Linear independence of a finite set of time-frequency shifts

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L² function. Firstly, background and motivation for the conjecture are provided. Secondly, the main progress of this linear independence in the past twenty five years is reviewed. Finally, the partial results of the high dimensional case and other cases for the conjecture are briefly presented.     

Split Orthogonal Arrays and Maximum Independent Resilient Systems of Functions

A system of (Boolean) functions in variables is called randomized if the functions preserve the property of their variables to be independent and uniformly distributed random variables. Such a system is referred to as -resilient if for any substitution of constants for any variables, where 0 i t, the derived system of functions in variables will be also randomized. We investigate the problem of finding the maximum number of functions in variables of which any form a -resilient system. This problem is reduced to the minimization of the size of certain combinatorial designs, which we call split orthogonal arrays. We extend some results of design and coding theory, in particular, a duality in bounding the optimal sizes of codes and designs, in order to obtain upper and lower bounds on . In some cases, these bounds turn out to be very tight. In particular, for some infinite subsequences of integers they allow us to prove that , , , , . We also find a connection of the problem considered with the construction of unequal-error-protection codes and superimposed codes for multiple access in the Hamming channel.     

Invariance of the Spectrum for Hypoelliptic Operators

We show that the spectra of the -realizations for a class of hypoelliptic (pseudo-)differential operators are independent of in an interval around depending on the growth properties of the symbol. For elliptic operators we obtain the classical boundedness interval of Fefferman; in the general case we obtain a smaller interval which is as large as one can possibly expect it to be.

     

Pair-Copula Bayesian Networks

Pair-copula Bayesian networks (PCBNs) are a novel class of multivariate statistical models, which combine the distributional flexibility of pair-copula constructions (PCCs) with the parsimony of conditional independence models associated with directed acyclic graphs (DAGs). We are first to provide generic algorithms for random sampling and likelihood inference in arbitrary PCBNs as well as for selecting orderings of the parents of the vertices in the underlying graphs. Model selection of the DAG is facilitated using a version of the well-known PC algorithm that is based on a novel test for conditional independence of random variables tailored to the PCC framework. A simulation study shows the PC algorithm’s high aptitude for structure estimation in non-Gaussian PCBNs. The proposed methods are finally applied to modeling financial return data. Supplementary materials for this article are available online.     

Decompositions of the free additive convolution

We introduce and study a new type of convolution of probability measures, denoted μν and called the s-free additive convolution, which is defined by the subordination functions associated with the free additive convolution. We derive an alternating decomposition of μν for compactly supported μ and ν, using another convolution called orthogonal additive convolution. This decomposition leads to two types of ‘complete’ alternating decompositions of the free additive convolution μ?ν. More importantly, we develop an operatorial approach to the subordination property and introduce the associated notion of s-free independence. Moreover, we establish relations between convolutions associated with the main notions of noncommutative independence (free, monotone and boolean). Finally, our result leads to natural decompositions of the free product of rooted graphs.     

Diophantine approximation by algebraic hypersurfaces and varieties

Questions on rational approximations to a real number can be generalized in two directions. On the one hand, we may ask about ``approximation' to a point in by hyperplanes defined over the rationals. That is, we seek hyperplanes with small distance from the given point. On the other hand, following Wirsing, we may ask about approximation to a real number by real algebraic numbers of degree at most .

The present paper deals with a common generalization of both directions, namely with approximation to a point in by algebraic hypersurfaces, or more generally algebraic varieties defined over the rationals.

     

A characterization of (γ, i)‐trees

We characterize trees with equal domination and independent domination numbers in terms of the sets of vertices of the tree which are contained in all its minimum dominating and minimum independent dominating sets. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 277–292, 2000     

Ensemble-SISPLS近红外光谱变量选择方法

近红外光谱具有高维小样本的特点,变量选择是提高定量分析模型稳健性和可解释性的一种有效方法。确定独立筛选（SIS）是一种基于边际相关性的超高维数据变量选择方法,广泛用于基因微阵列数据的变量选择。SIS具有将数据维度降低至样本大小规模的能力,其降维能力与LASSO相当,在相当宽泛的近似条件下,由于具有安全筛选性质,所有重要变量被保留的概率趋于1。基于确定独立筛选偏最小二乘（SIS-SPLS）的变量选择是一种迭代式的SIS变量选择方法,首先利用SIS方法完成光谱重要变量的初选;然后根据重要变量的边际相关性大小进行逐步前向选择：建立偏最小二乘回归模型,依据贝叶斯信息准则（BIC）确定最终的变量选择结果。SIS-SPLS以逐步前向选择的方式实现对重要变量的增量式筛选,随着潜变量个数的增加及因变量残差的逐步减小,SIS-SPLS方法选择的变量个数将趋于稳定。然而仅以边际相关性对变量重要性进行评价,当光谱变量个数远大于样本数时,该方法也存在选择的变量过多、变量选择结果不够稳健等问题。为进一步提高小样本情况下变量选择的稳健性,将集成学习引入SIS-SPLS方法之中,提出了一种集成SIS-SPLS变量选择方法（Ensemble-SISPLS）。该方法首先对校正集样本进行自助重采样,对采样得到的每一个校正子集分别使用SIS-SPLS方法进行变量筛选,通过投票机制并设置频次阈值对所有校正子集的变量选择结果进行集成,选择出现频次大于给定阈值的变量并建立偏最小二乘回归模型,计算5折交叉验证均方根误差。对频次阈值和潜变量个数两个关键参数使用网格搜索法进行优选,根据子模型的交叉验证均方根误差和变量个数对子模型性能进行综合评价,以最优子模型包含的变量作为最终的变量选择结果。分别在Corn数据集和当归数据集上进行变量选择实验,比较Ensemble-SISPLS,SIS-SPLS和UVE-PLS三种变量选择方法的性能。其中当归数据集共77个样本,样本采自甘肃岷县和渭源县,使用Nicolet-6700型近红外光谱仪扫描得到所有样本的近红外光谱并对当归中的阿魏酸含量进行预测。Ensemble-SISPLS方法在Corn数据集上选择的变量个数、RMSEP和决定系数分别为22,0.000 8和0.999 8;SIS-SPLS方法在Corn数据集上选择的变量个数、RMSEP和决定系数分别为97,0.007 3和0.998 8。Ensemble-SISPLS方法在当归数据集上选择的变量个数、RMSEP和决定系数分别为24,0.018 1和0.996 3;SIS-SPLS方法在当归数据集上选择的变量个数、RMSEP和决定系数分别为38,0.022 6和0.994 3。结果表明,该方法进一步提高了变量选择结果的稳健性和预测能力。Ensemble-SISPLS变量选择方法有效结合了SIS-SPLS较强的变量选择能力和集成学习良好的泛化能力,提高了变量选择的稳健性。此外,由于在子模型的预测能力和变量个数之间进行了折中,一定程度上减少了选择变量的个数,提高了模型的可解释性。     

韧性金属Kaiser效应方向独立性的实验研究

Kaiser效应是声发射技术实际应用的理论基础。本文以韧性金属材料Q345不同方向拉伸实验为例,探究力的方向对韧性金属Kaiser效应的影响。实验结果表明,韧性金属材料在某一方向加载时,表现出了Kaiser效应;当改变加载方向时,呈现出与第一次加载方向类似的声发射特性,说明声发射现象对其他加载方向上所承受的最大载荷不具有记忆的能力,而仅与该方向承载历史有关,即韧性金属Kaiser效应与力的方向性间具有独立性。同时,利用细观损伤理论对这一现象产生的机理进行了定性的解释。     

A class of multivariate copulas with bivariate Fréchet marginal copulas

In this paper, we present a class of multivariate copulas whose two-dimensional marginals belong to the family of bivariate Fréchet copulas. The coordinates of a random vector distributed as one of these copulas are conditionally independent. We prove that these multivariate copulas are uniquely determined by their two-dimensional marginal copulas. Some other properties for these multivariate copulas are discussed as well. Two applications of these copulas in actuarial science are given.