Fisher-Shannon Measures Research Articles

Advanced Analysis of Temporal Data Using Fisher-Shannon Information: Theoretical Development and Application in Geosciences

Complex non-linear time series are ubiquitous in geosciences. Quantifying complexity and non-stationarity of these data is a challenging task, and advanced complexity-based exploratory tool are required for understanding and visualizing such data. This paper discusses the Fisher-Shannon method, from which one can obtain a complexity measure and detect non-stationarity, as an efficient data exploration tool. The state-of-the-art studies related to the Fisher-Shannon measures are collected, and new analytical formulas for positive unimodal skewed distributions are proposed. Case studies on both synthetic and real data illustrate the usefulness of the Fisher-Shannon method, which can find application in different domains including time series discrimination and generation of times series features for clustering, modeling and forecasting. The paper is accompanied with Python and R libraries for the non-parametric estimation of the proposed measures.

Reply to “Comments on ‘Statistical complexity and Fisher–Shannon information measure of [formula omitted]’ [Phys. Lett. A 372 (13) (2008) 2271–2273

Clarifications about the source of error in [Phys. Lett. A 371 (2008) 2271] of momentum-space information measures are given along with the revised estimates which satisfy all the points commented upon by Astashkevich. We note that the position-space quantities calculated previously remain unchanged. Our main conclusion, that the inter-nuclear distance, R, dependence of statistical complexity and Fisher–Shannon measure derived from the simple Coulson wave function contain significant information on molecular bonding in H2+, remains qualitatively valid in the light of calculations using more accurate wave functions as reported by Astashkevich.

The biparametric Fisher–Rényi complexity measure and its application to the multidimensional blackbody radiation

We introduce a biparametric Fisher–Rényi complexity measure for general probability distributions and we discuss its properties. This notion, which is composed of two entropy-like components (the Rényi entropy and the biparametric Fisher information), generalizes the basic Fisher–Shannon measure and the previous complexity quantifiers of Fisher–Rényi type. Then, we illustrate the usefulness of this notion by carrying out a information-theoretical analysis of the spectral energy density of a d-dimensional blackbody at temperature T. It is shown that the biparametric Fisher–Rényi measure of this quantum system has a universal character in the sense that it does not depend on temperature nor on any physical constant (e.g. Planck constant, speed of light, Boltzmann constant), but only on the space dimensionality d. Moreover, it decreases when d is increasing, but exhibits a non trivial behavior for a fixed d and a varying parameter, which somehow brings up a non standard structure of the blackbody d-dimensional density distribution.

Statistical measures of complexity for quantum systems with continuous variables

The Fisher–Shannon statistical measure of complexity is analyzed for a continuous manifold of quantum observables. It is shown that evaluating this measure only in the configuration or in the momentum spaces does not provide an adequate characterization of the complexity of some quantum systems. In order to obtain a more complete description of complexity two new measures, respectively based on the minimization and the integration of the usual Fisher–Shannon measure over all the parameter space, are proposed and compared. Finally, these measures are applied to the concrete case of a free particle in a box.

Information-theoretical complexity for the hydrogenic identity S N 2 exchange reaction

We investigate the complexity of the hydrogenic identity S N 2 exchange reaction by means of information-theoretic functionals such as disequilibrium (D), exponential entropy (L), Fisher information (I), power entropy (J) and joint information-theoretic measures, i.e., the I–D, D–L and I–J planes and the Fisher–Shannon (FS) and Lopez-Mancini-Calbet (LMC) shape complexities. The several information-theoretic measures of the one-particle density were computed in position (r) and momentum (p) spaces. The analysis revealed that the chemically significant regions of this reaction can be identified through most of the information-theoretic functionals or planes, not only the ones which are commonly revealed by the energy, such as the reactant/product (R/P) and the transition state (TS), but also those that are not present in the energy profile such as the bond cleavage energy region (BCER), the bond breaking/forming regions (B–B/F) and the charge transfer process (CT). The analysis of the complexities shows that the energy profile of the identity S N 2 exchange reaction bears no simple behavior with respect to the LMC and FS measures. Most of the chemical features of interest (BCER, B–B/F and CT) are only revealed when particular information-theoretic aspects of localizability (L or J), uniformity (D) and disorder (I) are considered.

Configuration complexities of hydrogenic atoms

The Fisher-Shannon and Cramer-Rao information measures, and the LMC-like or shape complexity (i.e., the disequilibrium times the Shannon entropic power) of hydrogenic stationary states are investigated in both position and momentum spaces. First, it is shown that not only the Fisher information and the variance (then, the Cramer-Rao measure) but also the disequilibrium associated to the quantum-mechanical probability density can be explicitly expressed in terms of the three quantum numbers (n, l, m) of the corresponding state. Second, the three composite measures mentioned above are analytically, numerically and physically discussed for both ground and excited states. It is observed, in particular, that these configuration complexities do not depend on the nuclear charge Z. Moreover, the Fisher-Shannon measure is shown to quadratically depend on the principal quantum number n. Finally, sharp upper bounds to the Fisher-Shannon measure and the shape complexity of a general hydrogenic orbital are given in terms of the quantum numbers.