对应分析法的改进与应用

本文对国内外流行的对应分析法的不足,在尽可能多地保留变量信息的原则下,提出了改进的对应分析优化模型,论证了R型的因子分析模型L的因子双重信息图是对应分析优化模型的解,并结合应用实例给出了该模型的应用步骤。     

地区恶性肿瘤死亡率的对应分析

目的—了解山东省某县2000-2002年恶性肿瘤的地区分布和肿瘤类型分布特征.方法—应用分组对应分析对该县恶性肿瘤死亡资料进行分析.结果—得到各地区和各肿瘤类型的公因子及其负荷系数,并根据第一、二因子负荷系数绘制因子负荷平面图,可以清楚看出恶性肿瘤死亡率的聚集性及其高发地与低发地的分布.结论—将变量与样本结合起来的对应分析是对因子分析的有益补充,它可以分析二维数据阵的行因素与列因素之关系,达到研究目的.     

对应分析在恶性肿瘤死亡率分析中的应用

应用对应分析对 1996年福建省主要恶性肿瘤与地区分布资料进行分析 ,通过因子载荷系数将样品与变量点画在同一张因子载荷图上 ,从而能直观看出这些地区的癌谱特点。在研究样品与变量间关系时 ,对应分析有着一定的优越性。     

稳健性因子分析在城镇居民家庭现金消费支出中的应用

利用面向对象的稳健性因子分析R软件包Robustfa,对2011年全国除港、澳、湾以外的31个省、市、自治区的城镇居民家庭现金消费支出的8个指标进行了因子分析.通过使残差矩阵的元素平方和达到最小,发现了一个组合一主因子法与稳健性Mve估计量.通过由稳健性Mve估计量计算的马氏距离大于临界值,我们发现共有10个异常点.用经典估计量和稳健性Mve估计量计算的样本相关阵、旋转后的因子载荷矩阵、因子对原始变量的贡献、贡献率、累积贡献率、样本相关阵的特征值的碎石图、前两个因子得分的散点图、因子得分、按因子得分排序等结果均有较大的不同.最后通过组合主因子法与稳健性Mve估计量将8个指标归结为两个因子:基础消费因子和消费倾向因子,根据每个省份的两个因子得分情况对该省份的家庭现金消费支出情况作出综合评价,并根据稳健性因子分析的结果给出了相应建议.     

多变量样本的图分析法(一)

<正> 图形是帮助人们思维和判断的重要工具,当样本只有两个特性(变量或指标)时,可以用通常的直角坐标在平面上点图,当样本有三个变量时,虽然可以在三维的笛卡儿坐标里点图,但也是很不方便的,当变量数大于三时,用通常的方法已不能点图了.在多元分析中,样本的变量数一般均大于三,探讨多变量的点图法是长期来一直为人们所关注的研究课题,这里介绍一些有关的方法,特别是近十年来发展的一些方法.     

星座图在环境统计分析中的应用

统计图是帮助人们理解、思维和判断的重要工具.在环境统计分析中,常用统计图’来阐明环境变化的发展趋势和评价环境质量.但是,当样品的指标(变量)多于三维时,用通常的方法已不能作图.因此,探讨多指标的统计图是长期以来一直为人们所关注的研究课题.近来,虽然出现了脸谱图、连接向量图、三角多项式图等多指标统计图,但这些图形需要在电子计算机的帮助下才能使用.本文介绍的星座图,它是将样品的各个指标先通过一定的数据变换再绘制在一个半圆内,根据星座分布的趋势来刻划每个样品特征的图示方法.这种图形的计算较为简单,可以人工实现,且具有聚…     

图表信息题归类解析

图形、图象、表格是数学信息简明而又直观的表现形式 ,并且大多数都以应用问题的面貌出现 .答题者通过阅读材料、观察图表、分析数据 ,从中获取相关信息 ,从而发现规律 ,找出解决问题的方法 .一、表格信息题表格信息题是指题目给出相关表格 ,通过阅读表格 ,捕捉解题信息 ,分析给定的图表中的相关数据、相关隐含量之间的关系 ,用表格中的数据建立数学模型 ,经过推理计算把问题解决 .例 1　 ( 2 0 0 3年天津市中考题 ) [注意 :为了使同学们更好地解答本题 ,我们提供了一种解题思路 ,你可以按照这个思路 ,填写表格 ,并完成本题的全过程 .如果你…     

基于问题解决的数据分析能力培养--以《统计表和条形统计图》教学为例

统计是一门收集数据、分析数据并根据数据进行推断的艺术和科学.统计的目的是解决问题,在解决问题的过程中掌握数据处理的知识和方法,养成用数据思考问题的意识和习惯.但传统的统计教学忽视了统计教学的育人价值、数据分析能力培养及用数据解决问题的体验.基于问题解决的数据分析能力的培养,要注重培养学生用数据解决问题的兴趣,感受数据解决问题的价值,体验数据解决问题的过程.     

基于主成分分析的单变量时间序列聚类方法

针对时间序列数据的高维特性,在进行理论分析的基础上,利用主成分分析法提出了一种单变量时间序列数据降维的新方法,进而提出了基于主成分分析的单变量时间序列聚类方法。其主要思想是在线性空间中的同一组基下,用系数之间的相似性来刻画对应时间序列之间相似性,在理论分析过程中,首先对单变量时间序列数据集进行主成分分析,其次分析了单变量时间序列数据集、样本协方差矩阵的特征向量与主成分之间的关系,并证明了由主成分构成的向量组线性无关。为了进一步验证理论分析结果的正确性和所提算法的有效性,分别利用仿真数据和真实的股票数据进行了数值实验。     

多动点问题分类解析

多动点问题是高考重点内容之一,同学们常感困难,甚至无从入手,本文结合实例介绍这类问题的常见类型及处理方法,供同学们学习参考.一、相依型此类型指动点之间相互依赖,当一个动点固定后,其余动点也随之确定.解决该类型问题的途径是先固定一个动点,选用某个参数(变量)表示该点坐标,其余各动点坐标也相应用此参数来表示,从而达到解决问题的目的.     

Contribution Biplots

To interpret the biplot, it is necessary to know which points—usually variables—are the ones that are important contributors to the solution, especially when there are many variables involved. This information can be calculated separately as part of the biplot's numerical results, but this means that a table has to be consulted along with the graphical display. We propose a new scaling of the display, called the contribution biplot, which incorporates this diagnostic information directly into the display itself, showing visually the important contributors and thus facilitating the biplot interpretation and often simplifying the graphical representation considerably. The contribution biplot can be applied to a wide variety of analyses, such as correspondence analysis, principal component analysis, log-ratio analysis, and various forms of discriminant analysis, and, in fact, to any method based on dimension reduction through the singular value decomposition. In the contribution biplot, one set of points, usually the rows of a data matrix, optimally represents the spatial positions of the cases or sample units, according to an appropriate distance measure. The other set of points, usually the columns of the data matrix, is represented by vectors that are related to their contributions to the low-dimensional solution. A fringe benefit is that often only one common scale for the row and column points is needed on the principal axes, thus avoiding the problem of enlarging or contracting the scale of one set of points to make the biplot legible. Furthermore, the contribution biplot also solves the problem in correspondence analysis and log-ratio analysis of low-frequency categories that are located on the periphery of the map, giving the false impression that they are important, when they are in fact contributing minimally to the solution. This article has supplementary materials online.     

Constrained principal component analysis of standardized data for biplots with unit-length variable vectors

Principal component analysis (PCA) of an objects ×  variables data matrix is used for obtaining a low-dimensional biplot configuration, where data are approximated by the inner products of the vectors corresponding to objects and variables. Borg and Groenen (Modern multidimensional scaling. Springer, New York, 1997) have suggested another biplot procedure which uses a technique for approximating data by projections of object vectors on variable vectors. This technique is formulated as constraining the variable vectors in PCA to be of unit length and can be called unit-length vector analysis (UVA). However, an algorithm for UVA has not yet been developed. In this paper, we present such an algorithm, discuss the properties of UVA solutions, and demonstrate the advantage of UVA in biplots for standardized data with homogeneous variances among variables. The advantage of UVA-based biplots is that the projections of object vectors onto variable vectors express the approximation of data in an easy way, while in PCA-based biplots we must consider not only the projections, but also the lengths of variable vectors in order to visualize approximations.     

Logistic biplot for nominal data

Classical biplot methods allow for the simultaneous representation of individuals (rows) and variables (columns) of a data matrix. For binary data, logistic biplots have been recently developed. When data are nominal, both classical and binary logistic biplots are not adequate and techniques such as multiple correspondence analysis (MCA), latent trait analysis (LTA) or item response theory (IRT) for nominal items should be used instead. In this paper we extend the binary logistic biplot to nominal data. The resulting method is termed “nominal logistic biplot”(NLB), although the variables are represented as convex prediction regions rather than vectors. Using the methods from computational geometry, the set of prediction regions is converted to a set of points in such a way that the prediction for each individual is established by its closest “category point”. Then interpretation is based on distances rather than on projections. We study the geometry of such a representation and construct computational algorithms for the estimation of parameters and the calculation of prediction regions. Nominal logistic biplots extend both MCA and LTA in the sense that they give a graphical representation for LTA similar to the one obtained in MCA.     

Biplots of fuzzy coded data

A biplot, which is the multivariate generalization of the two-variable scatterplot, can be used to visualize the results of many multivariate techniques, especially those that are based on the singular value decomposition. We consider data sets consisting of continuous-scale measurements, their fuzzy coding and the biplots that visualize them, using a fuzzy version of multiple correspondence analysis. Of special interest is the way quality of fit of the biplot is measured, since it is well known that regular (i.e., crisp) multiple correspondence analysis seriously under-estimates this measure. We show how the results of fuzzy multiple correspondence analysis can be defuzzified to obtain estimated values of the original data, and prove that this implies an orthogonal decomposition of variance. This permits a measure-of-fit to be calculated in the familiar form of a percentage of explained variance, which is directly comparable to the corresponding fit measure used in principal component analysis of the original data. The approach is motivated initially by its application to a simulated data set, showing how the fuzzy approach can lead to diagnosing nonlinear relationships, and finally it is applied to a real set of meteorological data.     

The Interval-Censored Biplot

The principal components biplot is a useful visualization tool for the exploration of a samples by variables data matrix. In several data analysis situations, the data values are interval censored so that only the interval of a data value is available, but not the value itself. For such data, we propose the interval-censored biplot (IC-Biplot), a new exploratory and graphical method that is an extension of the principal component analysis biplot. It provides not only a two-dimensional graphic representation of respondents and their attributes, but also point estimates for the data values that are constrained to be in their interval. Two applications of the IC-Biplot are discussed. The first application considers data on emergence times of permanent teeth focusing on the pattern of emergence. The IC-Biplot confirms rank orders suggested earlier in the literature. Goodness-of-fit measures show that the present model seems to fit these data very well. The second application discusses a regular sample by attribute matrix from the literature on characteristics of several types of oils. This article has online supplementary materials.     

一类不可微二次规划逆问题

本文求解了一类二次规划的逆问题,具体为目标函数是矩阵谱范数与向量无穷范数之和的最小化问题.首先将该问题转化为目标函数可分离变量的凸优化问题,提出用G-ADMM法求解.并结合奇异值阈值算法,Moreau-Yosida正则化算法,matlab优化工具箱的quadprog函数来精确求解相应的子问题.而对于其中一个子问题的精确求解过程中发现其仍是目标函数可分离变量的凸优化问题,由于其变量都是矩阵,所以采用适合多个矩阵变量的交替方向法求解,通过引入新的变量,使其每个子问题的解都具有显示表达式.最后给出采用的G-ADMM法求解本文问题的数值实验.数据表明,本文所采用的方法能够高效快速地解决该二次规划逆问题.     

Are middle school mathematics teachers able to solve word problems without using variable?

Many people consider problem solving as a complex process in which variables such as x,?y are used. Problems may not be solved by only using ‘variable.’ Problem solving can be rationalized and made easier using practical strategies. When especially the development of children at younger ages is considered, it is obvious that mathematics teachers should solve problems through concrete processes. In this context, middle school mathematics teachers' skills to solve word problems without using variables were examined in the current study. Through the case study method, this study was conducted with 60 middle school mathematics teachers who have different professional experiences in five provinces in Turkey. A test consisting of five open-ended word problems was used as the data collection tool. The content analysis technique was used to analyze the data. As a result of the analysis, it was seen that the most of the teachers used trial-and-error strategy or area model as the solution strategy. On the other hand, the teachers who solved the problems using variables such as x, a, n or symbols such as Δ, □, ○, * and who also felt into error by considering these solutions as without variable were also seen in the study.     

Sparsest factor analysis for clustering variables: a matrix decomposition approach

We propose a new procedure for sparse factor analysis (FA) such that each variable loads only one common factor. Thus, the loading matrix has a single nonzero element in each row and zeros elsewhere. Such a loading matrix is the sparsest possible for certain number of variables and common factors. For this reason, the proposed method is named sparsest FA (SSFA). It may also be called FA-based variable clustering, since the variables loading the same common factor can be classified into a cluster. In SSFA, all model parts of FA (common factors, their correlations, loadings, unique factors, and unique variances) are treated as fixed unknown parameter matrices and their least squares function is minimized through specific data matrix decomposition. A useful feature of the algorithm is that the matrix of common factor scores is re-parameterized using QR decomposition in order to efficiently estimate factor correlations. A simulation study shows that the proposed procedure can exactly identify the true sparsest models. Real data examples demonstrate the usefulness of the variable clustering performed by SSFA.     

The Distance Perspective of Generalized Biadditive Models: Scalings and Transformations

Recently two articles studied scalings in biplot models, and concluded that these have little impact on the interpretation. In this article again scalings are studied for generalized biadditive models and correspondence analysis, that is, special cases of the general biplot family, but from a different perspective. The generalized biadditive models, but also correspondence analysis, are often used for Gaussian ordination. In Gaussian ordination one takes a distance perspective for the interpretation of the relationship between a row and a column category. It is shown that scalings—but also nonsingular transformations—have a major impact on this interpretation. So, depending on the perspective one takes, the inner product or distance perspective, scalings and transformations do have (distance) or do not have (inner-product) impact on the interpretation. If one is willing to go along with the assumption of the author that diagrams are in practice often interpreted by a distance rule, the findings in this article influence all biplot models.