Showing papers on "K-distribution published in 1995"

A statistical description of polarimetric and interferometric synthetic aperture radar data

Abstract: This paper provides a nearly complete analysis of the key distributions encountered in single- and multi-look polarimetric synthetic aperture radar data under the bivariate Gaussian and K -distribution models. It contains new analytic results on the moments of the amplitude and phase difference in single look data and on the moments of the amplitude in multi-look data. As yet no analytic results for the moments of multi-look phase difference have been found, except in limiting cases. The maximum likelihood estimators of the covariance matrix elements of two jointly Gaussian channels are derived, together with their asymptotic variances. The problems in extending this analysis to the bivariate K distribution are also discussed.

Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation

Abstract: This paper deals with the trend to equilibrium of solutions to the spacehomogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has previously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of order 2+e, then the convergence to equilibrium in his metric is exponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to the Boltzmann equation, by proving that the Fourier-transformed solutions are at least as regular as the Fourier transform of the initial data. This is also used to prove that even if the initial data only possess a second moment, then ∫∣v∣>R f(v, t) ∣v∣2 dv→0 asR→∞, and this convergence is uniform in time.

Statistical models for polarimetric data: consequences, testing and validity

Abstract: Low resolution polarimetric data gathered by the AirSAR system over the Feltwell U.K. agriculturaltestsite reveals Gaussian behaviour at C band for all vegetation types, but clear evidence of texture at longer wavelengths. The measurements are compared with the predictions of a polarimetric texture model based on a multivariate K distribution (which includes the Gaussian distribution as a special case), from which distributions of the inphase component, amplitude, amplitude ratio, phase difference and real hermitian product between channels are derived. Kolmogorov-Smirnov fits to these marginal distributions verify that C, Land P band observations over a range of vegetation types are consistent with the model, but there is evidence of the model breaking down in cereal fields at P band. The departure from Gaussian behaviour with increasing wavelength is strongest for cereals; less marked trends are observed for root vegetables, while forest appears Gaussian at all wavelengths. These results are un...

Ordering claim size distributions and mixed Poisson probabilities

Abstract: We investigate various orderings between continuous distributions for severities, having the same first n moments. Such situations occur for instance when severity distributions are fitted by the method of moments. General results are derived which establish an ordering between such distributions, and these results are applied to compare the Gamma, the Inverse Gaussian and the lognormal distributions with equal means and variances. Finally, we consider the situation where such continuous distributions are used as mixing distributions in mixed Poisson models for claim numbers, and show that the order properties of the mixing distributions are inherited by the corresponding mixed Poisson distributions.

Alternative distributions for the multiplicative model in SAR images

Abstract: Another class of distributions is proposed to model the terrain backscatter, namely the class of the generalized inverse Gaussian distributions. Besides allowing the explicit calculation of the density of the random variable that models the radar return, these distributions have the remarkable property of having the gamma and other distributions as particular cases. The resulting distributions for complex, multilook intensity and multilook amplitude data are derived assuming the multiplicative model. These new densities yield to a more general model-than the classical one, which is given by the class of K distributions. Several plots are presented, showing the flexibility of these new distributions and its possible use as a model for SAR data.

Calculating the K-distribution by saddlepoint integration

Abstract: An important probability distribution for modelling non-Gaussian phenomenon is the K-distribution. Because of its widespread use in radar and underwater acoustics, a method for calculating the distribution is required which is both easy and efficient. The paper presents a method for calculating the K-distribution, in particular the K-Bessel function, using saddlepoint integration. The paper provides a detailed truncation-error analysis of the technique, and an implementation in MATLAB.

p-Adic stochastics and Dirac quantization with negative probabilities

Abstract: A new mathematical apparatus, ap-adic theory of probability, is applied to realize the hypothetical world based on negative probability distributions created by Dirac for the relativistic quantization of photons. Within thep-adic theory of probability, negative probability distributions are well defined (in the language of limits of relative frequencies, but with respect to ap-adic metric). We propose that the negative Gibbs distributions arising in relativistic quantization are described byp-adic Stochastics.

Radar clutter classification using autoregressive modelling, K-distribution and neural network

Abstract: This paper is concerned with the classification of radar returns including sea, ground and composite clutters. We first present an analysis of radar clutter recorded data allowing to validate the K amplitude distribution and the autoregressive modelling of the spectrum. Then, we briefly describe a classifier based on a multi-layer neural network. The inputs of which are the shape parameter of the K-distribution, the magnitude and the phase of the poles and the reflection coefficients calculated by means of the Burg's or multi-segment algorithm. Experimental results are presented to illustrate the performance of the proposed classifier.

Simulation of radar sea clutter using autoregressive modelling and K-distribution

Abstract: In the context of target detection by surface radars, the paper presents a new technique for modelling and simulating the sea clutter. This technique was derived from an analysis of real-life data obtained with an S-band radar. The study on experimental measurements has shown that the correlation properties, in terms of power spectrum, of the sea clutter can be well described by means of a low order autoregressive (AR) model, while its amplitude can be modelled in using the K-distribution. A procedure for simulating sea clutter with an AR model combined with a K amplitude distribution, is proposed and is tested by statistical tests and a comparison with real sea clutter data.

High resolution SAR clutter textural analysis

Abstract: Coherent images of natural scenes formed using synthetic aperture radar (SAR) often possess textural properties associated with the clutter. Due to an improvement in the Defence Research Agency (DRA) X-band SAR, very high resolution imagery is now available for analysis. This increase in resolution has visibly modified the textural properties of observed clutter forcing a re-examination of the statistical image properties. The results of the study are given. Areas of imagery that appear homogenous have their single point distribution properties measured. Comparisons with known distributions that often fit similar data are made and shown to give poor agreement. Reasons for poor agreement, such as inhomogeneity, are investigated through the use of clutter simulations. K and lognormal mixture distributions are shown to offer good agreement with the observed distributions and also validate the premise of homogeneity for the regions considered.

A Family of Joint Probability Models for Cost and Schedule Uncertainties

Abstract: Tiis paper discusses a family of analytical models from which the joint probability of total program cost and schedule can be calculated, analyzed, and presented to decision-makers Specifically, the classical bivariate normal and two lesser known joint distributions, the normal-lognormal and the bivariate lognormal distributions are discussed Experiences from Monte Carlo simulations suggest that this family of bivariate distributions are candidate models for computing joint and conditional cost and schedule probabilities In particular, the discussion on the nonnal-lognormal distribution as a joint cost-schedule probability model extends research on the applicability of the bivariate lognormal presented last year* Joint probability distributions enable analysts and decision-makers to determine joint and conditional probabilities of the types

15. Operator-Limit Distributions in Probability Theory

Abstract: The central theme of that monograph are limit distributions in a scheme (1) A n d-valued independent random vectors, A i 's are MATRICES (linear bounded operators) on R d and x i 's are deterministic R d shift vectors. If in (1) one assumes that the triangular array A n X k , 1 ≤ k ≤ n, n ≥ 1, is uniformly infinitesimal then limit distributions are called operator-selfdecomposable. This class includes the classical selfdecomposable measures. For independent and identically distributed X i 's limiting laws are called operator-stable measures. Among them we have the classical and well know stable measures.

Investigations of Reynolds-Averaged Turbulence Quantities

Abstract: The equations for the higher-order moments and probability density function of turbulent velocity fluctuations are considered. These are derived utilizing the basic hydrodynamic equations of fluid flow. Using truncated cumulant expansions as approximations for the probability density distributions of the corresponding turbulence quantities, an alternative set of equations for the moments is formulated that contains only velocity correlations. From these equations, interrelations between the higher-order moments are deduced. Several theoretically derived relationships between correlations of different orders are experimentally verified using data available in the literature and also data measured by the authors. In the paper, an attempt is made to reconstruct the entire probability density distributions from derived inter-relations between the higher-order moments.

K distribution model of ultrasound speckle: fractional order SNRs and log compression variance

Abstract: The K distribution has been shown to be a good statistical model for the ultrasound echo envelope in terms of the scatterer statistics of the medium. The authors use this model to study the properties of the envelope moments, the signal-to-noise ratio (SNR) function, and, the log compression in terms of the scatterer statistics.

A novel linear-quadratic technique for coherent radar detection in mixed clutter environment

Abstract: This paper introduces a sub-optimum procedure for the detection of a signal of known form in the presence of non Gaussian disturbance, that is assumed to be a mixture of coherent K-distributed and Gaussian distributed clutter sources. As a comparison, the optimum Neymann-Pearson (NP) strategy to detect the a priori known signal against the above mentioned disturbance is also presented. The optimum NP detection scheme is not simple to implement: it involves a heavy numerical integration with respect to the texture variable of the K distribution. Besides it strongly depends on the parameters v and ? of the texture distribution, so no predetermined threshold can be assigned to achieve a given probability of false alarm if they are unknown. Our idea is to estimate the texture variable, or rather the local power in the cell under test, and to use this estimate in the likelihood ratio. The resulting detection strategy is linear-quadratic and clutter distribution free. Moreover the performance is almost equal to that of the optimum receiver.

Universal estimation of the optimal probability distributions for data compression of discrete memoryless sources with fidelity criterion

Abstract: Output probability distributions of the test channels play important roles in data compression of discrete memoryless sources with fidelity criterion. In this paper a universal algorithm for estimating the output probability distributions is proposed. Sample size required by the algorithm is evaluated under a criterion of estimation similar to that of PAC learning in the computational learning theory.