Non-linear statistical technique applied to data from baboon articular cartilage

Pattern recognition software was developed and applied together with statistical techniques to articular cartilage data from the knee joint of the baboon. The standard statistical method used for comparison was ANOVA which indicates linear discrimination. In addition a Karhunen-Loève expansion was performed to reduce the dimensionality of the data and provide independent uncorrelated variables. Nearest neighbour analysis, a non-linear method, when combined with bionomial probabilities gave discrimination that was not obtained by ANOVA. Use of pattern recognition and related techniques can improve and extend the analysis of biological data to include non-linear discrimination and classification.     

On thea priori model reduction: Overview and recent developments

Summary Karhunen-Loève expansion and snapshot POD are based on principal component analysis of series of data. They provide basis vectors of the subspace spanned by the data. All the data must be taken into account to find the basis vectors. These methods are not convenient for any improvement of the basis vectors when new data are added into the data base. We consider the data as a state evolution and we propose an incremental algorithm to build basis functions for the decomposition of this state evolution. The proposed algorithm is based on the APHR method (A Priori Hyper-Reduction method). This is an adaptive strategy to build reduced order model when the state evolution is implicitely defined by non-linear governing equations. In case of known state evolutions the APHR method is an incremental Karhunen-Loève decomposition. This approach is very convenient to expand the subspace spanned by the basis functions. In the first part of the present paper the main concepts related to the “a priori” model reduction technique are revisited, as a previous task to its application in the cases considered in the next sections. Some engineering problems are defined in domains that evolve in time. When this evolution is large the present and the reference configurations differ significantly. Thus, when the problem is formulated in the total Lagrangian framework frequent remeshing is required to avoid too large distortions of the finite element mesh. Other possibility for describing these models lies in the use of an updated formulation in which the mesh is conformed to each intermediate configuration. When the finite element method is used, then frequent remeshing must be carried out to perform an optimal meshing at each intermediate configuration. However, when the natural element method, a novel meshless technique, is considered, whose accuracy does not depend significantly on the relative position of the nodes, then large simulations can be performed without any remeshing stage, being the nodal position at each intermediate configuration defined by the transport of the nodes by the material velocity or the advection terms. Thus, we analyze the extension of the “a priori” model reduc tion, based on the use in tandem of the Karhunen-Loève decomposition (that extracts significant information) and an approximation basis enrichment based on the use of the Krylov's subspaces, previously proposed in the framework of fixed mesh simulation, to problems defined in domains evolving in time. Finally, for illustrating the technique capabilities, the “a priori” model reduction will be applied for solving the kinetic theory model which governs the orientation of the fibers immersed in a Newtonian flow.     

Pattern recognition by polynomial canonical regression

A pattern is conceived as an equivalence class of samples in a criterion space. Two cases are distinguished: (1) The number of classes is very small in comparison to the number of samples; (2) it approximates the number of samples.Prediction of class membership by means of predictor variables in the first case is replaced by identification of a small vicinity of a sample in the second one.Linear and approximations of nonlinear canonical regression functions are proposed for both issues. Preprocessing of data by Karhunen-Loève expansion is discussed.Empirical results concerning classification of schematic faces and identification of photographic portraits are presented.     

分布参数系统的时空ARX建模及预测控制

本文针对一类可由抛物型偏微分方程描述的分布参数系统,研究了一种基于输入输出数据的建模与控制方法.首先利用Karhunen-Loève（K-L）分解提取系统的一组主导空间基函数,并以此对系统输出进行时空分解,随后由时空分解得到的时间系数部分以及系统激励构成输入输出信息,利用最小二乘法辨识出时域ARX模型,最后针对该模型设计了广义预测控制器.仿真结果表明,上述控制方法能够对分布参数系统取得良好的控制效果.     

A new entropic method of dimensionality reduction specially designed for multiclass discrimination (DIRECLADIS)

This is a straightforward generalization of the Karhunen-Loève method making it possible to extract those variables which specifically serve the purpose of discriminating classes.     

Stochastic meshfree method for elasto-plastic damage analysis

This paper presents a stochastic meshfree method for solving boundary-value problems in damage mechanics under elasto-plastic conditions. Isotropic ductile damage evolution law is used to model the coupled elasto-plastic damage growth. Uncertainty associated with initial damage in materials is considered as random field. Moving least squares shape function method and Karhunen Loève expansion method are used for random field discretization. Statistical parameters of the response quantities are computed using perturbation method. The proposed method involves a new stochastic stress update procedure to solve the nonlinear equations in terms of discretized random variables arising from perturbation of equilibrium equations system. Numerical examples comprising of one and two dimensional problems are presented to illustrate the effectiveness of proposed method.     

Numerical Studies of Three-dimensional Stochastic Darcy’s Equation and Stochastic Advection-Diffusion-Dispersion Equation

Solute transport in randomly heterogeneous porous media is commonly described by stochastic flow and advection-dispersion equations with a random hydraulic conductivity field. The statistical distribution of conductivity of engineered and naturally occurring porous material can vary, depending on its origin. We describe solutions of a three-dimensional stochastic advection-dispersion equation using a probabilistic collocation method (PCM) on sparse grids for several distributions of hydraulic conductivity. Three random distributions of log hydraulic conductivity are considered: uniform, Gaussian, and truncated Gaussian (beta). Log hydraulic conductivity is represented by a Karhunen-Loève (K-L) decomposition as a second-order random process with an exponential covariance function. The convergence of PCM has been demonstrated. It appears that the accuracy in both the mean and the standard deviation of PCM solutions can be improved by using the Jacobi-chaos representing the truncated Gaussian distribution rather than the Hermite-chaos for the Gaussian distribution. The effect of type of distribution and parameters such as the variance and correlation length of log hydraulic conductivity and dispersion coefficient on leading moments of the advection velocity and solute concentration was investigated.     

高耸柔性结构振动模态阻尼比参数识别仿真

针对当前结构振动模态阻尼比参数识别相关方法存在识别率低等问题,结合柔性结构振动形态复杂度高等特点,提出基于ERA的高耸柔性结构振动模态阻尼比参数识别方法。通过标定相机的内外参数,使用两个相机同时对柔性结构的动态进行拍摄生成结构测点图像,采用三维动态重构方法对高耸柔性结构振动情况进行预处理,利用Harris角点检测法确定振动位移,并采集位移数据,得到模态阻尼比参数集合。基于数据采集结果,通过ERA方法对高耸柔性结构振动模态阻尼比参数进行识别。实验结果表明,上述方法具有较强的识别性能、柔性结构振动形态简单的特点。     

随机子空间方法在齿轮系统故障诊断中的应用

在齿轮系统故障诊断中.采用何种有效的方法对随机动态信号进行分析和特征提取是关键所在;相对于普通的分析方法而言结构模态参数辨识领域先进的时域识别方法随机子空间能更准确地识别出环境激励下结构系统的模态参数;将这一方法引入到运转的齿轮系统中进行了动态特性识别,对齿轮系统发生的故障进行特征提取、区分与诊断,并与正常齿轮系统对比,实验结果表明,该方法可以有效地对齿轮传动系统故障进行识别与诊断。     

Hybrid linear matrix factorization for topic-coherent terms clustering

Topic-coherent term clustering is the foundation of document organization, corpus summarization and document classification. It is especially useful in solving the emerging problem of big data. However, a term clustering method that can cope with high-dimension data with variable length and topics and meanwhile achieve high topic coherence is an ongoing request. It is a challenging problem in research. This paper proposes a hybrid linear matrix factorization method to identify the topic-coherent terms from documents to form a thesaurus for clustering. Starting from an analog Karhunen–Loève transformation from PCA scores fully into FA's factor coefficients space (loadings), the high-dimension of the full set of PCA scores is reduced and topic-coherent terms are classified by the main factors of FA which could be topics. Karhunen–Loève transformation reduces the total mean square error to increase topic coherence. The optimization of the initial transformation is carried out further in a manner of Karhunen–Loève expansion based on stochastic Wiener process. The optimal topic coherent bags of terms are found to build a more topic-coherent model. This approach is experimented on the CISI, MedSH and Tweets dataset in different sizes and number of topics. It achieves outstanding results better than the methods in comparison.     

An efficient coupling of FORM and Karhunen–Loève series expansion

The topic of this paper is the solution of reliability problems where failure is influenced by the spatial random fluctuations of loads and material properties. Homogeneous random fields are used to model this kind of uncertainty. The first step of the investigation is the random field discretization, which transforms a random field into a finite set of random variables. The second step is the reliability analysis, which is performed using the FORM in this paper. A parametric analysis of the reliability index is usually performed with respect to the random field discretization accuracy. This approach requires several independent reliability analyses. A new and efficient approach is proposed in this paper. The Karhunen–Loève series expansion is combined with the FEM for the discretization of the random fields. An efficient solution of the reliability problem is proposed to predict the reliability index as the discretization accuracy increases.     

一种用于汽车发动机故障诊断的贝叶斯网络模型

在汽车发动机故障诊断领域,由于设备内部的复杂性和导致故障的不确定因素,使得解决不确定性问题成为目前发动机故障诊断的首要问题;文章提出了一种用于解决不确定性问题的贝叶斯网络模型,该模型的网络结构学习采用了基于簇的搜索算法;为了获得更高准确率的故障诊断结果,模型加入了对当前信息集的采用,进行结构和参数的在线学习,改进了网络结构,网络通过概率传播算法,推理出产生故障的原因节点;在实例中表明,该模型能准确有效地解决发动机故障诊断中存在的不确定性问题,并与专家系统故障诊断模型做出比较,验证了基于该算法的贝叶斯网络模型在信息不确定性条件下能够提高诊断的准确率。     

Application of hierarchical matrices for computing the Karhunen–Loève expansion

Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen–Loève expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented.     

Response variability of laminate composite plates due to spatially random material parameter

Young’s modulus is at the center of attention in the stochastic finite element analysis since the parameter plays an important role in determining structural behavior. However, the other material parameter of Poisson’s ratio is another independent material parameter that governs the behavior of structural systems. Accordingly, the independent estimation of the influence of this parameter on the uncertain response of a system is of importance from the perspective of stochastic analysis. To this end, we propose a formulation to determine the response variability in laminated composite plates due to the spatial randomness of Poisson’s ratio. To filter out the independent contribution of random Poisson’s ratio, a decomposition of the constitutive matrix into several sub-matrices by using the Taylor’s expansion is needed, which makes the random Poisson’s ratio simple enough to be included in the formulation. To validate the adequacy of the proposed formulation, several examples are chosen and the results are compared with those given by Monte Carlo analysis. By means of the formulation suggested here, it is expected that an extension of the formulation to include the effect of correlations between random Poisson’s ratio and other structural and/or geometrical parameters will be achieved with ease, resulting in a more practical estimation of the response variability of laminated composite plates.     

Treatment of uncertain material interfaces in compressible flows

To treat uncertain interface position is an important issue for complex applications. In this paper, we address the characterization of randomly perturbed interfaces between fluids thanks to stochastic modeling and uncertainty quantification through the 2D Euler system. The perturbed interface is modeled as a random field and represented by a Karhunen–Loève expansion. The stochastic 2D Euler system is solved applying Polynomial Chaos theory through the Intrusive Polynomial Moment Method (IPMM). This stochastic resolution method is fully explained and studied (theoretically and numerically). Stochastic Richtmyer–Meshkov unstable flows are solved and presented for several configurations of the uncertain interface (different rugosities) between the fluids. The probability density functions of the mass density of the fluid in the vicinity of the interface are computed built and compared for the different simulations: the system exhibits strong sensitivity with respect to the stochastic initially leading modes.     

基于虚拟仪器试验系统的电子机柜模态试验

根据电子机柜模态试验的特点,在LabVIEW平台上开发出一套集数据采集、分析和参数识别为一体的虚拟仪器测试系统(VITS).通过与传统的测试仪器的测试结果比较表明,该测试系统能够实现对模态参数的计算和分析,计算结果准确,效率较高,满足电子机柜模态试验的要求.运用该虚拟仪器系统对电子机柜进行模态试验,获得其各阶固有频率.     

Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data

This paper is devoted to the identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data. The experimental data sets correspond to partial experimental data made up of an observation vector which is the response of a stochastic boundary value problem depending on the tensor-valued random field which has to be identified. So an inverse stochastic problem has to be solved to carry out the identification of the random field. A complete methodology is proposed to solve this challenging problem and consists in introducing a family of prior probability models, in identifying an optimal prior model in the constructed family using the experimental data, in constructing a statistical reduced order optimal prior model, in constructing the polynomial chaos expansion with deterministic vector-valued coefficients of the reduced order optimal prior model and finally, in constructing the probability distribution of random coefficients of the polynomial chaos expansion and in identifying the parameters using experimental data. An application is presented for which several millions of random coefficients are identified solving an inverse stochastic problem.     

Real-time deformable models of non-linear tissues by model reduction techniques

In this paper we introduce a new technique for the real-time simulation of non-linear tissue behavior based on a model reduction technique known as proper orthogonal (POD) or Karhunen-Loève decompositions. The technique is based upon the construction of a complete model (using finite element modelling or other numerical technique, for instance, but possibly from experimental data) and the extraction and storage of the relevant information in order to construct a model with very few degrees of freedom, but that takes into account the highly non-linear response of most living tissues. We present its application to the simulation of palpation a human cornea and study the limitations and future needs of the proposed technique.     

Optimization of shell buckling incorporating Karhunen-Loève-based geometrical imperfections

The optimization of shell buckling is performed considering peak normal force and absorbed internal energy in the presence of geometrical imperfections implemented through Karhunen-Loève expansions. Initially, the mass of a shell is minimized in the presence of random initial imperfections by allowing cutouts in the material, subject to constraints on the average peak force and average internal energy. Then, robustness is considered by minimizing the coefficient of variation of the normal peak force while constraining the average peak force and average internal energy. LS-OPT® is used both to generate an experimental design and to perform a Monte Carlo simulation (96 runs) using LS-DYNA® at each of the experimental design points. The effect of imperfections when minimizing the mass is not large, but when considering robustness, however, the optimal design has a substantially increased hole size and increased shell thickness, resulting in a heavier design with maximal robustness within the constraints.