Showing papers on "Probability density function published in 1993"

Maximum likelihood, consistency and data envelopment analysis: a statistical foundation

Abstract: This paper provides a formal statistical basis for the efficiency evaluation techniques of data envelopment analysis (DEA). DEA estimators of the best practice monotone increasing and concave production function are shown to be also maximum likelihood estimators if the deviation of actual output from the efficient output is regarded as a stochastic variable with a monotone decreasing probability density function. While the best practice frontier estimator is biased below the theoretical frontier for a finite sample size, the bias approaches zero for large samples. The DEA estimators exhibit the desirable asymptotic property of consistency, and the asymptotic distribution of the DEA estimators of inefficiency deviations is identical to the true distribution of these deviations. This result is then employed to suggest possible statistical tests of hypotheses based on asymptotic distributions.

Comparison of Smoothing Parameterizations in Bivariate Kernel Density Estimation

Abstract: The basic kernel density estimator in one dimension has a single smoothing parameter, usually referred to as the bandwidth. For higher dimensions, however, there are several options for smoothing parameterization of the kernel estimator. For the bivariate case, there can be between one and three independent smoothing parameters in the estimator, which leads to a flexibility versus complexity trade-off when using this estimator in practice. In this article the performances of the different possible smoothing parameterizations are compared, using both the asymptotic and exact mean integrated squared error. Our results show that it is important to have independent smoothing parameters for each of the coordinate directions. Although this is enough for many situations, for densities with high amounts of curvature in directions different to those of the coordinate axes, substantial gains can be made by allowing the kernel mass to have arbitrary orientations. The “sphering” approaches to choosing this o...

Non-Gaussian random vector identification using spherically invariant random processes

Abstract: With the modeling of non-Gaussian radar clutter in mind, elegant and tractable techniques are presented for characterizing the probability density function (PDF) of a correlated non-Gaussian radar vector. The need for a library of multivariable correlated non-Gaussian PDFs in order to characterize various clutter scenarios is discussed. Specifically,. the theory of spherically invariant random processes (SIRPs) is examined in detail. Approaches based on the marginal envelope PDF and the marginal characteristic function have been used to obtain several multivariate non-Gaussian PDFs. An important result providing the PDF of the quadratic form of a spherically invariant random vector (SIRV) is presented. This result enables the problem of distributed identification of a SIRV to be addressed. >

Joint Sequential Simulation of MultiGaussian Fields

Abstract: The sequential simulation algorithm can be used for the generation of conditional realizations from either a multiGaussian random function or any non-Gaussian random function as long as its conditional distributions can be derived. The multivariate probability density function (pdf) that fully describes a random function can be written as the product of a set of univariate conditional pdfs. Drawing realizations from the multivariate pdf amounts to drawing sequentially from that series of univariate conditional pdfs. Similarly, the joint multivariate pdf of several random functions can be written as the product of a series of univariate conditional pdfs. The key step consists of the derivation of the conditional pdfs. In the case of a multiGaussian fields, these univariate conditional pdfs are known to be Gaussian with mean and variance given by the solution of a set of normal equations also known as simple cokriging equations. Sequential simulation is preferred to other techniques, such as turning bands, because of its ease of use and extreme flexibility.

On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems

Abstract: The finite element method is applied to the solution of the transient Fokker-Planck equation for several often cited nonlinear stochastic systems accurately giving, for the first time, the joint probability density function of the response for a given initial distribution. The method accommodates nonlinearity in both stiffness and damping as well as both additive and multiplicative excitation, although only the former is considered herein. In contrast to the usual approach of directly solving the backward Kolmogorov equation, when appropriate boundary conditions are prescribed, the probability density function associated with the first passage problem can be directly obtained. Standard numerical methods are employed, and results are shown to be highly accurate. Several systems are examined, including linear, Duffing, and Van der Pol oscillators.

Distribution shape of two-dimensional DCT coefficients of natural images

Abstract: The univariate distribution of DCT coefficients of natural images is investigated. The probability density function (PDF) of the coefficients is modelled with the generalised Gaussian function (GGF) which includes the Gaussian and the Laplacian PDF as special cases. The shape parameter of the GGF is estimated according to the maximum likelihood principle, χ2 tests of fit showed that GGFs model the distribution of DCT coefficients more accurately than Laplacian PDFs.

A Fourier Approach to Nonparametric Deconvolution of a Density Estimate

Abstract: We consider the problem of constructing a nonparametric estimate of a probability density function h from independent random samples of observations from densities a and f, when a represents the convolution of h and f. Our approach is based on truncated Fourier inversion, in which the truncation point plays the role of a smoothing parameter. We derive the asymptotic mean integrated squared error of the estimate and use this formula to suggest a simple practical method for choosing the truncation point from the data. Strikingly, when the smoothing parameter is chosen in this way then in many circumstances the estimator behaves, to first order, as though the true f were known

A non-parametric and scale-independent method for cluster analysis – I. The univariate case

Abstract: The detection and analysis of structure and substructure in systems of galaxies is a well-known problem. Several methods of analysis exist with different ranges of applicability and giving different results. The aim of the present paper is to describe a general procedure of wide applicability that is based on a minimum number of general assumptions and gives an objective, testable, scale-independent and non-parametric estimate of the clustering pattern of a sample of observational data. The method follows the idea that the presence of a cluster in a data sample is indicated by a peak in the probability density underlying the data. There are two steps: the first is estimation of the probability density and the second is identification of the clusters

Weakly Non-Linear Gaussian Fluctuations and the Edgeworth Expansion

Abstract: We calculate the cosmological evolution of the 1-point probability distribution function (PDF), using an analytic approximation that combines gravitational perturbation theory with the Edgeworth expansion of the PDF. Our method applies directly to a smoothed mass density field or to the divergence of a smoothed peculiar velocity field, provided that rms fluctuations are small compared to unity on the smoothing scale, and that the primordial fluctuations that seed the growth of structure are Gaussian. We use this `Edgeworth approximation' to compute the evolution of $ $ and $ $; these measures are similar to the skewness and kurtosis of the density field, but they are less sensitive to tails of the probability distribution, so they may be more accurately estimated from surveys of limited volume. We compare our analytic calculations to cosmological N-body simulations in order to assess their range of validity. When $\sigma \ll 1$, the numerical simulations and perturbation theory agree precisely, demonstrating that the N-body method can yield accurate results in the regime of weakly non-linear clustering. We show analytically that `biased' galaxy formation preserves the relation $ \propto ^2$ predicted by second-order perturbation theory, provided that the galaxy density is a local function of the underlying mass density.

Evolution of One-Point Distributions from Gaussian Initial Fluctuations

Abstract: We study the quasilinear evolution of the one-point probability density functions (PDFs) of the smoothed density and velocity fields in a cosmological gravitating system beginning with Gaussian initial fluctuations. Our analytic results are based on the Zel'dovich approximation and laminar flow. A numerical analysis extends the results into the multistreaming regime using the smoothed fields of a CDM N-body simulation. We find that the PDF of velocity, both Lagrangian and Eulerian, remains Gaussian under the laminar Zel'dovich approximation, and it is almost indistinguishable from Gaussian in the simulations. The PDF of mass density deviates from a normal distribution early in the quasilinear regime and it develops a shape remarkably similar to a lognormal distribution with one parameter, the \rms density fluctuation $\sigma$. Applying these results to currently available data we find that the PDFs of the velocity and density fields, as recovered by the \pot\ procedure from observed velocities assuming $\Omega=1$, or as deduced from a redshift survey of \iras\ galaxies assuming that galaxies trace mass, are consistent with Gaussian initial fluctuations.

On the use of Bayesian probability theory for analysis of exponential decay data: an example taken from intravoxel incoherent motion experiments.

Abstract: Traditionally, the method of nonlinear least squares (NLLS) analysis has been used to estimate the parameters obtained from exponential decay data. In this study, we evaluated the use of Bayesian probability theory to analyze such data; specifically, that resulting from intravoxel incoherent motion NMR experiments. Analysis was done both on simulated data to which different amounts of Gaussian noise had been added and on actual data derived from rat brain. On simulated data, Bayesian analysis performed substantially better than NLLS under conditions of relatively low signal-to-noise ratio. Bayesian probability theory also offers the advantages of: a) not requiring initial parameter estimates and hence not being susceptible to errors due to incorrect starting values and b) providing a much better representation of the uncertainty in the parameter estimates in the form of the probability density function. Bayesian analysis of rat brain data was used to demonstrate the shape of the probability density function from data sets of different quality.

Smoothing Spline Density Estimation: Theory

Abstract: In this article, a class of penalized likelihood probability density estimators is proposed and studied. The true log density is assumed to be a member of a reproducing kernel Hilbert space on a finite domain, not necessarily univariate, and the estimator is defined as the unique unconstrained minimizer of a penalized log likelihood functional in such a space. Under mild conditions, the existence of the estimator and the rate of convergence of the estimator in terms of the symmetrized Kullback-Leibler distance are established. To make the procedure applicable, a semiparametric approximation of the estimator is presented, which sits in an adaptive finite dimensional function space and hence can be computed in principle. The theory is developed in a generic setup and the proofs are largely elementary. Algorithms are yet to follow.

Bayesian quality control using multivariate normal distributions

Abstract: An expression for the probability density of any distribution of observed values (given background values of known accuracy) is derived from the properties of multivariate normal distributions. This is used in the quality control of observations—‘good’ and ‘bad’ observations are assumed to have errors from a normal distribution and from a distribution giving no useful information respectively. Three Methods of quality control are presented and compared; two of these are based on the probability density derived above, and the third is based on a related maximum probability analysis. They differ in the optimality principal used: Individual Quality Control finds the most likely quality (i.e. good or bad) for each observation, given information from all the others; Simultaneous quality Control finds the most likely combination of qualities; while Variational Quality Control is based on a variational analysis which finds the most likely true values. The quality control should be considered as part of the ‘analysis’ process of using the observations; these approaches to quality control are considered as approximations to a system giving the ‘best’ analysis, based on minimizing a Bayesian loss function. Approximation are also necessary in their practical implementations; the effect of these on various operational schemes is discussed. The multi-observation framework used includes the ‘background’ check as a special case, and it is extended to deal with observations with common sources of gross error. Applications to multi-level checks, bias checks and checks for known error patterns are sketched. As a by-product the standard statistical interpolation formulae are derived from the properties of normal distributions, thus demonstrating the implicit dependence of statistical interpolation on the normal distribution.

Moments of the Counts Distribution in the 1.2 Jansky IRAS Galaxy Redshift Survey

Abstract: We have measured the count probability distribution function (CPDF) in a series of 10 volume-limited subsamples of a deep redshift survey of IRAS galaxies. The CPDF deviates significantly from both the Poisson and Gaussian limits in all but the largest volumes. We derive the volume-averaged 2-, 3-, 4-, and 5 point correlation functions from the moments of the CPDF and find them all to be reasonably well described by power laws. Weak systematic effects with the sample size provide evidence for stronger clustering of galaxies of higher luminosity on small scales. Nevertheless, remarkably tight relationships hold between the correlation functions of different order

Rotational transfer, an angular momentum model

Abstract: We have re‐examined critical experiments on collision induced rotational transfer (RT) and conclude that the probability of RT is controlled by the factors that control the probability of angular momentum (AM) change. The probability of energy change seems less important in this respect. In the light of this we suggest a model for RT in which the probability of AM change is calculated directly and present a formalism for this purpose. We demonstrate that such a calculation leads to an exponential‐like fall of RT probabilities with transferred AM, a consequence of the radial dependence of the repulsive part of the intermolecular potential. Thus in this AM model, the exponential gap law has a simple physical origin. The AM model we describe may be used as the basis of an inversion routine through which it is possible to convert RT data into a probability density of the repulsive anisotropy. Through this model therefore it is possible to relate experimental RT data directly to the forces that are responsible for rotational transfer. The hard ellipse model is used in this work to relate calculated anisotropies to a form that includes an isotropic component. The result is a representation of the intermolecular potential through which new insights into the RT process are gained.

The statistical distribution of swash maxima on natural beaches

Abstract: Cartwright and Longuet-Higgins (1956) describe the statistical distribution of maxima that would result from the linear superposition of random, Gaussian waves. The distribution function depends solely upon the relative width of the power spectrum and root-mean-square value of the process time series. Runup field data from three experiments are presented to determine the extent to which the distribution of swash maxima can be approximated using the Cartwright and Longuet-Higgins probability density function. The model is found to be satisfactory for describing various distribution statistics including the average maxima, the proportion of negative maxima, and the elevation at which one third of the swash maxima are exceeded. However, systematic discrepancies that scale as a function of time series skewness are observed in the statistics describing the upper tail of the distributions. Although we conclude that the linear model is incapable of delineating these apparent nonlinearities in the swash time series, the extent of the deviation can be estimated empirically for the purpose of constraining nonlinear models and nearshore engineering design.

Some two and three-dimensional persistent random walks

Abstract: We formulate non-Markovian versions of the persistent random walk in two and three dimensions and in continuous time. These models can be regarded as being generalizations of the original Pearson random walk and of the freely jointed chain which has been of some importance in polymer physics. Solutions for the probability density of the displacement of the random walker can be furnished for a restricted (essentially Markovian) set of these models. It is shown that in two dimensions the solution to one of these models is equivalent to the solution to an inhomogeneous telegrapher's equation. It does not appear to be possible, starting from a similar model in three dumensions, to find any form of the telegrapher's equation that follows from our solution of the equations in the Fourier-Laplace transform domains.

Stationary probability density functions: An exact result

Abstract: An exact expression is obtained for the probability density function (pdf) of any quantity measured in a general stationary process, in terms of conditional expectations of time derivatives of the signal. This expression indicates that the conditional expectations of both the time derivative squared and of the second time derivative influence the shape of the pdf, including its tails. A previous result of Ching [Phys. Rev. Lett. 70, 283 (1993)] for temperature measurements in turbulent flows corresponds to the particular case when the latter quantity is linear.

Smoothing Spline Density Estimation: A Dimensionless Automatic Algorithm

Abstract: As a sequel to an earlier article by Gu and Qiu, this article describes and illustrates a dimensionless automatic algorithm for nonparametric probability density estimation using smoothing splines. The algorithm is designed to calculate an adaptive finite dimensional solution to the penalized likelihood problem, which was shown by Gu and Qiu to share the same asymptotic convergence rates as the nonadaptive infinite dimensional solution. The smoothing parameter is updated jointly with the estimate in a performance-oriented iteration via a cross-validation performance estimate, where the performance is measured by proxies of the symmetrized Kullback-Leibler distance between the true density and the estimate. Simulations of limited scale are conducted to examine the relative effectiveness of the automatic smoothing parameter selection procedure and to assess the practical statistical performance of the methodology in general. The method is also applied to some real data sets. The algorithm is implem...

Wavelet-based spectral analysis of 1/f processes

Abstract: The authors attempt to show how and why a time-scale-based spectral estimation naturally suits the nature of 1/f processes, characterized by a power spectral density proportional to mod nu mod /sup - alpha /. They show that a time-scale approach allows an unbiased estimation of the spectral exponent alpha and interpret this result in terms of matched tilings of the time-frequency plane. They derive explicitly the probability density function of the estimated value of alpha . From this analysis, they find that there exists an optimum number of scales to use in a discrete wavelet scheme for obtaining a minimum variance estimator and that an improved procedure can be designed by making use of weighted least-squares in the estimation. >

Chord‐length and free‐path distribution functions for many‐body systems

Abstract: We study fundamental morphological descriptors of disordered media (e.g., heterogeneous materials, liquids, and amorphous solids): the chord‐length distribution function p(z) and the free‐path distribution function p(z,a). For concreteness, we will speak in the language of heterogeneous materials composed of two different materials or ‘‘phases.’’ The probability density function p(z) describes the distribution of chord lengths in the sample and is of great interest in stereology. For example, the first moment of p(z) is the ‘‘mean intercept length’’ or ‘‘mean chord length.’’ The chord‐length distribution function is of importance in transport phenomena and problems involving ‘‘discrete free paths’’ of point particles (e.g., Knudsen diffusion and radiative transport). The free‐path distribution function p(z,a) takes into account the finite size of a simple particle of radius a undergoing discrete free‐path motion in the heterogeneous material and we show that it is actually the chord‐length distribution fu...

Upper bounds to the error probability of decision feedback equalization

Abstract: The exact computation of the symbol error probability of systems using decision feedback equalization is difficult due to the propagation of errors. New upper bounds to the error probability of decision feedback equalization, which take error propagation into account, are developed. The derivations of the bounds assume a causal channel response, independent data symbols, and independent noise samples. The bounds are valid for any noise process that has a symmetric and unimodal probability density function. In some cases, the new bounds are significantly tighter than a well-known upper bound of D.L. Duttweiler et al. (1974). >

Probability densities of turbulent temperature fluctuations.

Abstract: Starting from an assumption of universality in turbulence and building upon the work in Y. G. Sinai and V. Yakhot, we construct a closed-form expression for the probability density function of temperature fluctuations. This result is found to be in good correspondence to experimental data obtained at Chicago and Yale. By extending this method, we obtain a similar expression for the probability density function of temperature differences between two times. Again the result is checked to hold very well, except for very short time separations

The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions

Abstract: Recent experimental and computational observations demonstrate the occurrence of large‐scale intermittency for diffusing passive scalars, as manifested by broader than Gaussian probability distribution functions. Here, a family of explicit exactly solvable examples is developed which demonstrates these effects of large‐scale intermittency at any positive time through simple formulas for the higher flatness factors without any phenomenological approximations. The exact solutions involve advection–diffusion with velocity fields involving a uniform shear flow perturbed by a random fluctuating uniform shear flow. Through an exact quantum mechanical analogy, the higher‐order statistics for the scalar in these models are solved exactly by formulas for the quantum‐harmonic oscillator. These explicit formulas also demonstrate that the large time asymptotic limiting probability distribution function for the normalized scalar can be either broader than Gaussian or Gaussian depending on the relative strength of the mean flow and the fluctuating velocity field.