基于岭估计的岭马田系统在复共线性数据中的应用

在多变量模式识别领域,变量间经常会存在复共线性,复共线性不仅会影响参数估计的效果,也会使变量的敏感性出现显著异常.马田系统是以马氏距离作为测量尺度的多变量模式识别方法,复共线性会通过马氏距离影响马田系统变量筛选的效果和判别的准确率.基于岭估计提出了一种新的测量尺度—岭马氏距离,利用岭迹法确定岭参数,将其引入马田系统使得马田系统对病态数据具有更好的耐受性.通过案例验证了岭马氏距离可以很好的克服复共线性,并提高马田系统的判别准确率.     

协方差阵奇异时马氏距离的一种改进

马氏(Mahalanobis)距离在数据分析中具有广泛应用,但目前对协方差矩阵奇异时马氏距离的定义和几何解释却不尽相同,导致距离值不唯一,影响了它的应用.当使用p×p协方差矩阵M的Moore-Penrose广义逆矩阵代替它的逆矩阵M~(-1)时,一个p维样本向量x到多维正态分布N(μ,M)(M的秩rp)的马氏距离依赖于x与μ的前r维分量,从而导致x携带信息的损失.为充分利用样本信息,组合马氏距离和欧氏距离给出M奇异时马氏距离的一种计算方法,新方法具有明确的几何解释.最后给出协方差矩阵奇异时计算广义马氏距离的几何解释和一个算例.     

改进的TOPSIS决策方法在供应商选择中的应用

运用马氏距离替代欧式距离改进传统的TOPSIS方法,解决当属性间存在线性相关时欧式距离失效的缺陷;充分考虑对立集合并引入联系向量距离,解决可能存在的方案距离正理想解和负理想解距离都近的缺陷.然后通过决策者偏好系数将马氏距离和联系向量距离所得结果合成新的相对贴近度,从而同时克服传统TOPSIS方法的以上两个缺陷.最后通过供应商选择的实例来验证方法的有效性.     

基于马氏距离的TOPSIS决策方法及其应用

针对决策指标之间的相关性问题,将马氏距离引入传统TOPSIS方法,提出了基于马氏距离的TOPSIS方法.在此基础上,分析了基于马氏距离改进后贴近度的性质,并以投资决策方案选择为例加以说明.结果表明,基于马氏距离改进的TOPSIS方法对决策数据的非奇异线性变换具有不变性.协方差矩阵体现了决策指标之间的相关性,因而可以有效避免指标的相关性对决策效果的影响.     

马氏距离聚类分析中协方差矩阵估算的改进

本文考虑了变量权重和样本类别的影响,建立了马氏距离聚类过程中评估协方差矩阵的迭代法。以Fisher的iris数据为样本,运用欧氏距离一般聚类、主成分聚类、改进前后的马氏距离聚类方法,进行实证分析和比较,结果表明本文所提出的新方法准确率至少提高了6.63%。最后,运用该方法对35个国家的相关指标数据进行聚类分析,确定了各国的卫生保健状况等级。     

总体协差阵为单位阵的最小距离判别

本引进最小距离域定义,得到了确定最小距离域的计算方法。在马氏距离判别定理的基础上,确定了最小距离判别规则,得到了利用该规则的判别方法。     

马氏环境中马氏链函数加权和的强收敛性

给出了状态有限的单无限马氏环境中马氏链泛函加权和的强收敛性,得到了状态有限的单无限马氏环境中马氏链泛函加权和的强收敛性成立的一系列充分条件.     

“一带一路”战略下江苏外向型经济发展能力研究——基于马氏距离的灰色TOPSIS评价模型

江苏作为"一带一路"战略的交汇点,有必要探究其各地区外向型经济发展能力.马氏距离具备消除指标间的相关性且不受量纲影响,代替TOPSIS中的欧氏距离,运用灰色关联度来判断指标的关联性,建立基于马氏距离、灰色关联度的TOPSIS外向型经济发展能力评价模型.以江苏13个地级城市为研究对象,进行实证研究.研究表明,外向型经济发展能力评价模型有助于综合判断各城市外向经济发展能力,发现短板并加以整改,促进"一带一路"建设.     

基于Pena距离的加权最小二乘估计的影响分析

利用Pena距离对加权线性最小二乘估计的影响问题进行讨论,得到加权最小二乘估计的Pena距离的表达式,对其性质进行讨论,从而得到高杠异常点的判别方法.文中对Pena距离与Cook距离的性能进行了对比,得到在一定条件下Pena距离优于Cook距离的结论.并通过数值实验对此方法的有效性进行验证.     

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本文研究了双无限环境中马氏链函数加权和的极限定理, 得到了双无限环境中马氏链函数加权和强收敛性成立的一系列充分条件.     

一种提高SVM训练速度的新方法

SVM解决两分类问题时,在大规模数据上训练速度很慢,利用数据提取的方法可以减少训练样本数目,加快训练速度。本文利用马氏距离和"aσ-方法"提出新的数据提取方法,根据样本点到训练集的马氏距离来确定样本点与样本集的位置关系,只提取对于建立超平面有作用的样本点,避免了以往数据提取方法的随机性;并考虑提取的数据占原来总样本集数目的比例,通过调整a的值,控制数据提取的数量,避免提取后训练样本集的数据太多或太少,从而加快SVM的训练速度。     

多因素资产定价模型的统计分析

本以2000年沪市A股的股票作为研究对象，利用聚类分析将其分类并抽取部分作为样本，应用纲目数据统计分析方法，对下周收益率的影响因素作了Logistic回归，得出了股票的收益率具有短期反转的特点。     

Contour approximation of data: A duality theory

Given a dataset D partitioned in clusters, the joint distance function (JDF) J(x) at any point x is the harmonic mean of the distances between x and the cluster centers. The JDF is a continuous function, capturing the data points in its lower level sets (a property called contour approximation), and is a useful concept in probabilistic clustering and data analysis. In particular, contour approximation allows a compact representation of the data: for a dataset in Rⁿ with N points organized in K clusters, the JDF requires K centers and covariances (if Mahalanobis distances are used), for a total of Kn(n+3)/2 parameters, and a considerable reduction of storage if N?K,n. The JDF of the whole dataset, J(D)?∑{J(x):x∈D}, is a measure of the classifiability of the data, and can be used to determine the “right” number of clusters for D. A duality theory for the JDF J(D) is given, in analogy with Kuhn’s geometric duality theory for the Fermat-Weber location problem. The JDF J(D) is the optimal value of a primal problem (P), for which a dual problem (D) is given, with a sharp lower bound on J(D).     

An ellipsoidal distance-based search strategy of ants for nonlinear single and multiple response optimization problems

Various continuous ant colony optimization (CACO) strategies are proposed by researchers to resolve continuous single response optimization problems. However, no such work is reported which also verifies suitability of CACO in case of both single and multiple response situations. In addition, as per literature survey, no variant of CACO can balance simultaneously all the three important aspects of an efficient search strategy, viz. escaping local optima, balancing between intensification and diversification scheme, and handling correlated variable search space structure. In this paper, a variant of CACO, so-called ‘CACO-MDS’ is proposed, which attempts to address all these three aspects. CACO-MDS strategy is based on a Mahalanobis distance-based diversification, and Nelder–Mead simplex-based intensification search scheme. Mahalanobis distance-based diversification search ensures exact measure of multivariate distance for correlated structured search space. The proposed CACO-MDS strategy is verified using fourteen single and multiple response multimodal function optimization test problems. A comparative analysis of CACO-MDS, with three different metaheuristic strategies, viz. ant colony optimization in real space (ACO_R), a variant of local-best particle swarm optimization (SPSO) and simplex-simulated annealing (SIMPSA), also indicates its superiority in most of the test situations.     

Can the bounds in the multivariate Chebyshev inequality be attained?

Chebyshev’s inequality was recently extended to the multivariate case. In this paper we prove that the bounds in the multivariate Chebyshev’s inequality for random vectors can be attained in the limit. Hence, these bounds are the best possible bounds for this kind of regions.     

The Distribution of Robust Distances

Mahalanobis-type distances in which the shape matrix is derived from a consistent, high-breakdown robust multivariate location and scale estimator have an asymptotic chi-squared distribution as is the case with those derived from the ordinary covariance matrix. For example, Rousseeuw's minimum covariance determinant (MCD) is a robust estimator with a high breakdown. However, even in quite large samples, the chi-squared approximation to the distances of the sample data from the MCD center with respect to the MCD shape is poor. We provide an improved F approximation that gives accurate outlier rejection points for various sample sizes.     

Partial Influence Functions

In this paper we extend the definition of the influence function to functionals of more than one distribution, that is, for estimators depending on more than one sample, such as the pooled variance, the pooled covariance matrix, and the linear discriminant analysis coefficients. In this case the appropriate designation should be “partial influence functions,” following the analogy with derivatives and partial derivatives. Some useful results are derived, such as an asymptotic variance formula. These results are then applied to several estimators of the Mahalanobis distance between two populations and the linear discriminant function coefficients.