Multivariate Cauchy Distribution Research Articles

Directional distribution of the pseudo intensity vector in anisotropic late reverberation.

The pseudo intensity vector (PIV) is often used to analyze the directional properties of spatial room impulse responses. In the early part of the response, it is capable of estimating the directions of individual reflections. However, thus far, its behaviour in the late field is unclear. Specifically, it is unknown whether anisotropy, i.e., a direction-dependent energy distribution, is captured by the directional estimates. In this study, a closed-form expression of the directional distribution of the pressure-normalized pseudo intensity vector contingent on a general stochastical model of anisotropic fields was analytically derived. This paper shows that the probability density function of this PIV is a multivariate Cauchy distribution, which does indeed depend on the energy distribution of the field, yet the directional distribution has very limited degrees of freedom. The derived distribution is compared to the results of Monte Carlo simulations and fields captured with a microphone array in a real room. These results facilitate better understanding of the behaviour of parametric spatial room impulse response methods and may enable improved directional estimators for anisotropic fields.

A Generic Formula and Some Special Cases for the Kullback-Leibler Divergence between Central Multivariate Cauchy Distributions.

This paper introduces a closed-form expression for the Kullback–Leibler divergence (KLD) between two central multivariate Cauchy distributions (MCDs) which have been recently used in different signal and image processing applications where non-Gaussian models are needed. In this overview, the MCDs are surveyed and some new results and properties are derived and discussed for the KLD. In addition, the KLD for MCDs is showed to be written as a function of Lauricella D-hypergeometric series . Finally, a comparison is made between the Monte Carlo sampling method to approximate the KLD and the numerical value of the closed-form expression of the latter. The approximation of the KLD by Monte Carlo sampling method are shown to converge to its theoretical value when the number of samples goes to the infinity.

Analytical solution of stochastic real‐time dispatch incorporating wind power uncertainty characterized by Cauchy distribution

Real-time power dispatch can coordinate wind farms, automatic generation control units and non-automatic generation control units. In real-time power dispatch, the probable wind power forecast errors should be appropriately formulated to ensure system security with high probability and minimize operational cost. Previous studies and the authors' onsite tests show that Cauchy distribution effectively fits the “leptokurtic” feature of small-timescale wind power forecast errors distributions. In this paper, a chance-constrained real-time dispatch model with the wind power forecast errors represented by multivariate Cauchy distribution is proposed. Since the Cauchy distribution is stable and has promising mathematical characteristics, the proposed chance-constrained real-time dispatch model can be analytically transformed to a convex optimization problem considering the dependence among wind farms’ outputs. Moreover, the proposed model incorporates an affine control strategy compatible with automatic generation control systems. This strategy makes the chance-constrained real-time dispatch adaptively take into account both the potential power ramping requirement and power variation on transmission lines caused by the generation adjustment to offset the wind power forecast errors in real-time power dispatch stage. Numerical test results show that the proposed method is reliable and effective. Meanwhile it is very efficient and suitable for real-time application.

On -characteristic functions and applications to asymptotic statistical inference

ABSTRACTIn this paper, we give emphasis to a method to do statistical inference and to study properties of random variables, whose probability density functions (pdfs) do not possess good regularity, decay, and integrability properties. The main tool will be what we will call -characteristic function, a generalization of the classical characteristic function that is basically a measurable transform of pdfs. In this perspective and using this terminology, we will restate and prove theorems, such as the law of large numbers and the central limit theorem that now, after this measurable transform, apply to basically every distribution, upon the correct choice of a free parameter α. We apply this theory to hypothesis testing and to the construction of confidence intervals for location parameters. We connect the classical parameters of a distribution to their related 1/α-counterparts, that we will call -momenta. We treat in detail the case of the multivariate Cauchy distribution for which we compute explicitly all the -expected values and -variances in dimension n = 1 and for which we construct an approximate confidence interval for the location parameter μ, by means of asymptotic theorems in the -context. Among the other things and to illustrate the usefulness of this point of view, we prove some new characterizations of the Poisson distribution, the uniform discrete, and the uniform continuous distribution.

A Robust Multiplicative Watermark Detector for Color Images in Sparse Domain

In recent years, digital watermarking has facilitated the protection of copyright information through embedding hidden information into the digital content. In this brief, for the first time, a blind multichannel multiplicative color image watermarking scheme in the sparse domain is proposed. In order to take into account the cross correlation between the coefficients of the color bands in the sparse domain, a statistical model based on the multivariate Cauchy distribution is used. The statistical model is then used to derive an efficient closed-form decision rule for the watermark detector. Experimental results and theoretical analysis are presented to validate the proposed watermark detector. The performance of the proposed detector is compared with that of the other detectors. The results demonstrate the improved detection rate and high robustness against the commonly used attacks such as JPEG compression, salt and pepper noise, median filtering, and Gaussian noise.

Fast simulated annealing with a multivariate Cauchy distribution and the configuration’s initial temperature

We propose a multi-dimensional fast simulated annealing method based on a multivariate Cauchy probability distribution and an initial temperature estimated from the configuration’s variation. While conventional multi-dimensional fast simulated annealing adopts the product of onedimensional random variables generated by a univariate Cauchy distribution, the proposed method generates a random vector from a multivariate Cauchy distribution. In this way, fast simulated annealing for a multi-dimensional problem maintains the same annealing schedule as that for the one-dimensional case. The proposed method also utilizes the initial temperature estimated from the configuration’s variation to generate a candidate state in addition to the conventional initial temperature derived from the variation of the objective function for the acceptance probability. The proposed method is shown not only to guarantee a fast annealing schedule but also to enhance the search capability. The proposed method was tested against the optimization of real-valued functions. We empirically found that the configuration’s initial temperature, together with multivariate Cauchy distribution, is more suitable than the conventional scheme for a fast annealing schedule. Moreover, the proposed method outperforms the conventional one in optimization problems having many variables.

Explosive strong periodic autoregression with multiplicity one

It is well known that the asymptotic distribution of the ordinary least squares estimate (OLSE) for a strong periodic autoregression (PAR) driven by an independent innovation is Gaussian under periodic stationarity and functional of the Brownian motion under periodic integration. This paper studies the asymptotic distribution of OLSE for strong pth order PAR(p) models in the explosive case with multiplicity one, i.e. when a certain autoregressive parameter lies outside both the periodic stationarity domain and its boundary, while all remaining parameters are inside the periodic stationarity region. It will be shown that the OLSE for PAR(1), scaled by a function of the monodromy parameter, converges in distribution to a random vector whose distribution reduces under the normality assumption to the standard multivariate Cauchy distribution. Furthermore, for a PAR(p) model, the OLSE, scaled by a function of the design matrix of the Gaussian model, is asymptotically Gaussian with independent components. Thus, the knife edge effect known for linear time-invariant AR models is still valid in the strong periodically time-varying case.

A Clarification of the Cauchy Distribution

We define a multivariate Cauchy distribution using a probability density function; subsequently, a Ferguson's definition of a multivariate Cauchy distribution can be viewed as a characterization theorem using the characteristic function approach. To clarify this characterization theorem, we construct two dependent Cauchy random variables, but their sum is not Cauchy distributed. In doing so the proofs depend on the characteristic function, but we use the cumulative distribution function to obtain the exact density of their sum. The derivation methods are relatively straightforward and appropriate for graduate level statistics theory courses.

Convex and star-shaped sets associated with multivariate stable distributions, I: Moments and densities

It is known that each symmetric stable distribution in R d is related to a norm on R d that makes R d embeddable in L p ( [ 0 , 1 ] ) . In the case of a multivariate Cauchy distribution the unit ball in this norm is the polar set to a convex set in R d called a zonoid. This work interprets symmetric stable laws using convex or star-shaped sets and exploits recent advances in convex geometry in order to come up with new probabilistic results for multivariate symmetric stable distributions. In particular, it provides expressions for moments of the Euclidean norm of a stable vector, mixed moments and various integrals of the density function. It is shown how to use geometric inequalities in order to bound important parameters of stable laws. Furthermore, covariation, regression and orthogonality concepts for stable laws acquire geometric interpretations.

Angular Gaussian and Cauchy estimation

This paper proposes a unified treatment of maximum likelihood estimates of angular Gaussian and multivariate Cauchy distributions in both the real and the complex case. The complex case is relevant in shape analysis. We describe in full generality the set of maxima of the corresponding log-likelihood functions with respect to an arbitrary probability measure. Our tools are the convexity of log-likelihood functions and their behaviour at infinity.

A note on the maximum likelihoood estimators for the location and scatter parameters of a multivariate cauchy distribution

The likelihood function of a multivariate Cauchy distribution with unknown location and scatter parameters in ρ dimensions has a unique maximum provided the sample is in general position and is of size n > p + 1 (see Kent and Tyler 1991 and Kent et al., 1994). In this paper, we show that when n=p+l, the maximum of the locationscatter Cauchy likelihood function occurs when the parameters lie on a ρ–dimensional surface.

Local convergence rates of simple evolutionary algorithms with Cauchy mutations

The standard choice for mutating an individual of an evolutionary algorithm with continuous variables is the normal distribution; however other distributions, especially some versions of the multivariate Cauchy distribution, have recently gained increased popularity in practical applications. Here the extent to which Cauchy mutation distributions may affect the local convergence behavior of evolutionary algorithms is analyzed. The results show that the order of local convergence is identical for Gaussian and spherical Cauchy distributions, whereas nonspherical Cauchy mutations lead to slower local convergence. As a by-product of the analysis, some recommendations for the parametrization of the self-adaptive step size control mechanism can be derived.

Multivariate cauchy distributions as locally gaussian distributions

In the present paper, we propose a definition of locally Gaussian probability distributions of random vectors based on the linearization of their conditional quantiles. We prove that the Cauchy distribution inR n is locally Gaussian and give explicit formulas for the vectors of expectations and covariance matrices of locally Gaussian approximations. We show that locally Gaussian approximations with different dimensionalities are in some sense compatible: all of them have equal corresponding correlation coefficients. For the Cauchy distribution in a Hilbert space we prove a limit theorem on the convergence of squared finite-dimensional conditional quantiles to the stable Lévy distribution.

Inference Based on Ranks for the Multiple-Design Multivariate Linear Model

Abstract Multivariate models more general than the standard multivariate linear model have received considerable attention in both the statistical and econometric literature; see Srivastava (1966, 1967, 1968) and Kleinbaum (1973). The multiple-design multivariate (MDM) linear model generalizes the standard multivariate linear model in the sense that a different design matrix is used for each response (dependent) variable. Important applications of the MDM linear model include changing covariates in repeated measurement designs and p-variate regression systems (p > 1) with different regressors for each of the p response variables. The latter situation has been described as “seemingly unrelated regression equations” in the econometric literature (Zellner 1962). In this article an approach to statistical inference is presented under the MDM linear model based on multivariate rank and aligned rank statistics. In Section 1 estimation and hypothesis testing for the MDM linear model are reviewed under parametric assumptions about the underlying multivariate distribution, such as a multivariate normal distribution or a continuous multivariate distribution that has a finite positive-definite variance—covariance matrix. In Sections 2 and 3 an approach to statistical inference based on ranks for the MDM linear model is presented. In contrast to the parametric assumptions, this approach requires only that the continuous multivariate distribution has a finite Fisher information matrix, which is less restrictive than a finite variance—covariance matrix. Thus, unlike the parametric analysis, heavy-tailed distributions without finite moments pose no problems in the rank approach. In particular, a multivariate linear rank statistic is constructed, and its asymptotic multivariate normality under a simple null hypothesis and a sequence of contiguous alternatives is established. These results lead to an asymptotic test of a simple null hypothesis and an expression for its asymptotic power. An R estimate of the parameter vector is derived via aligned rank statistics, and its asymptotic multivariate normality is established. A Wald-type statistic for an asymptotic test of a general linear hypothesis is constructed, and an expression for its asymptotic power based on a sequence of contiguous alternatives is developed. In Section 4 an example is presented in which the parametric and rank approaches are contrasted. The example consists of a pharmaceutical study with a four-way crossover design. To illustrate the nonrobustness of the parametric approach, the data set is reexamined after the introduction of random errors from multivariate normal and multivariate Cauchy distributions.

Best Median-Unbiased Estimation in Linear Regression with Bounded Asymmetric Loss Functions

We first show that the Generalized Least Squares estimator is the best median unbiased estimator of the regression parameters for quite general loss functions, when the parameter space is unrestricted. Of note is the fact that this result holds without moment restrictions. Thus, the errors may have multivariate Cauchy distribution. Next, we show that a restricted GLS estimator is best median unbiased for a linear combination of the regression parameters, when that linear combination is restricted to lie in an interval. Certain other linear combinations of the parameter vector may be subject to arbitrary additional restrictions. The paper then presents best median unbiased estimators of the error variance sigma-squared, as well as monotone functions of sigma-squared, when the errors are normally distributed. If sigma-squared is constrained to lie in a finite interval, the best estimator is a censored version of its unconstrained counterpart. When sigma-square is constrained only to be positive, the best median unbiased estimator is always larger than the best mean unbiased estimator s-squared, and is approximately equal to s-squared calculated with its degrees of freedom reduced by .66.

Some properties of mutivariate distributions with pdf's constant on ellipsoids

For random variables X1...., XK+1 with unbounded support, having a joint pdf constant on ellipsoids, the joint distribution of the ratios Yi = Xi / Xk+1(1 ≤ i ≤ k) is obtained. If the Xi's have a spherically symmetric pdf. the Yi's have a multivariate Cauchy distribution. The condition of unbounded support of the Xi's is not necessary, although it cannot always be relaxed.