The exploratory analysis of qualitative variables by means of three-way analysis of two types of quantification matrices

A comparison is made between a number of techniques for the exploratory analysis of qualitative variables. The paper mainly focuses on a comparison between multiple correspondence analysis (MCA) and Gower's principal co-ordinates analysis (PCO), applied to qualitative variables. The main difference between these methods is in how they deal with infrequent categories. It is demonstrated that MCA solutions can be dominated by infrequent categories, and that, especially in such cases, PCO is a useful alternative to MCA, because it tends to downweight the influence of infrequent categories. Apart from studying the difference between MCA and PCO, other alternatives for the analysis of qualitative variables are discussed, and compared to MCA and PCO.     

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.     

Regular polygons in taxicab geometry

A polygon of n sides will be called regular in taxicab geometry if it has n equal angles and n sides of equal taxicab length. This paper will show that there are no regular taxicab triangles and no regular taxicab pentagons. The sets of taxicab rectangles and taxicab squares will be shown to be the same, respectively, as the sets of Euclidean rectangles and Euclidean squares. A method of construction for a regular taxicab 2n-gon for any n will be demonstrated.     

Orthogonal rotation in PCAMIX

Kiers (Psychometrika 56:197–212, 1991) considered the orthogonal rotation in PCAMIX, a principal component method for a mixture of qualitative and quantitative variables. PCAMIX includes the ordinary principal component analysis and multiple correspondence analysis (MCA) as special cases. In this paper, we give a new presentation of PCAMIX where the principal components and the squared loadings are obtained from a Singular Value Decomposition. The loadings of the quantitative variables and the principal coordinates of the categories of the qualitative variables are also obtained directly. In this context, we propose a computationally efficient procedure for varimax rotation in PCAMIX and a direct solution for the optimal angle of rotation. A simulation study shows the good computational behavior of the proposed algorithm. An application on a real data set illustrates the interest of using rotation in MCA. All source codes are available in the R package “PCAmixdata”.     

Multi-criteria analysis and the resolution of sustainable development dilemmas: A stakeholder management approach

We demonstrate that stakeholder-oriented multi-criteria analysis (MCA) can adequately address a variety of sustainable development dilemmas in decision-making, especially when applied to complex project evaluations involving multiple objectives and multiple stakeholder groups. Such evaluations are typically geared towards satisfying simultaneously private economic goals, broader social objectives and environmental targets. We show that, under specific conditions, a variety of stakeholder-oriented MCA approaches may be able to contribute substantively to the resolution or improved governance of societal conflicts and the pursuit of the public good in the form of sustainable development. We contrast the potential usefulness of these stakeholder-oriented approaches – in terms of their ability to contribute to sustainable development – with more conventional MCA approaches and social cost–benefit analysis.     

A comparison of multiple criteria analysis techniques for water resource management

Multiple criteria analysis (MCA) is a framework for evaluating decision options against multiple criteria. Numerous techniques for solving an MCA problem are available. This paper applies MCA to six water management decision problems. The MCA methods tested include weighted summation, range of value, PROMTHEE II, Evamix and compromise programming. We show that different MCA methods were in strong agreement with high correlations amongst rankings. In the few cases where strong disagreement between MCA methods did occur it was due to presence of mixed ordinal-cardinal data in the evaluation matrix. The results suggest that whilst selection of the MCA technique is important more emphasis is needed on the initial structuring of the decision problem, which involves choosing criteria and decision options.     

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.     

On the taxicab distance sum function and its applications in discrete tomography

Periodica Mathematica Hungarica - Let a finite set $$F\subset \mathbb {R}^n$$ be given. The taxicab distance sum function is defined as the sum of the taxicab distances from the elements (focuses)...     

Interpreting multiple correspondence analysis

The practical aspects of interpreting multiple correspondence analysis are reviewed. The geometric concepts associated with simple correspondence analysis are shown to be inadequate for multiple correspondence analysis. An alternative approach, called joint correspondence analysis, is shown to be a more natural generalization of the simple case. The practical interpretation of homogeneity analysis is also discussed and a compromise is proposed between classical multiple correspondence analysis and joint correspondence analysis, preserving the optimal scaling properties of homogeneity analysis. A simple example is used throughout the discussion to illustrate the issues involved.     

同步N—策略多重休假M／M／c排队

研究了具有同步N-策略多重休假的M／M／c排队系统．在休假时间服从相型（PH）分布的假设下，给出了系统的稳态指标．证明在已知服务台全忙并且系统中顾客数大于或等于N的条件下，条件随机变量可分解成独立随机变量之和，其中一个是无休假经典M／M／c系统中的对应条件变量，另一个是休假引起的附加随机变量     

On the Average Taxicab Distance Function and Its Applications

Generalized conics are subsets in the space all of whose points have the same average distance from a given set of points (focal set). The function measuring the average distance is called the average distance function (or the generalized conic function). In general it is a convex function satisfying a kind of growth condition as the preliminary results of Sect. 2 show. Therefore any sublevel set is convex and compact. We can also conclude that such a function has a global minimizer.

The paper is devoted to the special case of the average taxicab distance function given by integration of the taxicab distance on a compact subset of positive Lebesgue measure in the Euclidean coordinate space.

The first application of the average taxicab distance function is related to its minimizer. It is uniquely determined under some natural conditions such as, for example, the connectedness of the integration domain. Geometrically, the minimizer bisects the measure of the integration domain in the sense that each coordinate hyperplane passing through the minimizer cuts the domain into two parts of equal measure. The convexity and the Lipschitzian gradient property allow us to use the gradient descent algorithm that is formulated in terms of a stochastic algorithm (Sect. 3) to find the bisecting point of a set in \(\mathbb{R}^{n}\).

Example 1 in Sect. 4 shows the special form of the average taxicab distance function of a convex polygon. The level curves (generalized conics) admit semidefinite representations as algebraic curves in the plane because the average taxicab distance function is piecewise polynomial of degree at most three.

Some applications in geometric tomography are summarized as our main motivation to investigate the concept of the average taxicab distance function. Its second order partial derivatives give the coordinate \(X\)-rays of the integration domain almost everywhere and vice versa: the average taxicab distance function can be expressed in terms of the coordinate \(X\)-rays. Therefore the reconstruction of the sets given by their coordinate \(X\)-rays can be based on the average taxicab distance function instead of the direct comparison of the \(X\)-rays. In general (especially, in some classes of non-convex sets), the convergence property of the average taxicab distance function with respect to the Hausdorff convergence of the integration domain is better than the convergence of the \(X\)-rays (see regular and \(X\)-regular convergences in Sect. 4.1) and we can apply a standard approximation paradigm (Footnote 1) to solve the problem.

In the last section we prove that any compact convex body is uniquely determined by the diagonalization of its covariogram function measuring the average taxicab distance of the points from the intersection of the body with the translates of its axis parallel bounding box.

     

A cutting plane algorithm for semi-definite programming problems with applications to failure discriminant analysis

We will propose a new cutting plane algorithm for solving a class of semi-definite programming problems (SDP) with a small number of variables and a large number of constraints. Problems of this type appear when we try to classify a large number of multi-dimensional data into two groups by a hyper-ellipsoidal surface. Among such examples are cancer diagnosis, failure discrimination of enterprises. Also, a certain class of option pricing problems can be formulated as this type of problem. We will show that the cutting plane algorithm is much more efficient than the standard interior point algorithms for solving SDP.     

The taxicab locus of Apollonius: promoting exploratory routes to the punchline using rich undergraduate tasks

ABSTRACT

A key motivational tactic in undergraduate mathematics teaching is to launch topics with fundamental questions that originate from surprising or remarkable phenomena. Nonetheless, constructing a sequence of tasks that promotes students' own routes to resolving such questions is challenging. This note aims to address this challenge in two ways. First, to illustrate the motivational tactic, the taxicab manifestation of a locus attributed to Apollonius is introduced and a natural question arising from comparison with the analogous Euclidean locus is considered, namely, does the taxicab locus of Apollonius ever coincide with a taxicab circle? Second, a companion sequence of rich undergraduate tasks is elaborated using theoretical design principles, with the tasks culminating in this fundamental geometric question. This note therefore provides a design approach that can be replicated in undergraduate teaching contexts based around similarly motivating mathematical phenomena.     

因子双重信息图在识别城市重金属污染中的应用

对国外流行的Beozecri对应分析法,这里用变量型数据阵指出该方法很大程度改变了数据阵的特征,不能达到对应分析目的,以致不能解决问题.为此,这里用因子双重信息图解决问题,通过比较,因子双重信息图优良地图示了数据阵中:变量之间、样品之间、样品与变量之间的关系,达到了对应分析目的,方法直接且简便,因子双重信息图较适应变量型数据阵这类问题的对应分析.     

S-PLUS code for simple and multiple correspondence analysis

An exploration of the application of S-PLUS code designed to perform classical correspondence analysis is made in this paper. This code allows for the “classical” analysis to be performed on categorical data consisting of two or more variables. For multi-way contingency tables, correspondence analysis can be performed by considering either the indicator matrix or the Burt matrix. The code also allows the user to incorporate into the analysis confidence circles (to identify categorical responses that deviate from the hypothesis of independence) and for an asymmetrical analysis to be performed. The function includes various warnings and stoppages to help the user properly analyse their data.     

On the classical limit of quantum mechanics,fundamental graininess and chaos: Compatibility of chaos with the correspondence principle

The aim of this paper is to review the classical limit of Quantum Mechanics and to precise the well known threat of chaos (and fundamental graininess) to the correspondence principle. We will introduce a formalism for this classical limit that allows us to find the surfaces defined by the constants of the motion in phase space. Then in the integrable case we will find the classical trajectories, and in the non-integrable one the fact that regular initial cells become “amoeboid-like”. This deformations and their consequences can be considered as a threat to the correspondence principle unless we take into account the characteristic timescales of quantum chaos. Essentially we present an analysis of the problem similar to the one of Omnès (1994,1999), but with a simpler mathematical structure.     

Computing Bounds on the Expected Maximum of Correlated Normal Variables

We compute upper and lower bounds on the expected maximum of correlated normal variables (up to a few hundred in number) with arbitrary means, variances, and correlations. Two types of bounding processes are used: perfectly dependent normal variables, and independent normal variables, both with arbitrary mean values. The expected maximum for the perfectly dependent variables can be evaluated in closed form; for the independent variables, a single numerical integration is required. Higher moments are also available. We use mathematical programming to find parameters for the processes, so they will give bounds on the expected maximum, rather than approximations of unknown accuracy. Our original application is to the maximum number of people on-line simultaneously during the day in an infinite-server queue with a time-varying arrival rate. The upper and lower bounds are tighter than previous bounds, and in many of our examples are within 5% or 10% of each other. We also demonstrate the bounds’ performance on some PERT models, AR/MA time series, Brownian motion, and product-form correlation matrices.     

Relation between the correspondence chromatic number and the Alon–Tarsi number

We study the relation between the correspondence chromatic number, also known as the DP-chromatic number, and the Alon–Tarsi number, both upper bounds on the list chromatic number of a graph. There are many graphs with Alon–Tarsi number greater than the correspondence chromatic number. We present here a family of graphs with arbitrary Alon–Tarsi number, with correspondence chromatic number one larger.     

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.     

Morphological Component Analysis and Inpainting on the Sphere: Application in Physics and Astrophysics

Morphological Component Analysis (MCA) is a new method which takes advantage of the sparse representation of structured data in large overcomplete dictionaries to separate features in the data based on the diversity of their morphology. It is an efficient technique in such problems as separating an image into texture and piecewise smooth parts or for inpainting applications. The MCA algorithm consists of an iterative alternating projection and thresholding scheme, using a successively decreasing threshold towards zero with each iteration. In this article, the MCA algorithm is extended to the analysis of spherical data maps as may occur in a number of areas such as geophysics, astrophysics or medical imaging. Practically, this extension is made possible thanks to the variety of recently developed transforms on the sphere including several multiscale transforms such as the undecimated isotropic wavelet transform on the sphere, the ridgelet and curvelet transforms on the sphere. An MCA-inpainting method is then directly extended to the case of spherical maps allowing us to treat problems where parts of the data are missing or corrupted. We demonstrate the usefulness of these new tools of spherical data analysis by focusing on a selection of challenging applications in physics and astrophysics.