Fisher Information Measure Research Articles

Information–Theoretic Analysis of Visibility Graph Properties of Extremes in Time Series Generated by a Nonlinear Langevin Equation

In this study, we examined how the nonlinearity α of the Langevin equation influences the behavior of extremes in a generated time series. The extremes, defined according to run theory, result in two types of series, run lengths and surplus magnitudes, whose complex structure was investigated using the visibility graph (VG) method. For both types of series, the information measures of the Shannon entropy measure and Fisher Information Measure were utilized for illustrating the influence of the nonlinearity α on the distribution of the degree, which is the main topological parameter describing the graph constructed by the VG method. The main finding of our study was that the Shannon entropy of the degree of the run lengths and the surplus magnitudes of the extremes is mostly influenced by the nonlinearity, which decreases with with an increase in α. This result suggests that the run lengths and surplus magnitudes of extremes are characterized by a sort of order that increases with increases in nonlinearity.

Fisher Information-Based Optimization of Mapped Fourier Grid Methods

The mapped Fourier grid method (mapped-FGM) is a simple and efficient discrete variable representation (DVR) numerical technique for solving atomic radial Schrödinger differential equations. It is set up on equidistant grid points, and the mapping, a suitable coordinate transformation to the radial variable, deals with the potential energy peculiarities that are incompatible with constant step grids. For a given constrained number of grid points, classical phase space and semiclassical arguments help in selecting the mapping function and the maximum radial extension, while the energy does not generally exhibit a variational extremization trend. In this work, optimal computational parameters and mapping quality are alternatively assessed using the extremization of (coordinate and momentum) Fisher information. A benchmark system (hydrogen atom) is employed, where energy eigenvalues and Fisher information are traced in a standard convergence procedure. High-precision energy eigenvalues exhibit a correlation with the extrema of Fisher information measures. Highly efficient mapping schemes (sometimes classically counterintuitive) also stand out with these measures. Same trends are evidenced in the solution of Dalgarno–Lewis equations, i.e., inhomogeneous counterparts of the radial Schrödinger equation occurring in perturbation theory. A detailed analysis of the results, implications on more complex single valence electron Hamiltonians, and future extensions are also included.

Why are information-theoretic descriptors powerful predictors of atomic and molecular polarizabilities.

We rationalize the excellent performance of information-theoretic descriptors for predicting atomic and molecular polarizabilities. It seems that descriptors which capture information about the change in valence-shell structure, especially the relative Fisher information measures, are particularly useful. Using this, we can rationalize why the G3 form of the relative Fisher information, which measures the deviation of effective nuclear charge between an atom-in-a-molecule and the reference pro-atom, is especially effective as a predictor of molecular polarizability. There are no methods used in this paper, which relies on mathematical derivation and analysis.

Asymptotic distribution of entropies and Fisher information measure of ordinal patterns with applications

We present the asymptotic distribution of the Rényi and Tsallis/Havrda–Charvát entropies and the Fisher information measure of ordinal patterns embedding their serial correlation. We study the convergence behavior of the asymptotic variance for some types of dynamics and the permutation entropy to the limit distribution. These results lead to tests for comparing the underlying dynamics of two time series. We apply these tests to discriminate uniform white noise, logistic maps with Gaussian noise, fractional Brownian motion, and f−k noise, with k∈{0.5,1,1.5,2,2.5}. We also applied these tests to cryptocurrency open prices data, with favorable results. We provide the R code that implements the functions.

Fisher and Bayes-Fisher information measures for finite mixture distributions

. In this work, we consider Fisher information and Bayes-Fisher information measures for the mixing parameter vector of a finite mixture density function and develop some associated results for this model. We provide several interesting connections between these measures and some known informational measures, such as chi-square divergence, Shannon entropy, Kullback-Leibler, Jeffreys, and Jensen-Shannon divergences. Finally, to demonstrate the usefulness of the Bayes-Fisher information measure, we apply it to a real example in image processing and present some numerical results. Our findings in this regard show that this information measure is an effective criteria for quantifying the similarity between two images.

Investigating the Impact of Xylella Fastidiosa on Olive Trees by the Analysis of MODIS Terra Satellite Evapotranspiration Time Series by Using the Fisher Information Measure and the Shannon Entropy: A Case Study in Southern Italy

Xylella Fastidiosa has been recently detected for the first time in southern Italy, representing a very dangerous phytobacterium capable of inducing severe diseases in many plants. In particular, the disease induced in olive trees is called olive quick decline syndrome (OQDS), which provokes the rapid desiccation and, ultimately, death of the infected plants. In this paper, we analyse about two thousands pixels of MODIS satellite evapotranspiration time series, covering infected and uninfected olive groves in southern Italy. Our aim is the identification of Xylella Fastidiosa-linked patterns in the statistical features of evapotranspiration data. The adopted methodology is the well-known Fisher–Shannon analysis that allows one to characterize the time dynamics of complex time series by means of two informational quantities, the Fisher information measure (FIM) and the Shannon entropy power (SEP). On average, the evapotranspiration of Xylella Fastidiosa-infected sites is characterized by a larger SEP and lower FIM compared to uninfected sites. The analysis of the receiver operating characteristic curve suggests that SEP and FIM can be considered binary classifiers with good discrimination performance that, moreover, improves if the yearly cycle, very likely linked with the meteo-climatic variability of the investigated areas, is removed from the data. Furthermore, it indicated that FIM exhibits superior effectiveness compared to SEP in discerning healthy and infected pixels.

Some insights on the COVID-19 pandemic from Fisher information

We explored the application of Fisher information to the study of pandemics and illustrated the insights that can be gained using the COVID-19 pandemic, as a test case. To do so, we applied the Fisher information theory previously applied to periodic systems, to non-periodic dynamic systems. The resulting mathematical machinery was then used to compute the Fisher information measure, as the amount of information extracted from the time series for COVID-19 confirmed infections and deaths. The analysis was performed for the World as a whole and five nation-states: India, USA, Japan, Germany, and Chile. Several insights resulted from the study: (1) the information content of the time series varied widely for different time periods, over the course of the pandemic, (2) it is advisable not to fit model parameters or make policy decisions based on data from time periods with low Fisher information, (3) the most information about a wave of infections comes towards the end of the wave where the time series data has the most information about the dynamics of the pandemic, and (4) the quality of the time series data significantly affects the Fisher information value, and, therefore, what can be learned from studying the time series.

Daily Streamflow of Argentine Rivers Analysis Using Information Theory Quantifiers.

This paper analyzes the temporal evolution of streamflow for different rivers in Argentina based on information quantifiers such as statistical complexity and permutation entropy. The main objective is to identify key details of the dynamics of the analyzed time series to differentiate the degrees of randomness and chaos. The permutation entropy is used with the probability distribution of ordinal patterns and the Jensen-Shannon divergence to calculate the disequilibrium and the statistical complexity. Daily streamflow series at different river stations were analyzed to classify the different hydrological systems. The complexity-entropy causality plane (CECP) and the representation of the Shannon entropy and Fisher information measure (FIM) show that the daily discharge series could be approximately represented with Gaussian noise, but the variances highlight the difficulty of modeling a series of natural phenomena. An analysis of stations downstream from the Yacyretá dam shows that the operation affects the randomness of the daily discharge series at hydrometric stations near the dam. When the station is further downstream, however, this effect is attenuated. Furthermore, the size of the basin plays a relevant role in modulating the process. Large catchments have smaller values for entropy, and the signal is less noisy due to integration over larger time scales. In contrast, small and mountainous basins present a rapid response that influences the behavior of daily discharge while presenting a higher entropy and lower complexity. The results obtained in the present study characterize the behavior of the daily discharge series in Argentine rivers and provide key information for hydrological modeling.

THE FRACTIONAL DISEN–FISHER PLANE: AN EFFECTIVE APPROACH TO DISTINGUISH COMPLEX TIME SERIES

With the explosive growth of data quantity and rapid development of nonlinear dynamics as well as the growing demand for complex data classification in the field of artificial intelligence and machine learning, the research of complex time series, generated by complex systems, has attracted enormous interests. However, how to simultaneously distinguish different types of time series data and extract more accurate and detailed information from them in the light of localized and global scale perspectives remains significant and needs to be tackled. Thus, in this paper, we propose the fractional DisEn–Fisher plane, which is innovatively constructed by the fractional form of dispersion entropy and Fisher information measure. These are both effective tools to diagnose the essential properties of complex time series, to analyze the complexity of systems and to depict the contained statistical information with higher accuracy and effectiveness. Several classical entropy plane methods are selected as a comparison to design simulation experiments by simulated data and three real-world datasets. Comparative experimental results show that this method is a feasible and reliable improvement method, which will help to provide more additional information in time series recognition and dynamic characterization. It may not only provide new insights for further improvement of complex time series analysis, but also have important implications for developing complex data clustering methods.

Fisher-Shannon Investigation of the Effect of Nonlinearity of Discrete Langevin Model on Behavior of Extremes in Generated Time Series.

Diverse forms of nonlinearity within stochastic equations give rise to varying dynamics in processes, which may influence the behavior of extreme values. This study focuses on two nonlinear models of the discrete Langevin equation: one with a fixed diffusion function (M1) and the other with a fixed marginal distribution (M2), both characterized by a nonlinearity parameter. Extremes are defined according to the run theory with thresholds based on percentiles. The behavior of inter-extreme times and run lengths is examined by employing Fisher's Information Measure and the Shannon Entropy. Our findings reveal a clear relationship between the entropic and informational measures and the nonlinearity of model M1-these measures decrease as the nonlinearity parameter increases. Similar relationships are evident for the M2 model, albeit to a lesser extent, even though the background data's marginal distribution remains unaffected by this parameter. As thresholds increase, both the values of Fisher's Information Measure and the Shannon Entropy also increase.

Jeffreys Divergence and Generalized Fisher Information Measures on Fokker-Planck Space-Time Random Field.

In this paper, we present the derivation of Jeffreys divergence, generalized Fisher divergence, and the corresponding De Bruijn identities for space-time random field. First, we establish the connection between Jeffreys divergence and generalized Fisher information of a single space-time random field with respect to time and space variables. Furthermore, we obtain the Jeffreys divergence between two space-time random fields obtained by different parameters under the same Fokker-Planck equations. Then, the identities between the partial derivatives of the Jeffreys divergence with respect to space-time variables and the generalized Fisher divergence are found, also known as the De Bruijn identities. Later, at the end of the paper, we present three examples of the Fokker-Planck equations on space-time random fields, identify their density functions, and derive the Jeffreys divergence, generalized Fisher information, generalized Fisher divergence, and their corresponding De Bruijn identities.

Deng–Fisher information measure and its extensions: Application to Conway’s Game of Life

The purpose of this work is to introduce Deng–Fisher information (DFI), Deng–Fisher information distance (DFID) and Jensen–Deng–Fisher (JDF) information distance measures based on the basic probability assignment concept. We also present results associated with these proposed information measures and examine DFI and DFID measures for escort of basic probability assignment functions. For illustrative purpose, we examined Conway’s Game of Life cellular automaton and present numerical results in terms of the proposed information measures. Results indicate that JDF information distance measures living cell population dynamics along time.

Urban and Peri-Urban Vegetation Monitoring Using Satellite MODIS NDVI Time Series, Singular Spectrum Analysis, and Fisher–Shannon Statistical Method

The purpose of this work was to evaluate the potential of Singular Spectrum Analysis (SSA) and the Fisher–Shannon method to analyse NDVI MODIS time series and to capture and estimate inner vegetation anomalies in forest covers. In particular, the Fisher–Shannon method allows to calculate two quantities, the Fisher Information Measure (FIM) and the Shannon entropy power (SEP), which are used to characterise the complexity of a time series in terms of organisation/disorder. Pilot sites located both in urban (Milano, Torino, and Roma) and peri-urban areas (Appia Park, Castel Porziano, and Castel Volturno) were selected. Among the six sites, Roma, Castel Porziano, and Castel Volturno are affected by the parasite Toumeyella parvicornis. The time series was analysed using the products available in Google Earth Engine. To explore and characterise long-term vegetation dynamics, the time series was analysed using a multistep processing chain based on the (i) normalisation of the satellite time series, (ii) removal of seasonality and any other periodical cycles using SSA, (iii) analysis of the de-trended data using the Fisher–Shannon statistical method, and (iv) validation through comparison with independent data and ancillary information. Our findings point out to a clear discrimination between healthy and unhealthy sites, being the first (Milano, Torino, Appia) characterised by a larger FIM (lower SEP) and the second (Roma, Castel Porziano, Castel Volturno) by a lower FIM (larger SEP). The results of the investigations showed that the use of the SSA and Fisher–Shannon statistical methods coupled with the NDVI time series of the MODIS satellite made it possible to effectively identify and characterise subtle but physically significant signals veiled by seasonality and annual cycles.

Jensen–Fisher information and Jensen–Shannon entropy measures based on complementary discrete distributions with an application to Conway’s game of life

Several information and divergence measures existing in the literature assist in measuring the knowledge contained in sources of information. Studying an information source from both positive and negative aspects will result in more accurate and comprehensive information. In many cases, extracting information through the positive approach could be not an easy task while it may be feasible when dealing with the negative aspect. Negation is a new perspective and direction to quantify the information or knowledge in a given system from the negative approach. In this work, we study some new information measures, such as Fisher information, Fisher information distance, Jensen–Fisher information and Jensen–Shannon entropy measures, based on complementary distributions. We then show that the proposed Jensen–Fisher information measure can be expressed based on Fisher information distance measure. We have further shown that the Jensen–Shannon entropy measure has two representations in terms of Kullback–Leibler divergence and Jensen–extropy measures. Some illustrations related to complementary distribution of Bernoulli and Poisson random variables are then presented. Finally, for illustrative purpose, we have examined a real example on Conway’s game of life and have presented some numerical results in terms of the proposed information measures.

Discerning Xylella fastidiosa-Infected Olive Orchards in the Time Series of MODIS Terra Satellite Evapotranspiration Data by Using the Fisher–Shannon Analysis and the Multifractal Detrended Fluctuation Analysis

Xylella fastidiosa is a phytobacterium able to provoke severe diseases in many species. When it infects olive trees, it induces the olive quick decline syndrome that leads the tree to a rapid desiccation and then to the death. This phytobacterium has been recently detected in olive groves in southern Italy, representing an important threat to the olive growing of the area. In this paper, in order to identify patterns revealing the presence of Xylella fastidiosa, several hundreds pixels of MODIS satellite evapostranspiration covering infected and healthy olive groves in southern Italy were analyzed by means of the Fisher–Shannon method and the multifractal detrended fluctuation analysis. The analysis of the receiver operating characteric curve indicates that the two informational quantities (the Fisher information measure and the Shannon entropy) and the three multifractal parameters (the range of generalized Hurst exponents and the width and the maximum of the multifractal spectrum) are well suited to discriminate between infected and healthy sites, although the maximum of the multifractal spectrum performs better than the others. These results could suggest the use of both the methods as an operational tool for early detection of plant diseases.

Fisher information and its extensions based on infinite mixture density functions

In this work, we consider the Fisher information and some of its well-known extended versions and then establish some results based on infinite mixture density functions for the proposed information measures. Specifically we introduce the Jensen-type of Fisher information (parametric-type and density-based) and generalized Fisher information measures based on infinite mixture density functions. We then show that the proposed Jensen–Fisher information measure have two representations in terms of Fisher information distance in the both parametric-type and density-based cases. We have further shown that the generalized Fisher information of an infinite mixture density can be stated based on Pearson–Vajda χk divergence. Finally, we examine the convexity property of q−Fisher information measure based on finite mixture density functions under a mild condition. Some examples related to scale mixture of skew-normal family of distributions, with emphasis on skew-normal density, are illustrated within the paper.

Scientific progress in information theory quantifiers

The Bandt and Pompe method (BPM) has been successfully applied to estimate the information theory quantifiers. The most significant limitation of the BPM is that this approach is based on point estimation. This research provides scientific progress into the several applications of the information theory quantifiers. In this way, we propose a new statistic confidence interval estimation method for the Permutation Entropy, Fisher Information Measure, and Macroeconophysics Indicator of Efficiency (MIEE). Therefore, we examine quarterly Gross Domestic Product (GDP) time series for 25 countries members of the Organization for Economic Co-operation and Development (OECD). The periods cover more than 59 years, from Q1-1960 to Q3-2019, with 238 data points. For each GDP time series, we employ the non-parametric method Local Block Bootstrap to provide a stylized fact about an empirical distribution for these data taking into account the information theory quantifiers. Based on the values of the empirical distribution of the information theory quantifiers, we construct the Shannon-Fisher causality plane (SFCP), which allows us to quantify the disorder and evaluate randomness present in the time series of quarterly GDP in each country. In addition, we use information theory quantifiers to rank the most efficient countries in allocating the resources arising from economic growth to generate a virtuous cycle of growth. Also, we apply the Principal Component Analysis (PCA) to check the results of the empirical distribution of the MIEE. We find the PCA promotes the same result as the MIEE. Our results suggest an alternative form of clustering using information theory quantifiers linked to the similarity of countries’ GDPs and the mutual influences of one country’s economic growth on the others.

ASSESSMENT OF SECTOR BOND, EQUITY INDICES AND GREEN BOND INDEX USING INFORMATION THEORY QUANTIFIERS AND CLUSTERS TECHNIQUES

Green bonds are financial assets similar to classic debt securities used to finance sustainable investments. Given this, they are a long-term investment alternative that effectively contributes to the planet’s future by preserving the environment and encouraging sustainable development. This research encompasses a rich dataset of equity and bond sectors, general indices, and the S&P Green Bond Index. We estimate the permutation entropy [Formula: see text], an appropriate statistical complexity measure [Formula: see text], and Fisher Information measure [Formula: see text]. Therefore, we employ these complexity measures to construct two 2D maps, the complexity-entropy causality plane ([Formula: see text] ×[Formula: see text]) and the Shannon–Fisher causality plane ([Formula: see text] ×[Formula: see text]). Also, we use the information theory quantifiers to rank these indices’ efficiency analogous to the complexity hierarchy. From a mathematical point of view, the complexity-entropy causality plane (CECP) is a map that considers the global analysis, while the SFCP is a map that simultaneously feels the global and local analysis. Our findings reveal that both 2D maps indicated the most efficient (b_info_tech) and least efficient (b_energy) assets. There are peculiarities in the ranking performed considering the information theory quantifiers used to build each map due to the mathematical distinction that underlies the construction of each map. Moreover, we applied two clustering approaches ([Formula: see text]-means and Hierarchical cluster) that categorically converged in the indication of four distinct groups, which allowed us to verify that, in an overview, equities present a unique dynamic when compared to bonds and the Green bond index.

Space-time analysis of informational properties of GPS time series recorded at the Campi Flegrei caldera (Italy)

In this work the time dynamics of GPS time series recorded at the Campi Flegrei caldera from 2000 to 2019 was investigated by using the Fisher Information Measure (FIM) and the Shannon entropy power (SEP), two informational methods that allow the detection of changes in the dynamical behavior of a complex system, like the volcanic one, quantifying respectively the order/organization and the disorder/uncertainty. By jointly analyzing the FIM and the SEP through the Fisher-Shannon Information Plane (FSIP) it was shown that the SEP discriminates quite well the vertical displacement time series, while the FIM discriminates better the horizontal ones, which are featured by a more complex behavior. Globally, the most striking change in the SEP is found in 2012, while at local scale abrupt loss of order (indicated by low values of FIM) well correlate with the occurrence of seismic swarms. A further informational quantity, the complexity C (given by the product between SEP and FIM), has been revealed to be sensitive to the baseline deformation level increases and to the small-amplitude modulation of deformation signals within the caldera.

Exploring Long-Term Anomalies in the Vegetation Cover of Peri-Urban Parks Using the Fisher-Shannon Method.

The main goal of this study was to evaluate the potential of the Fisher-Shannon statistical method applied to the MODIS satellite time series to search for and explore any small multiyear trends and changes (herein also denoted as inner anomalies) in vegetation cover. For the purpose of our investigation, we focused on the vegetation cover of three peri-urban parks close to Rome and Naples (Italy). For each of these three areas, we analyzed the 2000-2020 time variation of four MODIS-based vegetation indices: evapotranspiration (ET), normalized difference vegetation index (NDVI), leaf area index (LAI), and enhanced vegetation index (EVI). These data sets are available in the Google Earth Engine (GEE) and were selected because they are related to the interactions between soil, water, atmosphere, and plants. To account for the great variability exhibited by the seasonal variations while identifying small multiyear trends and changes, we devised a procedure composed of two steps: (i) application of the Singular Spectrum Analysis (SSA) to each satellite-based time series to detect and remove the annual cycle including the seasonality and then (ii) analysis of the detrended signals using the Fisher-Shannon method, which combines the Shannon entropy and the Fisher Information Measure (FIM). Our results indicate that among all the three pilot test areas, Castel Volturno is characterized by the highest Shannon entropy and the lowest FIM that indicate a low level of order and organization of vegetation time series. This behaviour can be linked to the degradation phenomena induced by the parasite (Toumeyella parvicornis) that has affected dramatically the area in recent years. Our results were nicely confirmed by the comparison with in situ analyzed and independent data sets revealing the existence of subtle, small multiyear trends and changes in MODIS-based vegetation indices.