On the dynamical properties of an elliptical–oval billiard with static boundary

Some dynamical properties for a classical particle confined inside a closed region with an elliptical–oval-like shape are studied. The dynamics of the model is made by using a two-dimensional nonlinear mapping. The phase space of the system is of mixed kind and we have found the condition that breaks the invariant spanning curves in the phase space. We have discussed also some statistical properties of the phase space and obtained the behaviour of the positive Lyapunov exponent.     

P-algorithm based on a simplicial statistical model of multimodal functions

A well-recognized one-dimensional global optimization method is generalized to the multidimensional case. The generalization is based on a multidimensional statistical model of multimodal functions constructed by generalizing computationally favorable properties of a popular one-dimensional model—the Wiener process. A simplicial partition of a feasible region is essential for the construction of the model. The basic idea of the proposed method is to search where improvements of the objective function are most probable; a probability of improvement is evaluated with respect to the statistical model. Some results of computational experiments are presented.     

Optimizing Markovian modeling of chaotic systems with recurrent neural networks

In this paper, we propose a methodology for optimizing the modeling of an one-dimensional chaotic time series with a Markov Chain. The model is extracted from a recurrent neural network trained for the attractor reconstructed from the data set. Each state of the obtained Markov Chain is a region of the reconstructed state space where the dynamics is approximated by a specific piecewise linear map, obtained from the network. The Markov Chain represents the dynamics of the time series in its statistical essence. An application to a time series resulted from Lorenz system is included.     

Invariant integral and statistical equations of paraxial light beam transmission in free space

The statistical behavior of arbitrary paraxial light beams propagating in free space is investigated by using the Hermite-Gaussian expansion method and Fock’s representation. A series of equivalent Gaussian parameters for paraxial beam and the statistical equations for these parameters are presented. The optical transmission problem in quasi-far field region is studied. The so-called general Hermite-Gaussian beam is defined. Project partly supported by the National Hi-tech Inertial Confinement Fusion Committee, the Natural Science Foundation of Guangdong Province, the Postdoctoral Foundation of China and Guangdong Province.     

Tolerance intervals in risk theory

Up to now, the main aim of credibility theory has been to provide statistical models which allow for estimating (net) risk premiums appropriately. In the present note, a simple credibility model based on the percentile principle is introduced. It turns out that there are close connections between the resulting credibility premiums and statistical tolerance limits.     

A modified power series distribution

Summary A modification of the usual power series distribution is proposed and a statistical treatment of data generated by this probability model is indicated. It is shown that this modified power series distribution belongs to a multiparameter discrete exponential family and standard theory is then applied to obtain complete sufficient statistics of the parameters, sampling distribution of these statistics, maximum likelihood and minimum variance unbiased estimators, uniformly most powerful unbiased tests, uniformly most accurate confidence intervals and expected cover tolerance regions. CSIRO, Division of Mathematical Statistics     

Structural Balance: A Dynamic Perspective

The paper presents results derived from a series of simulations of signed networks (i.e., networks containing positive and negative links) subject to a balancing process. The dynamics are followed in a space defined by variations in the proportion of positive links introduced and the tolerance to imbalance. The authors claim to have discovered boundaries in the space separating it into three distinct regions. One region probably displays the characteristics of self-organized criticality, which may have implications for the theory of extreme events.     

解线性约束优化问题的新锥模型信赖域法

本文提出了一个解线性等式约束优化问题的新锥模型信赖域方法.论文采用零空间技术消除了新锥模型子问题中的线性等式约束,用折线法求解转换后的子问题,并给出了解线性等式约束优化问题的信赖域方法.论文提出并证明了该方法的全局收敛性,并给出了该方法解线性等式约束优化问题的数值实验.理论和数值实验结果表明新锥模型信赖域方法是有效的,这给出了用新锥模型进一步研究非线性优化的基础.     

多元正态分布百分位面和容忍限的估计

针对实践中多个指标的联合分布是多元正态分布或近似多元正态分布的情况,将一元正态分布的置信限及其容忍限的估计推广到多元正态分布的场合,提出了百分位面的置信限和分布的容忍限和及容忍域的估计方法,在一定程度上克服了以往点估计的局限性,该方法具有更大的应用价值.     

β-Expectation tolerance regions for a generalized multivariate model with normal error variables

In this paper a procedure of construction of β-expectation tolerance regions in the framework of the structural method of inference has been developed. The procedure has been applied to the generalized multivariate model and the β-expectation tolerance region for this case has been constructed assuming the normal distribution for the error variables of the model.     

非线性半参数统计模型估计函数的几何方法

将非线性半参数统计模型的概率密度函数族视为统计流形,利用微分几何方法,建立非线性半参数统计模型相对应的Hilbert空间,进而研究非线性半参数统计模型的估计函数问题.利用两类得分函数张成的子空间对Hilbert空间进行正交分解,进而讨论估计函数所在的集合,以及如何选取最优估计函数的问题.最后,通过实例分析来验证此方法的有效性.     

On modeling left-truncated loss data using mixtures of distributions

A new statistical methodology is developed for fitting left-truncated loss data by using the G-component finite mixture model with any combination of Gamma, Lognormal, and Weibull distributions. The EM algorithm, along with the emEM initialization strategy, is employed for model fitting. We propose a new grid map which considers the model selection criterion (AIC or BIC) and risk measures at the same time, by using the entire space of models under consideration. A simulation study validates our proposed approach. The application of the proposed methodology and use of new grid maps are illustrated through analyzing a real data set that includes left-truncated insurance losses.     

Spatial functional normal mixed effect approach for curve classification

This paper proposes a spatial functional formulation of the normal mixed effect model for the statistical classification of spatially dependent Gaussian curves, in both parametric and state space model frameworks. Fixed effect parameters are represented in terms of a functional multiple regression model whose regression operators can change in space. Local spatial homogeneity of these operators is measured in terms of their Hilbert–Schmidt distances, leading to the classification of fixed effect curves in different groups. Assuming that the Gaussian random effect curves obey a spatial autoregressive dynamics of order one [SARH(1) dynamics], a second functional classification criterion is proposed in order to detect local spatially homogeneous patterns in the mean quadratic functional variation of Gaussian random effect curve increments. Finally, the two criteria are combined to detect local spatially homogeneous patterns in the regression operators and in the functional mean quadratic variation, under a state space approach. A real data example in the financial context is analyzed as an illustration.     

Step length problem for trimming curve approximation in tessellating trimmed surfaces

A trimmed parametric surface is mainly composed of a surface together with trimming curves lying in D, the parametric space of the surface. By investigating the interrelation between surface tessellation and trimming curve approximation, we point out some problems on trimming curve approximation in existing trimmed surface tessellation algorithms. Counter examples are presented to show that a valid approximation of trimming curves in D together with the refinement imposed by surface tessellation does not necessarily generate a valid linear approximation in 3D space. To assure the 3D derivation tolerance, we propose two novel step-length estimation methods such that a piecewise linear interpolant of the trimming curve based on the proposed step lengths will result in a valid linear approximation in 3D space. The first method exploits the triangle inequality and takes the derivation tolerance in 3D space into account to compute the effective step length. Our second method is based on segmenting the trimming curve into subcurves first and then approximates each subcurve according to the derivation tolerance in 3D space. Moreover, several empirical tests are given to demonstrate the correctness of our step length estimations.     

Order statistics and region-based evolutionary computation

Trust region algorithms are well known in the field of local continuous optimization. They proceed by maintaining a confidence region in which a simple, most often quadratic, model is substituted to the criterion to be minimized. The minimum of the model in the trust region becomes the next starting point of the algorithm and, depending on the amount of progress made during this step, the confidence region is expanded, contracted or kept unchanged. In the field of global optimization, interval programming may be thought as a kind of confidence region approach, with a binary confidence level: the region is guaranteed to contain the optimum or guaranteed to not contain it. A probabilistic version, known as branch and probability bound, is based on an approximate probability that a region of the search space contains the optimum, and has a confidence level in the interval [0,1]. The method introduced in this paper is an application of the trust region approach within the framework of evolutionary algorithms. Regions of the search space are endowed with a prospectiveness criterion obtained from random sampling possibly coupled with a local continuous algorithm. The regions are considered as individuals for an evolutionary algorithm with mutation and crossover operators based on a transformation group. The performance of the algorithm on some standard benchmark functions is presented.     

Marangoni-driven liquid films rising out of a meniscus onto a nearly-horizontal substrate

We revisit the situation of a thin liquid film driven up an inclined substrate by a thermally induced Marangoni shear stress against the opposing parallel component of gravity. In contrast to previous studies, we focus here on the meniscus region, in a case where the substrate is nearly horizontal. Our numerical simulations show that the time-dependent lubrication model for the film profile can reach a steady state in the meniscus region that is unlike the monotonic solutions investigated earlier. A systematic investigation of the steady states of the lubrication model is carried out by studying the phase space of the corresponding third-order ODE system. We find a rich structure of the phase space including multiple non-monotonic solutions with the same far-field film thickness. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parameter Estimation of Multivariate Regression Model with Fuzzy Random Errors

In many real-world problems, observations are usually described by approximate values due to fuzzy uncertainty, unlikeprobabilistic uncertainty that has nothing to do with experimentation. The combination of statistical model and fuzzy set theory is helpful to improve the identification and analysis of complex systems. As an extension ofstatistical techniques, this study is an investigation of the relationship between fuzzy multiple explanatory variables and fuzzy response with numeric coefficients and the fuzzy random error term. In this work we describe a parameter estimation procedure carrying out the least-squares method in a complete metric space of fuzzy numbers to determine the coefficients based on the extension principle. We demonstrate how the fuzzy least squares estimators present large sample statistical properties, including asymptotic normality, strong consistency and confidence region. The estimators are also examined via asymptotic relative efficiency concerning traditional least squares estimators. Different from the construction of error term in Kim et al.\cite{21}, it is more reasonable in the proposed model since the problems of inconsistency in referring to fuzzy variable and producing the negative spreads may be avoided. The experimental study verifies that the proposed fuzzy least squares estimators achieve the meaning consistent with the theory identification for large sample data set and better generalization regarding one single variable model.     

Reducing quadratic programming problem to regression problem: Stepwise algorithm

Quadratic programming is concerned with minimizing a convex quadratic function subject to linear inequality constraints. The variables are assumed to be nonnegative. The unique solution of quadratic programming (QP) problem (QPP) exists provided that a feasible region is non-empty (the QP has a feasible space).A method for searching for the solution to a QP is provided on the basis of statistical theory. It is shown that QPP can be reduced to an appropriately formulated least squares (LS) problem (LSP) with equality constraints and nonnegative variables. This approach allows us to obtain a simple algorithm to solve QPP. The applicability of the suggested method is illustrated with numerical examples.     

Minimum classification error training in example based speech and pattern recognition using sparse weight matrices

The Minimum Classification Error (MCE) criterion is a well-known criterion in pattern classification systems. The aim of MCE training is to minimize the resulting classification error when trying to classify a new data set. Usually, these classification systems use some form of statistical model to describe the data. These systems usually do not work very well when this underlying model is incorrect. Speech recognition systems traditionally use Hidden Markov Models (HMM) with Gaussian (or Gaussian mixture) probability density functions as their basic model. It is well known that these models make some assumptions that are not correct. In example based approaches, these statistical models are absent and are replaced by the pure data. The absence of statistical models has created the need for parameters to model the data space accurately. For this work, we use the MCE criterion to create a system that is able to work together with this example based approach. Moreover, we extend the locally scaled distance measure with sparse, block diagonal weight matrices resulting in a better model for the data space and avoiding the computational load caused by using full matrices. We illustrate the approach with some example experiments on databases from pattern recognition and with speech recognition.