Showing papers on "K-distribution published in 2008"

Fisher Distribution for Texture Modeling of Polarimetric SAR Data

Abstract: The multilook polarimetric synthetic aperture radar (PolSAR) covariance matrix is generally modeled by a complex Wishart distribution. For textured areas, the product model is used, and the texture component is modeled by a Gamma distribution. In many cases, the assumption of Gamma-distributed texture is not appropriate. The Fisher distribution does not have this limitation and can represent a large set of texture distributions. As an example, we examine its advantage for an urban area. From a Fisher-distributed texture component, we derive the distribution of the complex covariance matrix for multilook PolSAR data. The obtained distribution is expressed in terms of the KummerU confluent hypergeometric function of the second kind. Those distributions are related to the Mellin transform and second-kind statistics (Log-statistics). The new KummerU-based distribution should provide in many cases a better representation of textured areas than the classic K distribution. Finally, we show that the new model can discriminate regions with different texture distribution in a segmentation experiment with synthetic textured PolSAR images.

Using SAR Images to Detect Ships From Sea Clutter

Abstract: An innovative constant false alarm rate (CFAR) algorithm was studied for ship detection using synthetic aperture radar (SAR) images of the sea. Two advances were achieved. An alpha-stable distribution rather than a traditional Weibull or -distribution was used to model the distribution of sea clutter. The distribution of sea clutter in a SAR image was typically heterogeneous, caused mainly by variable wind and current conditions. Image segmentation was carried out to improve the homogeneity of the distribution in each subimage or region. In comparison with ship detection using the CFAR algorithms based on the Weibull or K -distribution, our algorithm detected the most number of ships with the smallest number of false alarms.

Classification With a Non-Gaussian Model for PolSAR Data

Abstract: In this paper, we present a generalized Wishart classifier derived from a non-Gaussian model for polarimetric synthetic aperture radar (PolSAR) data. Our starting point is to demonstrate that the scale mixture of Gaussian (SMoG) distribution model is suitable for modeling PolSAR data. We show that the distribution of the sample covariance matrix for the SMoG model is given as a generalization of the Wishart distribution and present this expression in integral form. We then derive the closed-form solution for one particular SMoG distribution, which is known as the multivariate K-distribution. Based on this new distribution for the sample covariance matrix, termed as the K -Wishart distribution, we propose a Bayesian classification scheme, which can be used in both supervised and unsupervised modes. To demonstrate the effect of including non-Gaussianity, we present a detailed comparison with the standard Wishart classifier using airborne EMISAR data.

Lagrangian Probability Distributions

Abstract: * Lagrange Bio * Dedication * Foreword * Preface * List of Tables * Abbreviations * Preliminary Information * Lagrangian Probability Distributions * Properties of General Lagrangian Distributions * Quasi-Probability Models * Some Urn Models * Development of Models and Applications * Modified Power Series Distributions * Some Basic Lagrangian Distributions * Generalized Poisson Distribution * Generalized Negative Binomial Distribution * Generalized Logarithmic Series Distribution * Lagrangian Katz Distribution * Random Walks and Jumps Models * Bivariate Lagrangian Distributions * Multivariate Lagrangian Distributions * Computer Generation of Lagrangian Variables * References * Index

Outage probability and ergodic capacity of free-space optical links over strong turbulence

Abstract: The performance of a free-space optical (FSO) communication system under strong turbulence regime that follows the K distribution is evaluated. Some useful channel statistics are derived in closed form and the outage probability and ergodic capacity for a single-input single-output FSO link are evaluated. Numerical examples are further presented to verify the accuracy of the derived mathematical expressions.

Classification With a Non-Gaussian Model for PolSAR Data

Abstract: -In this paper, we present a generalized Wishart classifier derived from a non-Gaussian model for polarimetric synthetic aperture radar (PolSAR) data. Our starting point is to demonstrate that the scale mixture of Gaussian (SMoG) distribution model is suitable for modeling PolSAR data. We show that the distribution of the sample covariance matrix for the SMoG model is given as a generalization of the Wishart distribution and present this expression in integral form. We then derive the closed-form solution for one particular SMoG distribution, which is known as the multivariate K-distribution. Based on this new distribution for the sample covariance matrix, termed as the K-Wishart distribution, we propose a Bayesian classification scheme, which can be used in both supervised and unsupervised modes. To demonstrate the effect of including non-Gaussianity, we present a detailed comparison with the standard Wishart classifier using airborne EMISAR data.

Coherent states and Bayesian duality

Abstract: We demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states, using the concept of a reproducing kernel Hilbert space. Each family of coherent states is shown to contain, in a sort of duality, which resembles an analogous duality in Bayesian statistics, a discrete probability distribution and a discretely parametrized family of continuous distributions. It turns out that nonlinear coherent states, of the type widely studied in quantum optics, are a particularly useful class of coherent states from this point of view, in that they contain many of the standard statistical distributions. We also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions.

On the Correlated $K$ -Distribution With Arbitrary Fading Parameters

Abstract: The correlated bivariate K-distribution with arbitrary and not necessarily identical parameters is introduced and analyzed. Novel infinite series expressions for the joint probability density function and moments are derived for the general case where the associated bivariate distributions, i.e., Rayleigh and gamma, are both arbitrary correlated. These expressions generalize previously known analytical results obtained for identical parameter cases. Furthermore, considering independent gamma distributions, the cumulative distribution and characteristic functions are analytically obtained. Although the derived expressions can be used in a wide range of applications, this letter focuses on the performance analysis of dual branch diversity receivers. Specifically, the outage performance of dual selection diversity receivers operating over correlated K fading/shadowing channels is analytically evaluated. Moreover, for low normalized outage threshold values, closed-form expressions are obtained.

Quantifying uncertainty in the estimation of probability distributions.

Abstract: We consider ordinary least squares parameter estimation problems where the unknown parameters to be estimated are probability distributions. A computational framework for quantification of uncertainty (e.g., standard errors) associated with the estimated parameters is given and sample numerical findings are presented.

A new look at q-exponential distributions via excess statistics

Abstract: Q-exponential distributions play an important role in nonextensive statistics. They appear as the canonical distributions, i.e. the maximum generalized q-entropy distributions under mean constraint. Their relevance is also independently justified by their appearance in the theory of superstatistics introduced by Beck and Cohen. In this paper, we provide a third and independent rationale for these distributions. We indicate that q-exponentials are stable by a statistical normalization operation, and that Pickands’ extreme values theorem plays the role of a CLT-like theorem in this context. This suggests that q-exponentials can arise in many contexts if the system at hand or the measurement device introduces some threshold. Moreover we give an asymptotic connection between excess distributions and maximum q-entropy. We also highlight the role of Generalized Pareto Distributions in many applications and present several methods for the practical estimation of q-exponential parameters.

On Sample Size in Using Central Limit Theorem for Gamma Distribution

Abstract: A general criterion in using the central limit theorem is based on the sample size n > 30, no matter what the population is. Such only one generalized criterion may not be suitable for various shapes of probability distributions. This study is to check gamma distribution, one of asymmetric continuous distributions, how fit that criterion by computer simulation techniques, and finds out the least required sample sizes that satisfy the central limit theorem under different parameters of the gamma distribution.

A parameterized statistical sonar image texture model

Abstract: Single-point statistical properties of envelope-detected data such as signal returns from synthetic aperture radar and sonar have traditionally been modeled via the Rayleigh distribution and more recently by the K-distribution. Two-dimensional correlations that occur in textured non-Gaussian imagery are more difficult to model and estimate than Gaussian textures due to the nonlinear transformations of the time series data that occur during envelope detection. In this research, textured sonar imagery is modeled by a correlated K-distribution. The correlated K-distribution is explained via the compound representation of the one-dimensional K-distribution probability density function. After demonstrating the model utility using synthetically generated imagery, model parameters are estimated from a set of textured sonar images using a nonlinear least-squares fit algorithm. Results are discussed with regard to texture segmentation applications.

Joint probability distributions and multipoint correlations of the continuous-time random walk.

Abstract: We present an efficient method to determine the Fourier-Laplace transform of the joint n-point probability distribution of a continuous-time random walk for arbitrary finite n. Additionally, we devise a recursive procedure with which it is possible to calculate the Laplace transforms of the multipoint correlation functions without having to determine the joint probability distributions first. The methods are used on several examples with both independent and dependent distributions for the waiting time and the spatial step size.

Semi-self-decomposable distributions on Z+

Abstract: We present a notion of semi-self-decomposability for distributions with support in Z+. We show that discrete semi-self-decomposable distributions are infinitely divisible and are characterized by the absolute monotonicity of a specific function. The class of discrete semi-self-decomposable distributions is shown to contain the discrete semistable distributions and the discrete geometric semistable distributions. We identify a proper subclass of semi-self-decomposable distributions that arise as weak limits of subsequences of binomially thinned sums of independent Z+-valued random variables. Multiple semi-self-decomposability on Z+ is also discussed.

Transformation of bimodal probability distributions into possibility distributions

Abstract: At the application level, it is important to be able to define around the measurement result an interval which will contain an important part of the distribution of the measured values, that is, a coverage interval. This practice acknowledged by the ISO Guide is a major shift from the probabilistic representation. It can be viewed as a probability-possibility transformation by viewing possibility distribution as encoding coverage intervals. In this paper, we extend previous works on unimodal distributions by proposing a possibility representation of bimodal probability distributions. Indeed, U-shaped distributions or Gaussian mixture distribution are not so rare in a context of physical measurements.

On Certain Mixture Distributions Based on Lagrangian Probability Models

Abstract: In this paper, some bivariate probability distributions for a discrete random variable and a continuous random variable are defined by using the Lagrangian probability distributions. The covariance between the variables of the bivariate probability distribu- tion is obtained. From the bivariate probability distributions, we derive some mixture distributions. The moments of the mixture distributions are also discussed. Finally, we give some examples of the bivariate probability distributions and their corresponding mixture distributions.

Asymptotic Concentration Behaviors of Linear Combinations of Weight Distributions on Random Linear Code Ensemble

Abstract: Asymptotic concentration behaviors of linear combinations of weight distributions on the random linear code ensemble are presented. Many important properties of a binary linear code can be expressed as the form of a linear combination of weight distributions such as number of codewords, undetected error probability and upper bound on the maximum likelihood error probability. The key in this analysis is the covariance formula of weight distributions of the random linear code ensemble, which reveals the second-order statistics of a linear function of the weight distributions. Based on the covariance formula, several expressions of the asymptotic concentration rate, which indicate the speed of convergence to the average, are derived.

Application of the theory of truncated probability distributions to studying minimal river runoff: Normal and gamma distributions

Abstract: The possibility of applying unilaterally truncated probability distributions of minimal water flow is considered. Relationships between moment estimates of truncated and full distributions are considered for the case of normal and gamma distributions.

The Distributions of Sums, Products and Ratios of Inverted Bivariate Beta Distribution 1

Abstract: In this paper we derive the exact distributions of sums, products, and ratios of two random variables when they follow the bivariate inverted beta distribution. Forms of the probability density functions of these distributions are presented. The moments of theses distributions are derived. We provide extensive tabulation of the percentiles points associated with the distributions obtained.

Classes of probability distributions and their applications

Abstract: The aim of this paper is a nontrivial application of certain classes of probability distribution functions with further establishing the bounds for the least root of the functional equation x = b G(μ − μx), where b G(s) is the Laplace-Stieltjes transform of an unknown probability distribution function G(x) of a positive random variable having the first two moments g1 and g2, and μ is a positive parameter satisfying the condition μg1 > 1. The additional information characterizing G(x) is that it belongs to the special class of distributions such that the difference between two elements of that class in the Kolmogorov (uniform) metric is not greater than κ. The obtained result is then used to establish the lower and upper bounds for loss probabilities in certain loss queueing systems with large buffers as well as continuity theorems in large M/M/1/n queueing systems.

Distributions and Applications Moment Information for Probability Distributions, Without Solving the Moment Problem. I: Where is the Mode?

Abstract: How much information does a small number of moments carry about the unknown distribution function? Is it possible to explicitly obtain from these moments some useful information, e.g., about the support, the modality, the general shape, or the tails of a distribution, without going into a detailed numerical solution of the moment problem? In this paper a theoretical result of Johnson and Rogers is generalized to be valid for all moment problems and is exploited to demonstrate that a few moments are able to provide us with valuable information about the position of the mode of an unknown (unimodal) distribution.

Amplitude statistical analysis of Ku-Band SAR ground clutter data

Abstract: The objective of this paper is to provide an analysis of the amplitude statistics of Ku-Band SAR ground clutter data using Weibull, Gamma, K and lognormal distribution. Their goodness-of-fit have been carried out by Kolmogorov-Smirnov (K-S) test after the estimation of the distribution parameters. The results show that Gamma distribution approximates much better the value of the empirical probability density function(PDF) for the agricultural terrain;K distribution approximates much better the value of the empirical PDF for the grass terrains; Lognormal distribution approximates much better the value of the empirical PDF for the building and hurst blocks.

Conflations of Probability Distributions: An Optimal Method for Consolidating Data from Different Experiments

Abstract: The conflation of a finite number of probability distributions P_1,..., P_n is a consolidation of those distributions into a single probability distribution Q=Q(P_1,..., P_n), where intuitively Q is the conditional distribution of independent random variables X_1,..., X_n with distributions P_1,..., P_n, respectively, given that X_1= ... =X_n. Thus, in large classes of distributions the conflation is the distribution determined by the normalized product of the probability density or probability mass functions. Q is shown to be the unique probability distribution that minimizes the loss of Shannon Information in consolidating the combined information from P_1,..., P_n into a single distribution Q, and also to be the optimal consolidation of the distributions with respect to two minimax likelihood-ratio criteria. When P_1,..., P_n are Gaussian, Q is Gaussian with mean the classical weighted-mean-squares reciprocal of variances. A version of the classical convolution theorem holds for conflations of a large class of a.c. measures.

Some Skew-Symmetric Double Inverted Distributions

Abstract: We define skew-symmetric distributions based on the double inverted gamma, double inverted Weibull and double inverted compound gamma distributions, all of which have symmetric density about zero. Expressions are derived for the probability density function (pdf), cumulative distribution function (cdf) and the moments of these distributions. However, some of these quantities could not be evaluated in closed forms and we used special functions to express them.

Comparing and interpolating distributions on manifold

Abstract: We are interested in comparing probability distributions defined on Riemannian manifold. The traditional approach to study a distribution relies on locating its mean point and finding the dispersion about that point. On a general manifold however, even if two distributions are sufficiently concentrated and have unique means, a comparison of their covariances is not possible due to the difference in local parametrizations. To circumvent the problem we associate a covariance field with each distribution and compare them at common points by applying a similarity invariant function on their representing matrices. In this way we are able to define distances between distributions. We also propose new approach for interpolating discrete distributions and derive some criteria that assure consistent results. Finally, we illustrate with some experimental results on the unit 2-sphere.

A new parameter estimation for K distribution clutter

Abstract: K distribution is one of the most widely applied statistical models for SAR images clutter. Now there are several approaches to estimate the parameters of K distribution. Based on Raghavanpsilas method, a new method named piecewise-approximation is given in this paper, where gamma and beta-prime distributions are applied to piecewisely approximate K distribution. The experiment results show that the new method broadens the applicable range of parameter, and has a higher accuracy than the method of moment.