Miguel A. Ré

Generalization of entropy based divergence measures for symbolic sequence analysis.

Entropy based measures have been frequently used in symbolic sequence analysis. A symmetrized and smoothed form of Kullback-Leibler divergence or relative entropy, the Jensen-Shannon divergence (JSD), is of particular interest because of its sharing properties with families of other divergence measures and its interpretability in different domains including statistical physics, information theory and mathematical statistics. The uniqueness and versatility of this measure arise because of a number of attributes including generalization to any number of probability distributions and association of weights to the distributions. Furthermore, its entropic formulation allows its generalization in different statistical frameworks, such as, non-extensive Tsallis statistics and higher order Markovian statistics. We revisit these generalizations and propose a new generalization of JSD in the integrated Tsallis and Markovian statistical framework. We show that this generalization can be interpreted in terms of mutual information. We also investigate the performance of different JSD generalizations in deconstructing chimeric DNA sequences assembled from bacterial genomes including that of E. coli, S. enterica typhi, Y. pestis and H. influenzae. Our results show that the JSD generalizations bring in more pronounced improvements when the sequences being compared are from phylogenetically proximal organisms, which are often difficult to distinguish because of their compositional similarity. While small but noticeable improvements were observed with the Tsallis statistical JSD generalization, relatively large improvements were observed with the Markovian generalization. In contrast, the proposed Tsallis-Markovian generalization yielded more pronounced improvements relative to the Tsallis and Markovian generalizations, specifically when the sequences being compared arose from phylogenetically proximal organisms.

Superdiffusion in decoupled continuous time random walks

Abstract Continuous time random walk models with decoupled waiting time density are studied. When the spatial one-jump probability density belongs to the Levy distribution type and the total time transition is exponential a generalized superdiffusive regime is established. This is verified by showing that the square width of the probability distribution (appropriately defined) grows as t2/γ with 0

Transient Behaviour in the Absorption Probability Distribution in the Presence of a Non-Markovian Dynamic Trap

We find the Absorption Probability Distribution in the presence of a dynamic trap. The results are exact for every switching-time density of the trap. The deterministic and Markovian cases can also be obtained. Siegerts result is reobtained in the static limit. Monte Carlo simulations are compared with the inverse Laplace of our solution, finding excellent agreement at all times.

Imperfect Trapping in a Random Walk with Both Species Mobile

It is presented here a continuous time random walk model for diffusion mediated reactions with both species mobile. The random walk is carried out over an infinite homogeneouos lattice. They are calculated the probability density for the time of reaction of a pair, the reaction rate and the time evolution of the concentration of the majority species. Analytical results are obtained in the Fourier-Laplace transform representation. Known results for a fixed trap are reobtained with appropriate marginal probabilities. It is thus justified Smoluchowski’s original approximation considering the trap at a fixed position and the majority species diffusing with a coefficient sum of the individual coefficients. The results obtained are illustrated by a one dimensional model with bias.

Lévy decoupled random walks

A diffusion model based on a continuous time random walk scheme with a separable transition probability density is introduced. The probability density for long jumps is proportional to x−1−γ (a Levy-like probability density). Even when the probability density for the walker position at time t,P(x;t), has not a finite second moment when 0<γ<2, it is possible to consider alternative estimators for the width of the distribution. It is then found that any reasonable width estimator will exhibit the same long-time behaviour, since in this limit P(x;t) goes to the distribution Lγ(x/tα), a Levy distribution. The scaling property is verified numerically by means of Monte Carlo simulations. We find that if the waiting time density has a finite first moment then α=1/γ, while for densities with asymptotic behaviour t−1−β with 0<β<1 (“long tail” densities) it is verified that α=β/γ. This scaling property ensures that any reasonable estimator of the distribution width will grow as tα in the long-time limit. Based on this long-time behaviour we propose a generalized criterion for the classification in superdiffusive and subdiffusive processes, according to the value of α.