The Stochastic Quasi-chemical Model for Bacterial Growth: Variational Bayesian Parameter Update

We develop Bayesian methodologies for constructing and estimating a stochastic quasi-chemical model (QCM) for bacterial growth. The deterministic QCM, described as a nonlinear system of ODEs, is treated as a dynamical system with random parameters, and a variational approach is used to approximate their probability distributions and explore the propagation of uncertainty through the model. The approach consists of approximating the parameters’ posterior distribution by a probability measure chosen from a parametric family, through minimization of their Kullback–Leibler divergence.     

一类非离散非连续的随机变量的概率分布

本文定义了一类非离散非连续的随机变量的概率分布,使其概率分布与离散型和连续型随机变量的概率分布表示保持一致,并举例求出考研中涉及过的该类型的随机变量的概率分布.     

High-Dimensional Mixed Graphical Models

While graphical models for continuous data (Gaussian graphical models) and discrete data (Ising models) have been extensively studied, there is little work on graphical models for datasets with both continuous and discrete variables (mixed data), which are common in many scientific applications. We propose a novel graphical model for mixed data, which is simple enough to be suitable for high-dimensional data, yet flexible enough to represent all possible graph structures. We develop a computationally efficient regression-based algorithm for fitting the model by focusing on the conditional log-likelihood of each variable given the rest. The parameters have a natural group structure, and sparsity in the fitted graph is attained by incorporating a group lasso penalty, approximated by a weighted lasso penalty for computational efficiency. We demonstrate the effectiveness of our method through an extensive simulation study and apply it to a music annotation dataset (CAL500), obtaining a sparse and interpretable graphical model relating the continuous features of the audio signal to binary variables such as genre, emotions, and usage associated with particular songs. While we focus on binary discrete variables for the main presentation, we also show that the proposed methodology can be easily extended to general discrete variables.     

A stochastic dynamic model of train-track-bridge coupled system based on probability density evolution method

An innovative stochastic dynamic model of a 3D train-track-bridge coupled system (TTBS) with refined wheel/rail interaction is established for a high-speed railway based on the random theory of probability density evolution method (PDEM). The multi-coupling effect of excitations can be simultaneously input into the new model, e.g. random track irregularity, random vehicle loads, stochastic system parameters, et al. Moreover, a new approach, named “Number theoretic method of multi-target probability functions” (NTM-mp), is developed to obtain the discrete point sets of multidimensional random parameters in hypercube space, aims to solve the point design of system uncertainty. The stochastic harmonic function (SHF) is applied to generate representative random track irregularity samples. The results of TTBS got by PDEM are verified with several typical case studies for its efficiency and reliability, which are the deterministic results in the representative publication, the Monte Carlo method (MCM) results, and the field testing results on the high-speed railway. At last, a typical case study of TTBS on a high-speed railway is presented for numerical analysis. Discussions and significant conclusions on the random dynamic responses are presented.     

Error analysis of a fully discrete projection method for magnetohydrodynamic system

In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi-implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H ( curl ) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.     

Properties of graphical regression models for multidimensional categorical data

We propose a two-component graphical chain model, the discrete regression distribution, where a set of discrete random variables is modeled as a response to a set of categorical and continuous covariates. The proposed model is useful for modeling a set of discrete variables measured at multiple sites along with a set of continuous and/or discrete covariates. The proposed model allows for joint examination of the dependence structure of the discrete response and observed covariates and also accommodates site-to-site variability. We develop the graphical model properties and theoretical justifications of this model. Our model has several advantages over the traditional logistic normal model used to analyze similar compositional data, including site-specific random effect terms and the incorporation of discrete and continuous covariates.     

A probabilistic theory of random maps

We present a probabilistic theory of random maps with discrete time and continuous state. The forward and backward Kolmogorov equations as well as the FPK equation governing the evolution of the probability density function of the system are derived. The moment equations of arbitrary order are derived, and the reliability and first passage time problem are also studied. Examples are presented to demonstrate the application of the theoretical development. Numerical solutions including the time histories of moment evolution, steady state probability density function, reliability and first passage time probability density function for time discrete random maps are included. The present work compliments the existing theory of continuous time stochastic processes.     

Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach

We present an approach for the transition from convex risk measures in a certain discrete time setting to their counterparts in continuous time. The aim of this paper is to show that a large class of convex risk measures in continuous time can be obtained as limits of discrete time-consistent convex risk measures. The discrete time risk measures are constructed from properly rescaled (‘tilted’) one-period convex risk measures, using a d-dimensional random walk converging to a Brownian motion. Under suitable conditions (covering many standard one-period risk measures) we obtain convergence of the discrete risk measures to the solution of a BSDE, defining a convex risk measure in continuous time, whose driver can then be viewed as the continuous time analogue of the discrete ‘driver’ characterizing the one-period risk. We derive the limiting drivers for the semi-deviation risk measure, Value at Risk, Average Value at Risk, and the Gini risk measure in closed form.     

On computing distributions of products of non-negative independent random variables

We introduce a new functional representation of probability density functions (PDFs) of non-negative random variables via a product of a monomial factor and linear combinations of decaying exponentials with complex exponents. This approximate representation of PDFs is obtained for any finite, user-selected accuracy. Using a fast algorithm involving Hankel matrices, we develop a general numerical method for computing the PDF of the sums, products, or quotients of any number of non-negative independent random variables yielding the result in the same type of functional representation. We present several examples to demonstrate the accuracy of the approach.     

A methodology for stochastic inventory models based on a zero‐adjusted Birnbaum‐Saunders distribution

The Birnbaum–Saunders (BS) distribution is receiving considerable attention. We propose a methodology for inventory logistics that allows demand data with zeros to be modeled by means of a new discrete–continuous mixture distribution, which is constructed by using a probability mass at zero and a continuous component related to the BS distribution. We obtain some properties of the new mixture distribution and conduct a simulation study to evaluate the performance of the estimators of its parameters. The methodology for stochastic inventory models considers also financial indicators. We illustrate the proposed methodology with two real‐world demand data sets. It shows its potential, highlighting the convenience of using it by improving the contribution margins of a Chilean food industry. Copyright © 2015 John Wiley & Sons, Ltd.     

A practical non-parametric copula algorithm for system reliability with correlations

System reliability analysis involving correlated random variables is challenging because the failure probability cannot be uniquely determined under the given probability information. This paper proposes a system reliability evaluation method based on non-parametric copulas. The approximated joint probability distribution satisfying the constraints specified by correlations has the maximal relative entropy with respect to the joint probability distribution of independent random variables. Thus the reliability evaluation is unbiased from the perspective of information theory. The estimation of the non-parametric copula parameters from Pearson linear correlation, Spearman rank correlation, and Kendall rank correlation are provided, respectively. The approximated maximum entropy distribution is then integrated with the first and second order system reliability method. Four examples are adopted to illustrate the accuracy and efficiency of the proposed method. It is found that traditional system reliability method encodes excessive dependence information for correlated random variables and the estimated failure probability can be significantly biased.     

离散型与连续型分布互为对偶的根源

互为对偶的离散型分布与连续型分布,可以看作是由同一个函数——源函数产生的。源函数的正线性组合、乘积和负导数,仍然是源函数。源函数揭示了互为对偶的分布的分布函数之间的相互关系,并能用来求随机变量的数字特征、特征函数、概率母函数、分布的最大值和参数的极大似然估计.     

Stability analysis for a fully discrete spectral scheme for Boussinesq systems

In this paper we perform a stability analysis of a fully discrete numerical method for the solution of a family of Boussinesq systems, consisting of a Fourier collocation spectral method for the spatial discretization and a explicit fourth order Runge–Kutta (RK4) scheme for time integration. Our goal is to determine the influence of the parameters, associated to this family of systems, on the efficiency and accuracy of the numerical method. This analysis allows us to identify which regions in the parameter space are most appropriate for obtaining an efficient and accurate numerical solution. We show several numerical examples in order to validate the accuracy, stability and applicability of our MATLAB implementation of the numerical method.     

The Bergström–Grigelionis asymptotic expansions

We consider a triangular array of independent identically distributed discrete random variables. We assume that the probability distribution of sums satisfies the necessary and sufficient conditions for the weak convergence to the compound Poisson distribution. The first known result (the case where random variables take only integer values) is due to B. Grigelionis, who estimated the convergence rate to the compound Poisson distribution. We extend the summation of random variables by including the variables taking discrete values and by using the Grigelionis ideas to obtain “lengthy” asymptotic expansions. These expansions are based on the well-known Bergstrom identity [H. Bergstrom, On asymptotic expansions of probability functions, Scand. Actuarial J., 34(1):1–33, 1951].     

Statistical maps and generalized random walks

We continue our study of statistical maps (equivalently, fuzzy random variables in the sense of Gudder and Bugajski). In the realm of fuzzy probability theory, statistical maps describe the transportation of probability measures on one measurable space into probability measures on another measurable space. We show that for discrete probability spaces each statistical map can be represented via a special matrix the rows of which are probability functions related to conditional probabilities and the columns are related to fuzzy n-partitions of the domain. Discrete statistical maps sending a probability measure p to a probability measure q can be represented via conditional distributions and correspond to joint probabilities on the product. The composition of statistical maps provide a tool to describe and to study generalized random walks and Markov chains.     

Chernoff bounds for discriminating between two markov processes

We study a statistical hypothesis testing problem, where a sample function of a Markov process with one of two sets of known parameters is observed over a finite time interval. When a log likelihood ratio test is used to discriminate between the two sets of parameters, we give bounds on the probability of choosing an incorrect hypothesis, and on the total probability of error, for both discrete and continuous time parameter, and discrete and continuous state space. The asymptotic behavior of the bounds is examined as the observation interval becomes infinite.     

Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. The recent surge of transport maps offers a mathematical foundation and new insights for tackling this challenge by coupling intractable random variables with tractable reference random variables. This paper generalises the functional tensor-train approximation of the inverse Rosenblatt transport recently developed by Dolgov et al. (Stat Comput 30:603–625, 2020) to a wide class of high-dimensional non-negative functions, such as unnormalised probability density functions. First, we extend the inverse Rosenblatt transform to enable the transport to general reference measures other than the uniform measure. We develop an efficient procedure to compute this transport from a squared tensor-train decomposition which preserves the monotonicity. More crucially, we integrate the proposed order-preserving functional tensor-train transport into a nested variable transformation framework inspired by the layered structure of deep neural networks. The resulting deep inverse Rosenblatt transport significantly expands the capability of tensor approximations and transport maps to random variables with complicated nonlinear interactions and concentrated density functions. We demonstrate the efficiency of the proposed approach on a range of applications in statistical learning and uncertainty quantification, including parameter estimation for dynamical systems and inverse problems constrained by partial differential equations.

     

An approximation of known accuracy for single server queues with inhomogeneous arrival rate and continuous service time distribution

A practical method for obtaining approximate results for single server queues with inhomogeneous queues and continuous service time distribution is presented. The method is based on a discrete approximation to the continuous service time distribution. Exact results can be obtained for the corresponding queueing system with discrete service time distribution. These results are then corrected, and the likely accuracy of the corrected results is estimated. Four measures of performance are considered, idleness probability, mean and variance of number of customers in the system and virtual waiting time.     

The maximum earthquake in future T years: Checking by a real catalog

The studies of disaster statistics are being largely carried out in recent decades. Some recent achievements in the field can be found in Pisarenko and Rodkin (2010). An important aspect in the seismic risk assessment is the using historical earthquake catalogs and the combining historical data with instrumental ones since historical catalogs cover very long time periods and can improve seismic statistics in the higher magnitude domain considerably. We suggest the new statistical technique for this purpose and apply it to two historical Japan catalogs and the instrumental JMA catalog. The main focus of these approaches is on the occurrence of disasters of extreme sizes as the most important ones from practical point of view. Our method of statistical analysis of the size distribution in the uppermost range of extremely rare events was suggested, based on maximum size M_max(τ) (e.g. earthquake energy, ground acceleration caused by earthquake, victims and economic losses from natural catastrophes, etc.) that will occur in a prescribed time interval τ. A new approach to the problem discrete data that we called “the magnitude spreading” is suggested. This method reduces discrete random value to continuous ones by addition a small uniformly distributed random components. We analyze this method in details and apply it to verification of parameters derived from two historical catalogs: the Usami earthquake catalog (599–1884) and the Utsu catalog (1885–1925). We compare their parameters with ones derived from the instrumental JMA catalog (1926–2014). The results of this verification are following: The Usami catalog is incompatible with the instrumental one, whereas parameters estimated from the Utsu catalog are statistically compatible in the higher magnitude domain with sample of M_max(τ) derived from the JMA catalog.