Arbitrary Probability Distribution Research Articles

Data reconciliation of nonnormal observations with nonlinear constraints

ABSTRACTThis paper presents a new method for the reconciliation of data described by arbitrary continuous probability distributions, with the focus on nonlinear constraints. The main idea, already applied to linear constraints in a previous paper, is to restrict the joint prior probability distribution of the observed variables with model constraints to get a joint posterior probability distribution. Because in general the posterior probability density function cannot be calculated analytically, it is shown that it has decisive advantages to sample from the posterior distribution by a Markov chain Monte Carlo (MCMC) method. From the resulting sample of observed and unobserved variables various characteristics of the posterior distribution can be estimated, such as the mean, the full covariance matrix, marginal posterior densities, as well as marginal moments, quantiles, and HPD intervals. The procedure is illustrated by examples from material flow analysis and chemical engineering.

A Bayesian Filter for Sound Environment System with Quantized Observation<sup>*</sup>

In the real sound environment, the observation data are usually contaminated by additional background noise of arbitrary distribution type. In order to estimate several evaluation quantities for specific signal based on the observed noisy data, it is fundamental to estimate the fluctuating wave form of the specific signal. On the other hand, the observation data are very often measured in a digital level form at discrete times. This is because some signal processing methods by utilizing a digital computer are indispensable for extracting exactly various kinds of statistical evaluation for the specific signal based on the quantized level data. In this study, a Bayesian filter matched to the complicated sound environment system is derived. First, in the real situation where the sound environment system is affected by background noise of arbitrary probability distribution, a stochastic system model with quantized observation is established. Next, two types of the recursive algorithm of Bayesian filter to estimate the unknown specific signal are theoretically proposed in the quantized level form. Finally, the effectiveness of the proposed theory is experimentally confirmed by applying it to the estimation problem of real sound environment.

Applying deep bidirectional LSTM and mixture density network for basketball trajectory prediction

Data analytics helps basketball teams to create tactics. However, manual data collection and analytics are costly and ineffective. Therefore, we applied a deep bidirectional long short-term memory (BLSTM) and mixture density network (MDN) approach. This model is not only capable of predicting a basketball trajectory based on real data, but it also can generate new trajectory samples. It is an excellent application to help coaches and players decide when and where to shoot. Its structure is particularly suitable for dealing with time series problems. BLSTM receives forward and backward information at the same time, while stacking multiple BLSTMs further increases the learning ability of the model. Combined with BLSTMs, MDN is used to generate a multi-modal distribution of outputs. Thus, the proposed model can, in principle, represent arbitrary conditional probability distributions of output variables. We tested our model with two experiments on three-pointer datasets from NBA SportVu data. In the hit-or-miss classification experiment, the proposed model outperformed other models in terms of the convergence speed and accuracy. In the trajectory generation experiment, eight model-generated trajectories at a given time closely matched real trajectories.

A mixed integer nonlinear model for closed-loop supply chains with incentive and proximity based product returns

Electronics firms in different locations are being required to collect used products due to environmental and health hazards. In order to meet environmental requirements, firms carry out collection activities and provide incentive offers to attract product returns. A mixed integer non-linear programming model for a Closed-Loop Supply Chain including decisions for collection activities, incentive offers and recovery options is formulated and validated. Quantity is modeled as a function of incentive offers and distance between the collection centers (distribution/retail centers) and consumers. The quality of product returns follows an arbitrary probability distribution, which is shifted by incentive level. Quality of product returns dictates the possible recovery options, which these can undergo. The model is subjected to scenario analysis. This is comprised of conditions wherein rebate, or discount incentives is preferred and when low or high incentive levels are favored. High stockout cost to secondary consumers encouraged the model to perform more cash rebate activities to stimulate more product returns to satisfy secondary consumer demand. In another situation, high cost of activities while having high stockout cost to secondary consumers induced the model to carryout discount activities as this would generate sales rather than the cash rebate which simply incentivizes the participation in the takeback program.

Computing Entropies with Nested Sampling

The Shannon entropy, and related quantities such as mutual information, can be used to quantify uncertainty and relevance. However, in practice, it can be difficult to compute these quantities for arbitrary probability distributions, particularly if the probability mass functions or densities cannot be evaluated. This paper introduces a computational approach, based on Nested Sampling, to evaluate entropies of probability distributions that can only be sampled. I demonstrate the method on three examples: a simple Gaussian example where the key quantities are available analytically; (ii) an experimental design example about scheduling observations in order to measure the period of an oscillating signal; and (iii) predicting the future from the past in a heavy-tailed scenario.

Distinguishability notion based on Wootters statistical distance: Application to discrete maps.

We study the distinguishability notion given by Wootters for states represented by probability density functions. This presents the particularity that it can also be used for defining a statistical distance in chaotic unidimensional maps. Based on that definition, we provide a metric d¯ for an arbitrary discrete map. Moreover, from d¯, we associate a metric space with each invariant density of a given map, which results to be the set of all distinguished points when the number of iterations of the map tends to infinity. Also, we give a characterization of the wandering set of a map in terms of the metric d¯, which allows us to identify the dissipative regions in the phase space. We illustrate the results in the case of the logistic and the circle maps numerically and analytically, and we obtain d¯ and the wandering set for some characteristic values of their parameters. Finally, an extension of the metric space associated for arbitrary probability distributions (not necessarily invariant densities) is given along with some consequences. The statistical properties of distributions given by histograms are characterized in terms of the cardinal of the associated metric space. For two conjugate variables, the uncertainty principle is expressed in terms of the diameters of the associated metric space with those variables.

Arbitrary Polynomial Chaos for Uncertainty Propagation of Correlated Random Variables in Dynamic Systems

Dynamic simulation of stochastic systems requires uncertainty propagation. Traditional sample-based uncertainty propagation methods are often computationally intractable for online optimization-based estimation and control applications. Generalized polynomial chaos (gPC) is an efficient uncertainty propagation method that has been used for solving various nonlinear estimation and optimal control problems. However, gPC requires knowledge of the exact probability distribution of the random variables, and does not explicitly account for correlations between these random variables. This paper demonstrates the use of arbitrary polynomial chaos (aPC) for propagation of correlated multivariate random variables. aPC constructs orthogonal polynomial basis functions from only the raw moments of the random variables. Thus, aPC can be used for propagation of uncertainties with arbitrary probability distributions, even if their functional forms are unknown. The main contributions of this paper consist of presenting an algorithm for generating a set of orthogonal polynomial basis functions for correlated multivariate random variables and applying the Galerkin projection to compute closed-form expressions for the dynamics of the aPC expansion coefficients. An algorithm is also presented for efficient calculation of inner products between polynomial basis functions needed in the Galerkin projection. The error convergence properties of aPC are investigated and compared to that of gPC and Monte Carlo using a dynamic simulation case study.

Software for Complete Mode Structure Analysis of a Light Field

We present a software package aimed at simulating photon-number probability distributions of a range of classical and non-classical states of light. This software can generate arbitrary probability distributions from user-defined mode structure of a light field. It can also solve the reverse problem, i.e., reconstructing the mode structure of a light field from a given probability distribution. The mode structure fully defines a light field. Thus it provides information about the source of light without having to directly access the source. The multimode fields simulated by this software include those comprised of a number of thermal modes, a Poisson mode, and single-photon modes. In addition, conjugated multimode sources (such as those created via parametric downconversion (PDC) or four-wave mixing (FWM)) can be simulated. This software provides for a nearly-perfect reconstruction of multimode fields comprised of several correlated and uncorrelated thermal and Poisson modes. The code is particularly effective at characterizing mesoscopic quantum states of light. This software can be modified to include other types of modes (i.e., governed by other statistical distributions), as needed.

Optimal Network Control With Stochastic Link Variations and Indefinite Quadratic Constraints

In this paper, we consider the problem of optimal control of a network with stochastic links modeled as multiplicative noise with given second moments and arbitrary probability distributions. Additional quadratic constraints are imposed, where the quadratic forms may be indefinite and, thus, not necessarily convex. We show that the problem can be transformed to a semidefinite program. The optimization problem is over a certain variable serving as the covariance matrix of the state and the controller. We show that affine controllers are optimal and depend on the optimal covariance matrix. Furthermore, we show that optimal controllers are linear if all of the quadratic forms are convex in the control variable. The solutions are presented for the finite and infinite horizon cases. We give a necessary and sufficient condition for mean square stabilizability of the dynamical system with additive and multiplicative noise. The condition is a Lyapunov-like condition whose solution is again given by the covariance matrix of the state and the control variable. The results are illustrated with an example.

The Statistics of Radio Astronomical Polarimetry: Disjoint, Superposed, and Composite Samples

Abstract A statistical framework is presented for the study of the orthogonally polarized modes of radio pulsar emission via the covariances between the Stokes parameters. To accommodate the typically heavy-tailed distributions of single-pulse radio flux density, the fourth-order joint cumulants of the electric field are used to describe the superposition of modes with arbitrary probability distributions. The framework is used to consider the distinction between superposed and disjoint modes, with particular attention to the effects of integration over finite samples. If the interval over which the polarization state is estimated is longer than the timescale for switching between two or more disjoint modes of emission, then the modes are unresolved by the instrument. The resulting composite sample mean exhibits properties that have been attributed to mode superposition, such as depolarization. Because the distinction between disjoint modes and a composite sample of unresolved disjoint modes depends on the temporal resolution of the observing instrumentation, the arguments in favor of superposed modes of pulsar emission are revisited, and observational evidence for disjoint modes is described. In principle, the four-dimensional covariance matrix that describes the distribution of sample mean Stokes parameters can be used to distinguish between disjoint modes, superposed modes, and a composite sample of unresolved disjoint modes. More comprehensive and conclusive interpretation of the covariance matrix requires more detailed consideration of various relevant phenomena, including temporally correlated subpulse modulation (e.g., jitter), statistical dependence between modes (e.g., covariant intensities and partial coherence), and multipath propagation effects (e.g., scintillation and scattering).

Hybrid Petri nets with general one-shot transitions

A hybrid Petri net formalism that allows deterministic, and fluid transitions is extended by generally distributed transitions that moves discrete tokens. Models in this formalism can be analyzed with Parametric Reachability Analysis, by computing all reachable locations, and by separating the deterministic and the stochastic evolution of the system. Several performance metrics, such as the distribution of fluid over time, can be derived by deconditioning according to arbitrary continuous probability distributions. This efficient concept allows for the analysis of an arbitrary number of fluid places, as opposed to classical stochastic hybrid Petri net approaches. Moreover, validation of our results against a FSPN tool shows that parametric reachability analysis provides more accurate results. A case study motivates and shows the potential of our approach.

Location distribution of a mobile terminal and its application to paging cost reduction and minimization

Reducing the cost of dynamic mobility management in wireless communication networks has been an interesting and important issue. It is well known that by using a selective paging method, both costs for location update and terminal paging can be reduced significantly. However, an efficient selective paging method needs the information of the location distribution of a mobile terminal. Based on our previous results on random walks among rings of cell structures, we analyze the location distribution of a mobile terminal in a paging area when a phone call arrives, where the inter-call time and the cell residence time can have arbitrary probability distributions. Our analysis is conducted for both distance-based and movement-based location management schemes, and for two different call handling models, i.e., the call plus location update model and the call without location update model. Together with our earlier results on location distribution in time-based location management schemes, for several selective paging methods, including progressive paging methods, ring paging methods, and cell paging methods, we are able to obtain their expected costs of paging for distance-based, movement-based, and time-based location management schemes. We find that a progressive paging method with very small time delay can reduce the terminal paging cost dramatically, while further increasing the time delay does not result in noticeable reduction of terminal paging cost. Our work reported in this paper significantly extends our understanding of cost reduction and minimization of dynamic location management in wireless communication networks.

Física Moderna e técnicas computacionais: como "ver" o átomo de Hidrogênio

http://dx.doi.org/10.5007/2175-7941.2016v33n2p667 Este artigo busca construir uma ligação entre dois temas que têm recebido cada vez mais atenção nos anos finais da educação básica e nos anos iniciais do ensino superior nas áreas de ciência e tecnologia: a introdução à física moderna e a introdução a técnicas básicas de programação. Como pretexto, o artigo descreve como gerar figuras que representam, através de um pontilhado ou uma escala de cores, as densidades de probabilidade para o elétron em um átomo de hidrogênio. Isso é feito apresentando brevemente a física do problema de base e comentando algoritmos que (a) implementam as equações do problema, (b) permitem gerar valores aleatórios a partir de distribuições de probabilidades arbitrárias e (c) efetivamente produzem as figuras. O objetivo fundamental é mostrar ao leitor como técnicas básicas de modelagem e visualização de dados são efetivamente implementadas utilizando um tema que em princípio é de seu interesse. A implementação é feita em HTML/CSS/JavaScript, uma tecnologia extremamente versátil e acessível que constitui a base de todo o conteúdo na internet hoje, incluindo muitos ambientes educacionais virtuais.

Gaussian process model-based approach for uncertainty quantification of natural frequencies of bridge

Uncertainty in bridge structural parameters will unavoidably lead to uncertainty in natural frequencies of bridge. Uncertainty quantification of natural frequencies of bridge enables to provide more accuarte information of strucutral dynamic characteristics for the bridge design. Mean and variance, two importance order statistics, can be used to characterize the uncertainty of natural frequencies. This paper aims at developing an efficient and reliable appraoch to compute the mean and varaince of natural frequencies of bridge. Bridge is usually greatly complex and large scale, so the traditional Monte Carlo simulation method is infeasible due to the issue of high computational cost. In this paper, Gaussian process model (GPM) is adopted to replace the complex FEM of bridge, aiming at mapping the relationship between the uncertain parameters and natural frequencies. Within GPM framework, the mean and variance can be efficiently obtained. The present GPM-based method is suitable for calculating mean and variance of physical system responses with arbitrary parameter probability distributions. And its feasibility is verified through a benchmark function with analytical mean and variance. Finally, the present GPM-based approach is applied to compute statistics of natural frequencies of Anqing Railway Yangtze River Bridge.

A surrogate method for density-based global sensitivity analysis

This paper describes an accurate and computationally efficient surrogate method, known as the polynomial dimensional decomposition (PDD) method, for estimating a general class of density-based f-sensitivity indices. Unlike the variance-based Sobol index, the f-sensitivity index is applicable to random input following dependent as well as independent probability distributions. The proposed method involves PDD approximation of a high-dimensional stochastic response of interest, forming a surrogate input–output data set; kernel density estimations of output probability density functions from the surrogate data set; and subsequent Monte Carlo integration for estimating the f-sensitivity index. Developed for an arbitrary convex function f and an arbitrary probability distribution of input variables, the method is capable of calculating a wide variety of sensitivity or importance measures, including the mutual information, squared-loss mutual information, and L1-distance-based importance measure. Three numerical examples illustrate the accuracy, efficiency, and convergence properties of the proposed method in computing sensitivity indices derived from three prominent divergence or distance measures. A finite-element-based global sensitivity analysis of a leverarm was performed, demonstrating the ability of the method in solving industrial-scale engineering problems.

An efficient metamodeling approach for uncertainty quantification of complex systems with arbitrary parameter probability distributions

Summary This paper proposes an efficient metamodeling approach for uncertainty quantification of complex system based on Gaussian process model (GPM). The proposed GPM-based method is able to efficiently and accurately calculate the mean and variance of model outputs with uncertain parameters specified by arbitrary probability distributions. Because of the use of GPM, the closed form expressions of mean and variance can be derived by decomposing high-dimensional integrals into one-dimensional integrals. This paper details on how to efficiently compute the one-dimensional integrals. When the parameters are either uniformly or normally distributed, the one-dimensional integrals can be analytically evaluated, while when parameters do not follow normal or uniform distributions, this paper adopts the effective Gaussian quadrature technique for the fast computation of the one-dimensional integrals. As a result, the developed GPM method is able to calculate mean and variance of model outputs in an efficient manner independent of parameter distributions. The proposed GPM method is applied to a collection of examples. And its accuracy and efficiency is compared with Monte Carlo simulation, which is used as benchmark solution. Results show that the proposed GPM method is feasible and reliable for efficient uncertainty quantification of complex systems in terms of the computational accuracy and efficiency. Copyright © 2016 John Wiley & Sons, Ltd.

Optimization results for a generalized coupon collector problem

Abstract In this paper we study a generalized coupon collector problem, which consists of analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one which stochastically maximizes the time needed to collect a fixed number of distinct coupons.

Robust Sequential Testing of Hypotheses on Discrete Probability Distributions

The paper is devoted to analysis of approximations of the models in sequential hypotheses testing and to construction of new robust sequential test under the distortions presented by mixtures of probability distributions. The previous results of the authors are extended to the case of arbitrary discrete probability distributions. The theory is illustrated by computer modelling.

Optimization Models for Production and Procurement Decisions Under Uncertainty

This paper studies an integrated optimization problem on production and procurement for a manufacturer who faces stochastic demands of multiple products made by multiple components. This paper first proposes a deterministic model for the problem when the demands of products are known. Then the model is extended to the case of uncertain demands. A stochastic programming model and a robust optimization model are proposed to handle the production and procurement decisions under uncertain demands. The stochastic programming model can cope with arbitrary probability distributions of products' random demands; while the robust optimization model applies to situations in which limited information about probability distributions is available. Some numerical experiments are performed to investigate the efficiency of the proposed models. We also compare the above two models and evaluate the potential benefits in practice.

Optimal planning of incentive-based quality in closed-loop supply chains

Electronic firms are being required to collect used products for environmental purposes. In order to meet requirements, these firms carry out collection activities and provide incentive offers to attract product returns. These product returns may then undergo recovery options such as refurbishing, remanufacturing, cannibalizing, and controlled disposal. A mixed integer nonlinear programming model for a closed-loop supply chain that includes decisions for collection activities, incentive offers, and recovery options is formulated and validated. Quantity is modeled as a function of incentive offers between the collection centers and consumers, while quality of product returns follows an arbitrary probability distribution based on the incentive level. Quality of product returns dictates the possible recovery options, which these products can undergo. The model is then subjected to scenario analysis, which identified conditions wherein rebate or discount incentives are preferred, and when low or high incentive levels are favored. High stockout costs to secondary consumers encouraged the model to perform more cash rebate activities to stimulate more product returns. Meanwhile, when both the costs of activities and stockouts are high, the model is induced to carry out discount activities as this would generate sales rather than the cash rebate which simply incentivizes the participation in the take-back program.