Showing papers on "K-distribution published in 2012"
TL;DR: A heavy-tailed CG model with an inverse Gaussian texture distribution is proposed and its distributional properties such as closed-form expressions for its probability density function (p.d.f.) as well as its amplitude p.
Abstract: The compound-Gaussian (CG) distributions have been successfully used for modelling the non-Gaussian clutter measured by high-resolution radars. Within the CG class, the complex K -distribution and the complex t-distribution have been used for modelling sea clutter which is often heavy-tailed or spiky in nature. In this paper, a heavy-tailed CG model with an inverse Gaussian texture distribution is proposed and its distributional properties such as closed-form expressions for its probability density function (p.d.f.) as well as its amplitude p.d.f., amplitude cumulative distribution function and its kurtosis parameter are derived. Experimental validation of its usefulness for modelling measured real-world radar lake-clutter is provided where it is shown to yield better fits than its widely used competitors.
110 citations
TL;DR: In this article, a two-parameter discrete gamma distribution is derived corresponding to the continuous two parameters gamma distribution using the general approach for discretization of continuous probability distributions and a few important distributional and reliability properties of the proposed distribution are examined.
Abstract: A two-parameter discrete gamma distribution is derived corresponding to the continuous two parameters gamma distribution using the general approach for discretization of continuous probability distributions. One parameter discrete gamma distribution is obtained as a particular case. A few important distributional and reliability properties of the proposed distribution are examined. Parameter estimation by different methods is discussed. Performance of different estimation methods are compared through simulation. Data fitting is carried out to investigate the suitability of the proposed distribution in modeling discrete failure time data and other count data.
98 citations
TL;DR: This paper introduces the generalized exponential-power series (GEPS) class of distributions, which is obtained by compounding generalized exponential and power series distributions and obtains several properties of the GEPS distributions such as moments, maximum likelihood estimation procedure via an EM-algorithm and inference for a large sample.
94 citations
01 Nov 2012
TL;DR: A unified analysis and general synthesis algorithms for point distributions that can generate distributions with given target characteristics, possibly extracted from an example point set, and introduce a unified characterization of distributions by mapping them to a space implied by pair correlations are proposed.
Abstract: Analyzing and synthesizing point distributions are of central importance for a wide range of problems in computer graphics. Existing synthesis algorithms can only generate white or blue-noise distributions with characteristics dictated by the underlying processes used, and analysis tools have not been focused on exploring relations among distributions. We propose a unified analysis and general synthesis algorithms for point distributions. We employ the pair correlation function as the basis of our methods and design synthesis algorithms that can generate distributions with given target characteristics, possibly extracted from an example point set, and introduce a unified characterization of distributions by mapping them to a space implied by pair correlations. The algorithms accept example and output point sets of different sizes and dimensions, are applicable to multi-class distributions and non-Euclidean domains, simple to implement and run in O(n) time. We illustrate applications of our method to real world distributions.
82 citations
TL;DR: In this paper, the probability distributions for quadratic quantum fields, averaged with a Lorentzian test function, were treated in four-dimensional Minkowski vacuum. But they are not able to give closed form expressions for the probability distribution, but rather use calculations of a finite number of moments to estimate the lower bounds, the asymptotic forms for large positive argument, and possible fits to the intermediate region.
Abstract: We treat the probability distributions for quadratic quantum fields, averaged with a Lorentzian test function, in four-dimensional Minkowski vacuum These distributions share some properties with previous results in two-dimensional spacetime Specifically, there is a lower bound at a finite negative value, but no upper bound Thus arbitrarily large positive energy density fluctuations are possible We are not able to give closed form expressions for the probability distribution, but rather use calculations of a finite number of moments to estimate the lower bounds, the asymptotic forms for large positive argument, and possible fits to the intermediate region The first 65 moments are used for these purposes All of our results are subject to the caveat that these distributions are not uniquely determined by the moments We apply the asymptotic form of the electromagnetic energy density distribution to estimate the nucleation rates of black holes and of Boltzmann brains
59 citations
Journal Article•
TL;DR: The Kummer beta generalized family as mentioned in this paper is a family of distributions that extends the normal, Weibull, gamma and Gumbel distributions, among several other well-known distributions.
Abstract: Ng and Kotz (1995) introduced a distribution that provides greater flexibility to extremes.We define
and study a new class of distributions called the Kummer beta generalized family to extend the
normal, Weibull, gamma and Gumbel distributions, among several other well-known distributions.
Some special models are discussed. The ordinary moments of any distribution in the new family
can be expressed as linear functions of probability weighted moments of the baseline distribution.
We examine the asymptotic distributions of the extreme values. We derive the density function
of the order statistics, mean absolute deviations and entropies. We use maximum likelihood
estimation to fit the distributions in the new class and illustrate its potentiality with an application
to a real data set.
49 citations
TL;DR: In this paper, a new method is proposed for generating discrete distributions, namely, the exponentiated-exponential-geometric distribution (E-GDF), which is a special class of the distributions, which contains the discrete analogues of continuous distributions.
38 citations
01 Jan 2012
TL;DR: In this paper, the best fitting distribution of wind speed data recorded in Islamabad on a daily basis for the years 2001 to 2003 was found using Maximum Likelihood (ML) method.
Abstract: In the present study an effort has been made to find out the best fitting distribution of wind speed data recorded in Islamabad on a daily basis for the years 2001 to 2003. For this purpose two parameter Gamma, Weibull, Lognormal, Rayleigh, three parameter Burr and Frechet distributions are fitted to data and parameters for each distribution are estimated using the Maximum Likelihood (ML) method. The performance of the distributions are evaluated using three Goodness of Fit (GOF) tests namely Chi-Squared (CS), Kolmogorove-Smirnove (KS) and Anderson Darling (AD) test, Further, the fitted graphics of Cumulative Distribution Function (CDF), Probability Distribution Function (PDF) and Probability Probability (PP) plots are used to confirm the GOF for the above six distributions. Finally, graphical and GOF results are compared suggesting that Burr, Lognormal and Gamma distributions are found to be most appropriate as compared to the Weibull, Rayleigh and Frechet distributions.
23 citations
TL;DR: In this article, a new lifetime class with decreasing failure rate was obtained by compounding truncated Poisson distribution and a lifetime distribution, where the compounding procedure follows same way that was previously carried out by Adamidis and Loukas(1998).
Abstract: A new lifetime class with decreasing failure rate which is obtained by compounding truncated Poisson distribution and a lifetime distribution, where the compounding procedure follows same way that was previously carried out by Adamidis and Loukas(1998). A general form of probability, distribution, survival and hazard rate functions as well as its properties will be presented for such a class. This new class of distributions generalizes several distributions which have been introduced and studied in the literature.
18 citations
22 Jul 2012
TL;DR: This paper presents statistical models for the sublook correlation magnitude (SCM), a test statistic for ship detection that can be produced from single-look complex (SLC) synthetic aperture radar (SAR) data, and validates the use of the K distribution and the Fisher distribution for sea clutter in SCM images.
Abstract: This paper presents statistical models for the sublook correlation magnitude (SCM), a test statistic for ship detection that can be produced from single-look complex (SLC) synthetic aperture radar (SAR) data. The SCM is extracted from the complex correlation between two subaperture images and provides enhanced contrast between coherent structures, such as marine vessels, and sea clutter. A modified SCM algorithm has been proposed, which introduces an antialiasing filter in order to allow overlapping sublook spectra. The consequences for the statistical modelling are discussed. We perform an empirical study which validates the use of the K distribution and the Fisher distribution as probability density functions for sea clutter in SCM images. This lays the groundwork for constant false alarm rate (CFAR) detection with SCM images. The fit of the models are assessed with real data.
15 citations
01 Jan 2012
TL;DR: The results for PML for distribution multiset estimation are extended to two related problems of estimating the parameterMultiset of multiple distributions or processes, including the problems of estimation the multisets of success probabilities of Bernoulli processes, and the mult iset of means of Poisson distributions.
Abstract: We derive competitive tests and estimators for several properties of discrete distributions, based on their i.i.d. sequences. We focus on symmetric properties that depend only on the multiset of probability values in the distributions and not on specific symbols of the alphabet that assume these values. Many applications of probability estimation, statistics and machine learning involve such properties. Our method of probability estimation, called profile maximum likelihood (PML), involves maximizing the likelihood of observing the profile of the given sequences, i.e., the multiset of symbol counts in the sequences. It has been used successfully for universal compression of large alphabet data sources, and has been shown empirically to perform well for other probability estimation problems like classification and distribution multiset estimation. We provide competitive estimation guarantees for the PML method for several such problems. For testing closeness of distributions, i.e., finding whether two given i.i.d. sequences of length n are generated by the same distribution or by two different ones, our schemes have an error probability of at most sqrt(delta) * exp(7n(2/3)) whenever the best possible error probability is delta 0 with error probability at most delta <= exp(-6n(1/2)), then the PML estimator is within a distance of 2 * epsilon with error probability at most delta * exp(6n(1/2)). Equivalently, the PML estimator approximates distributions to within a distance of 2 * epsilon with error probability delta using sequences of length n' = O(\max{n2/log2(1/4 delta),n}). Thus, this estimator is competitive with other estimators, including the one by Valiant et al. that approximates distributions of superlinear support size k = O(epsilon(2.1) * n * log(n)) to within a relative earthmover distance of epsilon and whose error probability can be shown to be at most exp(-n(0.9)). However, unlike the case of closeness testing, we do not yet have efficient schemes for computing the PML distribution. We extend the results for PML for distribution multiset estimation to two related problems of estimating the parameter multiset of multiple distributions or processes. These include the problems of estimating the multiset of success probabilities of Bernoulli processes, and the multiset of means of Poisson distributions
TL;DR: It is shown that by treating the cumulants as elements of the polynomial ring the authors can directly solve the unsupervised learning problem of finding the subspace on which several probability distributions agree, at a lower computational cost and with higher accuracy.
Abstract: We propose a novel algebraic algorithmic framework for dealing with probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of algebraic geometry, which we demonstrate in a compact proof for an identifiability criterion.
TL;DR: An extension of generalized lambda distributions (XGLD) model for estimating statistical distributions that has a considerable precision and high flexibility to fit more probability distribution functions with higher accuracy is presented.
Abstract: The family of four-parameter generalized lambda distributions (GLD) is known for its high flexibility. It provides an approximation of most of the usual statistical distributions (e.g., normal, uniform, lognormal, Weibull, etc.). Although GLD is used in many fields where precise data modeling is required, there are some statistical distributions that could not be estimated with high precision. The main objective of this article is to present an extension of generalized lambda distributions (XGLD) model for estimating statistical distributions. This new method has a considerable precision and high flexibility to fit more probability distribution functions with higher accuracy. Using the existing methods for calculation of GLD parameters, it provides methodology of calculating XGLD parameter measurement algorithmically. The XGLD estimations are computed for some well-known distributions and precision of estimations is compared with that of GLD.
Journal Article•
TL;DR: In this paper, a mixture model that combines two special cases of heavy-tailed Rayleigh distribution is proposed, and the performance of this model is strong compared with other models such as K distribution, G0 distribution, and heavy-tail Rayleigh models.
Abstract: We propose a novel mixture model that combines two special cases of heavy-tailed Rayleigh distribution. These two special families possess the only analytical forms of heavy-tailed Rayleigh distribution. As a consequence, the mixture model has an analytical form. Because heavy-tailed Rayleigh distribution is a member of spherically invariant random process, one can obtain the parameter estimation by the method-of-moments technique. Finally, the mixture model has been tested on various synthetic aperture radar images, and the performance of this model is strong compared with other models such as K distribution, G0 distribution, and heavy-tailed Rayleigh models.
Posted Content•
TL;DR: In this paper, a new Bayesian approach using Metropolis random walk chain and direct numerical integration is proposed to estimate the parameters of the stable distributions, which are a rich class of probability distributions that are widely used to model leptokurtic data.
Abstract: Stable distributions are a rich class of probability distributions that are widely used to model leptokurtic data. Since the probability density and distribution functions are not known in closed form, stable distributions are often specified by their characteristic functions. This paper reviews both the techniques used to compute the density functions and the methods used to estimate parameters of the stable distributions. A new Bayesian approach using Metropolis random walk chain and direct numerical integration is proposed. The performance of the method is examined by a simulation study.
Dissertation•
01 May 2012
MBDA1
TL;DR: The aim of this paper is to show that, given a couple of probability distributions with a single estimator, the evaluation of their Hellinger distance provides a metric for a Riemann manifold that implies that an estimator can always be found that makes the distance minimal.
Abstract: Hellinger distance provides a way to evaluate how far is a given probability distribution from another one. This kind of tool is well-suited e.g. for target recognition in a radar system. The aim of this paper is to show that, given a couple of probability distributions with a single estimator, the evaluation of their Hellinger distance provides a metric for a Riemann manifold that, being conformal, implies that an estimator can always be found that makes the distance minimal. So, in this case, the choice of the best distribution reduces simply to the computation of this estimator. Finally, applications in the area of target recognition can be devised.
22 Oct 2012
TL;DR: In this article, a prior distribution is introduced to the estimated term in the log-cumulant equation and closed-form Bayesian estimation is obtained, which leads to better sea clutter modeling in terms of more accurate constant false alarm rate (CFAR) control.
Abstract: In this paper, the problem of using the method of logarithmic cumulants (MoLC) for parameter estimation of the K distribution is addressed. Specifically, we have pointed out that the MoLC is likely to suffer from non-invertible equations. In order to overcome such difficulty, a prior distribution is introduced to the estimated term in the log-cumulant equation and closed-form Bayesian estimation is obtained. Numerical experiments demonstrate that this approach not only provides an always-solvable equation, but also universally improves the estimation accuracy. Finally, the application of the MoLC for ship detection in synthetic aperture radar (SAR) images is demonstrated. Experimental results with the RADARSAT-2 data show that the proposed method leads to better sea clutter modeling in terms of more accurate constant false alarm rate (CFAR) control.
22 Jul 2012
TL;DR: An accurate description of the ocean backscatter from a probability density function is proposed and the α-stable distributions are used which can take care the property of heavy tails.
Abstract: In this contribution, an accurate description of the ocean backscatter from a probability density function is proposed. The Elfouhaily spectrum has been used to generate a realistic sea surface. The scattering field will be computed by using the Physical Optics (PO). The K distribution has been already used to characterize the Radar Cross Section (RCS) of the sea surface. However, the probability density function of the RCS can have heavy tails. Consequently, we use the α-stable distributions which can take care the property of heavy tails. The probability density function is estimated with a least squared method. We finally compare the results obtained with each model by using the Kolmogorov-Smirnov test from several random surfaces and a statistical study is made by giving a boxplot of the estimated parameters of the α-stable distribution.
Patent•
13 Nov 2012
TL;DR: In this paper, a computerized sequential bounded estimation is performed on time-series data, where realistic prior probability distributions of pertinent variables are utilized, and time-dependent measurements and errors in measurements are received.
Abstract: Computerized sequential bounded estimation is performed on time-series data. Robust methods use bounds and probability distributions to estimate target parameters for time-dependent data, including but not limited to the location of objects or phenomena. Realistic prior probability distributions of pertinent variables are utilized, and time-dependent measurements and errors in measurements are received. Bounds and probability distributions can be obtained without making any assumption of linearity. The sequential methods used for location are applicable in other applications in which a function of the probability distribution is desired for variables that are related to measurements.
13 May 2012
TL;DR: This paper generalizes probability-possibility transformations to two-dimensional distributions with a particular care to the maximum specificity principle, so that joint probability distributions can be suitably transformed into maximally specific joint possibility distributions.
Abstract: In the recent years the possibility theory has been investigated by many Authors in the field of mathematics and engineering. A possibility distribution is, from the mathematical point of view, a generalization of a probability distribution, since it can represent a family of probability distributions. Given a probability distribution, different probability-possibility transformations have been defined, which transform the probability distribution into different possibility distributions. Probability-possibility transformations are useful in any problem where statistical data must be dealt within the possibility theory, together with other heterogeneous uncertain and imprecise data. This paper generalizes these transformations to two-dimensional distributions with a particular care to the maximum specificity principle, so that joint probability distributions can be suitably transformed into maximally specific joint possibility distributions.
15 May 2012
TL;DR: In this paper, error performance bounds are derived for coded FSO communication systems operating over atmospheric turbulence channels, which are modeled as K distribution conditions and a new approximate closed-form expression is derived for the pairwise error probability (PEP).
Abstract: In this paper, error performance bounds are derived for coded FSO communication systems operating over atmospheric turbulence channels, which are modeled as K distribution conditions. Existing upper bounds presented in the open technical literature demonstrate some discrepancy in the low signal-to-noise ratio (SNR). This limitation has its origin in the fact that the K probability density function (PDF) contains a modified Bessel function of the second kind, which precludes simple closed-form expressions for the error performance bounds. Recently, it is shown by the author that the probability density function (PDF) of the K distribution can be approximated accurately by a finite sum of weighted negative exponential PDFs. Based on this interesting result, in this paper, a new approximate closed-form expression is derived for the pairwise error probability (PEP). Compared with the previous published results, which are in the form of an infinite series or an upper bound that are accurate only for a certain range of signal-to-noise ratio (SNR), the presented PEP is in closed-for, and is more accurate both for small and high SNR values. The derived PEP is then applied to derive an upper bound to the bit-error probability for convolutional codes for FSO communication through the K channels. Numerical results are further demonstrated to confirm the analytical results and also to show the good accuracy of the derived expressions.
TL;DR: A system for simulating correlated gamma distributed fields of scattering coefficients, based on the requirement that in radar and other coherent imaging scenarios the locations of elemental scene scatterers must be random to achieve fully developed speckle in the image is presented.
Abstract: Simulation plays an important role in both the development and analysis of new radar imaging and processing systems and in the analysis of their data. We have developed a synthetic aperture radar (SAR) simulation system, cSAR, that simulates the raw signal data. An important component of this system is the simulation of clutter, or the random spatial fluctuations of backscatter. The prevailing statistical model for clutter is the K distribution. This model is founded on a gamma distributed scattering coefficient. In this paper we present a system for simulating correlated gamma distributed fields of scattering coefficients. Our system starts with the requirement that in radar and other coherent imaging scenarios the locations of elemental scene scatterers must be random to achieve fully developed speckle in the image. We start with a set of scatterers whose spatial location follows a uniform distribution. To generate a correlated random field with this random distribution of scatterers we use the turning ...
TL;DR: In this article, the joint distributions of order statistics of innid discrete random variables are expressed in the form of an integral and the results related to pf and df are given.
Abstract: In this study, the joint distributions of order statistics of innid discrete random variables are expressed. Also, the joint distributions are obtained in the form of an integral. Then, the results related to pf and df are given. MSC:62G30, 62E15.
03 May 2012
TL;DR: It is demonstrated that the Maximum Entropy principle in the context of Bayesian probability theory can be used to derive the probability distributions of those processes characterized by its scaling properties including multiscaling moments and geometric mean.
Abstract: We have demonstrated that the Maximum Entropy (ME) principle in the context of Bayesian probability theory can be used to derive the probability distributions of those processes characterized by its scaling properties including multiscaling moments and geometric mean We started from a proof-of-concept case of a power-law probability distribution, followed by the general case of multifractality aided by the wavelet representation of the cascade model The ME formalism leads to the probability distribution of the multiscaling parameter and those of incremental multifractal processes at different scales Compared to other methods, the ME method significantly reduces computational cost by leaving out unimportant details The ME distributions have been evaluated against the empirical histograms derived from the drainage area of river network, soil moisture and topography This analysis supports the assertion that the ME principle is a universal and unified framework for modeling processes governed by scale-in
Posted Content•
TL;DR: In this paper, a geometric formulation of fiducial probability was proposed based on the geometric properties of an ordinary two-variable function and its surface representation, which is an improvement over the usual pivotal quantity formulation.
Abstract: The geometric formulation of fiducial probability employed in this paper is an improvement over the usual pivotal quantity formulation. For a single parameter and single observation, the new formulation is based on the geometric properties of an ordinary two variable function and its surface representation.
The following theorem is proved: A fiducial distribution for the continuous parameter $\theta$ exists if and only if (i) the continuous random probability distributions of $x$ for different $\theta$'s are non-intersecting, and (ii) the random distributions are complete, i.e. at the extreme values of $\theta$ the limiting probability distributions are zero and one for all $x$.
The proof yields also a complete characterization of random distributions that lead to fiducial distributions.
The paper also treats intersecting distributions and non-intersecting incomplete distributions. The latter, which are frequently encountered in a null hypothesis, are shown to be associated with intersecting "composite" distributions.
An appendix compares the pivotal and geometric formulations.
01 Jan 2012
TL;DR: In this article, the authors apply the concepts and definitions given there to a number of practical distributions frequently met in physical science, such as the uniform, exponential, Cauchy, binomial, multinomial, Poisson and normal distributions.
Abstract: This chapter builds on the work of Chapter 3 and applies the concepts and definitions given there to a number of practical distributions frequently met in physical science. These are: the uniform, exponential, Cauchy, binomial, multinomial, Poisson and normal (Gaussian) distributions. Because of the great importance of the normal distribution, this is discussed in some detail for both the single variate and the multivariate cases, with a separate section on the bivariate case. The basic features and properties of these various distributions, and any relations between them, are derived.
TL;DR: The minimum error probability Bayes classifier is used to solve the problem of separating noise data which obey K distribution from the target which also obeys K, and achieves good results.
Abstract: Due to multipath noise pollution, SAS image consists of two parts : target and noise .They can be described by K + K mixture distribution . How to separate noise data which obey K distribution from the target which also obeys K is a hot topic in SAS image field. This paper used the minimum error probability Bayes classifier to solve this problem, and achieved good results. At the same time, this paper also studied the factors that affect the classification results, such as the absolute value difference of training sample parameters and K distribution parameters.
01 Jan 2012
TL;DR: In this article, a general form of continuous probability distribution is characterized through conditional expectation of contrast of order statistics, conditioned on a nonadjacent order statistics and some of its deductions are discussed.
Abstract: General form of continuous probability distribution is characterized through conditional expectation of contrast of order statistics, conditioned on a nonadjacent order statistics and some of its deductions are discussed. Mathematics Subject Classification: 62G30, 62E10