Showing papers on "Probability mass function published in 2000"

Scenario reduction in stochastic programming: An approach using probability metrics

Abstract: Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario reduction is stated as follows: Determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural (or canonical) probability metric. Arguments from stability analysis indicate that Fortet-Mourier type probability metrics may serve as such canonical metrics. Efficient algorithms are developed that determine optimal reduced measures approximately. Numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty. For instance, it turns out that after 50% reduction of the scenario tree the optimal reduced tree still has about 90% relative accuracy.

A First Look at Rigorous Probability Theory

Abstract: The need for measure theory probability triples further probabilistic foundations expected values inequalities and laws of large numbers distributions of random variables stochastic processes and gambling games discrete Markov chains some further probability results weak convergence characteristic functions decomposition of probability laws conditional probability and expectation Martingales introduction to other stochastic processes.

Concavity and Efficient Points of Discrete Distributions in Probabilistic Programming

Abstract: We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples.

On the Convergence Rate of Good-Turing Estimators

Abstract: Good-Turing adjustments of word frequencies are an important tool in natural language modeling. In particular, for any sample of words, there is a set of words not occuring in that sample. The total probability mass of the words not in the sample is the so-called missing mass. Good showed that the fraction of the sample consisting of words that occur only once in the sample is a nearly unbiased estimate of the missing mass. Here, we give a high-probability confidence interval for the actual missing mass. More generally, for 0, we give a confidence interval for the true probability mass of the set of words occuring times in the sample.

Small-World Networks: Links with long-tailed distributions

Abstract: Small-world networks (SWN), obtained by randomly adding to a regular structure additional links (AL), are of current interest. In this paper we explore (based on physical models) a new variant of SWN, in which the probability of realizing an AL depends on the chemical distance between the connected sites. We assume a power-law probability distribution and study random walkers on the network, focusing especially on their probability of being at the origin. We connect the results to L\'evy flights, which follow from a mean-field variant of our model.

On concentration of probability

Abstract: We give a survey of several methods to obtain sharp concentration results, typically with exponentially small error probabilities, for random variables occuring in combinatorial probability.

Improved Bounds and Simulation Procedures on the Value of the Multivariate Normal Probability Distribution Function

Abstract: Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function value are given in the paper. The author's variance reduction technique was based on the Bonferroni bounds involving the first two binomial moments only. The new variance reduction technique is adapted to the most refined new bounds developed in the last decade for the estimation the probability of union respectively intersection of events. Numerical test results prove the efficiency of the simulation procedures described in the paper.

IDEAs based on the normal kernels probability density function

Abstract: The IDEA framework is a general framework for iterated density estimation evolutionary algorithms. These algorithms use probabilistic models to guide the search in stochastic optimization. The estimation of densities for subsets of selected samples and the sampling from the resulting distributions, is the combination of the evolutionary recombination and mutation steps used in EAs. We investigate how the normal kernels probability density function can be used as the distribution of the problem variables in order to perform optimization using the IDEA framework. As a result, we present three probability density structure search algorithms, as well as a general estimation and samphng algorithm, all of which use the normal kernels distribution

Random spherical uncertainty in estimation and robustness

Abstract: A theorem is formulated that gives an exact probability distribution for a linear function of a random vector uniformly distributed over a ball in n-dimensional space. This mathematical result is illustrated via applications to a number of important problems of estimation and robustness under spherical uncertainty. These include parameter estimation, characterization of attainability sets of dynamical systems, and robust stability of affine polynomial families.

q-Probability: I. Basic Discrete Distributions

Abstract: For basic discrete probability distributions, − Bernoulli, Pascal, Poisson, hypergeometric, contagious, and uniform, − q-analogs are proposed.

Random fractals and probability metrics

Abstract: New metrics are introduced in the space of random measures and are applied, with various modifications of the contraction method, to prove existence and uniqueness results for self-similar random fractal measures. We obtain exponential convergence, both in distribution and almost surely, of an iterative sequence of random measures (defined by means of the scaling operator) to a unique self-similar random measure. The assumptions are quite weak, and correspond to similar conditions in the deterministic case. The fixed mass case is handled in a direct way based on regularity properties of the metrics and the properties of a natural probability space. Proving convergence in the random mass case needs additional tools, such as a specially adapted choice of the space of random measures and of the space of probability distributions on measures, the introduction of reweighted sequences of random measures and a comparison technique.

Degree of outlier calculation device, and probability density estimation device and forgetful histogram calculation device for use therein

Abstract: Degree of outlier of one input data is calculated by an amount of change in a learned probability density from that before learning as a result of taking in of the input data. This is because data largely differing in a tendency from a so far learned probability density function can be considered to have a high degree of outlier. More specifically, a function of a distance between probability densities before and after data input is calculated as a degree of outlier. Accordingly, a probability density estimation device appropriately estimates a probability distribution of generation of unfair data while sequentially reading a large volume of data and a score calculation device calculates and outputs a degree of outlier of each data based on the estimated probability distribution.

The probabilistic solutions to nonlinear random vibrations of multi-degree-of-freedom systems

Abstract: The probability density function of the responses of nonlinear random vibration of a multi-degree-of-freedom system is formulated in the defined domain as an exponential function of polynomials in state variables. The probability density function is assumed to be governed by Fokker-Planck-Kolmogorov (FPK) equation. Special measure is taken to satisfy the FPK equation in the average sense of integration with the assumed function and quadratic algebraic equations are obtained for determining the unknown probability density function. Two-degree-of-freedom systems are analyzed with the proposed method to validate the method for nonlinear multi-degree-of-freedom systems. The probability density functions obtained with the proposed method are compared with the obtainable exact and simulated ones. Numerical results showed that the probability density function solutions obtained with the presented method are much closer to the exact and simulated solutions even for highly nonlinear systems with both external and parametric excitations.

N-dimensional uniform probability distribution in nonlinear autoregressive filter structures

Abstract: In this brief, we consider a class of nonlinear discrete-time systems and prove conditions for the generation of signals with n-dimensional uniform probability distribution. To obtain this result we prove the invariance of the Frobenius-Perron-Operator on a finite-dimensional function space. Spectral properties of the generated signal, as well as the influence of an input signal on the signal characteristics are discussed.

Measurements of nondegenerate discrete observables

Abstract: Every measurement on a quantum system causes a state change from the system state just before the measurement to the system state just after the measurement conditional upon the outcome of measurement. This paper determines all the possible conditional state changes caused by measurements of nondegenerate discrete observables. For this purpose, the following conditions are shown to be equivalent for measurements of nondegenerate discrete observables: (i) The joint probability distribution of the outcomes of successive measurements depends affinely on the initial state. (ii) The apparatus has an indirect measurement model. (iii) The state change is described by a positive superoperator valued measure. (iv) The state change is described by a completely positive superoperator valued measure. (v) The output state is independent of the input state and the family of output states can be arbitrarily chosen by the choice of the apparatus. The implications to the measurement problem are discussed briefly.

Passage through a barrier with a slowly increasing control parameter

Abstract: A bistable system subjected to noise and a slow increase of the parameter controlling the instability is studied, with emphasis on the kinetics of the transitions across the barrier separating the stable states. It is shown that the presence of a ramp in the control parameter may considerably affect the distribution of probability mass on the two sides of the barrier as compared to the predictions of the classical Kramers theory.

Discrete Particle Distribution Model for Advection-Diffusion Transport

Abstract: A new methodology, named DisPar, based on a discrete probability distribution for a particle displacement, was developed to solve 1D advection-diffusion transport problems in water bodies. The discrete probability distribution for the particle displacement was developed as an average and variance function. These probabilities were used to predict the deterministic mass transfer between cells in one time step, and therefore the particle concentration in each cell was considered the state variable. The state equation was found to be similar to an explicit finite-difference formulation with a Eulerian grid. The model stability, positivity, and mass conservation are guaranteed by the probability distribution concept. DisPar produces solutions without numerical dispersion for constant velocity, diffusion coefficient, and cross-sectional area. In these conditions, DisPar was also developed as a function of space and time for an instantaneous mass spill. When the stability and positivity restrictions were respected, the model produced excellent results when compared to analytical solutions and other methods. The discrete particle displacement distribution concept differs from other numerical formulations, and therefore it represents a new modeling technique.

Efficient construction of reversible jump MCMC proposal dis- tributions

Abstract: The major implementational problem for reversible jump MCMC is that there is commonly no natural way to choose jump proposals since there is no Euclidean structure in the parameter space to guide our choice. In this paper we consider mechanisms for guiding the proposal choice. The first group of methods is based upon an analysis of acceptance probabilities for jumps. Essentially, these methods involve a Taylor series expansion of the acceptance probability around certain canonical jumps, and turns out to have close connections to Langevin algorithms. The second group of methods generalises the reversible jump algorithm using the so-called dual space approach. These allow the chain to retain some degree of memory so that when proposing to move from a smaller to a larger model, information is borrowed from the last time that the reverse move performed. The main motivation for this paper is that, in very high dimensional examples, the probability that the Markov chains moves between such spaces may be prohibitively small, as the probability mass is very thinly spread across the space. Finding reasonable jump proposals becomes extremely important. We will illustrate the procedure using several examples of reversible jump MCMC applications including the analysis of autoregressive time series, graphical Gaussian modelling and mixture modelling.

Practical implementation of possibilistic probability mass functions

Abstract: Probability assessments of events are often linguistic in nature. We model them by means of possibilistic probabilities (a version of Zadeh's fuzzy probabilities with a behavioural interpretation) with a suitable shape for practical implementation (on a computer). Employing the tools of interval analysis and the theory of imprecise probabilities we argue that the verification of coherence for these possibilistic probabilities, the corrections of non-coherent to coherent possibilistic probabilities and their extension to other events and gambles can be performed by finite and exact algorithms. The model can furthermore be transformed into an imprecise first-order model, useful for decision making and statistical inference.

Approximate Maximum Entropy Joint Feature Inference Consistent with Arbitrary Lower-Order Probability Constraints: Application to Statistical Classification

Abstract: We propose a new learning method for discrete space statistical classifiers. Similar to Chow and Liu (1968) and Cheeseman (1983), we cast classification/inference within the more general framework of estimating the joint probability mass function (p.m.f.) for the (feature vector, class label) pair. Cheeseman's proposal to build the maximum entropy (ME) joint p.m.f. consistent with general lower-order probability constraints is in principle powerful, allowing general dependencies between features. However, enormous learning complexity has severely limited the use of this approach. Alternative models such as Bayesian networks (BNs) require explicit determination of conditional independencies. These may be difficult to assess given limited data. Here we propose an approximate ME method, which, like previous methods, incorporates general constraints while retaining quite tractable learning. The new method restricts joint p.m.f. support during learning to a small subset of the full feature space. Classification gains are realized over dependence trees, tree-augmented naive Bayes networks, BNs trained by the Kutato algorithm, and multilayer perceptrons. Extensions to more general inference problems are indicated. We also propose a novel exact inference method when there are several missing features.

An historical overview of stable probability distributions in signal processing

Abstract: An historical, personal and idiosyncratic overview of stable probability distributions in signal processing is presented.

Continuous probability distributions from finite data.

Abstract: Recent approaches to the problem of inferring a continuous probability distribution from a finite set of data have used a scalar field theory for the form of the prior probability distribution. This paper presents a more general form for the prior distribution that has a geometrical interpretation and can improve the specificity of likely solutions. It is also demonstrated that a numerical sampling of the posterior probability distribution can be used as an alternative to a histogram for visualization and to make probabilistic inferences from the data.

A New Geometric Probability Technique for an N-dimensional Sphere and Its Applications to Physics

Abstract: A new formalism is presented for analytically obtaining the probability density function, \( P_{n}(s) \), for the distance between two random points in an \( n \)-dimensional sphere of radius \( R \). Our formalism allows \( P_{n}(s) \) to be calculated for a sphere having an arbitrary density distribution, and reproduces the well-known results for the case of a sphere with uniform density. The results find applications in stochastic geometry, probability distribution theory, astrophysics, nuclear physics, and elementary particle physics.

The GPS Integer Least-Squares Statistics

Abstract: Successful integer least-squares carrier phase ambiguity estimation is the key to fast and high precision GPS kinematic positioning. In order to describe the quality of the positioning results rigorously, one needs to know the probability distributions of both the integer and noninteger parameters in the GPS model. In this contribution these distributions are presented and discussed. The probability mass function of the integer least-squares ambiguities is needed to evaluate the ambiguity success rate and the joint distribution of the GPS baseline and atmospheric delays is needed to evaluate the relevant confidence regions.

Method and system for modeling a system

Abstract: The invention relates to a method and a system for detecting at least one partial model of a model pertaining to a system. A state of the system is described by state variables. At least on e of the state variables is a discrete state variable. Several value sets of the state variables are detected. A probability distribution for the state variables is detected by using the sets. The partial model of the system is detected using the sets and the probability distribution of the state variables and a statistical learning method. The partial model describes the system under the condition of the probability distribution for the state variables.

Distribution of processor-sharing customers for a large closed system with multiple classes

Abstract: A closed processor-sharing (PS) system with multiple customer classes is considered. The system consists of one infinite server (IS) station and one PS station. For a system with a large number of customers, a saturated PS station, and an arbitrary number of customer classes, asymptotic approximations to the stationary distribution of the total number of customers at the PS station are derived. The asymptotics for the probability mass function is described by a quasi-potential function, which defines the exponential decay for the distribution, and a state-dependent preexponential factor. Both functions have an explicit expression in terms of the solution at each point x of a polynomial equation whose order equals the number of classes and whose coefficients are explicit functions of x. The quasi-potential function at its minimum point provides the logarithmic asymptotics for the normalization constant, and the asymptotic approximation for the variance is inversely proportional to the second derivative of ...