未知方向密度估计的收敛率

设Ｘ为ｐ维随机向量,对于未知的投影方向θｏ（‖θｏ‖＝１）,本文利用θｏ的估计与核密度估计相结合的方法给出了θ＾Ｔ０Ｘ的密度（方向密度）的核型密度估计,获得了此估计的逐点渐近正态性,逐点精确强收敛率,一致精确强收敛率以及均方误差收敛率,所得结果与最优性与已知方向上的核密度估计完全一致。作为例子,对θｏ为Ｘ协方差阵的最大特征值所对应的特征方向,我们给出了θｏ的满足条件的估计极其方向密度估计。     

PPDE 中的一个极限定理

PP 方法是近年来兴起的一种处理高维数据的统计方法。它最早是由 Kruskal(1969,1972)提出来的。PP 的基本思想是把高维数据投影到低维空间中去。这样,通过对某些感兴趣的投影方向上的数据处理来解决高维问题,因而 PP 方法中最根本的问题是投影方向的选择以及选定方向后如何导出高维结构。PP 是 Projection Pursuit 的缩写,一般译为投影寻踪。     

The Asymptotics of Estimate of the Projection Directions Concerning Sliced inverse Regression

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基于投影寻踪模型的装备保障性目标评估

应用投影寻踪法,对保障性目标评估标准进行分析,利用在每个级别范围内随机产生的样本,建立了投影寻踪模型,确定了保障性目标评估指标的权重系数,得到了投影特征值变化区域与分类级别的对应关系,对某型航空装备使用初期的保障性试验数据进行分析,得到了其使用初期的级别和评估等级.为实现装备保障性评估目标评估拓展了新的途径,提供了科学依据.     

Projection Pursuit for Exploratory Supervised Classification

In high-dimensional data, one often seeks a few interesting low-dimensional projections that reveal important features of the data. Projection pursuit is a procedure for searching high-dimensional data for interesting low-dimensional projections via the optimization of a criterion function called the projection pursuit index. Very few projection pursuit indices incorporate class or group information in the calculation. Hence, they cannot be adequately applied in supervised classification problems to provide low-dimensional projections revealing class differences in the data. This article introduces new indices derived from linear discriminant analysis that can be used for exploratory supervised classification.     

企业竞争力评价的投影寻踪模型

建立了企业竞争能力评价的投影寻踪聚类模型,利用该模型可将企业竞争优势评价的多个指标转换为反映各指标综合信息的投影特征值,根据投影特征值的大小便可对企业竞争能力进行客观评价,整个过程无人为干扰。通过对5个煤炭企业竞争优势评价表明该方法操作简便,切实可行,为企业竞争能力综合评价的多因素影响问题提供了一条新途径。     

Minimax adjustment technique in a parameter restricted linear model

We consider an approach yielding a minimax estimator in the linear regression model with a priori information on the parameter vector, e.g., ellipsoidal restrictions. This estimator is computed directly from the loss function and can be motivated by the general Pitman nearness criterion. It turns out that this approach coincides with the projection estimator which is obtained by projecting an initial arbitrary estimate on the subset defined by the restrictions.     

投影寻踪模型在国民经济综合评价中的应用

文章针对区域国民经济发展的多属性,采用投影寻踪评价模型,用加速遗传算法寻找最佳投影方向,将多维属性指标转换为一维投影值.在此基础上,通过建立单属性指标的分类等级区间,给出区域国民经济发展水平分类的投影值区间,从而实现对区域国民经济发展水平的分类.应用表明,它是国民经济评价的一种计算过程简单、直观的新方法.     

投影寻踪等级评价模型在冰草草坪质量综合评价中的应用

针对以往草坪质量评价方法存在人为权重赋值误差,以及简便计算等方面的不足的问题,提出了草坪坪用质量指标评定的投影寻踪等级评价模型.该方法可以依据标准样本自身的数据特性寻求最佳投影方向,通过最佳投影方向与评价指标的线性投影得到投影指标值,通过这一指标可以对样本进行统一评价和分类,并可利用最佳投影方向判断各评价指标对综合评价目标的贡献大小.排除人为权重干扰,对哈尔滨地区引种的冰草(A gropyron elongatum)坪用质量进行了综合判定.     

基于投影寻踪和最优分割的企业信用评级模型

提出一种基于投影寻踪和最优分割的企业信用评级模型。该模型运用投影寻踪对样本企业进行信用综合评分,将信用综合得分由大到小排序,生成有序样品序列;利用最优分割法对有序样品进行聚类,得出明确的聚类结果;将最优分割点对应的信用综合得分作为划分信用等级的阈值,从而实现对样本企业的信用评级。应用实例证明了该模型的可行性和有效性。     

Projection Pursuit Indexes Based on Orthonormal Function Expansions

Abstract

Projection pursuit describes a procedure for searching high-dimensional data for “interesting” low-dimensional projections via the optimization of a criterion function called the projection pursuit index. By empirically examining the optimization process for several projection pursuit indexes, we observed differences in the types of structure that maximized each index. We were especially curious about differences between two indexes based on expansions in terms of orthogonal polynomials, the Legendre index, and the Hermite index. Being fast to compute, these indexes are ideally suited for dynamic graphics implementations.

Both Legendre and Hermite indexes are weighted L ² distances between the density of the projected data and a standard normal density. A general form for this type of index is introduced that encompasses both indexes. The form clarifies the effects of the weight function on the index's sensitivity to differences from normality, highlighting some conceptual problems with the Legendre and Hermite indexes. A new index, called the Natural Hermite index, which alleviates some of these problems, is introduced.

A polynomial expansion of the data density reduces the form of the index to a sum of squares of the coefficients used in the expansion. This drew our attention to examining these coefficients as indexes in their own right. We found that the first two coefficients, and the lowest-order indexes produced by them, are the most useful ones for practical data exploration because they respond to structure that can be analytically identified, and because they have “long-sighted” vision that enables them to “see” large structure from a distance. Complementing this low-order behavior, the higher-order indexes are “short-sighted.” They are able to see intricate structure, but only when they are close to it.

We also show some practical use of projection pursuit using the polynomial indexes, including a discovery of previously unseen structure in a set of telephone usage data, and two cautionary examples which illustrate that structure found is not always meaningful.     

A projection method of estimation for a subfamily of exponential families

Summary This paper is concerned with estimation for a subfamily of exponential-type, which is a parametric model with sufficient statistics. The family is associated with a surface in the domain of a sufficient statistic. A new estimator, termed a projection estimator, is introduced. The key idea of its derivation is to look for a one-to-one transformation of the sufficient statistic so that the subfamily can be associated with a flat subset in the transformed domain. The estimator is defined by the orthogonal projection of the transformed statistic onto the flat surface. Here the orthogonality is introduced by the inverse of the estimated variance matrix of the statistic on the analogy of Mahalanobis's notion (1936,Proc. Nat. Inst. Sci. Ind.,2, 49–55). Thus the projection estimator has an explicit representation with no iterations. On the other hand, the MLE and classical estimators have to be sought as numerical solutions by some algorithm with a choice of an initial value and a stopping rule. It is shown that the projection estimator is first-order efficient. The second-order property is also discussed. Some examples are presented to show the utility of the estimator.     

A new projection estimate for multivariate location with minimax bias

The maximum asymptotic bias of an estimator is a global robustness measure of its performance. The projection median estimator for multivariate location shows a remarkable behavior regarding asymptotic bias. In this paper we consider a modification of the projection median estimator which renders an estimate with better bias performance for point mass contaminations (the worst situation for the projection median estimator). Moreover, it achieves the lowest bound for an equivariant estimate for point mass contaminations.     

基于极大熵投影寻踪组合法的水资源预警评价模型

利用基于极大熵准则赋权和基于实数加速遗传算法的投影寻踪方法相结合的组合附权法确定了各预警指标的权重;采用层次分析法计算水资源可持续利用复合系统中各子系统所占权重;利用综合评价模型计算出哈尔滨市水资源可持续发展指数;最终得到哈尔滨市水资源可持续利用预警结果.     

加权平衡指数损失函数下的广义信度保费估计

在非寿险中,在索赔经历虽然相互独立,但有时会服从不同的分布.通过考虑保费的目标估计来对风险保费进行了研究,并采用正交投影的方法得到了目标问题的最优解,从而得到了加权平衡指数损失函数下的信度估计.此外,给出了结构参数的无偏估计,并给出了模拟.结果表明,在考虑目标保费的情况下,当选取一个合适的权重,可以得到未来保费的最优估计.     

Fuzzy projection pursuit density estimation by eigenvalue method

This paper focuses on the use of kernel method and projection pursuit regression for non-parametric probability density estimation. Direct application of the kernel method is not able to pick up characteristic features of multidimensional density function. We propose a fuzzy projection pursuit density estimation based on the membership function and the eigenvector of the covariance matrix. Marginal densities along the subspace spanned by the projection vector are estimated. The proposed projection pursuit is one of the methods which are able to bypass the ‘curse of dimensionality’ in multidimensional density estimation. An application to experimental design for machining accuracy of end milling with the tool in small diameter is presented to demonstrate its usefulness.     

Optimal fourier-hermite expansion for estimation

The purpose of the paper is to present a systematic method for developing an approximate recursive estimator which is optimal for the given structure and approaches the best estimate, when the order of approximation increases.The minimal variance estimate is projected onto the Hilbert subspace of all Fourier-Hermite (FH) series, driven by the observations, with the same given index set. The projection results in a system of linear algebraic equations for the FH coefficients, the parameters of the desired approximate estimator.The estimator consists of finitely many Wiener integrals of the observations and a memoryless nonlinear postprocessor. The postprocessor is an arithmetic combination of the Hermite polynomials evaluated at the Wiener integrals. A couple of recursive methods for calculating the Wiener integrals are included.     

Adaptive meshless Galerkin boundary node methods for hypersingular integral equations

Adaptive refinement techniques are developed in this paper for the meshless Galerkin boundary node method for hypersingular boundary integral equations. Two types of error estimators are derived. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two consecutive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the numerical solution itself and its projection. These error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization scheme is presented to accomodate the non-local property of hypersingular integral operators for the needed computable local error indicators. The convergence of the adaptive meshless techniques is verified theoretically. To confirm the theoretical results and to show the efficiency of the adaptive techniques, numerical examples in 2D and 3D with high singularities are provided.     

Grand Tour and Projection Pursuit

Abstract

The grand tour and projection pursuit are two methods for exploring multivariate data. We show how to combine them into a dynamic graphical tool for exploratory data analysis, called a projection pursuit guided tour. This tool assists in clustering data when clusters are oddly shaped and in finding general low-dimensional structure in high-dimensional, and in particular, sparse data. An example shows that the method, which is projection-based, can be quite powerful in situations that may cause grief for methods based on kernel smoothing. The projection pursuit guided tour is also useful for comparing and developing projection pursuit indexes and illustrating some types of asymptotic results.     

基于投影寻踪分类模型的各地区节能降耗评价研究

能源作为经济增长中重要的投入要素,对各国经济发展都起着决定性影响。国家"十一五"规划中明确提出2010年中国单位GDP能源消耗下降20%,这个宏伟目标的实现对于我国经济的可持续发展和改善人民生活环境具有重要意义。本文运用投影寻踪分类模型,对节能降耗综合评价指标体系进行分析,得到最佳投影值及各地区应承担的节能降耗责任大小。与主成分分析法结果的比较表明投影寻踪分类模型对这类指标评价问题的合理性和可行性。