Exact Probability Distribution Research Articles

A variant of Horn's problem and the derivative principle

Identifying the spectrum of the sum of two given Hermitian matrices with fixed eigenvalues is the famous Horn's problem. In this note, we investigate a variant of Horn's problem, i.e., we identify the probability density function (abbr. pdf) of the diagonals of the sum of two random Hermitian matrices with given spectra. We then use it to re-derive the pdf of the eigenvalues of the sum of two random Hermitian matrices with given eigenvalues via the derivative principle, a powerful tool used to get the exact probability distribution by reducing to the corresponding distribution of diagonal entries. We can also recover Jean-Bernard Zuber's recent results. Moreover, as an illustration, we derive the analytical expressions of eigenvalues of the sum of two random Hermitian matrices from GUE(n) or Wishart ensemble by the derivative principle, respectively. We also investigate the statistics of exponential of random matrices and connect them with Golden-Thompson inequality, and partially answer a question proposed by Forrester. Some potential applications in quantum information theory, such as uniform average quantum Jensen-Shannon divergence and average coherence of uniform mixture of two orbits, are discussed.

Local and global degree profiles of randomly grown self-similar hooking networks under uniform and preferential attachment

We investigate node degrees in a network grown from a seed by hooking self-similar components under two models of randomness: a uniform attachment model and a model based on preferential attachment. We study two degree profiles: a local profile tracking the evolution of the degree of a particular node over time, and a global profile concerned about counts of the number of nodes of a particular degree.For the local profile, under uniform attachment growth, we have the exact mean, variance and probability distribution in terms of standard combinatorial numbers like generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: The early nodes have an asymptotically normal distribution, intermediate nodes have a Poisson distribution and the late nodes have a degenerate distribution. In contrast, under preferential attachment, the moments of the degree of a node contain Stirling numbers of the second kind and (under appropriate scaling) the profile has a gamma-type limit law.As for the global profile, we use Pólya urns to derive strong laws. Four regimes arise according to the structure of the seed. Within these regimes, we identify a few degenerate cases. Barring these degenerate cases, we uncover an asymptotically normal joint multivariate distribution for nodes of very small degrees.

Continuous-time quantum walks on strongly regular graphs with loops and its application to spatial search for multiple marked vertices

The continuous-time quantum walk (CTQW) on the strongly regular graph is studied in this paper, and the exact transition probability distribution between any two vertices of the graph is provided by using the method of counting the walks between these two vertices. The CTQW is also considered on the perturbed strongly regular graph (SRG) by adding multiple loops on given vertices. Spatial search using CTQW can be regarded as a special case of CTQW on the perturbed SRG by adding loops. Combined with the approach of walk counting, the proper parameter settings and search time of spatial search for both a single marked vertex and two marked vertices are provided. The results show that both kinds of spatial search can be undertaken in $$O(\sqrt{N})$$ time in a degree k SRG with N vertices, by setting jump rate of CTQW to be 1 / k.

Distributionally robust project crashing with partial or no correlation information

AbstractCrashing is shortening the project makespan by reducing activity times in a project network by allocating resources to them. Activity durations are often uncertain and an exact probability distribution itself might be ambiguous. We study a class of distributionally robust project crashing problems where the objective is to optimize the first two marginal moments (means and SDs) of the activity durations to minimize the worst‐case expected makespan. Under partial correlation information and no correlation information, the problem is solvable in polynomial time as a semidefinite program and a second‐order cone program, respectively. However, solving semidefinite programs is challenging for large project networks. We exploit the structure of the distributionally robust formulation to reformulate a convex‐concave saddle point problem over the first two marginal moment variables and the arc criticality index variables. We then use a projection and contraction algorithm for monotone variational inequalities in conjunction with a gradient method to solve the saddle point problem enabling us to tackle large instances. Numerical results indicate that a manager who is faced with ambiguity in the distribution of activity durations has a greater incentive to invest resources in decreasing the variations rather than the means of the activity durations.

Modeling and analysis of composite laminates in the presence of uncertainties

The basic or primitive parameters of composite laminates, such as the constituent materials properties, the thickness of each ply, the ply orientations and the applied loads, exhibit variabilities, hence, the response of the laminated composites also exhibits some degree of variability. Thus, the accuracy and reliability of the laminates cannot be assured if the variabilities present in the basic parameters are ignored. In the past decades, the probabilistic approach has been used widely to simulate the uncertainties in the macromechanics of composite laminates. However, the exact probability distributions of the primitive parameters are not known in most of the applications. This work, for the first time, models the uncertainties in the macromechanics of composite laminates using the interval analysis and the universal grey system theory by representing the primitive parameters as intervals. Also, for comparison, a probabilistic approach is presented with plus/minus three standard deviations about the mean of the response. Due to the so-called dependency problem, the interval analysis predicts wider and inaccurate ranges. Thus, a truncation-based interval analysis procedure with a suitable truncation parameter is used to overcome the limitation associated with the original interval analysis. Specifically, in this work, the uncertainties in the in-plane and flexural engineering constants and laminae stresses are studied. The environmental effects on the laminae stresses are also investigated. Numerical examples are presented to demonstrate the application of the three interval-based uncertainty methods for the behavior of composite laminates by considering different stacking sequences of graphite/epoxy and glass/epoxy laminates.

Data-driven scheduling optimization under uncertainty using Renyi entropy and skewness criterion

Abstract In order to deal with the resource cost uncertainties, this paper introduces a Renyi mean-entropy-skewness (RMES) information criterion for the scheduling optimization problems in flexible manufacturing systems (FMSs). Motivated by potential limitations in the existing measures, this third-order information criterion is carefully integrated to be more general and more robust in representing the schedule dispersion under uncertainties. The RMES information criterion is estimated using data-driven techniques so that it does not rely on the assumptions of exact probability distributions which are usually unknown in practice. Modeled with Petri net (PN) and system state reachable graph (RG), the RG-based dynamic programming (DP) algorithm and an approximate dynamic programming (ADP) algorithm are presented to solve the proposed model. The effectiveness of the introduced information criterion is verified by both technical proofs and extensive simulation studies of systems with a wide range of scales and data types. A real stamping industrial case study is also conducted as a justification of the model’s practical applicability.

Distributionally robust scheduling on parallel machines under moment uncertainty

This paper investigates a distributionally robust scheduling problem on identical parallel machines, where job processing times are stochastic without any exact distributional form. Based on a distributional set specified by the support and estimated moments information, we present a min-max distributionally robust model, which minimizes the worst-case expected total flow time out of all probability distributions in this set. Our model doesn’t require exact probability distributions which are the basis for many stochastic programming models, and utilizes more information compared to the interval-based robust optimization models. Although this problem originates from the manufacturing environment, it can be applied to many other fields when the machines and jobs are endowed with different meanings. By optimizing the inner maximization subproblem, the min-max formulation is reduced to an integer second-order cone program. We propose an exact algorithm to solve this problem via exploring all the solutions that satisfy the necessary optimality conditions. Computational experiments demonstrate the high efficiency of this algorithm since problem instances with 100 jobs are optimized in a few seconds. In addition, simulation results convincingly show that the proposed distributionally robust model can hedge against the bias of estimated moments and enhance the robustness of production systems.

EDA on the asymptotic normality of the standardized sequential stopping times, Part-I: Parametric models

ABSTRACTIn sequential analysis, an experimenter may gather information regarding an unknown parameter by observing random samples in successive steps. We emphasize a number of specific parametric models under a variety of loss functions. The total number of observations collected at termination is a positive integer-valued random variable, customarily denoted by . The exact probability distribution of N is often hard to obtain. However, under a set of regulatory conditions, our standardized version of the stopping variable from Definition 2.1 in Section 2.2 would follow a normal distribution in the asymptotic sense. In this article, we first show how these regulatory conditions build upon one another in order to conclude the asymptotic normality of such standardized stopping variable, . We provide exploratory data analysis (EDA) via a number of interesting illustrations obtained through large-scale simulation studies. We demonstrate the broad applicability of purely sequential methodologies included in this article along with the appropriateness of our conclusions regarding the asymptotic normality of the standardized stopping variable as a practical guideline.

Modeling a two-stage supply contract problem in a hybrid uncertain environment

This study examines the supply contract problem in supply chain management, in which the uncertain demand is affected by a variety of factors. Since the available historical data cannot adequately determine the exact probability distribution of future demand, the prediction of experienced experts about uncertain demand is usually required in the practical modeling process of supply contract problem. The main contributions of this study are to model the uncertain demand by both probability distribution and possibility distribution, and develop a new two-stage expected value optimization model for our supply contract problem, in which an options-futures contract is employed to reduce the risk of uncertain demand. In the first decision-making stage, the distributor signs an options-futures contract with the supplier to determine the futures and options ordering quantities. After knowing the realization of uncertain demand, the distributor takes the signed options as the recourse decision in the second stage. When the actual demand follows some common probability and possibility distributions, the proposed two-stage optimization model can be turned into its deterministic equivalent programming model so that it can be solved efficiently via commercial optimization software. To illustrate the effectiveness of the proposed optimization method, a practical natural gas supply contract problem is provided in our numerical experiments. The computational results demonstrate that the possibility distribution of uncertain demand does affect the ordering decisions, and modeling uncertain demand by both probability distribution and possibility distribution is more realistic in our natural gas supply contract problem. As a result, the proposed two-stage optimization method can guide distributors to make their informed decisions in practical supply contract problem when the probability distribution of uncertain demand is partially known.

Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension

We investigate the motion of a run-and-tumble particle (RTP) in one dimension. We find the exact probability distribution of the particle with and without diffusion on the infinite line, as well as in a finite interval. In the infinite domain, this probability distribution approaches a Gaussian form in the long-time limit, as in the case of a regular Brownian particle. At intermediate times, this distribution exhibits unexpected multi-modal forms. In a finite domain, the probability distribution reaches a steady-state form with peaks at the boundaries, in contrast to a Brownian particle. We also study the relaxation to the steady-state analytically. Finally we compute the survival probability of the RTP in a semi-infinite domain with an absorbing boundary condition at the origin. In the finite interval, we compute the exit probability and the associated exit times. We provide numerical verification of our analytical results.

Robust Shortest Path Problem With Distributional Uncertainty

Routing service considering uncertainty is at the core of intelligent transportation systems and has attracted increasing attention. Existing stochastic shortest path models require the exact probability distributions of travel times and usually assume that they are independent. However, the distributions are often unavailable or inaccurate due to insufficient data, and correlation of travel times over different links has been observed. This paper presents a robust shortest path (RSP) model that only requires partial distribution information of travel times, including the support set, mean, variance, and correlation matrix. We introduce a concept of robust mean-excess travel time to hedge against the risk from both the uncertainty of the random travel times and the uncertainty in their distributions. To solve the RSP problem, an equivalent dual formulation is derived and used to design tight lower and upper bound approximation methods, which adopt the scenario approach and semi-definite programming approach, respectively. To solve large problems, we further propose an efficient primal approximation method, which only needs to solve two deterministic shortest path problems and a mean-standard deviation shortest path problem, and analyze its approximation performance. Experiments validate the tightness of the proposed bounds and demonstrate the impact of uncertainty on the relative benefit and cost of robust paths.

Методика расчета распределения вероятностей значений симметричных аддитивно разделяемых статистик, приближенного к их точному распределению

Методика расчета распределения вероятностей значений симметричных аддитивно разделяемых статистик, приближенного к их точному распределению

Exact eigenfunction amplitude distributions of integrable quantum billiards

The exact probability distributions of the amplitudes of eigenfunctions, Ψ(x, y), of several integrable planar billiards are analytically calculated and shown to possess singularities at Ψ = 0; the nature of this singularity is shape-dependent. In particular, we prove that the distribution function for a rectangular quantum billiard is proportional to the complete elliptic integral, K(1 − Ψ2), and demonstrate its universality, modulo a weak dependence on quantum numbers. On the other hand, we study the low-lying states of nonseparable, integrable triangular billiards and find the distributions thereof to be described by the Meijer G-function or certain hypergeometric functions. Our analysis captures a marked departure from the Gaussian distributions for chaotic billiards in its survey of the fluctuations of the eigenfunctions about Ψ = 0.

Optimal Wind Farm Allocation in Multi-Area Power Systems Using Distributionally Robust Optimization Approach

This paper presents a distributionally robust planning model to determine the optimal allocation of wind farms in a multi-area power system, so that the expected energy not served (EENS) is minimized under uncertain wind power and generator forced outages. Unlike conventional stochastic programming approaches that rely on detailed information of the exact probability distribution, the proposed method attempts to minimize the expectation term over a collection of distributions characterized by accessible statistical measures, so it is more practical in cases where the detailed distribution data is unavailable. This planning model is formulated as a two-stage problem, where the wind power capacity allocation decisions are determined in the first stage, before the observation of uncertainty outcomes, and operation decisions are made in the second stage under specific uncertainty realizations. In this paper, the second-stage decisions are approximated by linear decision rule functions, so that the distributionally robust model can be reformulated into a tractable second-order cone programming problem. Case studies based on a five-area system are conducted to demonstrate the effectiveness of the proposed method

Algorithm for Calculating the Exact Amount of n-bit Sequences with at Least One Run of Length k (k ≤ n)

A run of length k in a sequence of n bits (0, 1) is an equal bits string preceded or succeeded by a different bit (or none). In this paper a computational algorithm for calculating the exact amount of binary sequences of length n, containing at least one run of length k (k ≤ n) is presented. The immediate application of this algorithm is to calculate the exact probability distribution of the number of n-bit sequences with containing at least one run of length k, and no one greater than k, for use for example as part of a test for randomness

Ambiguity and the historical equity premium

This paper assesses the quantitative impact of ambiguity on the historically observed equity premium. We consider a Lucas-tree pure‐exchange economy with a single agent where we introduce two key nonstandard assumptions. First, the agent’s beliefs about the dividend/consumption process is ambiguous, i.e., she is uncertain about the exact probability distribution governing the realization of future dividends and consumption. Second, the agent’s preferences are sensitive to this ambiguity, a property formalized using the smooth ambiguity model. The consumption and dividend process is assumed to evolve according to a hidden state model, popularized by Bansal and Yaron (2004), where a persistent latent state variable describes temporary shocks to the mean of consumption growth prospects. We further extend the model to allow for uncertainty about the magnitude of the persistence of the latent state. The agent’s beliefs are ambiguous due to the uncertainty about the conditional mean of the probability distribution on consumption and dividends in the next period. We show that in this model ambiguity is endogenously dynamic, for example, increasing during recessions. This results in an endogenously volatile and (counter-)cyclical equity premium. We calibrate the level of ambiguity aversion to match only the first moment of the risk-free rate in data, and ambiguity to match the uncertainty conditional on the historical growth path, and evaluate the model using moderate levels of risk aversion. We find that this simple modification of a Lucas-tree model accounts for a large part of the historical equity premium, both in terms of its level and variation over time.

You Are in a Drawdown. When Should You Start Worrying?

Trading strategies that were profitable in the past often degrade with time. Since unlucky streaks can also hit healthy strategies, how can one detect that something truly worrying is happening? It is intuitive that a drawdown that lasts too long or one that is too deep should lead to a downward revision of the assumed Sharpe ratio of the strategy. In this note, we give a quantitative answer to this question based on the exact probability distributions for the length and depth of the last drawdown for upward drifting Brownian motions. We also point out that both managers and investors tend to underestimate the length and depth of drawdowns consistent with the Sharpe ratio of the underlying strategy.

Statistical Monitoring of Process Capability Index Having One Sided Specification Under Repetitive Sampling Using an Exact Distribution

This paper proposes a control chart using repetitive sampling for monitoring process capability index having one-sided specification limit using an exact distribution. The performance of the proposed chart is evaluated in terms of average run lengths using the exact probability distribution of the process capability index. The result shows that the proposed chart is more efficient by indicating the quicker out-of-control signals than the one using single sampling.

Exact probability distribution function for the volatility of cumulative production

In this paper we study the volatility and its probability distribution function for the cumulative production based on the experience curve hypothesis. This work presents a generalization of the study of volatility in Lafond et al. (2017), which addressed the effects of normally distributed noise in the production process. Due to its wide applicability in industrial and technological activities we present here the mathematical foundation for an arbitrary distribution function of the process, which we expect will pave the future research on forecasting of the production process.

Inference in population genetics using forward and backward, discrete and continuous time processes

A central aim of population genetics is the inference of the evolutionary history of a population. To this end, the underlying process can be represented by a model of the evolution of allele frequencies parametrized by e.g., the population size, mutation rates and selection coefficients. A large class of models use forward-in-time models, such as the discrete Wright-Fisher and Moran models and the continuous forward diffusion, to obtain distributions of population allele frequencies, conditional on an ancestral initial allele frequency distribution. Backward-in-time diffusion processes have been rarely used in the context of parameter inference. Here, we demonstrate how forward and backward diffusion processes can be combined to efficiently calculate the exact joint probability distribution of sample and population allele frequencies at all times in the past, for both discrete and continuous population genetics models. This procedure is analogous to the forward-backward algorithm of hidden Markov models. While the efficiency of discrete models is limited by the population size, for continuous models it suffices to expand the transition density in orthogonal polynomials of the order of the sample size to infer marginal likelihoods of population genetic parameters. Additionally, conditional allele trajectories and marginal likelihoods of samples from single populations or from multiple populations that split in the past can be obtained. The described approaches allow for efficient maximum likelihood inference of population genetic parameters in a wide variety of demographic scenarios.