Showing papers on "K-distribution published in 1989"

Symmetric Multivariate and Related Distributions

Abstract: Part 1 Preliminaries: construction of symmetric multivariate distributions notation of algebraic entities and characteristics of random quantities the "d" operator groups and invariance dirichlet distribution problems 1. Part 2 Spherically and elliptically symmetric distributions: introduction and definition marginal distributions, moments and density marginal distributions moments density the relationship between (phi) and f conditional distributions properties of elliptically symmetric distributions mixtures of normal distributions robust statistics and regression model robust statistics regression model log-elliptical and additive logistic elliptical distributions multivariate log-elliptical distribution additive logistic elliptical distributions complex elliptically symmetric distributions. Part 3 Some subclasses of elliptical distributions: multiuniform distribution the characteristic function moments marginal distribution conditional distributions uniform distribution in the unit sphere discussion symmetric Kotz type distributions definition distribution of R(2) moments multivariate normal distributions the c.f. of Kotz type distributions symmetric multivariate Pearson type VII distributions definition marginal densities conditional distributions moments conditional distributions moments some examples extended Tn family relationships between Ln and Tn families of distributions order statistics mixtures of exponential distributions independence, robustness and characterizations problems V. Part 6 Multivariate Liouville distributions: definitions and properties examples marginal distributions conditional distribution characterizations scale-invariant statistics survival functions inequalities and applications.

Maximum-entropy models in science and engineering

Abstract: Partial table of contents: Maximum--Entropy Probability Distributions: Principles, Formalism and Techniques Maximum--Entropy Discrete Univariate Probability Distributions Maximum--Entropy Discrete Multivariate Probability Distributions Maximum--Entropy Continuous Multivariate Probability Distributions Maximum--Entropy Distributions in Statistical Mechanics Minimum Discrepancy Measures Concavity (Convexity) of Maximum--Entropy (Minimum Information) Functions Equivalence of Maximum--Entropy Principle and Gauss's Principle of Density Estimation Maximum--Entropy Principle and Contingency Tables Maximum--Entropy Principle and Statistics Maximum--Entropy Models in Regional and Urban Planning Maximum--Entropy Models in Marketing and Elections Maximum--Entropy Spectral Analysis Maximum--Entropy Image Reconstruction Maximum--Entropy Principle in Operations Research References Author Index Subject Index

K-distribution and polarimetric terrain radar clutter

Abstract: A multivariate K-distribution is proposed to model the statistics of fully polarimetric radar data from earth terrain with polarizations HH, HV, VH, and VV. In this approach, correlated polarizations of radar signals, as characterized by a covariance matrix, are treated as the sum of N n-dimensional random vectors; N obeys the negative binomial distribution with a parameter alpha and mean N-bar. Subsequently, an n-dimensional K-distribution, with either zero or nonzero mean, is developed in the limit of infinite N-bar or illuminated area. The probability density function (PDF) of the K-distributed vector normalized by its Euclidean norm is independent of the parameter alpha and is the same as that derived from a zero-mean Gaussian-distributed random vector.

Limiting probability distributions of a passive scalar in a random velocity field.

Abstract: Diffusion of a passive scalar in a random velocity field v(x,t) is considered. An exact, closed form of the limiting (t\ensuremath{\rightarrow}\ensuremath{\infty}) probability distribution of the normalized scalar is derived. The predictions of the theory are compared with the results of numerical simulations.

Multiply stochastic representations for K distributions and their Poisson transforms

Abstract: The K distribution is used in a number of areas of scientific endeavor. In optics, it provides a useful statistical description for fluctuations of the irradiance (and the electric field) of light that has been scattered or transmitted through random media (e.g., the turbulent atmosphere). The Poisson transform of the K distribution describes the photon-counting statistics of light whose irradiance is K distributed. The K-distribution family can be represented in a multiply stochastic (compound) form whereby the mean of a gamma distribution is itself stochastic and is described by a member of the gamma family of distributions. Similarly, the family of Poisson transforms of the K distributions can be represented as a family of negative-binomial transforms of the gamma distributions or as Whittaker distributions. The K distributions have heretofore had their origins in random-walk models; the multiply stochastic representations provide an alternative interpretation of the genesis of these distributions and their Poisson transforms. By multiple compounding, we have developed a new transform pair as a possibly useful addition to the K-distribution family. All these distributions decay slowly and are difficult to calculate accurately by conventional formulas. A recursion relation, together with a generalized method of steepest descent, has been developed to evaluate numerically the photon-counting distributions and their factorial moments with excellent accuracy.

A statistical theory for the distribution of energy dissipation in intermittent turbulence

Abstract: A new statistical theory involving the gamma distribution is presented for the local average rate of dissipation er. The gamma distribution for er leads to results for the high‐order moments of the velocity structure function that lie between those predicted by the lognormal model and β model and closely follow those of the multifractal model. Comparisons of results predicted by the gamma distribution are made with previously published experimental data, showing excellent agreement. Based on the gamma distribution for er, the one‐dimensional K distribution is developed as a plausible model for the distribution of energy dissipation e. Some laboratory measurements of velocity fluctuations and certain derived quantities from a modified airjet are also discussed. A good agreement is found by making comparisons of the measured cumulative probability distribution of the squared time derivative of the streamwise velocity component with that predicted by this new model.

Comparative Study of some Probability Distributions Applied to Liquid Sprays

Abstract: A comparative study of a few probability distributions generally used in describing various spray properties based on droplet size and velocity is described. Four different size distributions and three different coupled distributions, representing jointly size and velocity, are considered. A comparison of the results calculated for eight sets of data, taken from three different sources for the size distribution, and one data set for the joint size and velocity distribution, shows that the log-hyperbolic function presents the best choice among both one- and two-dimensional distributions. Considerations on one- and two-dimensional probability distributions are followed by a description of the conservation laws and their use in the prediction of the loss of momentum and energy of doplets. Finally, a proposal is made for how the sprays should be studied experimentally and analysed theoretically in order to obtain the maximum possible information. A few suggestions are then provided for further research in this direction.

Probability-density-functional description of photoelectron statistics

Abstract: We show that time distributions of photoelectrons can best be described by a generalized probability-density function which we call the probability-density functional. The probability-density functional gives complete information about the statistical properties of any random-point process, and provides the most natural definitions of the ensemble average and generating functional of a quantity distributed in time. Using this functional, we develop a new approach to photoelectron statistics that systematically relates various joint probability distributions of photoelectrons to one another and to the corresponding generating functionals. We demonstrate that this method can be used to derive previously obtained results and we use it to obtain new results. In particular, the amount of information about the statistical properties of photoelectrons that is contained in a probability distribution can be obtained by comparing the corresponding generating functional with the probability-density functional. Further, we examine time derivatives of the probability distribution for the number of counts and develop new relationships for probability distributions of triggered counting. Finally, we apply our results to obtain an ``inverse'' relationship between photon bunching (antibunching) and super-Poisson (sub-Poisson) statistics.

Continuous Probability Distributions

Abstract: In Chapter 3 we considered an example of a continuous variable, namely the height of a student, and we summarized the heights of 50 students by a histogram, Fig. 3.1, reproduced here as Fig. 7.1.

Analysis of probability-density functions for laser scintillations in a turbulent atmosphere

Abstract: We have reanalyzed a selection of data obtained from laser scintillation experiments in the light of recent criticisms of analysis techniques based on the moments of the data. In agreement with those criticisms, we find that comparing the higher moments of the data with those of model distributions can produce misleading results. We therefore use a least-squares fitting method to compare directly the experimental probability distributions with the models. The conclusion of the earlier experiments, that the K distribution is generally better than the log-normal distribution in the strong-scintillation regime, however, is not changed by the new analysis.

Robust large deviations performance analysis for large sample detectors

Abstract: Large deviations theory is used to analyze the exponential rate of decrease of error probabilities for a sequence of decisions based on a test statistics sequence (T/sub n/). It is assumed that (for a given statistical hypothesis) the distributions of T/sub n/ are determined by some unknown member of a class of probability distributions. The worst case, or least favorably exponential rate of error probability decrease over this class, is sought. It is shown that the Legendre-Fenchel transform of the maximized cumulant function yields a lower bound for the minimized large deviations rate function, and that in many cases this bound is tight. Application of the result is illustrated by a detailed consideration of i.i.d memoryless detection with an epsilon -contamination distribution family. >

Multi-resolution multi-frequency analysis of high range resolution sea reflections

Abstract: A description of the surface by a K distribution is of particular interest in that the K distribution can be decomposed into a Rayleigh distributed fast fluctuating amplitude component modulated by a chi distributed slow fluctuating amplitude component. Thus if the observed data can be described by a K distribution the components of the return can be analysed separately and related to physical aspects of the sea surface. The slow correlation characteristics of the backscatter can then be included in target detection strategies and radar performance calculations. Failure to allow for the correlation characteristics can lead to large errors in radar detection performance and radar false alarm rate. Agreement with a K distribution model has been demonstrated for a limited range of radar parameters. The K model has been shown to be valid at I Band and resolutions down to 4 m. The author describes measurements and analyses performed at the Marconi Research Centre to extend the observations to higher resolutions and a range of frequencies.

Probability Distributions Associated with Power Series Having Non-negative Coefficients

Abstract: Using an idea of Khinchin, we show how to associate to a power series with non-negative coefficients a one-parameter family of probability distributions. The limit theorems of probability yield asymptotic formulas for coefficients, sections, and other functionals of the power series. This generalizes results of Hayman and Edrei and has applications to the partition function. Given a sequence of normalized probability distributions, it is shown how to construct a power series whose associated distributions are asymptotic to the given distributions for a certain sequence of parameter values.

Experimental Verification Of The K Distribution Family For Scattering

Abstract: Experimental verification of theK distribution family for scatteringArthur R. WeeksandRonald L. PhillipsThe University of Central FloridaThe Center for Research in Electro- Opticsand Lasers (CREOL)P.O. Box 25000, Orlando, Florida 32816ABSTRACTThe intensity statistics for a laser beam propagating through a random phasescreen was compared to the H -K distribution. Since the H -K distribution was derivedfrom the conditionalization of the Rician distribution with the Gamma distribution,experimental intensity data was collected to verify these distributions. Theintensity data was first lowpass filtered to yield statistical moments that werefound to be in agreement with those of the gamma distribution. Next, the intensitydata was segmented into small time Intervals, in which all time segments withapproximately the same variance were grouped together. The statistical moments forthese new time series were then compared against the moments for the Riciandistribution. The moments for the segmented time series matched the moments for theRician distribution extremely well.1. INTRODUCTIONAs a laser beam propagates through a time varying random phase screen, itexperiences random fluctuations in phase. These time varying random phasefluctuations will manifest themselves as time variations in the intensity of thepropagating laser beam. If a point detector is used to detect this laser beam,variations in the output of the detector will be seen. The purpose of this paper isto verify the statistics of these fluctuations.A propane flame placed a distance below the center of a collimated laser beamwas used to generate these time varying phase fluctuations. The propane flameheated the air above the burner developing thermal mixing of the hot and cold air.This thermal mixing resulted in temperature variations across the laser beam's path.These variations in temperature caused the index of refraction of the air tofluctuate. Since the wave number of the electromagnetic wave from the laserbeam is proportional to the index of refraction, these fluctuations in therefractive index lead to variations in the laser beam's amplitude and phase.Predicting the fluctuations in the amplitude and phase requires the use ofprobability theory

Transform related to spherical distributions and its inversion

Abstract: For describing the envelope and the values of a narrowband random noise Roberts uses the probability distribution of the length of a vector independently distributed uniformly in direction and the probability distribution of the length of a projection of this vector. The latter distribution is here expressed in terms of the former, and a simple form is found for the inverse of this relationship.

Tests for the Gamma Distribution with Estimated Shape Parameters

Abstract: : It is well known that the distributions of the goodness -of-fit statistics W squared, U squared, A squared do not depend on location and scale parameters even when these parameters must be estimated. It has generally been assumed, however, that when shape parameters must be estimated the null distributions of these statistics would depend too severely on the unknown parameters to permit their use in practice. Exact asymptotic distributions of these statistics are presented for the two parameter gamma family distributions. Critical points are given in tables constructed so as to avoid any need for extrapolation. The results presented show that the tests can be expected to be useful in practice.

Inequalities for alternating branching processes

Abstract: A discrete time branching process is considered in which the offspring distribution alternates between two known probability distributions. There are two cases to consider, depending on which of the probability distributions initiates the process. It is shown, both graphically and analytically that for any pair of distributions, the probability of ultimate extinction is reduced if the process with the smaller individual probability of extinction starts first. An example is given in the case when the offspring distribution alternates between two geometric distributions.

Application of extreme value theory to the bec family of distributions

Abstract: Arnold and Strauss (1988) derived a family of bivariate life distributions having the property that the conditional distributions are exponential. Asymptotic distributions for the marginal and bivariate extremes for this family of distributions are derived employing the asymptotic theory of extreme order statistics.

Probability distributions for the integrated intensity and photocount associated with a K-distribution for laser intensity.

Abstract: A numerical approach is used for evaluating the probability density functions of the integrated intensity and photocount when the received laser intensity obeys a K-distribution. Different scattering strengths are considered for comparisons. Also, those probability distribution functions based on a generalized K-distribution and a lognormally modulated exponential distribution for the laser intensity are computed and compared with those based on the K-distribution. Some general trends are drawn. The accuracy of our numerical computations is also discussed.