Simple and multiple correspondence analysis for ordinal-scale variables using orthogonal polynomials

Correspondence analysis (CA) has gained a reputation for being a very useful statistical technique for determining the nature of association between two or more categorical variables. For simple and multiple CA, the singular value decomposition (SVD) is the primary tool used and allows the user to construct a low-dimensional space to visualize this association. As an alternative to SVD, one may consider the bivariate moment decomposition (BMD), a method of decomposition that involves using orthogonal polynomials to reflect the structure of ordered categorical responses. When the features of BMD are combined with SVD, a hybrid decomposition (HD) is formed. The aim of this paper is to show the applicability of HD when performing simple and multiple CA.     

碳排放IDA模型的算法比较及应用研究

在分析碳排放指数分解模型IDA的基本理论框架和各类型算法的结构、特点基础上,以中国1991—2014年相关数据的实证分析作为各算法应用的解读,并基于适用性、有效性综合评价提出算法选择的参考信息:LD算法分解的各因素作用易于理解,但存在分解余项问题;RLD与Shapley算法能够实现因素完全分解,且本质上具有一致性;GFI算法适合多因素效应完全分解,但计算过程复杂;AMDI与AWD算法受到分解余项和对数权重赋值限制的影响约束;LMDIⅠ算法具有灵活的分解形式及因素完全分解特征等。     

Explorative Data Analysis and CATANOVA for Ordinal Variables: An Integrated Approach

Categorical analysis of variance (CATANOVA) is a statistical method designed to analyse variability between treatments of interest to the researcher. There are well-established links between CATANOVA and the τ statistic of Goodman and Kruskal which, for the purpose of the graphical identification of this variation, is partitioned using singular value decomposition for Non-Symmetrical Correspondence Analysis (NSCA) (D'Ambra & Lauro, 1989). The aim of this paper is to show a decomposition of the Between Sum of Squares (BSS), measured both in CATANOVA framework and in the statistic τ, into location, dispersion and higher order components. This decomposition has been developed using Emerson's orthogonal polynomials. Starting from this decomposition, a statistical test and a confidence circle have been calculated for each component and for each modality in which the BSS was decomposed, respectively. A Customer Satisfaction study has been considered to explain the methodology.     

Estimation of systems of equations subject to curvature constraints

Two computationally manageable estimation procedures for maintaining curvature constraints are compared. Using an efficient nonlinear optimizing algorithm, the well-known Cholesky factorization is compared with the eigenvalue decomposition method for estimating a system of input demand and output supply equations. Statistical tests are provided. The Cholesky factorization is superior when efficient optimizers are used. The eigenvalue decomposition may become a better choice when the user is limited to less efficient algorithms.     

Incremental modelling for compositional data streams

ABSTRACT

Incremental modelling of data streams is of great practical importance, as shown by its applications in advertising and financial data analysis. We propose two incremental covariance matrix decomposition methods for a compositional data type. The first method, exact incremental covariance decomposition of compositional data (C-EICD), gives an exact decomposition result. The second method, covariance-free incremental covariance decomposition of compositional data (C-CICD), is an approximate algorithm that can efficiently compute high-dimensional cases. Based on these two methods, many frequently used compositional statistical models can be incrementally calculated. We take multiple linear regression and principle component analysis as examples to illustrate the utility of the proposed methods via extensive simulation studies.     

我国生产用能源消费变动的分解分析

摘　　要:能源消费分解是探讨能源消费变动影响因素的一种常用方法。在本文中介绍了目前研究中较为合理的一种分解方法——对数平均D氏指数法,并借助这一方法来对我国“六五”时期以来的生产用能源消费情况进行分解分析,探讨产业结构变动和产业内效率提高对能源消费和总体单位能耗的影响。根据测算结果,笔者认为产业内能源效率的提高是我国能源节约的主要因素;进一步建议政府构建“能源分解指数体系”作为制定能源政策的依据。     

Shrinkage and Orthogonal Decomposition

ABSTRACT. The problem of estimating the mean of a multivariate normal distribution when the parameter space allows an orthogonal decomposition is discussed. Risk functions and lower bounds for a class of shrinkage estimators that includes Stein's estimator are derived, and an improvement on Stein's estimator that takes advantage of the orthogonal decomposition is introduced. Uniform asymptotics related to Pinsker's minimax risk is derived and we give conditions for attaining the lower risk bound. Special cases including regression and analysis of variance are discussed.     

Stochastic Bounds with a Low Rank Decomposition

We investigate how we can bound a discrete time Markov chain (DTMC) by a stochastic matrix with a low rank decomposition. In the first part of the article, we show the links with previous results for matrices with a decomposition of size 1 or 2. Then we show how the complexity of the analysis for steady-state and transient distributions can be simplified when we take into account the decomposition. Finally, we show how we can obtain a monotone stochastic upper bound with a low rank decomposition.     

A Decomposition of Pearson–Fisher and Dzhaparidze–Nikulin Statistics and Some Ideas for a More Powerful Test Construction

An explicit decomposition on asymptotically independent distributed as chi-squared with one degree of freedom components of the Pearson–Fisher and Dzhaparidze–Nikulin tests is presented. The decomposition is formally the same for both tests and is valid for any partitioning of a sample space. Vector-valued tests, components of which can be not only different scalar tests based on the same sample, but also scalar tests based on components or groups of components of the same statistic are considered. Numerical examples illustrating the idea are presented.     

Decompositions by sources and by subpopulations of the Pietra index: two applications to professional football teams in Italy

In this paper two innovative procedures for the decomposition of the Pietra index are proposed. The first one allows the decomposition by sources, while the second one provides the decomposition by subpopulations. As special case of the latter procedure, the “classical” decomposition in two components (within and between) can be easily obtained. A remarkable feature of both the proposed procedures is that they permit the assessment of the contribution to the Pietra index at the smallest possible level: each source for the first one and each subpopulation for the second one. To highlight the usefulness of these procedures, two applications are provided regarding Italian professional football (soccer) teams.

     

A decomposition of the Bradley–Blackwood paired-samples omnibus test

We show that the Bradley–Blackwood simultaneous test for equal means and equal variances in paired-samples additively decomposes into separate tests of these hypotheses. The test of equal variances in the decomposition is the standard Pitman–Morgan procedure. The test of equal means in the decomposition is based on a t-ratio with (n ? 2) degrees of freedom and has the additional restriction that the variances are equal.     

All possible subset regressions using the triangular decomposition

We propose a new algorithm for carrying out all possible subset regressions using the triangular decomposition. This algorithm is simple but very efficient computationally when compared with the existing algorithms based on sweepings or the QR decomposition.     

Separating trend and cyclical dynamics in state space models with exogenous inputs

In the present study we compare three state rotation methods in modelling the impact of the US economy on the Finnish economy, i.e. Schur decomposition, eigenvalue analysis and singular value decomposition. Singular value decomposition is seen to provide a robust approximation of the state rotation in most cases studied, irrespective of whether the characteristic roots of the state transition matrix are complex. Thus, singular value decomposition seems to be a viable computational device not only in estimating the system matrices of the state space model, but also in state rotation, as compared to the more involved techniques based on eigenvalue analysis or Schur decomposition.     

A Note on the Factorization of a Matrix Inverse

The decomposition of a matrix as a product of a lower triangular with ones on the diagonal and an upper triangular matrix is useful for solving systems of linear equations. For a given non singular matrix, this type of decomposition is unique and algorithms exist to obtain the two factors. However, in certain problems the factorization of the inverse matrix may be of interest. This note presents an algorithm for factoring the inverse matrix using simple operations of elements from the original matrix. As examples, we give factorizations for several well-known and widely used correlation matrices. The usefulness and practicality of these factorizations are provided in an application of statistical modeling using unbiased estimating equations.     

The generalized symmetric eigenproblem in multivariate statistical models

The solution of the generalized symmetric eigenproblem Ax = λBx is required in many multivariate statistical models, viz. canonical correlation, discriminant analysis, multivariate linear model, limited information maximum likelihoods. The problem can be solved by two efficient numerical algorithms: Cholesky decomposition or singular value decomposition. Practical considerations for implementation are also discussed.     

城镇劳动力市场上的性别工资差异问题研究

经济转型以来，中国城镇劳动力市场上女性工资与男性工资存在很大差距。文章利用Brown分解方法对中国城镇职工的性别工资差距进行了分解。分解结果显示，职业内工资差异与职业间工资差异分别为19．8％和18．4％，即存在比较严重的纵向和横向性别职业隔离。此外，受教育程度、工作年限等人力资本水平对男女工资差异有显著的影响，职业性别歧视也是导致城镇劳动力市场上男女工资差异的主要原因。     

On equivalence of fractional factorial designs based on singular value decomposition

The singular value decomposition of a real matrix always exists and is essentially unique. Based on the singular value decomposition of the design matrices of two general 2-level fractional factorial designs, new necessary and sufficient conditions for the determination of combinatorial equivalence or non-equivalence of the corresponding designs are derived. Equivalent fractional factorial designs have identical statistical properties for estimation of factorial contrasts and for model fitting. Non-equivalent designs, however, may have the same statistical properties under one particular model but different properties under a different model. Results extend to designs with factors at larger number of levels.     

Identifying Variables Contributing to Outliers in Phase I

When a process is monitored with a T ² control chart in a Phase II setting, the MYT decomposition is a valuable diagnostic tool for interpreting signals in terms of the process variables. The decomposition splits a signaling T ² statistic into independent components that can be associated with either individual variables or groups of variables. Since these components are T ² statistics with known distributions, they can be used to determine which of the process variable(s) contribute to the signal. However, this procedure cannot be applied directly to Phase I since the distributions of the individual components are unknown. In this article, we develop the MYT decomposition procedure for a Phase I operation, when monitoring a random sample of individual observations and identifying outliers. We use a relationship between the T ² statistic in Phase I with the corresponding T ² statistic resulting when an observation is omitted from this sample to derive the distributions of these components and demonstrate the Phase I application of the MYT decomposition.     

Modeling Financial Return Dynamics via Decomposition

While the predictability of excess stock returns is detected by traditional predictive regressions as statistically small, the direction-of-change and volatility of returns exhibit a substantially larger degree of dependence over time. We capitalize on this observation and decompose the returns into a product of sign and absolute value components whose joint distribution is obtained by combining a multiplicative error model for absolute values, a dynamic binary choice model for signs, and a copula for their interaction. Our decomposition model is able to incorporate important nonlinearities in excess return dynamics that cannot be captured in the standard predictive regression setup. The empirical analysis of U.S. stock return data shows statistically and economically significant forecasting gains of the decomposition model over the conventional predictive regression.     

CATANOVA for two-way cross classified categorical data

In this article we develop an extension of categorical analysis of variance for one response and two factors, based on a partitioning of a measure of predictability for three-way contingency tables, known as Gray and Williams's index. At the first instance moment the decomposition of this multiple measure of association in partial association measures is shown. Finally, for ordinal-scale variables, we propose an extension of this decomposition using a particular set of orthogonal polynomials.