scholarly journals Heavy-tailed random matrices

2018 ◽  
Author(s):  
Vladimir Kravtsov
Keyword(s):  
Random Matrices ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Basin Of Attraction ◽  
Random Variables ◽  
Universal Properties ◽  
Random Matrix Ensembles ◽  
Free Random Variables ◽  
Heavy Tailed ◽  
Non Gaussian

This article considers non-Gaussian random matrices consisting of random variables with heavy-tailed probability distributions. In probability theory heavy tails of distributions describe rare but violent events which usually have a dominant influence on the statistics. Furthermore, they completely change the universal properties of eigenvalues and eigenvectors of random matrices. This article focuses on the universal macroscopic properties of Wigner matrices belonging to the Lévy basin of attraction, matrices representing stable free random variables, and a class of heavy-tailed matrices obtained by parametric deformations of standard ensembles. It first examines the properties of heavy-tailed symmetric matrices known as Wigner–Lévy matrices before discussing free random variables and free Lévy matrices as well as heavy-tailed deformations. In particular, it describes random matrix ensembles obtained from standard ensembles by a reweighting of the probability measure. It also analyses several matrix models belonging to heavy-tailed random matrices and presents methods for integrating them.

scholarly journals Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

2008 ◽  
Vol 45 (02) ◽  
pp. 531-541 ◽  
Author(s):  
A. G. Rossberg
Keyword(s):  
Laplace Transform ◽  
High Precision ◽  
Computational Efficiency ◽  
Generating Functions ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Random Variables ◽  
Numerical Examples ◽  
Pareto Distributions ◽  
The Laplace Transform

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.

Investigating rainfall and catchment attributes promoting heavy-tailed distributions of river flows

2021 ◽  
Author(s):  
Hsing-Jui Wang ◽  
Soohyun Yang ◽  
Ralf Merz ◽  
Stefano Basso
Keyword(s):  
Heavy Tails ◽  
Root Zone ◽  
Probability Distributions ◽  
River Basins ◽  
Model Soil ◽  
Rainfall Frequency ◽  
Heavy Tailed Distributions ◽  
Physical Attributes ◽  
Heavy Tailed ◽  
Stochastic Properties

<p>Heavy-tailed probability distributions of streamflow are frequently observed in river basins. They indicate sizable odds of extreme events in these catchments and thus signal the existence of enhanced hydrological perils. Notwithstanding their relevance for characterizing the hydrological hazard of river basins, identifying specific mechanisms which promote the emergence of heavy-tailed flow distributions has proved challenging due to the complex hydrological response of such dynamical systems exposed to highly variable rainfall inputs.</p><p>In this study we combine a continuous hydrological model grounded on the geomorphological theory of the hydrologic response with archetypical descriptions of the spatial and temporal distributions of rainfall inputs and catchment attributes to investigate physical mechanisms and stochastic features leading to the emergence of heavy tails.</p><p>In the model, soil moisture dynamics driven by the water balance in the root zone trigger superficial and subsurface runoff contributions, which are routed to the catchment outlet by means of a representation of transport by travel time distributions. The framework enables a parsimonious distributed description of hydrological processes, suitably considered with their stochastic character, and is thus fit for the goal of investigating manifold mechanisms promoting heavy-tailed streamflow distributions.</p><p>A set of archetypical spatial and temporal variabilities of rainfall inputs and catchment attributes (e.g., localized versus uniform rainfall in the catchment, lumped versus distributed catchment attributes, mainly upstream versus downstream source areas, high versus low rainfall frequency) are finally imposed in the model and their capability (or not) to affect the tail of the streamflow distribution is investigated.</p><p>The proposed framework provides a way to disentangle physical attributes of river catchments and stochastic properties of hydroclimatic variables which control the emergence of heavy-tailed streamflow distributions and thus identify the key drivers of the inherent hydrological hazard of river basins.</p>

scholarly journals From Synaptic Interactions to Collective Dynamics in Random Neuronal Networks Models: Critical Role of Eigenvectors and Transient Behavior

2020 ◽  
Vol 32 (2) ◽  
pp. 395-423 ◽  
Author(s):  
E. Gudowska-Nowak ◽  
M. A. Nowak ◽  
D. R. Chialvo ◽  
J. K. Ochab ◽  
W. Tarnowski
Keyword(s):  
Neural Networks ◽  
Critical Role ◽  
Random Variables ◽  
Transient Behavior ◽  
Interaction Matrix ◽  
Synaptic Interaction ◽  
Human Connectome Project ◽  
Free Random Variables ◽  
Heavy Tailed ◽  
The Stability

The study of neuronal interactions is at the center of several big collaborative neuroscience projects (including the Human Connectome Project, the Blue Brain Project, and the Brainome) that attempt to obtain a detailed map of the entire brain. Under certain constraints, mathematical theory can advance predictions of the expected neural dynamics based solely on the statistical properties of the synaptic interaction matrix. This work explores the application of free random variables to the study of large synaptic interaction matrices. Besides recovering in a straightforward way known results on eigenspectra in types of models of neural networks proposed by Rajan and Abbott ( 2006 ), we extend them to heavy-tailed distributions of interactions. More important, we analytically derive the behavior of eigenvector overlaps, which determine the stability of the spectra. We observe that on imposing the neuronal excitation/inhibition balance, despite the eigenvalues remaining unchanged, their stability dramatically decreases due to the strong nonorthogonality of associated eigenvectors. This leads us to the conclusion that understanding the temporal evolution of asymmetric neural networks requires considering the entangled dynamics of both eigenvectors and eigenvalues, which might bear consequences for learning and memory processes in these models. Considering the success of free random variables theory in a wide variety of disciplines, we hope that the results presented here foster the additional application of these ideas in the area of brain sciences.

Fisher Information and Uncertainty Principle for Skew-Gaussian Random Variables

2021 ◽  
pp. 2150039
Author(s):  
Javier E. Contreras-Reyes
Keyword(s):  
Fisher Information ◽  
Shannon Entropy ◽  
Uncertainty Principle ◽  
Gaussian Distribution ◽  
Heavy Tails ◽  
Random Variables ◽  
Gaussian Random Variables ◽  
Evolving Systems ◽  
Non Gaussian ◽  
Estimate System

Fisher information is a measure to quantify information and estimate system-defining parameters. The scaling and uncertainty properties of this measure, linked with Shannon entropy, are useful to characterize signals through the Fisher–Shannon plane. In addition, several non-gaussian distributions have been exemplified, given that assuming gaussianity in evolving systems is unrealistic, and the derivation of distributions that addressed asymmetry and heavy–tails is more suitable. The latter has motivated studying Fisher information and the uncertainty principle for skew-gaussian random variables for this paper. We describe the skew-gaussian distribution effect on uncertainty principle, from which the Fisher information, the Shannon entropy power, and the Fisher divergence are derived. Results indicate that flexibility of skew-gaussian distribution with a shape parameter allows deriving explicit expressions of these measures and define a new Fisher–Shannon information plane. Performance of the proposed methodology is illustrated by numerical results and applications to condition factor time series.

scholarly journals Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

2008 ◽  
Vol 45 (2) ◽  
pp. 531-541 ◽  
Author(s):  
A. G. Rossberg
Keyword(s):  
Laplace Transform ◽  
High Precision ◽  
Computational Efficiency ◽  
Generating Functions ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Random Variables ◽  
Numerical Examples ◽  
Pareto Distributions ◽  
The Laplace Transform

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.

scholarly journals MIXED CAUSAL-NONCAUSAL AR PROCESSES AND THE MODELLING OF EXPLOSIVE BUBBLES

2019 ◽  
Vol 35 (6) ◽  
pp. 1234-1270 ◽  
Author(s):  
Sébastien Fries ◽  
Jean-Michel Zakoian
Keyword(s):  
Conditional Distribution ◽  
Stable Distribution ◽  
Financial Time Series ◽  
Domain Of Attraction ◽  
Real Data ◽  
Autoregressive Processes ◽  
Financial Time ◽  
Causal Representation ◽  
Heavy Tailed ◽  
Non Gaussian

Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and, therefore, provide a convenient framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. Extending the study of the noncausal AR(1) model by Gouriéroux and Zakoian (2017), we show that the conditional distribution in direct time is lighter-tailed than the errors distribution, and we emphasize the presence of ARCH effects in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a causal AR representation with non-i.i.d. errors can be consistently estimated by classical least-squares. We derive a portmanteau test to check the validity of the estimated AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal, or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.

Centered error entropy Kalman filter with application to satellite attitude determination

2021 ◽  
pp. 014233122110198
Author(s):  
Baojian Yang ◽  
Lu Cao ◽  
Dechao Ran ◽  
Bing Xiao
Keyword(s):  
Kalman Filter ◽  
Gaussian Noise ◽  
Attitude Determination ◽  
Complexity Analysis ◽  
Iteration Algorithm ◽  
Robust Algorithms ◽  
Convergence Conditions ◽  
Satellite Attitude ◽  
Heavy Tailed ◽  
Non Gaussian

Due to unavoidable factors, heavy-tailed noise appears in satellite attitude estimation. Traditional Kalman filter is prone to performance degradation and even filtering divergence when facing non-Gaussian noise. The existing robust algorithms have limited accuracy. To improve the attitude determination accuracy under non-Gaussian noise, we use the centered error entropy (CEE) criterion to derive a new filter named centered error entropy Kalman filter (CEEKF). CEEKF is formed by maximizing the CEE cost function. In the CEEKF algorithm, the prior state values are transmitted the same as the classical Kalman filter, and the posterior states are calculated by the fixed-point iteration method. The CEE EKF (CEE-EKF) algorithm is also derived to improve filtering accuracy in the case of the nonlinear system. We also give the convergence conditions of the iteration algorithm and the computational complexity analysis of CEEKF. The results of the two simulation examples validate the robustness of the algorithm we presented.

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