Showing papers on "Probability density function published in 1990"
TL;DR: The authors examine the relative entropy distance D/sub n/ between the true density and the Bayesian density and show that the asymptotic distance is (d/2)(log n)+c, where d is the dimension of the parameter vector.
Abstract: In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. The authors examine the relative entropy distance D/sub n/ between the true density and the Bayesian density and show that the asymptotic distance is (d/2)(log n)+c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate D/sub n//n converges to zero at rate (log n)/n. The constant c, which the authors explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stock-market portfolio selection. >
517 citations
01 Jan 1990
TL;DR: The probability function is derived axiomatically the probability function that should be used to make decisions given any form of underlying uncertainty.
Abstract: Summary We derive axiomatically the probability function that should be used to make decisions given any form of underlying uncertainty.
511 citations
TL;DR: It is shown that as the temperature approaches zero, the algorithm becomes the basic ISODATA algorithm and the method is independent of the initial choice of cluster means.
393 citations
TL;DR: Set-membership techniques for estimating parameters from uncertain data are reviewed and a suitable characterization of the set of all parameter vectors is found consistent with the model structure, data, and bounds on the errors.
312 citations
TL;DR: In this paper, a maximum-likelihood method for estimating the model parameters of a model used to represent the spectral features in a solar oscillation power spectrum is proposed, which accounts for redistribution of spectral power caused by gaps in the data string, by convolving the model with the power spectrum of the observed window function.
Abstract: To produce accurate estimates of the line-profile parameters of a model used to represent the spectral features in a solar oscillation power spectrum, it is necessary to (1) select the appropriate probability density function when deriving the maximum-likelihood function to be employed for the parameter estimation and (2) allow for the redistribution of spectral power caused by gaps in the data string. This paper describes a maximum-likelihood method for estimating the model parameters (based on the observed power spectrum statistics) that accounts for redistribution of spectral power caused by gaps in the data string, by convolving the model with the power spectrum of the observed window function. The accuracy and reliability of the method were tested using both artificial and authentic solar oscillation power spectrum data. A comparison of this method with various least-squares techniques is also presented.
276 citations
TL;DR: In this article, a method is presented for forming both a point estimate and a confidence set of semiparametric densities for a plausible range of smoothing parameters, where the boundaries of the smoothing parameter are determined by a nonparametric goodness-of-fit test based on the sample spacings.
Abstract: A method is presented for forming both a point estimate and a confidence set of semiparametric densities. The final product is a three-dimensional figure that displays a selection of density estimates for a plausible range of smoothing parameters. The boundaries of the smoothing parameter are determined by a nonparametric goodness-of-fit test that is based on the sample spacings. For each value of the smoothing parameter our estimator is selected by choosing the normal mixture that maximizes a function of the sample spacings. A point estimate is selected from this confidence set by using the method of cross-validation. An algorithm to find the mixing distribution that maximizes the spacings functional is presented. These methods are illustrated with a data set from the astronomy literature. The measurements are velocities at which galaxies in the Corona Borealis region are moving away from our galaxy. If the galaxies are clustered, the velocity density will be multimodal, with clusters correspond...
263 citations
IBM1
TL;DR: An experiment in which sandpiles are built up to a critical size and then perturbed by the dropping of individual grains of sand onto the pile demonstrates that real, finite-size sandpile may be described by models of self-organized criticality, but it is found that this description breaks down in the limit of largeSandpiles.
Abstract: We have carried out an experiment in which sandpiles are built up to a critical size and then perturbed by the dropping of individual grains of sand onto the pile. After each grain is added, the size of the resulting avalanche, if any, is recorded. For sufficiently small sandpiles, the observed mass fluctuations are scale invariant and the probability distribution of avalanches shows finite-size scaling. This demonstrates that real, finite-size sandpiles may be described by models of self-organized criticality. However, we also find that this description breaks down in the limit of large sandpiles.
253 citations
Abstract: A new stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence is proposed. The model is three-dimensional and its formulation takes account of recent improvements in the understanding of one-particle models. In particular the model is designed so that if the particle pairs are initially well mixed in the fluid, they will remain so. In contrast to previous models, the new model leads to a prediction for the particle separation probability density function which is in qualitative agreement with inertial subrange theory. The values of concentration variance from the model show encouraging agreement with experimental data. The model results suggest that, at large times, the intensity of concentration fluctuations (i.e. standard deviation of concentration divided by mean concentration) tends to zero in stationary conditions and to a constant in decaying turbulence.
250 citations
TL;DR: A formalism is developed to obtain two different types of nearest-neighbor probability density functions and closely related quantities, such as their associated cumulative distributions and conditional pair distributions, for many-body systems of {ital D}-dimensional spheres.
Abstract: The probability of finding a nearest neighbor at some given distance from a reference point in a many-body system of interacting particles is of importance in a host of problems in the physical as well as biological sciences. We develop a formalism to obtain two different types of nearest-neighbor probability density functions ({ital void} and {ital particle} probability densities) and closely related quantities, such as their associated cumulative distributions and conditional pair distributions, for many-body systems of {ital D}-dimensional spheres. For the special case of impenetrable (hard) spheres, we compute low-density expansions of each of these quantities and obtain analytical expressions for them that are accurate for a wide range of sphere concentrations. Using these results, we are able to calculate the mean nearest-neighbor distance for distributions of {ital D}-dimensional impenetrable spheres. Our theoretical results are found to be in excellent agreement with computer-simulation data.
249 citations
TL;DR: It is shown, both by analytic theory and by analog simulations, that the density of residence times has a detailed structure reflective of the inherent symmetries of the system.
Abstract: In this paper a periodically driven, bistable system with additive noise is considered in the overdamped limit. Here we have adopted the probability density of residence times as the tool for dynamical studies on the system. We contrast this to the body of previous work, in the area now known as ``stochastic resonance,'' wherein the power spectral density was the preferred physical quantity. It is shown, both by analytic theory and by analog simulations, that the density of residence times has a detailed structure reflective of the inherent symmetries of the system. Closed-form expressions are developed for the distribution function as well as for several averaged quantities of interest. It is emphasized that all our analytic results predict observable physical quantities, which are then demonstrated with measurements on the analog simulator.
180 citations
TL;DR: In this article, a model for the joint pdf of velocity and dissipation following a fluid particle is developed by constructing stochastic models for the velocity and dissolution following a particle.
Abstract: In probability density function (pdf) methods, statistics of inhomogeneous turbulent flow fields are calculated by solving a modeled transport equation for a one‐point joint probability density function. The method based on the joint pdf of velocity and fluid compositions is particularly successful since the most important processes—convection and reaction—do not have to be modeled. However, this joint pdf contains no length‐scale or time‐scale information that can be used in the modeling of other processes. This deficiency can be remedied by considering the joint pdf of velocity, dissipation, and composition. In this paper, by reference to the known properties of homogeneous turbulence, a modeled equation for the joint pdf of velocity and dissipation is developed. This is achieved by constructing stochastic models for the velocity and dissipation following a fluid particle.
TL;DR: In this paper, the asymptotic normality of kernel estimators of the multivariate density of stationary random fields indexed by ZN is established and appropriate choices of the bandwiths are found.
TL;DR: The relationship between the weighted distributions and the parent distributions in the context of reliability and life testing depends on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions.
Abstract: C. R. Rao pointed out that “The role of statistical methodology is to extract the relevant information from a given sample to answer specific questions about the parent population” and raised the question “What population does a sample represent”? Wrong specification can lead to invalid inference giving rise to a third kind of error. Rao introduced the concept of weighted distributions as a method of adjustment applicable to many situations. In this paper, we study the relationship between the weighted distributions and the parent distributions in the context of reliability and life testing. These relationships depend on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions. As special cases, the length biased distribution, the equilibrium distribution of the backward and forward recurrence times and the residual life distribution, which frequently arise in practice, are studied and their relationships with the original di...
TL;DR: In this article, a method for the simulation of the composite power system is proposed for the purpose of evaluating the probability distribution function of circuit flows and bus voltage magnitudes, which consists of two steps.
Abstract: A method for the simulation of the composite power system is proposed for the purpose of evaluating the probability distribution function of circuit flows and bus voltage magnitudes. The method consists of two steps. First, given the probabilistic electric load model, the probability distribution function of the total generation of generation buses is computed. Second, circuit flows and bus voltage magnitudes are expressed as linear combinations of power injections at generation buses. This relationship allows the computation of the distribution functions of circuit flows and bus voltage magnitudes. The method incorporates major operating practices such as economic dispatch and nonlinearities resulting from the power flow equations. Validation of the method is performed via Monte Carlo simulation. Typical results are presented, showing that the proposed method matches the results obtained with the Monte Carlo simulations very well. Potential applications of the proposed method are: composite power system reliability analysis and transmission loss evaluation. >
TL;DR: In this article, the authors discuss how to estimate the underlying distribution from a differential geometric viewpoint, assuming that the manifold is closed and that the distribution is (sufficiently) smooth.
Abstract: Supposing a given collection $y_1, \cdots, y_N$ of i.i.d. random points on a Riemannian manifold, we discuss how to estimate the underlying distribution from a differential geometric viewpoint. The main hypothesis is that the manifold is closed and that the distribution is (sufficiently) smooth. Under such a hypothesis a convergence arbitrarily close to the $N^{-1/2}$ rate is possible, both in the $L_2$ and the $L_\infty$ senses.
TL;DR: The Lyapunov number partition function method is used to calculate the spectra of generalized dimensions and of scaling indices for these attractors and special attention is devoted to the numerical implementation of the method and the evaluation of statistical errors due to the finite number of sample orbits.
Abstract: We consider qualitative and quantitative properties of ``snapshot attractors'' of random maps. By a random map we mean that the parameters that occur in the map vary randomly from iteration to iteration according to some probability distribution. By a ``snapshot attractor'' we mean the measure resulting from many iterations of a cloud of initial conditions viewed at a single instant (i.e., iteration). In this paper we investigate the multifractal properties of these snapshot attractors. In particular, we use the Lyapunov number partition function method to calculate the spectra of generalized dimensions and of scaling indices for these attractors; special attention is devoted to the numerical implementation of the method and the evaluation of statistical errors due to the finite number of sample orbits. This work was motivated by problems in the convection of particles by chaotic fluid flows.
TL;DR: The relative range error is proposed as a better way of quantifying the range resolution of a stereo imaging system than the percent range error when the depths in the scene lie within a narrow range.
Abstract: The probability density function of the range estimation error and the expected value of the range error magnitude are derived in terms of the various design parameters of a stereo imaging system. In addition, the relative range error is proposed as a better way of quantifying the range resolution of a stereo imaging system than the percent range error when the depths in the scene lie within a narrow range. >
TL;DR: In this article, an effort is made to determine relationships between reflectivity (Z) and rain rate (R) which are tuned to the local climatology, where the relation is tuned so that the probability distribution of reflectivity, P(Z), replicates that of R over some predetermined space-time climatic domain.
Abstract: An effort is made to determine relationships between reflectivity (Z) and rain rate (R) which are tuned to the local climatology. The development of such relations was motivated by the need to understand the role of precipitation in controlling general circulation and in affecting such phenomena as ENSO. Attention is given to methods of deriving such relations and how they are linked to area integral rainfall measurements. In essence, the relation is tuned so that the probability distribution of reflectivity, P(Z), replicates that of R over some predetermined space-time climatic domain. Thus, the accurate measurement of the average R over any smaller domain depends on how closely the sampled P(Z) approximates the climatic P(Z). The probability matching method used is a modification of the approach of Calheiros and Zawadzki (1987) and Rosenfeld (1980). The technique is applied to data from Germany and the eastern tropical Atlantic (GATE).
TL;DR: In this article, the statistical distribution of zero-crossing wave heights is considered within the context of a previous theory proposed by the writer some years ago and the underlying model, definitions, and assumptions are reexamined systematically to develop asymptotic approximations to the probability density, exceedance probability, and statistical moments of wave heights larger than the mean wave height.
Abstract: The statistical distribution of zero-crossing wave heights is considered within the context of a previous theory proposed by the writer some years ago. The underlying model, definitions, and assumptions are reexamined systematically to develop asymptotic approximations to the probability density, exceedance probability, and statistical moments of wave heights larger than the mean wave height. The asymptotic results have closed forms, and thus are easier to use in practical applications than the original theory, which requires numerical integration. Comparisons to empirical data are given to show that the present asymptotic theory produces the observed statistics of large wave heights faithfully to within 1%. Further, comparisons with other relevant theories also reveal that if one remains true to the theoretical definitions, then the present theory is the most accurate in predicting the exceedance distribution of large wave heights. Finally, the asymptotic theory is coupled with the statistics of wave periods to derive a theoretical expression for the joint distribution of large wave heights and associated periods. The predictive utility of this last result remains to be explored.
TL;DR: In this paper, it is shown that measurements of the statistical properties of the concentration distributions of dispersing scalars taken from many different turbulent shear flows have a great number of common features.
Abstract: It is shown that measurements of the statistical properties of the concentration distributions of dispersing scalars taken from many different turbulent shear flows have a great number of common features. In particular the same simple relationship between the mean concentration and the mean-square fluctuation is shown to hold in all the flows, and this relationship is derived theoretically from well-known results for the unreal case when there is no molecular diffusion by a natural hypothesis about the effects of molecular diffusion. Application of the hypothesis to the higher moments and shape parameters gives results that agree reasonably well with the data (given the unavoidable experimental errors). The hypothesis should be subjected to further experimental analysis, and could simplify the application of turbulence closures and similar models. Extensions of the ideas to the probability density function of the scalar concentration suggest that it becomes self-similar. A final conclusion is that more attention to experimental errors due to instrument smoothing is highly desirable.
TL;DR: It is concluded that in most cases, the values of angles and distances are being altered only slightly by the imaging process, and they can still serve as a strong cue for model-based recognition.
Abstract: Two novel probabilistic models for viewed angles and distances are derived using an observability sphere method. The method, which is based on the assumption that the prior probability density is isotropic for all viewing orientations, can be used for the computation of observation probabilities for object's aspects, features, and probability densities of their quantitative attributes. Using the sphere, it is discovered that the probability densities of viewed angles, distances, and even projected curvature have sharp peaks at their original values. From this peaking effect, it is concluded that in most cases, the values of angles and distances are being altered only slightly by the imaging process, and they can still serve as a strong cue for model-based recognition. The probabilistic models for 3-D object recognition from monocular images are used. To form the angular elements that are needed, the objects are represented by their linear features and specific points primitives. Using the joint density model of angles and distances, the probabilities of initial matching hypotheses and mutual information coefficients are estimated. These results are then used for object recognition by optimal matching search and stochastic labeling schemes. Various synthetic and real objects are recognized by this approach. >
TL;DR: In this article, an adaptation of least squares cross-validation is proposed for bandwidth choice in the kernel estimation of the derivatives of a probability density, which is demonstrated by an example and a simulation study.
Abstract: An adaptation of least squares cross-validation is proposed for bandwidth choice in the kernel estimation of the derivatives of a probability density. The practicality of the method is demonstrated by an example and a simulation study. Theoretical justification is provided by an asymptotic optimality result
01 Jan 1990
TL;DR: In general, our opinion is not expressible in precise numerical probabilities, but in some cases it is, and in the case of a small field of propositions, my opinion may be exactly represented by a single probability function as mentioned in this paper.
Abstract: In general, our opinion is certainly not expressible in precise numerical probabilities. But in some cases it is, and in the case of a small field of propositions—e.g. a field generated by a single proposition—my opinion may be exactly represented by a single probability function. Therefore I shall begin with the fiction that it is always so.
TL;DR: In this article, the authors derived the equation for the gravity anomaly of an asymmetrical trapezoidal model, assuming a quadratic density function, and developed methods of interpretation, using the Marquardt algorithm.
Abstract: The decrease of density contrast with depth in many sedimentary basins can be approximated by a quadratic function. The interpretation of gravity anomalies over sedimentary strata can be more realistic if variable density contrasts, rather than constant density contrasts, are assumed. I derive the equation for the gravity anomaly of an asymmetrical trapezoidal model, assuming a quadratic density function, and I develop methods of interpretation, using the Marquardt algorithm. While interpreting a synthetic anomaly profile, the convergence of the algorithm is shown by plotting the values of the objective function, damping parameter, and various parameters of the model with respect to iteration number; the results are superior to those obtained when using constant density contrasts. In addition, I apply the method in the interpretation of gravity anomalies over the San Jacinto graben and the lower Godavari basin.
TL;DR: It is proved that the probability of n customers being in the queueing system Ck/Cm/s, when it is saturated is a linear combination of geometric terms, and an exact algorithm for finding the system-size distribution and the system's performance measures is proposed.
Abstract: We solve the queueing system Ck/Cm/s, where Ck is the class of Coxian probability density functions pdfs of order k, which is a subset of the pdfs that have a rational Laplace transform. We formulate the model as a continuous-time, infinite-space Markov chain by generalizing the method of stages. By using a generating function technique, we solve an infinite system of partial difference equations and find closed-form expressions for the system-size, general-time, prearrival, post-departure probability distributions and the usual performance measures. In particular, we prove that the probability of n customers being in the system, when it is saturated is a linear combination of geometric terms. The closed-form expressions involve a solution of a system of nonlinear equations that involves only the Laplace transforms of the interarrival and service time distributions. We conjecture that this result holds for a more general model. Following these theoretical results we propose an exact algorithm for finding the system-size distribution and the system's performance measures. We examine special cases and apply this method for numerically solving the C2/C2/s and Ek/C2/s queueing systems.
03 Apr 1990
TL;DR: A simple method is investigated, to re-estimate the vector quantization codebook without continuous probability density function assumptions, and preliminary experiments show that such reestimation methods are as effective as the semicontinuous model, especially when the continuous probabilitydensity function assumption is inappropriate.
Abstract: The semicontinuous hidden Markov model is used in a 1000-word speaker-independent continuous speech recognition system and compared with the continuous mixture model and the discrete model. When the acoustic parameter is not well modeled by the continuous probability density, it is observed that the model assumption problems may cause the recognition accuracy of the semicontinuous model to be inferior to the discrete model. A simple method based on the semicontinuous model is investigated, to re-estimate the vector quantization codebook without continuous probability density function assumptions. Preliminary experiments show that such reestimation methods are as effective as the semicontinuous model, especially when the continuous probability density function assumption is inappropriate. >
TL;DR: This probability representation was derived in the course of a structural analysis of selenobiotinyl streptavidin from MAD data and applications have also been made in the structure determinations of interleukin-1 alpha and a drug complex with brominated DNA.
Abstract: A probability distribution function, cast in the representation of Hendrickson & Lattman [Acta Cryst. (1970), B26, 136-143], has been derived for the phase information from measurements of multiwavelength anomalous diffraction (MAD). This probability function readily permits one to determine figure-of-merit weights similar to those used in isomorphous replacement, and the coefficients that characterize this distribution function facilitate the combining of MAD phasing with results from other sources of phase information. This probability representation was derived in the course of a structural analysis of selenobiotinyl streptavidin from MAD data and applications have also been made in the structure determinations of interleukin-1 alpha and a drug complex with brominated DNA.
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06 Dec 1990
TL;DR: In this paper, a method of assessing a link for a digital communication system and providing a value for a channel or link state parameter, particularly bit error rate, and apparatus for the same, is presented.
Abstract: A method of assessing a link for a digital communication system and providing a value for a channel or link state parameter, particularly bit error rate, and apparatus for the same, in which an estimate of the probability density function for the channel or link is obtained by categorising decision variables into threshold categories, and comparing the estimated probability density function with stored known probability density functions. With each stored probability density function the value of the channel parameter being monitored is stored. The value of the channel parameter being monitored is determined by selecting the value associated with the stored probability density function closest resembling the estimated probability density.
17 Jun 1990
TL;DR: The stochastic calculus and a Lyapunov argument prove that competitive synaptic vectors converge to centroids exponentially quickly and does not depend on a specific dynamical model of how neuronal activations change.
Abstract: The probabilistic foundations of competitive learning systems are developed. Continuous and discrete formulations of unsupervised, supervised, and differential competitive learning systems are studied. These systems estimate an unknown probability density function from random pattern samples and behave as adaptive vector quantizers. Synaptic vectors in feedforward competitive neural networks quantize the pattern space and converge to pattern class centroids or local probability maxima. The stochastic calculus and a Lyapunov argument prove that competitive synaptic vectors converge to centroids exponentially quickly. Convergence does not depend on a specific dynamical model of how neuronal activations change
TL;DR: The concepts of Bayes prediction analysis are used to obtain predictive distributions of the next time to failure of software when its past failure behavior is known and can show an improved predictive performance for some data sets even when compared with some more sophisticated software-reliability models.
Abstract: The concepts of Bayes prediction analysis are used to obtain predictive distributions of the next time to failure of software when its past failure behavior is known. The technique is applied to the Jelinski-Moranda software-reliability model, which in turn can show an improved predictive performance for some data sets even when compared with some more sophisticated software-reliability models. A Bayes software-reliability model is presented which can be applied to obtain the next time to failure PDF (probability distribution function) and CDF (cumulative distribution function) for all testing protocols. The number of initial faults and the per-fault failure rate are assumed to be s-independent and Poisson and gamma distributed respectively. For certain data sets, the technique yields better predictions than some alternative methods if the frequential likelihood and U-plot criteria are adopted. >