The maximization of determinants of block Toeplitz matrices with applications in optimal design theory

We consider the maximization of the determinant of the trigonometric moment matrix of a matrix measure on the unit circle. The results are analog to those obtained by Dette and Studden (2005) for moment matrices of matrix measures on a compact interval. A statistical application in optimal design theory is considered and we present several optimal designs for Fourier regression models.     

经济结构演变与中国服务业的发展

 内容提要：本文基于乘数分解方法和劳动力流量矩阵,利用我国1997、2002和2007年投入产出表以及OECD国家的统计数据与投入产出表,分析我国服务业发展的特点,并与美国、日本、德国为代表的西方发达国家进行比较,探讨了我国服务业发展的机制、面临的问题和发展路径。     

中国省际贸易矩阵的估计与应用

基于2012年中国31省市的投入产出表,借鉴国内资金流量表、投入产出分析方法和社会核算矩阵理论编制了中国省际贸易矩阵,估计和分析了各地区省际间贸易往来规模及特征,并进行了各地区贸易往来影响效应的比较。结果发现：北京、海南等省际贸易依存度较高,四川、山东等省内贸易比例较大,江苏、北京等省际贸易总量较大,而青海、西藏等省际贸易总量较小;北京、上海等发达省份之间的贸易往来较为密切,同时一些中部省份是贸易流入的主要地区;北京、上海等呈现较强的影响力和感应度,浙江、江苏等沿海省份有着较强的感应度和较弱的影响力,宁夏、陕西等西部省份呈现较强影响力和较弱的感应度。     

可达性矩阵的C语言程序实现

讨论了可达性矩阵的数学思想以及用计算机程序实现其复杂的求解过程。     

稳健高维协方差矩阵估计及其投资组合应用——基于中心正则化算法

高维协方差矩阵的估计问题现已成为大数据统计分析中的基本问题,传统方法要求数据满足正态分布假定且未考虑异常值影响,当前已无法满足应用需要,更加稳健的估计方法亟待被提出。针对高维协方差矩阵,一种稳健的基于子样本分组的均值-中位数估计方法被提出且简单易行,然而此方法估计的矩阵并不具备正定稀疏特性。基于此问题,本文引进一种中心正则化算法,弥补了原始方法的缺陷,通过在求解过程中对估计矩阵的非对角元素施加L1范数惩罚,使估计的矩阵具备正定稀疏的特性,显著提高了其应用价值。在数值模拟中,本文所提出的中心正则稳健估计有着更高的估计精度,同时更加贴近真实设定矩阵的稀疏结构。在后续的投资组合实证分析中,与传统样本协方差矩阵估计方法、均值-中位数估计方法和RA-LASSO方法相比,基于中心正则稳健估计构造的最小方差投资组合收益率有着更低的波动表现。     

A Ridge-Regression Sidelight

A useful matrix result is applied in ridge regression to cast light on the form of the ridge-regression residual sum of squares as a power series in the ridge parameter. An example illustrates that the series may diverge or else converge very slowly, depending on the value of the ridge parameter.     

Bayesian inference in a matrix normal dynamic linear model with unknown covariance matrices

In this paper, we consider the problem of estimating the parameters of a matrix normal dynamic linear model when the variance and covariance matrices of its error terms are unknown and can be changing over time. Given that the analysis is not conjugate, we use simulation methods based on Monte Carlo Markov chains to estimate the parameters of the model. This analysis allows us to carry out a dynamic principal components analysis in a set of multivariate time series. Furthermore, it permits the treatment of series with different lengths and with missing data. The methodology is illustrated with two empirical examples: the value added distribution of the firms operating in the manufacturing sector of the countries participating in the BACH project, and the joint evolution of a set of international stock-market indices.     

O(N 2)-Operation Approximation of Covariance Matrix Inverse in Gaussian Process Regression Based on Quasi-Newton BFGS Method

Gaussian process (GP) is a Bayesian nonparametric regression model, showing good performance in various applications. However, during its model-tuning procedure, the GP implementation suffers from numerous covariance-matrix inversions of expensive O(N³) operations, where N is the matrix dimension. In this article, we propose using the quasi-Newton BFGS O(N²)-operation formula to approximate/replace recursively the inverse of covariance matrix at every iteration. The implementation accuracy is guaranteed carefully by a matrix-trace criterion and by the restarts technique to generate good initial guesses. A number of numerical tests are then performed based on the sinusoidal regression example and the Wiener–Hammerstein identification example. It is shown that by using the proposed implementation, more than 80% O(N³) operations could be eliminated, and a typical speedup of 5–9 could be achieved as compared to the standard maximum-likelihood-estimation (MLE) implementation commonly used in Gaussian process regression.     

Asymmetrical Orthogonal Arrays With Run Size 100

Nowadays orthogonal arrays play important roles in statistics and other fields. Usual difference matrices are essential for the construction of many symmetrical or a few asymmetrical orthogonal arrays. But there are also asymmetrical orthogonal arrays which can not be obtained by the usual difference matrices. In order to construct these asymmetrical orthogonal arrays, a class of special matrices were discovered from the orthogonal decompositions of projection matrices. In this article, an interesting equivalent relationship between orthogonal arrays and the generalized difference matrices is presented. As an application, a lot of new orthogonal arrays of run size 100 have been constructed.     

Matrix variate Macdonald distribution

Abstract

In this article, we generalize the univariate Macdonald distribution to the matrix case and give its derivation using matrix variate gamma distribution. We study several properties such as cumulative distribution function, marginal distribution of submatrix, triangular factorization, moment generating function, and expected values of several functions of the Macdonald matrix. Some of these results are expressed in terms of special functions of matrix arguments and zonal polynomials.