基于模糊分类的多维定性数据的统计分析

在多维定性数据处理的问题中,不仅要研究变量之间的关系、样本之间的关系,还需研究样本与变量之间的关系,从而对潜在样本群子结构水平及其类型特征进行分析。本文采用对偶标度统计方法利用设计矩阵变换将原始多维定性数据进行类型特征分析,并在此基础上建立了基于模糊分类的直方图,表现出了潜在样本群的特征及其子结构之间的差异与联系。文章以分析儿童主体品质发展规律为例,说明此方法的有效性。     

直觉不确定纯语言信息的集结方法及在群决策中的应用

直觉不确定纯语言标度变量是直觉模糊数和不确定纯语言标度变量的拓展,本文定义了直觉不确定纯语言标度变量的运算法则,提出了一些基于直觉不确定纯语言评估标度及其运算法则的信息集结算子,在此基础上,给出了一种专家权重、属性权重及属性值均以语言标度形式给出的直觉不确定纯语言信息的集结方法,并将此方法应用到群决策中,通过实例分析说明了该方法的有效性和可行性。     

基于直觉纯语言集结算子的多属性群决策方法

定义了直觉纯语言集及其运算法则和直觉纯语言变量的数学期望和精确函数,提出了一些基于直觉纯语言评估标度及其运算法则的信息集结算子。在此基础上,给出了一种专家权重、属性权重及属性值均以语言标度形式给出的直觉纯语言信息集结方法,并将此方法应用到多属性群决策中,实例分析表明了该方法的有效性和可行性。     

时间标度上时滞脉冲复值神经网络的全局稳定性

研究了时间标度上具有时滞和脉冲影响的复值神经网络的全局稳定性问题.利用时间标度上的微积分理论,将连续时间型复值神经网络和离散时间型复值神经网络统一在同一个框架下进行研究.在不要求激励函数有界的条件下,运用同胚映射原理,建立了确保时滞复值神经网络平衡点存在性和唯一性的判定条件.通过构造合适的Lyapunov-Krasovskii泛函,并使用自由权矩阵方法和矩阵不等式技巧,获得了时间标度上具有时滞和脉冲影响的复值神经网络平衡点全局稳定性的充分条件.给出的判据是由复值线性矩阵表示的,易于MATLAB软件的YALMIP Toolbox实现.数值仿真实例验证了获得结果的有效性.     

四维Minkowski空间中类空超曲面的双曲高斯映射的奇点

介绍了四维Minkowski空间中类空超曲面的局部理论,定义了类空超曲面上的双曲高斯映射,双曲高度函数及距离平方函数,给出了一些定理的详细证明.介绍了一种证明高度函数是Morse族的新方法并应用Arnold等建立的Lagrange奇点理论对类空超曲面的双曲高斯映射的奇点进行了分类.     

基于标度理论的股指时间序列相似性分析

股指时间序列的相似性分析是当前金融学研究的热点之一。为了提高股指时间序列相似性分析的准确度,从标度不变性、多重分形及波动聚集性三个层面定义了标度理论的度量指标,并基于此对股指序列进行表示。将分割后的每一序列子区间看作时间点,则分割、表示后的不同股指序列构成一个多指标的面板数据。基于面板数据特征及指标相对重要性,提出了一种新型的多指标面板数据相似性度量函数——复合距离函数,用以度量股指时间序列的相似性。聚类结果表明,相较于其他两种方法,基于标度理论和复合距离函数的相似性度量方法能够显著提高相似性度量的准确度,同时具有较强的稳健性。     

一类新的多维标度法模型及其应用（英文）

本文指出了MDS（多维标度法）模型在实际应用中的缺陷，综合运用多目标决策技术、多元统计中的主成分分析方法给出了一类新的MDS的算法及应用步骤，将其运用于我国出口贸易发展策略的研究，为有关主管部门提供了有益的定量分析决策依据和政策建议．     

算子代数分类浅探

本文介绍半个多世纪以来算子代数主要是C~*-代数分类理论发展的概况;叙述了一系列当年的主要结果及相关的主要概念;也谈到了目前世界上最新的包括作者最近的关于单顺从C~*-代数的分类定理以及有关的应用,并列出了一些今后C~*-代数的发展课题.     

基于样本协方差矩阵的多维随机数生成方法

对于概率模型未知的多维数据样本容量扩充问题,根据主成分分析原理以及多维正态分布的性质,讨论并给出了与已知多维样本数据有相同协方差结构的模拟数据生成算法,并在此基础上给出了变量的离散化处理方法。实现了在小样本数据基础上不改变变量间协方差结构的样本容量扩充,为小样本条件下的数学建模、检验和分析提供样本数据支撑。     

层次分析方法进展

本文介绍层次分析方法的发展概况,并分析其主要优点与缺陷;据笔者查阅层次分析方法近十年来国内外400多篇论文,对层次分析方法的理论成果进行了分类,并分别作了相应介绍;提出了一些尚待解决的研究方向.     

Minimax optimal procedures for testing the structure of multidimensional functions

We present a novel method for detecting some structural characteristics of multidimensional functions. We consider the multidimensional Gaussian white noise model with an anisotropic estimand. Using the relation between the Sobol decomposition and the geometry of multidimensional wavelet basis we can build test statistics for any of the Sobol functional components. We assess the asymptotical minimax optimality of these test statistics and show that they are optimal in presence of anisotropy with respect to the newly determined minimax rates of separation. An appropriate combination of these test statistics allows to test some general structural characteristics such as the atomic dimension or the presence of some variables. Numerical experiments show the potential of our method for studying spatio-temporal processes.     

An improved tri-coloured rooted-tree theory and order conditions for ERKN methods for general multi-frequency oscillatory systems

This paper develops an improved tri-coloured rooted-tree theory for the order conditions for ERKN methods solving general multi-frequency and multidimensional second-order oscillatory systems. The bottleneck of the original tricoloured rooted-tree theory is the existence of numerous redundant trees. In light of the fact that the sum of the products of the symmetries and the elementary differentials is meaningful, this paper naturally introduces the so-called extended elementary differential mappings. Then, the new improved tri-coloured rooted tree theory is established based on a subset of the original tri-coloured rooted-tree set. This new theory makes all redundant trees disappear, and thus, the order conditions of ERKN methods for general multi-frequency and multidimensional second-order oscillatory systems are reduced greatly. Furthermore, with this new theory, we present some new ERKN methods of order up to four. Numerical experiments are implemented and the results show that ERKN methods can be competitive with other existing methods in the scientific literature, especially when comparatively large stepsizes are used.     

Microlocal analysis of edge flatness through directional multiscale representations

Edges and surface boundaries are often the most relevant features in images and multidimensional data. It is well known that multiscale methods including wavelets and their more sophisticated multidimensional siblings offer a powerful tool for the analysis and detection of such sets. Among such methods, the continuous shearlet transform has been especially successful. This method combines anisotropic scaling and directional sensitivity controlled by shear transformations in order to precisely identify not only the location of edges and boundary points but also edge orientation and corner points. In this paper, we show that this framework can be made even more flexible by controlling the scaling parameter of the anisotropic dilation matrix and considering non-parabolic scaling. We prove that, using ‘higher-than-parabolic’ scaling, the modified shearlet transform is also able to estimate the degree of local flatness of an edge or surface boundary, providing more detailed information about the geometry of edge and boundary points.     

Approximate Wavelets and the Approximation of Pseudodifferential Operators

This paper studiesapproximate multiresolution analysisfor spaces generated by smooth functions providing high-order semi-analytic cubature formulas for multidimensional integral operators of mathematical physics. Since these functions satisfy refinement equations with any prescribed accuracy, methods from wavelet theory can be applied. We obtain an approximate decomposition of the finest scale space into almost orthogonal wavelet spaces. For the example of the Gaussian function we study some properties of the analytic prewavelets and describe the projection operators onto the wavelet spaces. The multivariate wavelets retain the property of the scaling function to provide efficient analytic expressions for the action of important integral operators, which leads to sparse and semi-analytic representations of these operators.     

Method of “transition into space of derivatives”: 40 years of evolution

In 1975, the “method of transition into space of derivatives” was proposed. It is an efficiently verifiable frequency criterion for the existence of a nontrivial periodic solution in multidimensional models of automatic control systems with one differentiable nonlinear term. The method used the classical torus principle and refrained from any constructions in the phase space of the system under study. Moreover, the method allowed researchers to broaden the class of systems to which it could be applied. In this work, we give a survey of the results presenting generalization and expansion of the method. We also show the connection between the method of transition into space of derivatives, the well-known generalized Poincaré–Bendixson principle proposed by R. A. Smith, and the results of contemporary authors who are active in the theory of oscillations in multidimensional systems. In the recent years, the author obtained frequency criteria for the existence of orbitally stable cycles in multiinput multioutput (MIMO) control systems and methods for the construction of multidimensional systems having a unique equilibrium and an arbitrarily prescribed number of orbitally stable cycles, which are described in the paper. The extension of the generalized Poincaré–Bendixson principle to multidimensional systems with angular coordinate is presented. We show the application of described methods of investigation of oscillation processes in multidimensional dynamical systems to solving S. Smale’s problem in the chemical kinetics theory of biological cells and also to finding hidden attractors of the generalized Chua system and the minimal global attractor of a system with a polynomial nonlinear term. The publication is illustrated by numerous examples.     

Graphical Sensitivity Analysis for Multidimensional Scaling

This article introduces graphical sensitivity analysis for multidimensional scaling. This new technique is designed to combat two problems associated with multidimensional scaling analyses: The possibility of local minima and the uncertainty regarding sensitivity of the solution to changes in the parameters. Graphical sensitivity analysis is currently available in ViSta-MDS, a test bed for graphical model examination. By graphically manipulating points in the solution space, analysts may examine the sensitivity of the solution to changes in the model parameters. Furthermore, the analyst may search for alternative solutions that represent local minima. An example of graphical sensitivity analysis using ViSta-MDS is described.     

SPECTRAL APPROXIMATION ORDERS OF MULTIDIMENSIONAL NONSTATIONARY BIORTHOGONAL SEMI-MULTIRESOLUTION ANALYSIS IN SOBOLEV SPACE

Subdivision algorithm （Stationary or Non-stationary） is one of the most active and exciting research topics in wavelet analysis and applied mathematical theory. In multidimensional non-stationary situation, its limit functions are both compactly supported and infinitely differentiable. Also, these limit functions can serve as the scaling functions to generate the multidimensional non-stationary orthogonal or biorthogonal semi-multiresolution analysis （Semi-MRAs）. The spectral approximation property of multidimensional non-stationary biorthogonal Semi-MRAs is considered in this paper. Based on nonstationary subdivision scheme and its limit scaling functions, it is shown that the multidimensional nonstationary biorthogonal Semi-MRAs have spectral approximation order r in Sobolev space H^s（R^d）, for all r ≥ s ≥ 0.     

Dirichlet principle using computers

In this article we shall give practical and numerical solutions of the Laplace equation on multidimensional spaces and show the numerical experiments by using computers. Our method is based on the Dirichlet principle by combinations with generalized inverses, Tikhonov's regularization and the theory of reproducing kernels.     

Some aspects of holomorphic mappings: A survey

This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the reflection principle, the scaling method, and the Kobayashi–Royden metric. We sketch the proofs of certain principal results and discuss some recent achievements. Several open problems are also stated.