Rotation in Correspondence Analysis

In correspondence analysis rows and columns of a nonnegative data matrix are depicted as points in a, usually, two-dimensional plot. Although such a two-dimensional plot often provides a reasonable approximation, the situation can occur that an approximation of higher dimensionality is required. This is especially the case when the data matrix is large. In such instances it may become difficult to interpret the solution. Similar to what is done in principal component analysis and factor analysis the correspondence analysis solution can be rotated to increase the interpretability. However, due to the various scaling options encountered in correspondence analysis, there are several alternative options for rotating the solutions. In this paper we consider two options for rotation in correspondence analysis. An example is provided so that the benefits of rotation become apparent.     

中国图学名词溯源

图学是中国古代科技史的重要组成部分。图和图样名词的厘定映射了中国古代图学的科学成就。这些有关图学的名词,所涉学科甚多,门类甚广,表现了极为深远的图学社会化内容。文章以此为题,探讨了中国古代地图、建筑图、机械制图等图学名词的历史源流。     

Some Interpretative Tools for Non-Symmetrical Correspondence Analysis

Non-symmetrical correspondence analysis (NSCA) is a very practical statistical technique for the identification of the structure of association between asymmetrically related categorical variables forming a contingency table. This paper considers some tools that can be used to numerically and graphically explore in detail the association between these variables and include the use of confidence regions, the establishment of the link between NSCA and the analysis of variance of categorical variables, and the effect of imposing linear constraints on a variable. The authors would like to thank the anonymous referees for their comments and suggestions during the preparation of this paper.     

Correspondence Analysis with Linear Constraints of Ordinal Cross-Classifications

Within the non-iterative procedures for performing a correspondence analysis with linear constraints, a strategy is proposed to impose linear constraints in analyzing a contingency table with one or two ordered sets of categories. At the heart of the approach is the partition of the Pearson chi-squared statistics which involves terms that summarize the association between the nominal/ordinal variables using bivariate moments based on orthogonal polynomials. Linear constraints are then included directly in suitable matrices reflecting the most important components, overcoming also the problem of imposing linear constraints based on subjective decisions.     

Handling Missing Values with Regularized Iterative Multiple Correspondence Analysis

A common approach to deal with missing values in multivariate exploratory data analysis consists in minimizing the loss function over all non-missing elements, which can be achieved by EM-type algorithms where an iterative imputation of the missing values is performed during the estimation of the axes and components. This paper proposes such an algorithm, named iterative multiple correspondence analysis, to handle missing values in multiple correspondence analysis (MCA). The algorithm, based on an iterative PCA algorithm, is described and its properties are studied. We point out the overfitting problem and propose a regularized version of the algorithm to overcome this major issue. Finally, performances of the regularized iterative MCA algorithm (implemented in the R-package named missMDA) are assessed from both simulations and a real dataset. Results are promising with respect to other methods such as the missing-data passive modified margin method, an adaptation of the missing passive method used in Gifi’s Homogeneity analysis framework.     

Multiple Correspondence Analysis via Polynomial Transformations of Ordered Categorical Variables

We present an alternative approach to Multiple Correspondence Analysis (MCA) that is appropriate when the data consist of ordered categorical variables. MCA displays objects (individuals, units) and variables as individual points and sets of category points in a low-dimensional space. We propose a hybrid decomposition on the basis of the classical indicator super-matrix, using the singular value decomposition, and the bivariate moment decomposition by orthogonal polynomials. When compared to standard MCA, the hybrid decomposition will give the same representation of the categories of the variables, but additionally, we obtain a clear association interpretation among the categories in terms of linear, quadratic and higher order components. Moreover, the graphical display of the individual units will show an automatic clustering.     

塔斯基的真理论与符合论

塔斯基的语义真理论常被当作符合论.通过区分塔斯基式真定义的普遍特征和具体的定义形式,并对不同语言构造具有不同语义特征的定义实例,揭示出塔斯基所持的是一种语义工具论.塔斯基的真理论并不是符合论.     

李俨与史密斯的通信

美国哥伦比亚大学珍本与手稿图书馆史密斯(David Eugene Smith)档案和中国科学院自然科学史研究所图书馆藏有近20通李俨(1892-1963)和史密斯(1860-1944)通信的正本、副本及草稿.这些通信的主要内容是有关合作撰写英文本中国数学史.尽管他们的合作没有成功,但这些信件是研究中国数学史学史的重要材料.     

Optimal Variable Weighting for Ultrametric and Additive Trees and K-means Partitioning: Methods and Software

K -means partitioning. We also describe some new features and improvements to the algorithm proposed by De Soete. Monte Carlo simulations have been conducted using different error conditions. In all cases (i.e., ultrametric or additive trees, or K-means partitioning), the simulation results indicate that the optimal weighting procedure should be used for analyzing data containing noisy variables that do not contribute relevant information to the classification structure. However, if the data involve error-perturbed variables that are relevant to the classification or outliers, it seems better to cluster or partition the entities by using variables with equal weights. A new computer program, OVW, which is available to researchers as freeware, implements improved algorithms for optimal variable weighting for ultrametric and additive tree clustering, and includes a new algorithm for optimal variable weighting for K-means partitioning.     

图型形象思维的研究及应用前景

本文首次将形象思维具体到图型形象上,从而使形象思维的研究有了切实可行的方法,更有效地促进形象思维在人工智能领域的应用。     

A Property of the CHAID Partitioning Method for Dichotomous Randomized Response Data and Categorical Predictors

In this paper, we present empirical and theoretical results on classification trees for randomized response data. We considered a dichotomous sensitive response variable with the true status intentionally misclassified by the respondents using rules prescribed by a randomized response method. We assumed that classification trees are grown using the Pearson chi-square test as a splitting criterion, and that the randomized response data are analyzed using classification trees as if they were not perturbed. We proved that classification trees analyzing observed randomized response data and estimated true data have a one-to-one correspondence in terms of ranking the splitting variables. This is illustrated using two real data sets.     

The Metric Bridge Partition Problem: Partitioning of a Metric Space into Two Subspaces Linked by an Edge in Any Optimal Realization

Let G = (V,E,w) be a graph with vertex and edge sets V and E, respectively, and w:E → R ⁺ a function which assigns a positive weight or length to each edge of G. G is called a realization of a finite metric space (M,d), with M = { 1,...,n} if and only if { 1,...,n} ⫅ V and d(i,j) is equal to the length of the shortest chain linking i and j in G ∀ i,j = 1,...,n. A realization G of (M,d), is said optimal if the sum of its weights is minimal among all the realizations of (M,d). Consider a partition of M into two nonempty subsets K and L, and let e be an edge in a realization G of (M,d); we say that e is a bridge linking K with L if e belongs to all chains in G linking a vertex of K with a vertex of L. The Metric Bridge Partition Problem is to determine if the elements of a finite metric space (M,d) can be partitioned into two nonempty subsets K and L such that all optimal realizations of (M,d) contain a bridge linking K with L. We prove in this paper that this problem is polynomially solvable. We also describe an algorithm that constructs an optimal realization of (M,d) from optimal realizations of (K,d|_K) and (L,d|_L).     

李俨、严敦杰往来书信(1940—1941年)(三)

李俨致严敦杰信1941年敦杰先生大鉴:读一月二十三日大示及徐树勋刊<圜锥曲线>(内卷一缺末一页,请设法抄副见示,以便订成全帙)及二十日来电并收及.     

李俨、严敦杰往来书信(1940—1941年)(二)

李俨致严敦杰信1940年敦杰先生惠鉴: 二星期来未接来示,甚以为念. 兹阅及道光二十年<俄罗斯进呈书籍目录>有算书数种,兹抄上,请于参考完毕原纸见掷为荷. 此间日来阴雨,甚平静.兄处如何? 尊稿祖暅别传,因两星期来到外段勘工,未及抄出.一俟抄出,再行呈寄不误.     

李俨、严敦杰往来书信(1940—1941年)(一)

科学史研究在中国已有近百年的历史,从学术史的角度回顾这段历史也日渐成为迫切的研究课题,为此目的,本刊特开辟史家通信专栏,希冀通过对史料的发掘和整理,推动对中国科学史学科史的研究。     

世界军事科技发展对我军的挑战

本文从世界军事发展的必然趋势入手，深入分析我军当前科技人才匮乏、素质亟待提高，科技练兵科技含量不足和军事科研活动滞缓的机制及其文化成因，并提出管窥之见。     

休闲的生态哲学解析

休闲本身就是生态的。休闲的生态本质能促使人类走向生态文明。休闲不仅关系到个体人的生态生存与生态发展,也关系到人类社会生态文明的建设。休闲有助于维护人体生态系统的平衡;有助于促进人在社会生态系统中的全面发展;有助于满足人的精神系统的需要。休闲可以通过丰富人与人、人与社会的有机联系,促进社会全面平衡发展,最终促使社会走向生态文明。在休闲体验中能真正实现人与自然的互动,从而使人与自然共同走向生态文明。