偏态自适型投影深度

对常规投影深度进行了改进,得到了一类新的可自适应反映多元数据云偏态的统计投影深度.新的深度函数仍然具有凸的深度域,满足一般统计深度函数所应具备的全部基本特点.此外,还考虑了新投影深度函数的计算问题,证得在任意维空间里,它及由它诱导的深度域均是可以精确计算的,从技术角度讲,其处理比常规投影深度更加简便.最后,提供了一些数据示例用以展示此偏态自适型投影深度的有限样本性质.     

统计深度的几何推广

改进统计深度的定义,并将点的深度概念推广到直线与平面的深度,由此得到深度计算的基本定理和深度的一系列性质.最后讨论应用展望.     

一种基于高斯核的广义数据深度

本文提出了一个具有自适应等深面的广义数据深度函数.针对目前数据深度函数的中心区域都是凸的,造成无法有效处理具有ItrJ形结构数据的问题.本文从样本深度的角度,利用高斯核特征空间最小球原理,提出了一种广义的数据深度.它具有正交、尺度、平移不变性,满足弱单调性和无穷远处为0的性质.相比现有数据深度而言,它对凹形结构的数据具有更好的深度解释.更为重要的是它的计算比较简单,从而保证了它的实用性.     

基于群论的深度卷积网络分析

近年来深度学习已成为机器学习中处理大量复杂数据的有效方法,它通过多层次的结构从高维数据中提取特征,从而解决分类、回归等实际任务.文章首先回顾了深度卷积网络和自编码器的数学模型,然后引入群论中分析对称性的一些方法,对深度卷积网络在数据降维时的有效性进行了初步的讨论,最后根据深度卷积网络对称群的逐层关系提出了改进神经网络的一个原则.     

灰色绝对关联度的改进模型

基于刘思峰教授提出的灰色绝对关联度模型,提出了一点改进,给出了新的灰色绝对关联度模型,并讨论了新模型的性质及其算法.一方面保持了原绝对关联度模型的优点,另一方面改正了原模型在某些方面的不足.     

样本深度的基本性质

本文讨论由L2深度修正得到的(L)2深度相应的样本深度的性质,得到了样本深度的相合性和渐近正态性,并证明了它在任意紧集上的一致相合性.最后,基于上述性质简要讨论了样本深度等高的一些性质.     

基于顾客选择行为的租车需求修复方法研究

为了提高顾客对不同类车型租车需求量的预测精度,需要对历史约束数据进行修复.传统Spill模型能够在一定程度上修复受约束需求数据,但并没有考虑到顾客的主观选择行为.为此,设计顾客租车行为调查,利用多项Logit模型对数据进行处理,得到顾客偏好概率,在此基础上改进Spill模型.通过实例验证分析,改进Spill模型比原模型平均误差更小,说明改进模型更有效.     

回归深度函数的一般概念

作为可度量某数据点相对于给定多元数据集中心化程度的工具,统计深度函数在多元稳健数据分析中发挥着重要的作用.在过去数十年中,许多著名的深度函数被相继提出.然而,现有的这些深度函数主要用于位置情形下的描述性或推断性统计分析,尚难被用于回归情形下的类似分析.鉴于此,本文考虑了如何将现有的深度函数推广至回归情形,并提出了一类新的可助于此类推广的一般回归深度函数.本文最后通过示例展示了所提回归深度的相关性质.     

非参数固定设计回归模型中的Stahel-Donoho核估计

研究非参数固定设计回归模型中的稳健核估计. 提出了一种Stahel-Donoho核估计, 在此核估计中, 权重函数既依赖于数据深度, 又依赖于设计点和估计点之间的距离. 对不可直接计算的误差深度, 利用局部近似, 给出了一种近似计算方法, 使得新的估计是计算有效的. 新的估计获得较高的崩溃点值, 并有渐近正态和均方收敛等良好的大样本性质. 与参数模型中的深度加权估计不同的是,这种深度加权非参数估计有简单的方差结构,于是,人们可以比较新旧估计的有效性.数据模拟结果表明,新的方法可以平滑回归估计,并获得稳健性和有效性的良好平衡.     

Fusion框架系统的局部框架扰动的稳定性

该文在Hilbert空间中一般的框架序列扰动形式下,利用正交投影的性质和对偶框架的性质研究了原序列张成的闭子空间与扰动序列张成的闭子空间的关系,并探讨了局部框架的一般扰动对fusion框架系统稳定性的影响.这些结果推广和改进了由Casazza,Kutyniok和Li等得到的著名结果.     

~2样本深度的基本性质

本文讨论由L2深度修正得到的L2深度相应的样本深度的性质,得到了样本深度的相合性和渐近正态性,并证明了它在任意紧集上的一致相合性．最后,基于上述性质简要讨论了样本深度等高的一些性质．     

一类非线性波动方程位势井深度函数的连续性

本文研究一类非线性波动方程位势井深度函数的连续性.通过引入位势井深度函数并给出其性质,给出了位势井深度函数连续性的证明.而位势井深度函数连续性保证了在其基础上得到的位势井族有意义.     

Rayleigh projection depth

In this paper, a novel projection-based depth based on the Rayleigh quotient, Rayleigh projection depth (RPD), is proposed. Although, the traditional projection depth (PD) has many good properties, it is indeed not practical due to its difficult computation, especially for the high-dimensional data sets. Defined on the mean and variance of the data sets, the new depth, RPD, can be computed directly by solving a problem of generalized eigenvalue. Meanwhile, we extend the RPD as generalized RPD (GRPD) to make it suitable for the sparse samples with singular covariance matrix. Theoretical results show that RPD is also an ideal statistical depth, though it is less robust than PD.     

组模偏正则化及其应用

本文研究组模下偏正则最小化问题,证明了解的存在性,稀疏性.研究了零空间性质对最优解的刻画.仔细探讨了解的一种单调性,并应用这种单调性说明最优化问题的求解可以分解到各组中.最后给出了一个所证定理在地震反演的应用.     

Depth for complexes, and intersection theorems

This paper introduces a new notion of depth for complexes; it agrees with the classical definition for modules, and coincides with earlier extensions to complexes, whenever those are defined. Techniques are developed leading to a quick proof of an extension of the Improved New Intersection Theorem (this uses Hochster's big Cohen-Macaulay modules), and also a generalization of the “depth formula” for tensor product of modules. Properties of depth for complexes are established, extending the usual properties of depth for modules. Received May 6, 1997; in final form December 3, 1997     

On data depth in infinite dimensional spaces

The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite dimensional data, there have been some recent attempts to develop depth functions for data in infinite dimensional spaces. In this paper, we consider some notions of depth in infinite dimensional spaces and study their properties under various stochastic models. Our analysis shows that some of the depth functions available in the literature have degenerate behaviour for some commonly used probability distributions in infinite dimensional spaces of sequences and functions. As a consequence, they are not very useful for the analysis of data satisfying such infinite dimensional probability models. However, some modified versions of those depth functions as well as an infinite dimensional extension of the spatial depth do not suffer from such degeneracy and can be conveniently used for analyzing infinite dimensional data.     

Property A and Uniform Embeddability of Metric Spaces Under Decompositions of Finite Depth

Property A and uniform embeddability are notions of metric geometry which imply the coarse Baum-Connes conjecture and the Novikov conjecture. In this paper, the authors prove the permanence properties of property A and uniform embeddability of metric spaces under large scale decompositions of finite depth.     

Depth notions for orthogonal regression

Global depth, tangent depth and simplicial depths for classical and orthogonal regression are compared in examples, and properties that are useful for calculations are derived. The robustness of the maximum simplicial depth estimates is shown in examples. Algorithms for the calculation of depths for orthogonal regression are proposed, and tests for multiple regression are transferred to orthogonal regression. These tests are distribution free in the case of bivariate observations. For a particular test problem, the powers of tests that are based on simplicial depth and tangent depth are compared by simulations.     

An integro-differential equation for surface ocean waves with finite depth

An explicit integro-differential equation formulation is derived for surface ocean waves with finite depth. The equation involves only 2D surface variables. For this equation, we establish the stability and existence of solutions, and explain the effect of depth on surface wave properties.