Fisher Information Measure Research Articles

Information-theoretical complexity for the hydrogenic abstraction reaction

In this work, we have investigated the complexity of the hydrogenic abstraction reaction by means of information 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 and López–Mancini–Calbet (LMC) shape complexities. The analysis of the information-theoretical functionals of the one-particle density was computed in position (r) and momentum (p) space. The analysis revealed that all of the chemically significant regions can be identified from the information functionals and most of the information-theoretical planes, i.e. the reactant/product regions (R/P), the transition state (TS), including those that are not present in the energy profile such as the bond cleavage energy region (BCER), and the bond breaking/forming regions (B–B/F). The analysis of the complexities shows that, in position as well as in the joint space, the energy profile of the abstraction reaction bears the same information-theoretical features as the LMC and FS measures. We discuss why most of the chemical features of interest, namely the BCER and B–B/F, are lost in the energy profile and that they are only revealed when particular information-theoretical aspects of localizability (L or J), uniformity (D) and disorder (I) are considered.

Data fusion for reconstruction algorithms via different sensors in geophysical sensing

Information fusion via multimodal inverse problems and different sensors is addressed using a Fisher information analysis approach. The Fisher information measure is inherently additive, and it facilitates an appropriate weighting of the measurement data that is statistically optimal and can hence be useful with reconstruction algorithms in geophysical sensing. Given that there exists proper knowledge about the sensor noise statistics, correlations and spectral contents, as well as a correct forward model, the Fisher information is a natural measure of information because it is closely linked to the statistical maximum likelihood principle. To illustrate the concept of data correlation based on statistical Fisher information analysis, two simple and generic examples are employed in electrical resistivity and electromagnetic tomography, which are motivated by geophysical applications, such as tunnel detection. The examples demonstrate that a properly weighted data fusion can be of crucial importance for an ill-posed multimodal inverse problem.

Special features of the relation between Fisher information and Schrödinger eigenvalue equation

It is well known that a suggestive relation exists that links Schrödinger's equation (SE) to the information-optimizing principle based on Fisher's information measure. The connection entails the existence of a Legendre transform structure underlying the SE. Here, we show that appeal to this structure leads to a first order differential equation for the SE's eigenvalues that, in certain cases, can be used to obtain the eigenvalues without explicitly solving the SE. Complying with the above mentioned equation constitutes a necessary condition to be satisfied by an energy eigenvalue. We show that the general solution is unique.

Fisher Information at the Edge of Chaos in Random Boolean Networks

We study the order-chaos phase transition in random Boolean networks (RBNs), which have been used as models of gene regulatory networks. In particular we seek to characterize the phase diagram in information-theoretic terms, focusing on the effect of the control parameters (activity level and connectivity). Fisher information, which measures how much system dynamics can reveal about the control parameters, offers a natural interpretation of the phase diagram in RBNs. We report that this measure is maximized near the order-chaos phase transitions in RBNs, since this is the region where the system is most sensitive to its parameters. Furthermore, we use this study of RBNs to clarify the relationship between Shannon and Fisher information measures.

Inferring an optimal Fisher measure

It is well known that a suggestive relation exists that links the Schrödinger equation (SE) to the information-optimizing principle based on the Fisher information measure (FIM). We explore here an approach that will allow one to infer the optimal FIM compatible with a given amount of prior information without explicitly solving first the associated SE. This technique is based on the virial theorem and it provides analytic solutions for the physically relevant FIM, that which is minimal subject to the constraints posed by the prior information.

Discriminating quarry blasts from earthquakes in Vértes Hills (Hungary) by using the Fisher-Shannon method

The Fisher-Shannon information (FS) plane, defined by the Fisher information measure (FIM), and the Shannon entropy power, Nx, is a robust tool for investigating the complex dynamics of time series. In the FS plane, earthquakes and quarry blasts measured in Vertes Hills (Hungary) are significantly discriminated from each other. Furthermore, the results of the receiving-operating characteristics (ROC) analysis reveal that FIM is a classifier more efficient than Nx in discriminating earthquakes from quarry blasts.

Fisher Information and Steric Effect: Study of the Internal Rotation Barrier of Ethane

On the basis of a density-based quantification of the steric effect [Liu, S. B. J. Chem. Phys.2007, 126, 244103], the origin of the internal rotation barrier between the eclipsed and staggered conformers of ethane is systematically investigated in this work from an information-theoretical point of view by using the Fisher information measure in conjugated spaces. Two kinds of computational approaches are considered in this work: adiabatic (with optimal structure) and vertical (with fixed geometry). The analyses are performed systematically by following, in each case, the conformeric path by changing the dihedral angle from 0 to 180° . This is calculated at the HF, MP2, B3LYP, and CCSD(T) levels of theory and with several basis sets. Selected descriptors of the densities are utilized to support the observations. Our results show that in the adiabatic case the eclipsed conformer possesses a larger steric repulsion than the staggered conformer, but in the vertical cases the staggered conformer retains a larger steric repulsion. Our results verify the plausibility for defining and computing the steric effect in the post-Hartree-Fock level of theory according to the scheme proposed by Liu.

Complexity measures and information planes of x-ray astrophysical sources

The complex dynamics of x-ray astrophysical sources is investigated by analyzing theirdaily light curves in four different energy bands by using three information-theoreticmeasures: Fisher information measure, the Shannon entropy and the disequilibrium. Theobtained results suggest a universal complex behavior in the x-ray light, independent of thesource and of the energy band.

Fisher information description of the classical–quantal transition

We investigate the classical limit of the dynamics of a semiclassical system that represents the interaction between matter and a given field. The concept of Fisher Information measure ( F ) on using as a quantifier of the process, we find that it adequately describes the transition, detecting the most salient details of the changeover. Used in conjunction with other possible information quantifiers, such as the Normalized Shannon Entropy ( H ) and the Statistical Complexity ( C ) by recourse to appropriate planar representations like the Fisher Entropy ( F × H ) and Fisher Complexity ( F × C ) planes, one obtains a better visualization of the transition than that provided by just one quantifier by itself. In the evaluation of these Information Theory quantifiers, we used the Bandt and Pompe methodology for the obtention of the corresponding probability distribution.

Data Fusion for Electromagnetic and Electrical Resistive Tomography Based on Maximum Likelihood

This paper presents a maximum likelihood based approach to data fusion for electromagnetic (EM) and electrical resistive (ER) tomography. The statistical maximum likelihood criterion is closely linked to the additive Fisher information measure, and it facilitates an appropriate weighting of the measurement data which can be useful with multiphysics inverse problems. The Fisher information is particularly useful for inverse problems which can be linearized similar to the Born approximation. In this paper, a proper scalar product is defined for the measurements and a truncated Singular Value Decomposition (SVD) based algorithm is devised which combines the measurement data of the two imaging modalities in a way that is optimal in the sense of maximum likelihood. As a multiphysics problem formulation with applications in geophysics, the problem of tunnel detection based on EM and ER tomography is studied in this paper. To illustrate the connection between the Green's functions, the gradients and the Fisher information, two simple and generic forward models are described in detail regarding two-dimensional EM and ER tomography, respectively.

Analysis of dynamics in magnetotelluric data by using the Fisher–Shannon method

The Fisher–Shannon information (FS) plane, defined by the Fisher information measure and the Shannon entropy power, is used to investigate the complex dynamics of magnetotelluric data of three stations in Taiwan. In the FS plane the electric and magnetic components are significantly separated, characterized by different degrees of order. Further investigation shows that signals measured in areas with very high level of seismic activity are well discriminated.

Time-dependent Fisher Information Measure of volcanic tremor before the 5 April 2003 paroxysm at Stromboli volcano, Italy

The investigation of the time dynamics of volcanic tremor recorded at Stromboli volcano before the paroxysm occurred on April 5, 2003 was performed, on the base of a new approach, the Fisher Information Measure (FIM), which allows to detect changes in the dynamical behavior of a complex system. The particular observed pattern suggests that the signal varies between sets of disordered states (small FIM) and sets of ordered states (large FIM). Significant precursory changes in the temporal variation of the FIM were revealed at least 42 h before the paroxysm and lasting about 17 h. The timescales highlighted are compatible to those found by other authors and could qualify the FIM as a good detector of regime changes and possible precursors of anomalous volcanic activity.

Fisher Information Study in Position and Momentum Spaces for Elementary Chemical Reactions

The utility of the Fisher information measure is analyzed to detect the transition state, the stationary points of a chemical reaction, and the bond breaking/forming regions of elementary reactions such as the simplest hydrogen abstraction and the identity SN2 exchange ones. This is performed by following the intrinsic reaction path calculated at the MP2 and QCISD(T) levels of theory with a 6-311++G(3df, 2p) basis set. Selected descriptors of both position and momentum space densities are utilized to support the observations, such as the molecular electrostatic potential (MEP), the hardness, the dipole moment, along with geometrical parameters. Our results support the concept of a continuum of transient of Zewail and Polanyi for the transition state rather than a single state, which is also in agreement with reaction force analyses.

Fisher Information and Semiclassical Treatments

We review here the difference between quantum statistical treatments and semiclassical ones, using as the main concomitant tool a semiclassical, shift-invariant Fisher information measure built up with Husimi distributions. Its semiclassical character notwithstanding, this measure also contains abundant information of a purely quantal nature. Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well.

Fisher information and the thermodynamics of scale-invariant systems

We present a thermodynamic formulation for scale-invariant systems based on the minimization with constraints of the Fisher information measure. In such a way a clear analogy between these systems’ thermal properties and those of gases and fluids is seen to emerge in a natural fashion. We focus our attention on the non-interacting scenario, speaking thus of scale-free ideal gases (SFIGs) and present some empirical evidences regarding such disparate systems as electoral results, city populations and total citations in Physics journals, that seem to indicate that SFIGs do exist. We also illustrate the way in which Zipf’s law can be understood in a thermodynamical context as the surface of a finite system. Finally, we derive an equivalent microscopic description of our systems which totally agrees with previous numerical simulations found in the literature.

A simple mathematical model for anomalous diffusion via Fisher's information theory

Starting with the relative entropy based on a previously proposed entropy function S q [ p ] = ∫ d x p ( x ) × ( − ln p ( x ) ) q , we find the corresponding Fisher's information measure. After function redefinition we then maximize the Fisher information measure with respect to the new function and obtain a differential operator that reduces to a space coordinate second derivative in the q → 1 limit. We then propose a simple differential equation for anomalous diffusion and show that its solutions are a generalization of the functions in the Barenblatt–Pattle solution. We find that the mean squared displacement, up to a q-dependent constant, has a time dependence according to 〈 x 2 〉 ∼ K 1 / q t 1 / q , where the parameter q takes values q = 2 n − 1 2 n + 1 (superdiffusion) and q = 2 n + 1 2 n − 1 (subdiffusion), ∀ n ⩾ 1 .

Note on semiclassical uncertainty relations

An important manifestation of the Uncertainty Principle, one of the cornerstones of our present understanding of Nature, is that related to semiclassical localization in phase-space. We wish here to add some notes on the subject with reference to the canonical harmonic oscillator problem, with emphasis in the concepts of semiclassical Husimi distributions, the associated Wehrl entropy, escort distributions, and Fisher's information measure.

An Adaptable Image Retrieval System With Relevance Feedback Using Kernel Machines and Selective Sampling

This paper presents an adaptable content-based image retrieval (CBIR) system developed using regularization theory, kernel-based machines, and Fisher information measure. The system consists of a retrieval subsystem that carries out similarity matching using image-dependant information, multiple mapping subsystems that adaptively modify the similarity measures, and a relevance feedback mechanism that incorporates user information. The adaptation process drives the retrieval error to zero in order to exactly meet either an existing multiclass classification model or the user high-level concepts using reference-model or relevance feedback learning, respectively. To facilitate the selection of the most informative query images during relevance feedback learning a new method based upon the Fisher information is introduced. Model-reference and relevance feedback learning mechanisms are thoroughly tested on a domain-specific image database that encompasses a wide range of underwater objects captured using an electro-optical sensor. Benchmarking results with two other relevance feedback learning methods are also provided.

LMC-complexity and various chaotic regimes

A variant of the statistical complexity originally advanced by López-Ruiz et al. [R. López-Ruiz, H.L. Mancini, X. Calbet, Phys. Lett. A 209 (1995) 321], is here used in conjunction with Fisher's information measure so as to explore fine details of chaotic dynamics. As a main result we can easily distinguish between (i) periodicity and chaos or (ii) between distinct chaotic dynamics belonging to different attractors.

Relativistic effects on information measures for hydrogen-like atoms

Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r , but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p . Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac–Fock vs. Hartree–Fock complexity measures for many-electron neutral atoms.