Showing papers on "K-distribution published in 2007"

The κ-μ distribution and the η-μ distribution

Abstract: This paper presents two general fading distributions, the kappa-mu distribution and the eta-mu distribution, for which fading models are proposed. These distributions are fully characterized in terms of measurable physical parameters. The kappa-mu distribution includes the Rice (Nakagami-n), the Nakagami-m, the Rayleigh, and the one-sided Gaussian distributions as special cases. The eta-mu distribution includes the Hoyt (Nakagami-q), the Nakagami-m, the Rayleigh, and the one-sided Gaussian distributions as special cases. Field measurement campaigns were used to validate these distributions. It was observed that their fit to experimental data outperformed that provided by the widely known fading distributions, such as the Rayleigh, Rice, and Nakagami-m. In particular, the kappa-mu distribution is better suited for line-of-sight applications, whereas the eta-mu distribution gives better results for non-line-of-sight applications.

Multipeaked probability distributions of recurrence times.

Abstract: We determine probabilities of recurrence time into finite-sized, physically meaningful subsets of phase space. We consider three different autonomous chaotic systems: (i) scattering in a three-peaked potential, (ii) connected billiards, and (iii) Lorenz equations. We find multipeaked probability distributions, similar to the distributions found in (driven) stochastically resonant systems. In nondriven systems, such as ours, only monotonic decaying distributions (exponentials, stretched exponentials, power laws, and slight variations or combinations of these) have hitherto been reported. Discrete peaks in autonomous systems have as yet escaped attention in autonomous systems and correspond to specific trajectory subsets involving an integer number of loops.

Probability distributions generated by fractional diffusion equations 1

Abstract: Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these equations provide probability density functions, evolving on time or variable in space, which are related to the class of stable distributions. This property is a noteworthy generalization of what happens for the standard diffusion equation and can be relevant in treating financial and economical problems where the stable probability distributions play a key role.

On efficacy of Rayleigh‐inverse Gaussian distribution over K‐distribution for wireless fading channels

Abstract: For studying performance characteristics of radio channels, the knowledge about the probability density function (pdf) of fading–shadowing effects is essential. K-distribution corresponding to Rayleigh–gamma distribution (RGD) is widely used to approximate a more realistic Rayleigh–lognormal distribution (RLD) which does not have a closed form expression. A new composite Rayleigh-inverse Gaussian distribution (RIGD), an alternative to K-distribution, is analyzed with regards to its suitability and effectiveness in radio channels. Detailed investigations are made to study the performance characteristics of RIGD and K-distribution (RGD) in terms of Kullback–Leibler (KL) measure of divergence. Based on these investigations, it is found that RIGD is better suited for capturing fading–shadowing aspects of radio channels instead of K-distribution. Copyright © 2006 John Wiley & Sons, Ltd.

Probability Distributions of Correlation and Differentials in Block Ciphers

Abstract: We study the probability distributions of difference propagation probabilities and input- output correlations for functions and block ciphers of given dimensions, for several of them for the first time. We show that these parameters have distributions that are well-studied in the field of probability such as the normal, Poisson and extreme value distributions. The results of this paper can be used to estimate how much effort will be required to generate functions satisfying certain criteria. The distributions we derive for block ciphers illustrate the significant difference between fixed-key parameters and averaged parameters.

A Simple Approximation to the Convolution of Gamma Distributions (Revision of DP 2006-27)

Abstract: The exact expressions for the convolutions of gamma distributions with different scale parameters is quite complicated. The approximation by means of another gamma distribution is shown to be remarkably accurate for wide ranges of the parameter values, especially if more than two random variables are involved. The approximation is particularly good for the upper quantiles that play an important role in auditors' decisions.

Ill-Posed Problems in Probability And Stability of Random Sums

Abstract: The General Form of Quantitative Convergence Criteria Some Important New Classes of Probability Metrics Convergence in Weak and Strong Metrics Convergence to Prescribed Distributions Ill-Posed Problems in Computer Tomography Stable Probabilistic Schemes Central Pre-Limit Theorems Infinitely Divisible and Stable Distributions Geometric Stable Distributions on the Real Line Multivariate Geometric Stable Distributions Geometric Stable Laws on Banach Space Estimation and Empirical Issues for GS Distributions A Generalisation of Stable Laws Characterisations of Distributions in Reliability Index.

Angular Dependence of ${\cal K}$ -Distributed Sonar Data

Abstract: Backscattered signal statistics are widely used for target detection and seafloor characterization. The K-distribution shows interesting properties for describing experimental backscattered intensity statistics. In addition to the fact that its probability distribution function accurately fits actual sonar data, it advantageously provides a physical interpretation linked to the backscattering phenomenon. Sonar systems usually record backscattered signals from a wide angular range across the ship's track. In this context, previous studies have shown that backscatter statistics strongly depend on the incidence angle. In this paper, we propose an extension of previous works to model the angular evolution of the K-distribution shape parameter. This evolution is made clear and analyzed from experimental data recorded with two sonar systems: a 95-kHz multibeam echosounder and a 110-kHz sidescan sonar. Model fitting with data backscattered from six seafloor configurations shows the improvement provided by our extension as compared to two previous models

Extreme Value Distributions for the Skew-Symmetric Family of Distributions

Abstract: We derive the extreme value distribution of the skew-symmetric family, the probability density function of the latter being defined as twice the product of a symmetric density and a skewing function. We show that, under certain conditions on the skewing function, this extreme value distribution is the same as that for the symmetric density. We illustrate our results using various examples of skew-symmetric distributions as well as two data sets.

The Effect of Multipath on the Envelope Statistics of Bottom Clutter

Abstract: Active sonar systems operating in shallow-water environments are often faced with high numbers of false alarms, generically referred to as clutter, arising from among other sources bottom scattering that results in heavy tails in the matched filter envelope probability density function compared with the Rayleigh distribution. In this paper, the effect of multipath propagation on the envelope statistics (i.e., the disparity from the Rayleigh distribution) is modeled through the use of the -distribution where the shape and scale parameters are formed from the autocorrelation function of the transmit waveform, the multipath structure, and the strength and spatial density of the bottom scatterers. Use of the -distribution is justified by showing that it is the limiting distribution of the sum of independent but not identically distributed -distributed random variables, which is representative of multipath when the bottom produces -distributed backscatter. The shape parameter, which drives the clutter statistics, is seen to be inversely proportional to bandwidth at bandwidths low enough that the multipath is not resolved and again at bandwidths high enough that all of the paths are resolved. As has been previously reported by LePage [IEEE J. Ocean. Eng., vol. 29, no. 2, pp. 330-346, 2004], multipath is shown to make clutter statistics more Rayleigh-like, which in this analysis equates to an increase in the -distribution shape parameter. The model is used to evaluate the effect on clutter statistics of varying environmental characterizations and system configurations where it is seen that, for a constant sound-speed profile, increasing the vertical aperture of the sonar, the center frequency, or surface roughness can lead to less multipath and, therefore, a reduction in the -distribution shape parameter and an increase in the probability of false alarm.

Finite-state Markov Model for Lognormal, Chi-square (Central), Chi-square (Non-central), and K -distributions

Abstract: This paper formulates a finite-state Markov channel model to represent received signal-to-noise (SNR) ratios having lognormal, K-distribution, chi-square (central) and chi-square (non-central) distributions in a slow fading channel. The range of the SNRs is partitioned into a finite number of states following earlier works in literature. Performance measures like level crossing rates, steady-state probabilities, transition probabilities, and state-time durations are derived, and numerical results are plotted and discussed for the FSMC models for all the distributions.

High grazing angle X-band sea clutter distributions

Abstract: This paper investigates the statistical properties of sea clutter at grazing angles higher than traditionally used for airborne maritime radar surveillance, i.e. 10° - 45°. Specifically, we study the spatial distribution of X-band, high resolution and high grazing angle polarimetric sea clutter data. We found that among the VV, HV, and HH polarisations, the W data is least spiky and the K distribution usually provides a good fit. The HH data is spikiest and its distribution exhibits a sudden departure from the K distribution in the upper tail region, which usually requires the KA or similar distributions to achieve a better fit. Due to drawbacks of the KA distribution, this paper proposes a combination of two K distributions to fit the distribution of sea clutter with spikes. Referred to as the KK distribution model, it is found to provide the best overall fit. (5 pages)

Probability distributions in conservative energy exchange models of multiple interacting agents

Abstract: Herein we study energy exchange models of multiple interacting agents that conserve energy in each interaction. The models differ regarding the rules that regulate the energy exchange and boundary effects. We find a variety of stochastic behaviours that manifest energy equilibrium probability distributions of different types and interaction rules that yield not only the exponential distributions such as the familiar Maxwell‐Boltzmann‐Gibbs distribution of an elastically colliding ideal particle gas, but also uniform distributions, truncated exponential distributions, Gaussian distributions, Gamma distributions, inverse power law distributions, mixed exponential and inverse power law distributions, and evolving distributions. This wide variety of distributions should be of value in determining the underlying mechanisms generating the statistical properties of complex phenomena including those to be found in complex chemical reactions.

The wrapped Gamma distribution and wrapped sums and linear combinations of independent Gamma and Laplace distributions

Abstract: In this paper we first obtain an expression for the probability density function of the wrapped or circular Gamma distribution and then we show how it may be seen, both for integer and non-integer shape parameter, as a mixture of truncated Gamma distributions. Some other properties of the wrapped Gamma distribution are studied and it is shown how this distribution and mixtures of these distributions may be much useful tools in modelling directional data in biology and meteorology. Based on the results obtained, namely the ones concerning mixtures, and on some properties of the distributions of the sum of independent Gamma random variables, the wrapped versions of the distributions of such sums, for both integer and non-integer shape parameters are derived. Also the wrapped sum of independent generalized Laplace distributions is introduced as a particular case of a mixture of wrapped Gamma distributions. Among the particular cases of the distributions introduced there are symmetrical, slightly ske...

3D-5 Parameter Estimation of the Homodyned K Distribution Based on Signal to Noise Ratio

Abstract: The Homodyned K (H-K) distribution is very important in Ultrasound (US) imaging because it models the speckle in a general way, even if the number of scatterers per resolution cell is low and if a coherent component exists. Due to the analytical complexity of this distribution, only a few works based on the H-K have been reported and other simpler distributions have been used instead. Here we provide evidence that, once we have the signal to noise ratio (SNR) from a set of parameters of the H-K distribution, we can estimate the parameters of any given sample set in an easy computational way. In addition, we present a validation study for our estimation procedure.

Statistical Relationship between Three- and Two-Dimensional Spatial Distributions of Dispersed Phases

Abstract: We defined 3-dimensional local number, LN3D, 2-dimensional local number, LN2D, and their probability distribution to describe the spatial distribution of second phase particles, and then suggested the statistical relationship of probability distributions between LN2D and LN3D concerning uniform random and clustering spatial distributions. The relationship was validated by computer experiments using particles of overlap permissive spheres, and was applied to the real microstructures of Al-10 vol%SiC composites. Using the relationship, probability distributions of either LN3D or LN2D could be predicted from the measured relative frequency distributions of another dimension in computer experiments with satisfactory accuracy. Using the relationship, the probability distributions of LN3D was approximately predicted from the relative frequency distributions of LN2D that were obtained by measurements of spatial distributions of SiC particles in Al-SiC composites. [doi:10.2320/matertrans.MER2007137]

Applications and approximations of heterogeneous weighted and unweighted j-binomial probability distributions

Abstract: We illustrate several applications of probability distributions for N weighted Bernoulli trials sampled from J heterogeneous populations with unequal pi rates and wi weights. These distributions can be significantly non-binomial, non-normal, and underdispersed compared to their binomial counterparts, with corresponding consequences on probability calculations, control charts, sequential probability ratio tests, and statistical properties. Real-world applications, however, can have scores of nested convolutions and be computationally expensive to obtain exact solutions (CPU time > 20 minutes). Modified Kullback-Leibler, total absolute deviation, and variance ratio statistics therefore were used to evaluate several approximations, including normalizations, Monte Carlo routines, and a standardized GramCharlier expansion based on Hermite polynomials and lower order cumulants.

Skewed Bessel function distributions with application to rainfall data

Abstract: Skewed distributions have attracted significant attention in the last few years. In this article, a skewed Bessel function distribution with the probability density function (pdf) f(x)=2 g (x) G (λ x) is introduced, where g (·) and G (·) are taken, respectively, to be the (pdf) and the cumulative distribution function of the Bessel function distribution [McKay, A.T., 1932, A Bessel function distribution, Biometrica, 24, 39–44]. Several particular cases of this distribution are identified and various representations for its moments derived. Estimation procedures by the method of maximum likelihood are also derived. Finally, an application is provided to rainfall data from Orlando, Florida.

Gaussian Hypergeometric Probability Distributions for Fitting Discrete Data

Abstract: The distributions generated by the Gaussian hypergeometric function compose a tetraparametric family that includes many of the most common discrete distributions in the literature. In this article, probability aspects related to the whole family are reviewed and methods of estimation for fitting them to real data are developed. Several applied examples are also provided to illustrate the procedures and compare the methods of estimation.

Classical Broadcasting is Possible with Arbitrarily High Fidelity and Resolution

Abstract: We quantify the resolution with which any probability distribution may be distinguished from a displaced copy of itself in terms of a characteristic width. This width, which we call theresolution, is well defined for any normalizable probability distribution. We use this concept to study the broadcasting of classical probability distributions. Ideal classical broadcasting creates two (or more) output random variables each of which has the same distribution as the input random variable. We show that the universal broadcasting of probability distributions may be achieved with arbitrarily high fidelities for any finite � resolution. By restricting probability distributions to any finiteresolution we have therefore shown that the classical limit of quantum broadcasting is consistent with the actual classical case.

Fuzzy representation of incomplete knowledge about infinite support probability distributions in a measurement context

Abstract: This paper deals with a fuzzy/possibility representation of measurement uncertainty that often arises in physical domains. The construction of the possibility distribution is based on the stacking up of probability coverage intervals. The paper shows that the specificity of the possibility distribution depends on the nature of the a priori information available about the entity under measurement. In particular the following commonly occurring situations reflecting different amounts of a priori information are considered: only the mean, or the mode of the underlying infinite support continuous probability distribution is known; in addition, a dispersion parameter and/or shape information such as symmetry and unimodality are known. The associated possibility distributions are determined from probability inequalities and represent the maximal unpresumptive distribution consistent with available knowledge. They allow to determine the impact of each piece of information on the reduction of coverage interval lengths.

Bivariate Gamma Distributions for Multisensor Sar Images

Abstract: This paper addresses the problem of estimating the parameters of a family of bivariate gamma distributions whose margins have different shape parameters. These distributions are interesting to detect changes in two synthetic radar aperture (SAR) images acquired by different sensors and having different numbers of looks. The estimators based on the maximum likelihood method and the method of moments are studied for these distributions. An application to change detection is finally discussed.

Learning mixtures of distributions

Abstract: This thesis studies the problem of learning mixtures of distributions, a natural formalization of clustering. A mixture of distributions is a collection of distributions D = {D1, ... DT}, and mixing weights, {w1,..., wT} such that Σiwi = 1. A sample from a mixture is generated by choosing i with probability wi and choosing a sample from distribution Di. Given samples from a mixture of distributions, the problem of learning the mixture is that of finding the parameters of the distributions comprising D and grouping the samples according to source distribution. A common theoretical framework for addressing the problem also assumes that we are given a separation condition, which is a promise that any two distributions in the mixture are sufficiently different. In this thesis, we study three aspects of the problem. First, in Chapter 3, we focus on optimizing the separation condition while learning mixtures of distributions. The most common algorithms in practice are singular value decomposition based algorithms, which work when the separation is Θ pswmin p , where σ is the maximum directional standard deviation of any distribution in the mixture, and wmin is the minimum mixing weight. We show an algorithm which successfully learns mixtures of distributions with a separation condition that depends only logarithmically on the skewed mixing weights. In particular, it succeeds for a separation between the centers that is Θ pTlogLp , where T is the number of distributions, and Λ is polynomial in T and the imbalance in the mixing weights. We require that the distance between the centers be spread across Θ(T log Λ) coordinates. In addition, we show that if every vector in the subspace spanned by the centers has a small projection, of the order of 1TlogL on each coordinate vector, then, our algorithm succeeds for a separation of only O ps*TlogL p , where σ* is the maximum directional standard deviation in the space containing the centers. Our algorithm works for Binary Product Distributions and Axis-Aligned Gaussians. The spreading condition above is implied by the separation condition for binary product distributions, and is necessary for algorithms that rely on linear correlations. Motivated by the application in population genetics, in Chapter 4, we study the sample complexity of learning mixtures of binary product distributions. In this thesis, we take a step towards learning mixtures of binary product distributions with optimal sample complexity by providing an algorithm which learns a mixture of two binary product distributions with uniform mixing weights and low sample complexity. Our algorithm clusters all the samples correctly with high probability, so long as d(μ1, μ 2) the square of the Euclidean distance between the centers of distributions is at least polylogarithmic in s, the number of samples and the following trade-off holds between the separation and the number of samples: sdd2pm1,m 2p≥adnlogslogpnsp for some constant a. Finally, in Chapter 5, we study the problem of learning mixtures of heavy-tailed product distributions. To this end, we provide an embedding from R n to {0, 1}n', which maps random samples from distributions with medians that are far apart to random samples from distributions on {0, 1}n', with centers that are far apart. The main application of our embedding is in designing an algorithm for learning mixtures of heavy-tailed distributions. We provide a polynomial-time algorithm, which learns mixtures of general product distributions, as long as the distribution of each coordinate satisfies two properties: symmetry about the median and ¾-radius upper-bounded by R. The separation required by our algorithm to correctly classify a 1–δ fraction of the samples is that the distance between the medians of any two distributions in the mixture should be O pRTlogL+R TlogTd p , and this distance should be spread across O(T log Λ + T log Td ) coordinates. A second application of our embedding is in designing algorithms for learning mixtures of distributions with finite variance, which work under a separation requirement of O ps*TlogL p and a spreading requirement of O(T log Λ + T log Td ). This algorithm does not require the more stringent spreading condition needed by the algorithm which offers similar guarantees in Chapter 3.

Non-Rayleigh echo PDF's for broadband acoustic scattering by patches of discrete targets with applications to fish

Abstract: Echoes from patches of fish fluctuate significantly from ping to ping as the sonar beam is swept across the patches. The fluctuations can be strongly non-Rayleigh because 1) there can be a small number of targets in the beam at a time, 2) the distribution of fish can be inhomogeneous, or "patchy", and 3) the echoes are weighted by the non-uniform response of the sonar beam. We have previously identified two distributions to describe the statistical behavior of non-Rayleigh echoes from fish - the K-distribution for patches of multiple unresolved fish and a power law distribution for individual resolved fish. The Redistribution has been shown in previous studies (Abraham and Lyons, IEEE J. Ocean. Eng. 27: 800-813, 2002) to describe the statistics of targets with a Gamma distribution of echo amplitude. The power law distribution is based on the method proposed by Ehrenberg et al. for circular aperture transceivers (J. Acoust. Soc. Am. 69: 955-962, 1981). In this paper, we provide a more general physical interpretation to the R-distribution and develop a generalized power-law distribution to include beampattern effect. In addition, we compare the data collected in the 2-10 kHz range with Atlantic herring with the model predictions for two types of groupings of echoes - within patches and across patches. The Rullback-Leibler (RL) distances between the observed and theoretically predicted distributions showed that for echoes within patches, Rayleigh distribution can reasonably describe the fish echoes amplitude while for echoes across patches, the echo amplitude can be characterized by either the R-distribution or a mixed distribution that involves the power law distribution of sonar beam, depending on whether the echoes can be resolved.

Sea Spike Characteristics Studies and Simulation Analyses

Abstract: A compound model of sea-clutter radar cross-section (RCS) based on sea spike was introduced and a definition of spike with the profiles of time, attenuation and frequency based on principle of stationary surge was proposed C band real radar sea-clutter data were analyzed and the modified mean square difference test for analyzing the “tail” of clutter was constructed The statistic characteristics of spike under the specified false-alarm probability based on K-distributed model were studied The results indicate that both binomial distribution and Poisson distribution well fit the statistic characteristics of sea spike under lower false-alarm probability