Showing papers on "K-distribution published in 2013"

A new method for generating families of continuous distributions

Abstract: In this paper, a new method is proposed for generating families of continuous distributions. A random variable $$X$$ , “the transformer”, is used to transform another random variable $$T$$ , “the transformed”. The resulting family, the $$T$$ - $$X$$ family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function of the random variable $$X$$ . Many new distributions, which are members of the family, are presented. Several known continuous distributions are found to be special cases of the new distributions.

On the Method of Logarithmic Cumulants for Parametric Probability Density Function Estimation

Abstract: Parameter estimation of probability density functions is one of the major steps in the area of statistical image and signal processing. In this paper we explore several properties and limitations of the recently proposed method of logarithmic cumulants (MoLC) parameter estimation approach which is an alternative to the classical maximum likelihood (ML) and method of moments (MoM) approaches. We derive the general sufficient condition for a strong consistency of the MoLC estimates which represents an important asymptotic property of any statistical estimator. This result enables the demonstration of the strong consistency of MoLC estimates for a selection of widely used distribution families originating from (but not restricted to) synthetic aperture radar image processing. We then derive the analytical conditions of applicability of MoLC to samples for the distribution families in our selection. Finally, we conduct various synthetic and real data experiments to assess the comparative properties, applicability and small sample performance of MoLC notably for the generalized gamma and K families of distributions. Supervised image classification experiments are considered for medical ultrasound and remote-sensing SAR imagery. The obtained results suggest that MoLC is a feasible and computationally fast yet not universally applicable alternative to MoM. MoLC becomes especially useful when the direct ML approach turns out to be unfeasible.

Bivariate distribution models using copulas for reliability analysis

Abstract: The modeling of joint probability distributions of correlated variables and the evaluation of reliability under incomplete probability information remain a challenge that has not been studied extensively. This article aims to investigate the effect of copulas for modeling dependence structures between variables on reliability under incomplete probability information. First, a copula-based method is proposed to model the joint probability distributions of multiple correlated variables with given marginal distributions and correlation coefficients. Second, a reliability problem is formulated and a direct integration method for calculating probability of failure is presented. Finally, the reliability is investigated to demonstrate the effect of copulas on reliability. The joint probability distribution of multiple variables, with given marginal distributions and correlation coefficients, can be constructed using copulas in a general and flexible way. The probabilities of failure produced by different copulas...

On φ-Families of Probability Distributions

Abstract: We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a φ-function, resulting in a φ-family of probability distributions. We show how φ-families are constructed. In a φ-family, the analogue of the cumulant-generating function is a normalizing function. We define the φ-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback–Leibler divergence. A formula for the φ-divergence where the φ-function is the Kaniadakis κ-exponential function is derived.

Probability Distributions of Ellipticity

Abstract: In this paper we derive an exact full expression for the 2D probability distribution of the ellipticity of an object measured from data, only assuming Gaussian noise in pixel values. This is a generalisation of the probability distribution for the ratio of single random variables, that is well-known, to the multivariate case. This expression is derived within the context of the measurement of weak gravitational lensing from noisy galaxy images. We find that the third flattening, or epsilon-ellipticity, has a biased maximum likelihood but an unbiased mean; and that the third eccentricity, or normalised polarisation chi, has both a biased maximum likelihood and a biased mean. The very fact that the bias in the ellipticity is itself a function of the ellipticity requires an accurate knowledge of the intrinsic ellipticity distribution of the galaxies in order to properly calibrate shear measurements. We use this expression to explore strategies for calibration of biases caused by measurement processes in weak gravitational lensing. We find that upcoming weak lensing surveys like KiDS or DES require calibration fields of order of several degrees and 1.2 magnitudes deeper than the wide survey in order to correct for the noise bias. Future surveys like Euclid will require calibration fields of order 40 square degree and several magnitude deeper than the wide survey. We also investigate the use of the Stokes parameters to estimate the shear as an alternative to the ellipticity. We find that they can provide unbiased shear estimates at the cost of a very large variance in the measurement. The python code used to compute the distributions presented in the paper and to perform the numerical calculations are available on request.

Review of Envelope Statistics Models for Quantitative Ultrasound Imaging and Tissue Characterization

Abstract: The homodyned K-distribution and the K-distribution, viewed as a special case, as well as the Rayleigh and the Rice distributions, viewed as limiting cases, are discussed in the context of quantitative ultrasound (QUS) imaging. The Nakagami distribution is presented as an approximation of the homodyned K-distribution. The main assumptions made are: (1) the absence of log-compression or application of non-linear filtering on the echo envelope of the radiofrequency signal; (2) the randomness and independence of the diffuse scatterers. We explain why other available models are less amenable to a physical interpretation of their parameters. We also present the main methods for the estimation of the statistical parameters of these distributions. We explain why we advocate the methods based on the X-statistics for the Rice and the Nakagami distributions, and the K-distribution. The limitations of the proposed models are presented. Several new results are included in the discussion sections, with proofs in the appendix.

Image watermark detection in the wavelet domain using Bessel K densities

Abstract: In this study, the authors propose a wavelet domain still image watermark detection method which uses the Bessel K probability density function to describe the distribution of wavelet coefficients. In this study, watermark detection is formulated as a binary statistical decision problem which is to detect a signal submerged in the noise that follows a Bessel K distribution. Using this formulation, an optimal watermark detector using likelihood ratio test is proposed. The experimental results of the proposed method in a variety of situations demonstrate that the proposed method has a robust detection performance for additive spread spectrum watermarks.

Second-Order Asymptotics of Conversions of Distributions and Entangled States Based on Rayleigh-Normal Probability Distributions

Abstract: We discuss the asymptotic behavior of conversions between two independent and identical distributions up to the second-order conversion rate when the conversion is produced by a deterministic function from the input probability space to the output probability space. To derive the second-order conversion rate, we introduce new probability distributions named Rayleigh-normal distributions. The family of Rayleigh-normal distributions includes a Rayleigh distribution and coincides with the standard normal distribution in the limit case. Using this family of probability distributions, we represent the asymptotic second-order rates for the distribution conversion. As an application, we also consider the asymptotic behavior of conversions between the multiple copies of two pure entangled states in quantum systems when only local operations and classical communications (LOCC) are allowed. This problem contains entanglement concentration, entanglement dilution and a kind of cloning problem with LOCC restriction as special cases.

Joint probability distributions and conditional probabilities in the tomographic representation of quantum states

Abstract: The tomographic-probability distributions associated with quantum states of photons and qudits are generalized and considered as joint probability distributions and conditional probability distributions. Entropic inequalities such as the subadditivity condition as well as nonnegativity of Shannon entropy are applied for obtaining new analogues of entropic uncertainty relations.

On Generalized Gamma Convolution Distributions

Abstract: We present here characterizations of certain families of generalized gamma convolution distributions of L. Bondesson based on a simple relationship between two truncated moments. We also present a list of well-known random variables whose distributions or the distributions of certain functions of them belong to the class of generalized gamma convolutions. Mathematics Subject Classification 2000: 46F10

A generalisation of the Rayleigh distribution with applications in wireless fading channels

Abstract: The signal received in a mobile radio environment exhibits rapid signal level fluctuations which are generally Rayleigh-distributed. These result from interference by multiple scattered radio paths between the base station and the mobile receptor. Fading-shadowing effects in wireless channels are usually modelled by means of the Rayleigh–Lognormal distribution (RL), which has a complicated integral form. The K-distribution (K) is similar to RL but it has a simpler form and its probability density function admits a closed form; however, due to the Bessel function, parameter estimates are not direct. Another possible approach is that of the Rayleigh-inverse Gaussian distribution (RIG). In this paper, an alternative is presented, a generalisation of the Rayleigh distribution which is simpler than the RL, K and RIG distributions, and thus more suitable for the analysis and design of contemporary wireless communication systems. Closed-form expressions for the bit error rate (BER) for differential phase-shift keying (DPSK) and minimum shift keying (MSK) modulations with the proposed distribution are obtained. Theoretical results based on statistically well-founded distance measurements validate the new distribution for the cases analysed. Copyright © 2011 John Wiley & Sons, Ltd.

Evaluation of run-length distribution for CUSUM charts under gamma distributions

Abstract: Numerical evaluation of run-length distributions of CUSUM charts under normal distributions has received considerable attention. However, accurate approximation of run-length distributions under non-normal or skewed distributions is challenging and has generally been overlooked. This article provides a fast and accurate algorithm based on the piecewise collocation method for computing the run-length distribution of CUSUM charts under skewed distributions such as gamma distributions. It is shown that the piecewise collocation method can provide a more robust approximation of the run-length distribution than other existing methods such as the Gaussian quadrature-based approach, especially when the process distribution is heavily skewed. Some computational aspects including an alternative formulation based on matrix decomposition and geometric approximation of run-length distribution are discussed. Design guidelines of such a CUSUM chart are also provided.

Compounding: an R package for computing continuous distributions obtained by compounding a continuous and a discrete distribution

Abstract: In this manuscript we introduce R package Compounding for dealing with continuous distributions obtained by compounding continuous distributions with discrete distributions. We demonstrate its use by computing values of cumulative distribution function, probability density function, quantile function and hazard rate function, generating random samples from a population with compounding distribution, and computing mean, variance, skewness and kurtosis of a random variable with a compounding distribution. We consider 24 discrete distributions which can be compounded with any continuous distribution implemented in R.

The analysis of sea clutter statistics characteristics based on the observed sea clutter of Ku-band Radar

Abstract: The model of GTI, TSC, NRL distribution the most fundament characteristic of sea clutter, as used in radar performance evaluation. The model of Rayleigh, LogNormal, Weibull and K distribution radar clutter are analyzed and modeled and some kinds of radar clutter, such as ground, weather, chaff and sea clutter are modeled and simulated. The analysis focuses on amplitude characteristic of sea clutter. The analysis would contribute to designing and implementation of radar filter and increasing the ability of suppressing sea clutter and ensuring the detection ability of radar itself.

Modeling multivariate distributions using Monte Carlo simulation for structural reliability analysis with complex performance function

Abstract: The simulation of multivariate distributions has not been investigated extensively. This article aims to propose Monte Carlo simulation (MCS)-based procedures for modeling the joint probability dis...

Approximating Continuous Probability Distributions Using the 10th, 50th, and 90th Percentiles

Abstract: In economic decision analyses, continuous uncertainties are often represented by discrete probability distributions. In this article, we analyze the ability of discretizations based on the 10th, 50th, and 90th percentiles to match the mean, variance, skewness, and kurtosis of a wide range of distributions in the Johnson distribution system. In addition, we develop new discretization methods that improve upon current practice. Finally, we demonstrate that all of these methods are special cases from a continuum of weightings and show under which conditions each is most appropriate. Our results provide guidelines for the methods’ applications and limits to their usefulness.

Order Statistics for Discrete Distributions

Abstract: Some properties of order statistics based on discrete distributions are discussed. Conditional and unconditional distributions of discrete order statistics are found. Some problems are solved for order statistics, which are based on geometric distributions. We also consider sampling without replacement from finite populations and consider the distributions and properties of the corresponding order statistics.

On ϕ-Families of Probability Distributions

Abstract: We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a ϕ-function, resulting in a ϕ-family of probability distributions. We show how ϕ-families are constructed. In a ϕ-family, the analogue of the cumulant-generating function is a normalizing function. We de- fine the ϕ-divergence as the Bregman divergence associated to the normalizing func- tion, providing a generalization of the Kullback-Leibler divergence. A formula for the ϕ-divergence where the ϕ-function is the Kaniadakis κ-exponential function is derived.

Extension of the Fourth Moment Theorem to invariant measures of diffusions

Abstract: We characterize the convergence in distribution of a sequence of random variables in a Wiener chaos of a fixed order to a probability distribution which is the invariant measure of a diffusion process. This class of target distributions includes the most known continuous probability distributions. Our results are given in terms of the Malliavin calculus and of the coefficients of the associated diffusion process and extend the standard Fourth Moment Theorem by Nualart and Peccati. In particular we prove that, among the distributions whose associated squared diffusion coefficient is a polynomial of second degree (with some restrictions on its coefficients), the only possible limits of sequences of multiple integrals are the Gaussian and the Gamma laws.

On the application of copula theory for determination of probabilistic characteristics of springflood

Abstract: Considered is the possibility of using copula theory for creating joint probability distributions of springflood peak discharges and flow volumes taking account of the relations between discharges and flow volumes. For approximation of marginal distributions, Gumbel distribution was used for peak discharges, and two-parameter gamma distribution, for flow volumes. Joint two-dimensional distribution was built as a marginal distribution function which was set as one of the three one-parameter Archimedean copulas using different ways of determining their parameters. The best results were obtained for Gumbel-Hougaard copula using the method of maximum likelihood to determine its parameters. Major flood risk estimates determined from one- and two-dimensional probability distributions of their characteristics were compared with each other. Demonstrated are the benefits of using two-dimensional probability distributions of flood characteristics as compared with one-dimensional distributions for probabilistic estimation of floods. The data on springflood peak discharges and flow volumes in the Belaya and Vyatka rivers were used for this study.

SCS-CN Method Revisited Using Entropy Theory

Abstract: . The SCS-CN method is one of the most popular methods for computing runoff from small watersheds (agricultural, forest, rural, and urban) for individual rainfall events. This study revisits the method using the entropy theory, which provides insights into the structure of the method and permits derivation of the probability distributions of the variables (CN = curve number, S = maximum soil moisture retention, P = precipitation, J = cumulative infiltration, I a = initial abstraction, and Q = surface runoff) inherent in the method if they are assumed random. If the variables are continuous, then the derivation of the distributions is based on the maximization of the Shannon entropy, subject to given constraints, and the derived distributions are non-parametric. If the variables are discrete, then the derivation is based on the maximization of cross entropy, subject to fractile constraints, wherein prior probability distributions are derived by Shannon entropy maximizing. The derived distributions are tested using field data. It is found that the SCS-CN method requires no information for the probability distribution of runoff associated with it, other than obeying the total probability law. Employing four statistical measures, including Akaike information criterion (AIC), Bayesian information criterion (BIC), bias (BIAS), and root mean square error (RMSE), to determine the goodness-of-fit of probability distributions to 100-CN, Q, P, S, Q/(P-I a ), and J/S, it is found that the gamma distribution, on the whole, is the preferred distribution.

The new unified representation of multivariate skewed distributions

Abstract: There are so many proposals in construction skewed distributions, and it is worth finding an overall class which covers all of these proposals. We introduce a new unified representation of multivariate skewed distributions. We will show that this new unified multivariate form of skewed distributions includes all of the continuous multivariate skewed distributions in the literature. This new unified representation is based on the multivariate probability integral transformation and can be decomposed into one factor that is original multivariate symmetric probability density function (pdf) f on ℜ k and skewed factor defined by a pdf p on [0, 1] k . This decomposition leads us to prove some useful properties of this new unified form. Stochastic representations and basic properties of this new form are also investigated in this article. Our work is motivated by considering the different skewing mechanisms which lead to different skewed distributions and show that all of these common-used distributions can be ...

A Family of Skew-Slash Distributions and Estimation of its Parameters via an EM Algorithm

Abstract: In this paper, a family of skew-slash distributions is defined and investigated. We define the new family by the scale mixture of a skew-elliptically distributed random variable with the power of a uniform random variable. This family of distributions contains slash-elliptical and skew-slash distributions. We obtain the moments and some distri- butional properties of the new family of distributions. In the special case of slash skew-t distribution, an EM-type algorithm is presented to estimate the parameters. Some applications are provided for illustra- tions.

Adaptive detection in compound-gaussian clutter with inverse gaussian texture

Abstract: This paper mainly deals with the problem of target detection in the presence of Compound-Gaussian (CG) clutter with the Inverse Gaussian (IG) texture and the unknown Power Spectral Density (PSD). The traditional CG distributions, in particular the K distribution and the complex multivariate t distribution, are widely used for modeling the real clutter data from the High-Resolution (HR) radars. Recently, the novel CG distribution with the IG texture is described as the IG-CG distribution and validated to provide the better flt with the recorded data of the HR clutter than the mentioned two competitors. Within the IG-CG framework, the detector is flrstly proposed here in terms of the two-step Generalized Likelihood Ratio Test (GLRT) criterion, and the empirical estimation method is resorted to estimate the unknown PSD in order to adapt the realistic scenario. The proposed detector is tested on the real-life HR clutter data, in comparison with the Adaptive Normalized Matched Filter (ANMF) processor, and the detection results illustrate that it outperforms the ANMF.

Shortest Path Problem with Gamma Probability Distribution Arc Length

Abstract: We propose a dynamic program to find the shortest path in a network having gamma probability distributions as arc lengths. Two operators of sum and comparison need to be adapted for the proposed dynamic program. Convolution approach is used to sum two gamma probability distributions being employed in the dynamic program.