Discrete Probability Distributions Research Articles

A Quantum Algorithm Framework for Discrete Probability Distributions With Applications to Rényi Entropy Estimation

A Quantum Algorithm Framework for Discrete Probability Distributions With Applications to Rényi Entropy Estimation

Spectral decomposition of [formula omitted] and Poincaré inequality on a compact interval — Application to kernel quadrature

Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form ∫abf(x)dμ(x)=∑i=1nwif(xi) where f belongs to H1(μ). Here, μ belongs to a class of continuous probability distributions on [a,b]⊂R and ∑i=1nwiδxi is a discrete probability distribution on [a,b]. We show that H1(μ) is a reproducing kernel Hilbert space with a continuous kernel K, which allows to reformulate the quadrature question as a kernel (or Bayesian) quadrature problem. Although K has not an easy closed form in general, we establish a correspondence between its spectral decomposition and the one associated to Poincaré inequalities, whose common eigenfunctions form a T-system (Karlin and Studden, 1966). The quadrature problem can then be solved in the finite-dimensional proxy space spanned by the first eigenfunctions. The solution is given by a generalized Gaussian quadrature, which we call Poincaré quadrature.We derive several results for the Poincaré quadrature weights and the associated worst-case error. When μ is the uniform distribution, the results are explicit: the Poincaré quadrature is equivalent to the midpoint (rectangle) quadrature rule. Its nodes coincide with the zeros of an eigenfunction and the worst-case error scales as b−a23n−1 for large n. By comparison with known results for H1(0,1), this shows that the Poincaré quadrature is asymptotically optimal. For a general μ, we provide an efficient numerical procedure, based on finite elements and linear programming. Numerical experiments provide useful insights: nodes are nearly evenly spaced, weights are close to the probability density at nodes, and the worst-case error is approximately O(n−1) for large n.

A Rényi-type quasimetric with random interference detection

AbstractThis paper introduces a new dissimilarity measure between two discrete and finite probability distributions. The followed approach is grounded jointly on mixtures of probability distributions and an optimization procedure. We discuss the clear interpretation of the constitutive elements of the measure under an information-theoretical perspective by also highlighting its connections with the Rényi divergence of infinite order. Moreover, we show how the measure describes the inefficiency in assuming that a given probability distribution coincides with a benchmark one by giving formal writing of the random interference between the considered probability distributions. We explore the properties of the considered tool, which are in line with those defining the concept of quasimetric—i.e. a divergence for which the triangular inequality is satisfied. As a possible usage of the introduced device, an application to rare events is illustrated. This application shows that our measure may be suitable in cases where the accuracy of the small probabilities is a relevant matter.

Poisson-New Quadratic-Exponential Distribution

This proposed distribution is a discrete compound probability distribution with only one parameter. To get this distribution, Poisson distribution has been mixed with the New Quadratic-Exponential distribution of Sah (2022). Hence, it is named as “Poisson-New Quadratic-Exponential Exponential Distribution (PNLED)”. The important statistical characteristics needed to check the validity of this distribution have been derived and clearly explained. To check the validity of the theoretical works of this distribution, while using goodness of fit on some over-dispersed count data, what we have been found that this distribution seems a better alternative of Poisson-Lindley distribution (PLD) of Sankaran (1970), Poisson Mishra distribution (PMD) of Sah (2017) and Poisson-Modified Mishra distribution (PMMD) of Sah and Sahani (2023).

Particle filtering supported probability density estimation of mobility patterns

This paper presents a methodology that aims to enhance the accuracy of probability density estimation in mobility pattern analysis by integrating prior knowledge of system dynamics and contextual information into the particle filter algorithm. The quality of the data used for density estimation is often inadequate due to measurement noise, which significantly influences the distribution of the measurement data. Thus, it is crucial to augment the information content of the input data by incorporating additional sources of information beyond the measured position data. These other sources can include the dynamic model of movement and the spatial model of the environment, which influences motion patterns. To effectively combine the information provided by positional measurements with system and environment models, the particle filter algorithm is employed, which generates discrete probability distributions. By subjecting these discrete distributions to exploratory techniques, it becomes possible to extract more certain information compared to using raw measurement data alone. Consequently, this study proposes a methodology in which probability density estimation is not solely based on raw positional data but rather on probability-weighted samples generated through the particle filter. This approach yields more compact and precise modeling distributions. Specifically, the method is applied to process position measurement data using a nonparametric density estimator known as kernel density estimation. The proposed methodology is thoroughly tested and validated using information-theoretic and probability metrics. The applicability of the methodology is demonstrated through a practical example of mobility pattern analysis based on forklift data in a warehouse environment.

Premium Linear-Exponential Mixture of Poisson Distribution

The proposed distribution is a discrete probability distribution with only one parameter and it has been derived by compounding the Poisson distribution with the Premium Linear-exponential distribution (PLED) of Sah (2022). So, it is named as Premium Linear-Exponential Mixture of Poisson Distribution (PLEMPD). The required statistical characteristics have been derived and explained to give the theoretical concept of this distribution. The main purpose of proposing this distribution is to provide a better goodness of fit for over-dispersed count data than the previously derived one parameter continuous mixtures of Poisson distribution of same nature by others. Taking some secondary over-dispersed count data and using the theoretical concept of this distribution, it has been observed that this distribution is more suitable for statistical modeling of over-dispersed count data than Poisson-Lindley distribution (PLD) of Sankaran (1970) and Poisson-Mishra distribution (PMD) of Sah (2017).

이산확률분포의 표본 분위수 계산 방법에 관한 연구

In this paper, we studied the methods of computing sample quantiles for the case of discrete probability distributions. There are two main methods which are introduced in statistics textbooks. We compared these two methods by simulation from practical and educational necessity. We considered some cases of Poisson, binomial, geometric, negative binomial, and discrete uniform distributions by setting up the parameters, and for each distribution we performed 10,000 times of simulation of drawing random samples of size 20 and 50. At each time of simulation, we computed the difference between the population quantile and the sample quantile obtained by each method. We compared the two methods by using two criteria: one is the mean square error over 10,000 times of simulation, and the other is the frequency of obtaining closer sample quantile to the population quantile than the competing method. We also obtained the estimated probabilities of exact estimation of population quantiles for the two methods, and performed the tests of homogeneity on the probability distribution of under-estimation, exact estimation, and over-estimation of the population quantiles.

A Dirichlet process mixture regression model for the analysis of competing risk events

We develop a regression model for the analysis of competing risk events. The joint distribution of the time to these events is flexibly characterized by a random effect which follows a discrete probability distribution drawn from a Dirichlet Process, explaining their variability. This entails an additional layer of flexibility of this joint model, whose inference is robust with respect to the misspecification of the distribution of the random effects. The model is analysed in a fully Bayesian setting, yielding a flexible Dirichlet Process Mixture model for the joint distribution of the time to events. An efficient MCMC sampler is developed for inference. The modelling approach is applied to the empirical analysis of the surrending risk in a US life insurance portfolio previously analysed by Milhaud and Dutang (2018). The approach yields an improved predictive performance of the surrending rates.

Nonparametric clustering of discrete probability distributions with generalized Shannon’s entropy and heatmap

This article proposes a non-parametric clustering heatmap for non-ordinal samples based on their underlying discrete probability distributions. The proposed heatmap is a preliminary tool to extract immediate information from non-ordinal data without making any parametric assumption.

A discrete extension of the Burr-Hatke distribution: Generalized hypergeometric functions, different inference techniques, simulation ranking with modeling and analysis of sustainable count data

<abstract><p>The intertwining relationship between sustainability and discrete probability distributions found its significance in decision-making processes and risk assessment frameworks. Count data modeling and its practical applications have gained attention in numerous research studies. This investigation focused on a particular discrete distribution characterized by a single parameter obtained through the survival discretization method. Statistical attributes of this distribution were accurately explicated using generalized hypergeometric functions. The unveiled characteristics highlighted its suitability for analyzing data displaying "right-skewed" asymmetry and possessing extended "heavy" tails. Its failure rate function effectively addressed scenarios marked by a consistent decrease in rates. Furthermore, it proved to be a valuable tool for probabilistic modeling of over-dispersed data. The study introduced various estimation methods such as maximum product of spacings, Anderson-Darling, right-tail Anderson-Darling, maximum likelihood, least-squares, weighted least-squares, percentile, and Cramer-Von-Mises, offering comprehensive explanations. A ranking simulation study was conducted to evaluate the performance of these estimators, employing ranking techniques to identify the most effective estimator across different sample sizes. Finally, real-world sustainability engineering and medical datasets were analyzed to demonstrate the significance and application of the newly introduced model.</p></abstract>

UNIFORMLY CONVEX AND STARLIKE PROBABILITY DISTRIBUTION

The purpose of the present paper is to introduce k− uniformly convex and k− uniformly starlike discrete probability distributions and obtain some results regarding moments, factorial moments and moment generating functions for these distributions.

Phase-type stress-strength reliability models under progressive type-II right censoring

. The study of stress-strength reliability estimation based on phase-type distribution helps to gather results on estimation of stress-strength reliability with any probability distribution that is defined on the non negative real numbers as any discrete or continuous probability distributions on the positive real line can be represented as phase-type. The matrix representation of the parameters of phase-type distributions helps in their flexible evaluation and easy manipulation. Also in many of the experimental studies, it is very convenient and useful to apply progressive type-II right censoring mechanism in the process of data collection. In this article, we consider the estimation of stress-strength reliability (R) based on phase-type distribution under progressive type-II right censoring mechanism. Both stress and strength random variables are assumed to follow either continuous phase-type or discrete phase-type distribution. We have developed the algorithm for computing Maximum likelihood estimate (MLE) of R based on the expectation maximization (EM) method and the Bayes estimate of R using Markov Chain Monte Carlo technique. A detailed numerical illustration using simulated data/ real data sets are carried out.

On Testing of Multiple Hypotheses of Continuous Probability Distributions Arranged into Two Groups

The optimal Neyman-Pearson procedure of detection is investigated for models characterized by four continuous probability distributions arranged into two groups considered as hypotheses. It is worthy to note that the case of three discrete probability distributions arranged in two groups was studied by Haroutunian and Yesayan in [1]. The Neyman-Pearson theorem holds immense importance when it comes to solving problems that demand decision making or conclusions to a higher accuracy.

Mapping time series into signed networks via horizontal visibility graph

Time series could be mapped into complex networks through the visibility or horizontal visibility algorithms, and the properties of the constructed network reflect the nonlinear dynamics of the time series. When horizontal visibility algorithm is directly applied to climate anomaly time series, in which both local maximum and local minimum are equally important, local minimum might be “overlooked”. In this paper, we propose a new method that maps climate anomaly time series into signed networks. Positive and negative data values of climate anomaly time series are classified into two types and mapped as nodes of signed networks. Links connecting nodes of the same type are assigned positive signs, while links connecting neighboring nodes of different types are assigned negative signs. This method is also applicable to time series those are assumed to be “stationary” or with no significant trends. Four kinds of degree as well as the degree distributions of the signed networks have been defined. Specifically, the degree and degree distribution could be partly derived analytically for periodic and uncorrelated random time series. The theoretical predictions for periodic and uncorrelated random time series have also been verified by extensive numerical simulations. Based on the entropy of the distribution of net degree, we propose a new complexity measure for chaotic time series. Compared to some previous complexity measures, the new complexity measure is an objective measure without transforming continuous values into discrete probability distributions but still has higher accuracy and sensitivity. Moreover, correlation information of stochastic time series can also be extracted via a topological parameter, the mean of ratio degree, of the signed networks. The extraction of serial correlation has been illustrated through numerical simulations and verified through an empirical climate time series.

Towards optimal running timesfor optimal transport

We provide faster algorithms for approximating the optimal transport distance, e.g. earth mover's distance, between two discrete probability distributions on n elements. We present two algorithms which compute couplings between marginal distributions with an expected transportation cost that is within an additive ϵ of optimal in time O˜(n2/ϵ); one algorithm is straightforward to parallelize and implementable in depth O˜(1/ϵ). Further, we show that additional improvements on our results must be coupled with breakthroughs in algorithmic graph theory.

Fuzzy multicriteria evaluation and trends of asset management performance: A case study of Spanish buildings

Asset management has an increasingly important role in organisations, as it can affect productivity and cost effectiveness, the useful life of facilities, the quality of processes, compliance with standards and safety and environmental laws, and more recently, to ensure more sustainable operations and use of resources. However, despite its importance in the provision of services to users, the buildings sector is deficient in contributions in the literature that: a) use an objective multicriteria model, and those developed in a fuzzy environment are even rarer; b) analyse the evolution of asset management in a country over an extended period of time; c) use weightings for each factor analysed, taken from judgements provided by experts in the buildings sector; d) consider all the full data for the sector, rather than average values; e) analyse the buildings complete sector and according to the size of the building. This paper, therefore, sets out a fuzzy multicriteria model for analysing the evolution of the asset management performance of the buildings sector by size (small, medium and large) in Spain from 2000 to 2020. The weightings of criteria and sub-criteria will be calculated based on the judgements of four experts in Spanish maintenance. 229 surveys carried out by the Spanish Maintenance Association are used, looking at all the information provided, rather than average values, developing discrete probability distribution functions to allow the uncertainty of the historical evolution of the asset management of buildings in Spain to be calculated. The average utility variation over the period 2000–2020 in the sector is 1.36 %. This value, although positive, is very low to cover a period of 20 years, and it seems that maintenance departments are not adapted to the new challenges of Industry 4.0.

Mathematical bounds on Shannon entropy given the abundance of the ith most abundant taxon

The measurement of diversity is a central component of studies in ecology and evolution, with broad uses spanning multiple biological scales. Studies of diversity conducted in population genetics and ecology make use of analogous concepts and even employ equivalent mathematical formulas. For the Shannon entropy statistic, recent developments in the mathematics of diversity in population genetics have produced mathematical constraints on the statistic in relation to the frequency of the most frequent allele. These results have characterized the ways in which standard measures depend on the highest-frequency class in a discrete probability distribution. Here, we extend mathematical constraints on the Shannon entropy in relation to entries in specific positions in a vector of species abundances, listed in decreasing order. We illustrate the new mathematical results using abundance data from examples involving coral reefs and sponge microbiomes. The new results update the understanding of the relationship of a standard measure to the abundance vectors from which it is calculated, potentially contributing to improved interpretation of numerical measurements of biodiversity.

Comparison between Two Algorithms for Computing the Weighted Generalized Affinity Coefficient in the Case of Interval Data

From the affinity coefficient between two discrete probability distributions proposed by Matusita, Bacelar-Nicolau introduced the affinity coefficient in a cluster analysis context and extended it to different types of data, including for the case of complex and heterogeneous data within the scope of symbolic data analysis (SDA). In this study, we refer to the most significant partitions obtained using the hierarchical cluster analysis (h.c.a.) of two well-known datasets that were taken from the literature on complex (symbolic) data analysis. h.c.a. is based on the weighted generalized affinity coefficient for the case of interval data and on probabilistic aggregation criteria from a VL parametric family. To calculate the values of this coefficient, two alternative algorithms were used and compared. Both algorithms were able to detect clusters of macrodata (aggregated data into groups of interest) that were consistent and consonant with those reported in the literature, but one performed better than the other in some specific cases. Moreover, both approaches allow for the treatment of large microdatabases (non-aggregated data) after their transformation into macrodata from the huge microdata.

Probabilistic robotic logic programming with hybrid Boolean and Bayesian inference

AbstractBayesian inference provides a probabilistic reasoning process for drawing conclusions based on imprecise and uncertain data that has been successful in many applications within robotics and information processing, but is most often considered in terms of data analysis rather than synthesis of behaviours. This paper presents the use of Bayesian inference as a means by which to perform Boolean operations in a logic programme while incorporating and propagating uncertainty information through logic operations by inference. Boolean logic operations are implemented in a Bayesian network of Bernoulli random variables with tensor-based discrete distributions to enable probabilistic hybrid logic programming of a robot. This enables Bayesian inference operations to coexist with Boolean logic in a unified system while retaining the ability to capture uncertainty by means of discrete probability distributions. Using a discrete Bayesian network with both Boolean and Bayesian elements, the proposed methodology is applied to navigate a mobile robot using hybrid Bayesian and Boolean operations to illustrate how this new approach improves robotic performance by inclusion of uncertainty without increasing the number of logic elements required. As any logical system could be programmed in this manner to integrate uncertainty into decision-making, this methodology can benefit a wide range of applications that use discrete or probabilistic logic.

Logic + probabilistic programming + causal laws.

Probabilistic planning attempts to incorporate stochastic models directly into the planning process, which is the problem of synthesizing a sequence of actions that achieves some objective for a putative agent. Probabilistic programming has rapidly emerged as a key paradigm to integrate probabilistic concepts with programming languages, which allows one to specify complex probabilistic models using programming primitives like recursion and loops. Probabilistic logic programming aims to further ease the specification of structured probability distributions using first-order logical artefacts. In this article, we briefly discuss the modelling of probabilistic planning through the lens of probabilistic (logic) programming. Although many flavours for such an integration are possible, we focus on two representative examples. The first is an extension to the popular probabilistic logic programming language PROBLOG, which permits the decoration of probabilities on Horn clauses-that is, prolog programs. The second is an extension to the popular agent programming language GOLOG, which permits the logical specification of dynamical systems via actions, effects and observations. The probabilistic extensions thereof emphasize different strengths of probabilistic programming that are particularly useful for non-trivial modelling issues raised in probabilistic planning. Among other things, one can instantiate planning problems with growing and shrinking state spaces, discrete and continuous probability distributions, and non-unique prior distributions in a first-order setting.