Practical estimation of high dimensional stochastic differential mixed-effects models

Stochastic differential equations (SDEs) are established tools for modeling physical phenomena whose dynamics are affected by random noise. By estimating parameters of an SDE, intrinsic randomness of a system around its drift can be identified and separated from the drift itself. When it is of interest to model dynamics within a given population, i.e. to model simultaneously the performance of several experiments or subjects, mixed-effects modelling allows for the distinction of between and within experiment variability. A framework for modeling dynamics within a population using SDEs is proposed, representing simultaneously several sources of variation: variability between experiments using a mixed-effects approach and stochasticity in the individual dynamics, using SDEs. These stochastic differential mixed-effects models have applications in e.g. pharmacokinetics/pharmacodynamics and biomedical modelling. A parameter estimation method is proposed and computational guidelines for an efficient implementation are given. Finally the method is evaluated using simulations from standard models like the two-dimensional Ornstein-Uhlenbeck (OU) and the square root models.     

Statistical control of control-affine nonlinear systems with nonquadratic cost functions: HJB and verification theorems

In statistical control, the cost function is viewed as a random variable and one optimizes the distribution of the cost function through the cost cumulants. We consider a statistical control problem for a control-affine nonlinear system with a nonquadratic cost function. Using the Dynkin formula, the Hamilton-Jacobi-Bellman equation for the nth cost moment case is derived as a necessary condition for optimality and corresponding sufficient conditions are also derived. Utilizing the nth moment results, the higher order cost cumulant Hamilton-Jacobi-Bellman equations are derived. In particular, we derive HJB equations for the second, third, and fourth cost cumulants. Even though moments and cumulants are similar mathematically, in control engineering higher order cumulant control shows a greater promise in contrast to cost moment control. We present the solution for a control-affine nonlinear system using the derived Hamilton-Jacobi-Bellman equation, which we solve numerically using a neural network method.     

Statistical analysis of water-quality data containing multiple detection limits: S-language software for regression on order statistics

Trace contaminants in water, including metals and organics, often are measured at sufficiently low concentrations to be reported only as values below the instrument detection limit. Interpretation of these “less thans” is complicated when multiple detection limits occur. Statistical methods for multiply censored, or multiple-detection limit, datasets have been developed for medical and industrial statistics, and can be employed to estimate summary statistics or model the distributions of trace-level environmental data.We describe S-language-based software tools that perform robust linear regression on order statistics (ROS). The ROS method has been evaluated as one of the most reliable procedures for developing summary statistics of multiply censored data. It is applicable to any dataset that has 0 to 80% of its values censored. These tools are a part of a software library, or add-on package, for the R environment for statistical computing. This library can be used to generate ROS models and associated summary statistics, plot modeled distributions, and predict exceedance probabilities of water-quality standards.     

Robust design with arbitrary distributions using Gauss-type quadrature formula

In this paper, we present the Gauss-type quadrature formula as a rigorous method for statistical moment estimation involving arbitrary input distributions and further extend its use to robust design optimization. The mathematical background of the Gauss-type quadrature formula is introduced and its relation with other methods such as design of experiments (DOE) and point estimate method (PEM) is discussed. Methods for constructing one dimensional Gauss-type quadrature formula are summarized and the insights are provided. To improve the efficiency of using it for robust design optimization, a semi-analytic design sensitivity analysis with respect to the statistical moments is proposed for two different multi-dimensional integration methods, the tensor product quadrature (TPQ) formula and the univariate dimension reduction (UDR) method. Through several examples, it is shown that the Gauss-type quadrature formula can be effectively used in robust design involving various non-normal distributions. The proposed design sensitivity analysis significantly reduces the number of function calls of robust optimization using the TPQ formulae, while using the UDR method, the savings of function calls are observed only in limited situations.     

Exact analysis of performance models by the Method of Moments

Performance evaluation of distributed systems and service-oriented architectures is often based on stochastic models, such as closed queueing networks which are commonly solved by the Mean Value Analysis (MVA) algorithm. However, the MVA is unable to solve models with hundreds or thousands of users accessing services of multiple classes, a configuration that is often useful to predict the performance of real-world applications. This paper introduces the Method of Moments (MoM), the first exact algorithm for solving closed queueing networks with large population sizes.Compared to the MVA algorithm, which is based on a recursive evaluation of mean queue-lengths, MoM defines a recursion on higher-order moments of queue-lengths that is solved at each step by a linear system of equations. This approach dramatically decreases the costs of an exact analysis compared to the MVA approach. We prove that MoM requires log-quadratic time and log-linear space in the total population size, whereas MVA complexity expressions grow combinatorially as the product of class populations. This extends the feasibility of exact methods to a much larger family of multiclass performance models than those that can be solved by the MVA algorithm.     

Stochastic sensitivity in metal forming of rigid-poroplastic materials

Stochastic sensitivity in metal forming process of rigid-poroplastic material is analyzed. The theoretical formulation for stochastic sensitivity is described which presents probabilistic distributions taking into account random initial and boundary conditions. The stochastic equations for metal forming process as well as for stochastic sensitivities are presented. Stochastic finite element equations for rigid-poroplastic materials are solved for the first two probabilistic moments. As the examples the sensitivity problems in the process of compression of rigid-poroplastic ring with random material parameters are discussed. The differences in deterministic and stochastic sensitivities are presented. The results derived can be used for the subsequent quantitative stochastic design as well as stochastic optimization.     

Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Most of SDEwMSs do not have explicit solutions so it is important to have numerical solutions. It is surprising that there are not any numerical methods established for SDEwMSs yet, although the numerical methods for stochastic differential equations (SDEs) have been well studied. The main aim of this paper is to develop a numerical scheme for SDEwMSs and estimate the error between the numerical and exact solutions. This is the first paper in this direction and the emphasis lies on the error analysis.     

A three-dimensional static torso model for the six human lumbar joints

A simple three-dimensional static torso model for the prediction of the force distributions at the six human lumbar levels in different activities was developed. There are two procedures involved in the model calculations, the intersegmental resultant joint forces and moments of the presumed 26 joints and the internal muscoloskeletal force distributions of the 6 intervertebral disc joints. In formulation of the joint force distribution problem, 6 muscle forces and 3 moment equilibrium equations at one of the six disc joints (e.g., L3/L4) with muscle stress (muscle force divided by physiological cross sectional area - PCSA) upper limits were used. The disc compressive force was minimized as the cost function of the linear programming to solve the force distribution problem. This solution of disc force distribution was further substituted into the next level. Such sequential substitutions of joint level's force equilibrium equations were used to solve for the disc forces at all 6 joint levels.     

Developing and distributing network based engineering solutions

The need to have easy access to the solutions of a variety of frequently occurring engineering problems, has resulted in the development of a wealth of tools ranging from engineering handbooks to sophisticated desktop engineering software. In this paper, we examine the use of internet/intranet-based methods for providing the tools that are commonly needed by engineers in their daily work. Using the World Wide Web as the distribution medium and the Internet browser as the execution environment, Sun Microsystem's Java technology provides the foundation for the development of an Engineer's Tool Box (ETB) that provides a framework whereby independent engineering software tools are linked, managed, and accessed globally via the Internet. Individual applications are developed to address specific engineering problems using simple, straightforward interfaces and are linked into the distributed ETB framework to solve more complex problems. The capabilities and limitations of the Java platform for developing and supporting such Internet based distributed engineering software tools are discussed herein.     

Multilevel design of experiments for statistical moment and probability calculation

A procedure for statistical moment estimation and reliability analysis using design of experiment (DOE) is proposed. A numerical method of finding the optimal levels and weights of DOE for statistical moment estimation is established and applied to three- and five-level cases. The four statistical moments of the system response function are then calculated from the full-factorial DOE, and the probability distribution of the system response function is obtained using the empirical distribution systems such as the Pearson system. The proposed method is tested through several examples and compared with other analysis methods, including the previous developments of a three-level full-factorial design. The results show that it relieves much of the difficulties met in the previous method and provides good accuracy compared to other methods for various input distributions.     

Estimating Components of Univariate Gaussian Mixtures Using Prony's Method

A new technique for estimating the component parameters of a mixture of univariate Gaussian distributions using the method of moments is presented. The method of moments basically involves equating the sample moments to the corresponding mixture moments expressed in terms of component parameters and solving these equations for the unknown parameters. These moment equations, however, are nonlinear in the unknown parameters, and heretofore, an analytic solution of these equations has been obtained only for two-component mixtures [2]. Numerical solutions also tend to be unreliable for more than two components, due to the large number of nonlinear equations and parameters to be solved for. In this correspondence, under the condition that the component distributions have equal variances or equal means, the nonlinear moment equations are transformed into a set of linear equations using Prony's method. The solution of these equations for the unknown parameters is analytically feasible and numerically reliable for mixtures with several components. Numerous examples using the proposed technique for two-, three-, and four-component mixtures are presented.     

Statistics of the morphological pattern-spectrum moments for a random-grain model

The morphological pattern spectrum (granulometric size distribution) results from iteratively opening an image and at each step recording the area of the opened image. Owing to the manner in which the size distribution is normalized, it defines a probability distribution function and possesses moments. If a binary image is considered as a random process, then the moments of the pattern spectrum are random variables. It is these random moments that are employed as shape and texture signatures in image classification and segmentation. Consequently, the statistics (moments) of these moments are important, and in the present paper these are studied for a grain model that has been used in various applications. A numerical procedure is developed to obtain approximate moment distributions, and both exact and asympotic methods are developed to express the mean and variance of the pattern-spectrum mean and variance. The general methods are applied to both normal- and gamma-distributed grain sizes.     

Towards Combining Probabilistic and Interval Uncertainty in Engineering Calculations: Algorithms for Computing Statistics under Interval Uncertainty, and Their Computational Complexity

In many engineering applications, we have to combine probabilistic and interval uncertainty. For example, in environmental analysis, we observe a pollution level x(t) in a lake at different moments of time t, and we would like to estimate standard statistical characteristics such as mean, variance, autocorrelation, correlation with other measurements. In environmental measurements, we often only measure the values with interval uncertainty. We must therefore modify the existing statistical algorithms to process such interval data.In this paper, we provide a survey of algorithms for computing various statistics under interval uncertainty and their computational complexity. The survey includes both known and new algorithms.     

Polymath软件在化学工程中非均相催化反应的模拟研究

Polymath软件作为一种科学与工程计算的数学软件包,用它所带的库函数可以高效、便捷地解决化学工程中许多数学问题,从而把人们从复杂的编程与调试中解脱出来.本文以化学工程中的气-固非均相催化反应为例,讨论Polymath在求解非线性微分方程组、插值计算以及图形绘制等方面的应用,结果表明:利用Polymath求解化学工程计算问题稳定性好、方便快捷,是化工设计、计算的好工具.     

变码率视频流的排队网络分析

作为多媒体服务的主要组成部分,视频数据传输的性能评估是多媒体传输系统性能评估的核心内容。文章首先介绍了泊松点过程驱动的随机微分方程的相关数学理论,然后以变码率(VBR)视频的排队网络为实际应用背景,使用这种数学工具建立系统的排队模型,并进行了针对队列一阶距的性能评价分析。通过深入的理论分析,文章给出了针对视频传输方案性能改进和性能评估的有益结论,并给出了一种可能的队列各阶距的估算方法。     

Solution of time-domain Maxwell equation with PML by using modified Laguerre polynomials

In this work, a stable numerical algorithm proposed by Chung et al. for the time-domain Maxwell equations is generalized. The time-domain Maxwell equations are solved by expressing the transient behaviors in terms of the modified Laguerre polynomials, and then the original equations of the initial value and boundary value can be transformed into a series of problems independent of the time variable. In this case the method of finite difference （FD）, the finite element method （FEM）, the method of moment （MoM）, etc. or the combination of these methods can be used to solve the problems. Finally, a numerical model is provided for the scattering problem with perfect matched layer （PML） by using FD. The comparison between the results of the proposed method and FDTD is presented to verify the proposed new method.     

Two alternative methods for solving the Vlasov Fokker–Planck equation

The expansion of moments technique for generating short time expansions for the moments with the distribution function is used to solve the reduced Vlasov Fokker–Planck equation. The obtained results are compared with those found by other theories such as the operator technique. The results obtained by expansion of moments confirm the correctness of those obtained by the operator method. The method is straightforward and concise, and its applications are promising and can be applied to other moment equations arising in physics.     

拖尾Rayleigh 分布: 基本性质及其应用

针对使用拖尾Rayleigh分布对合成孔径雷达(Synthetic aperture radar, SAR)幅值图像建模时遇到的问题, 本文讨论了拖尾Rayleigh分布的相关性质及其应用. 首先, 基于负数阶矩理论, 本文提出了拖尾Rayleigh分布的比值估计、对数矩估计和迭代对数矩估计三种参数估计方法, 并通过Monte Carlo仿真实验比较了它们的估计性能. 其次, 本文使用渐近级数计算拖尾Rayleigh分布的概率密度函数, 基于插值多项式拟合, 提出了高效计算密度函数的三步方法. 最后, 本文给出了SAR幅值图像基于拖尾Rayleigh分布的建模实例. 结果表明, 和一般的Rayleigh分布相比, 拖尾Rayleigh分布可以精确反映SAR幅值图像尖峰厚尾的统计特征, 因此它是SAR幅值图像建模的有效工具.     

Non-homothetic granulometric mixing theory with application to blood cell counting

A granulometry is a family of morphological openings by scaled structuring elements. As the scale increases, increasing image area is removed. Normalizing removed area by the total area yields the pattern spectrum of the image. The pattern spectrum is a probability distribution function and its moments are known as granulometric moments. Modeling the image as a random set, the pattern spectrum is a random function and its moments are random variables. The original granulometric mixing theory provides closed-form representation of the granulometric moments, shows that the distributions of the moments are asymptotically normal, and gives asymptotic expressions for the means and variances of the granulometric moments. The theory applies to random-set models formed as disjoint unions of randomly scaled image primitives (grains). The scaled grains are known as homothetics. The theory can be used in a method-of-moments fashion to estimate the parameters governing the random scaling factors and mixture proportions. Application is limited by the homothetic assumption. This paper drops the homothetic requirement and provides a mixing theory for a disjoint union of fully randomized primitives. Whereas the original asymptotic theory gives expressions for the moments themselves and is applied by taking expectations afterwards, the non-homothetic theory involves the granulometric size density, which is the mean of the original size distribution prior to normalization. Hence, the representation concerns expectations, not random variables. Nonetheless, a similar method-of-moments approach can be used to estimate mixture proportions. A large part of the paper is devoted to estimate blood cell proportions that correspond to cell-age categories. Each cell class is represented by a random grain, and the problem is to estimate the proportions of cells occurring in the various age categories. The random behavior of the cells in each category makes the non-homothetic theory appropriate. Because the estimation strategy leads to a system of nonlinear equations whose solution presents computational difficulties, the estimation is accomplished via a divide-and-conquer strategy in which the full mixture problem is partitioned into smaller problems, and the solutions of these problems are joined to solve the full problem.     

Numerical simulation of the motion of granular material using object-oriented techniques

The objective of this contribution is to present a numerical simulation method to model the motion of a packed bed on a moving grate or in a rotary kiln using object-oriented techniques. The packed bed can be described as granular material consisting of a large number of particles. The method chosen is the Lagrangian time-driven method and it uses the position, the orientation, the velocity and the angular velocity of particles as independent variables. These are obtained by time integration of the three-dimensional dynamics equations which were derived from the classical Newtonian mechanics approach based on the second law of Newton for the translation and rotation of each particle in the granular material. This includes keeping track of all forces and moments acting on each particle at every time-step. Particles are treated as contacting visco-elastic bodies which can overlap each other. Contact forces depend on the overlap geometry, material properties and dynamics of particles and include normal and tangential components of repulsion force with visco-elastic models for energy dissipation through internal and surface friction. The resulting equations of particle motion are solved by the Gear predictor–corrector scheme of fifth-order accuracy.The simulation method is based on object-oriented methodologies and programmed in the programming language C++. This approach supports objects which can be used for three-dimensional particles of various shapes and sizes and for walls as boundaries. The programming modules are implemented in the TOSCA (tools of object-oriented software for continuum mechanic applications) software package which allows for a high degree of flexibility and for shortening the duration of the software development process. As methods for particle motion may deal with particles of different sizes and materials, the approach allows to describe transport processes in technical applications.