoding sequences. This algorithm, which is not species-dependent, can be implemented with other techniques for rapidly and accurately locating relatively long coding regions in genomic sequences.

Statistical mechanics in biology: how ubiquitous are long-range correlations?

The purpose of this opening talk is to describe examples of recent progress in applying statistical mechanics to biological systems. We first briefly review several biological systems, and then focus on the fractal features characterized by the long-range correlations found recently in DNA sequences containing non-coding material. We discuss the evidence supporting the finding that for sequences containing only coding regions, there are no long-range correlations. We also discuss the recent finding that the exponent alpha characterizing the long-range correlations increases with evolution, and we discuss two related models, the insertion model and the insertion-deletion model, that may account for the presence of long-range correlations. Finally, we summarize the analysis of long-term data on human heartbeats (up to 10(4) heart beats) that supports the possibility that the successive increments in the cardiac beat-to-beat intervals of healthy subjects display scale-invariant, long-range anti-correlations (a tendency to beat faster is balanced by a tendency to beat slower later on). In contrast, for a group of subjects with severe heart disease, long-range correlations vanish. This finding suggests that the classical theory of homeostasis, according to which stable physiological processes seek to maintain constancy, should be extended to account for this type of dynamical, far from equilibrium, behavior.

Statistical properties of DNA sequences

We review evidence supporting the idea that the DNA sequence in genes containing non-coding regions is correlated, and that the correlation is remarkably long range--indeed, nucleotides thousands of base pairs distant are correlated. We do not find such a long-range correlation in the coding regions of the gene. We resolve the problem of the non-stationarity feature of the sequence of base pairs by applying a new algorithm called detrended fluctuation analysis (DFA). We address the claim of Voss that there is no difference in the statistical properties of coding and non-coding regions of DNA by systematically applying the DFA algorithm, as well as standard FFT analysis, to every DNA sequence (33301 coding and 29453 non-coding) in the entire GenBank database. Finally, we describe briefly some recent work showing that the non-coding sequences have certain statistical features in common with natural and artificial languages. Specifically, we adapt to DNA the Zipf approach to analyzing linguistic texts. These statistical properties of non-coding sequences support the possibility that non-coding regions of DNA may carry biological information.

Statistical and Linguistic Features of DNA Sequences

We present evidence supporting the idea that the DNA sequence in genes containing noncoding regions is correlated, and that the correlation is remarkably long range--indeed, base pairs thousands of base pairs distant are correlated. We do not find such a long-range correlation in the coding regions of the gene. We resolve the problem of the non-stationary feature of the sequence of base pairs by applying a new algorithm called Detrended Fluctuation Analysis (DFA). We address the claim of Voss that there is no difference in the statistical properties of coding and noncoding regions of DNA by systematically applying the DFA algorithm, as well as standard FFT analysis, to all eukaryotic DNA sequences (33 301 coding and 29 453 noncoding) in the entire GenBank database. We describe a simple model to account for the presence of long-range power-law correlations which is based upon a generalization of the classic Levy walk. Finally, we describe briefly some recent work showing that the noncoding sequences have certain statistical features in common with natural languages. Specifically, we adapt to DNA the Zipf approach to analyzing linguistic texts, and the Shannon approach to quantifying the redundancy of a linguistic text in terms of a measurable entropy function. We suggest that noncoding regions in plants and invertebrates may display a smaller entropy and larger redundancy than coding regions, further supporting the possibility that noncoding regions of DNA may carry biological information.

Statistical and linguistic features of noncoding DNA: A heterogeneous «Complex system»

SummaryWe present evidence supporting the idea that the DNA sequence in genes containing noncoding regions is correlated, and that the correlation is remarkably long range-indeed, base pairs thousands of base pairs distant are correlated. We do not find such a long-range correlation in the coding regions of the gene; we utilize this fact to build a Coding Sequence Finder algorithm, which uses statistical ideas to locate the coding regions of an unknown DNA sequence. We resolve the problem of the «non-stationarity» feature of the sequence of base pairs (that the relative concentration of purines and pyrimidines changes in different regions of the mosaic-like chain) by describing a new algorithm called Detrended Fluctuation Analysis (DFA). We address the claim of Voss that there is no difference in the statistical properties of coding and noncoding regions of DNA by systematically applying the DFA algorithm, as well as standard FFT analysis, to every DNA sequence (33 301 coding and 29 453 non-coding) in the entire GenBank database. We describe a simple model to account for the presence of long-range power law correlations (and the systematic variation of the scaling exponent α with evolution) which is based upon a generalization of the classic Lévy walk. Finally, we describe briefly some recent work showing that the noncoding sequences have certain statistical features in common with natural languages. Specifically, we adapt to DNA the Zipf approach to analyzing linguistic texts, and the Shannon approach to quantifying the «redundancy» of a linguistic text in terms of a measurable entropy function. We suggest that noncoding regions in eukaryotes may display a smaller entropy and larger redundancy than coding regions for plants and invertebrates, further supporting the possibility that noncoding regions of DNA may carry biological information.

Long-range correlations and generalized Lévy walks in DNA sequences

There is a mounting body of evidence suggesting that the noncoding regions of DNA are rather special for at least two reasons: n n1. n nThey display long-range power-law correlations, as opposed to previously-believed exponentially-decaying correlations. n n n n n2. n nThey display features common to hierarchically-structured languages-specifically, a linear Zipf plot and a non-zero redundancy.

Investigations of Financial Markets Using Statistical Physics Methods

We begin with a brief historical note concerning the growing interest of statistical physicists in the analysis and modeling of financial markets. We then briefly discuss the key concepts of arbitrage and efficient markets. We relate these concepts to apparently ‘universal’ aspects observed in the empirical analysis of stock price dynamics in financial markets. In particular, we consider (i) the empirical behavior of the probability density function for the return of an economic time series to where it started and (ii) the content of economic information in a financial time series.

An Introduction to Econophysics: Correlation and anticorrelation between stocks

One of the more appealing ideas in econophysics is that financial markets can be described along lines similar to successful descriptions of critical phenomena. Critical phenomena are physical phenomena that occur in space (real or abstract) and time. We have considered thus far only a single asset and its time evolution, but in this chapter we discuss an approach based on the simultaneous investigation of several stock-price time series belonging to a given portfolio. Indeed, the presence of cross-correlations (and anticorrelations) between pairs of stocks has long been known, and plays a key role in the theory of selecting the most efficient portfolio of financial goods [49, 115]. We show how relevant these correlations and anticorrelations are by discussing a study devoted to detect the amount of synchronization present in the dynamics of a pair of stocks traded in a financial market [107]. The specific properties of the covariance matrix of stock returns of a given portfolio of stocks have been investigated extensively. Also we briefly consider studies that aim (i) to detect the number of economic factors affecting the dynamics of stock prices in a given financial market [34, 154], and (ii) to evaluate the deviations observed between market data and the results expected from the theory of random matrices [63, 87, 134]. Simultaneous dynamics of pairs of stocks In financial markets, many stocks are traded simultaneously.