Correspondence Analysis Solution Research Articles

On the uniqueness of correspondence analysis solutions

In correspondence analysis (CA), the rows and columns of a contingency table are optimally represented in a k-dimensional approximation, where it is common to set k=3 (which includes a so-called trivial dimension). Since CA is a dimension reduction technique, we might expect that the k-dimensional approximation is not unique, i.e. there exist several contingency tables with the same k-dimensional approximation. Interestingly, Van de Velden et al. [17] find in their computational experiments that 3-dimensional CA solutions are unique up to rotation, which leads to the question whether this is always the case. We show that k-dimensional CA solutions are not necessarily unique. That is, two distinct contingency tables may have the same k-dimensional approximation. We present necessary and sufficient conditions for the non-uniqueness of CA solutions, which hold for any value of k. Based on our sufficient conditions, we present a procedure to generate contingency tables with the same k-dimensional solution. Finally, we note that it is difficult to satisfy the necessary conditions, which suggests that CA solutions are most likely unique in practice.

A new mixed integer programming approach for inverse correspondence analysis

Correspondence analysis (CA) is a dimension reduction technique for categorical data in a two-way contingency table. The low-dimensional CA solution optimally depicts the relationship between the categorical variables. We consider the inverse correspondence analysis (ICA) problem, where we use the low-dimensional representation in order to retrieve the original data table. We propose a mixed integer programming formulation for the ICA problem based on transition formulas, which link the row and column coordinates in a CA solution. We show that our formulation has better theoretical characteristics than the existing formulation in the literature and is able to model a generalised ICA problem, which requires less input. In addition, we introduce an iterative method, which uses the fit of individual points in the low-dimensional CA solution. By incorporating statistical information on the quality of CA solutions into our methodology, we are able to retrieve the original data for larger ICA instances.

A comparison of latent semantic analysis and correspondence analysis of document-term matrices

Abstract Latent semantic analysis (LSA) and correspondence analysis (CA) are two techniques that use a singular value decomposition for dimensionality reduction. LSA has been extensively used to obtain low-dimensional representations that capture relationships among documents and terms. In this article, we present a theoretical analysis and comparison of the two techniques in the context of document-term matrices. We show that CA has some attractive properties as compared to LSA, for instance that effects of margins, that is, sums of row elements and column elements, arising from differing document lengths and term frequencies are effectively eliminated so that the CA solution is optimally suited to focus on relationships among documents and terms. A unifying framework is proposed that includes both CA and LSA as special cases. We empirically compare CA to various LSA-based methods on text categorization in English and authorship attribution on historical Dutch texts and find that CA performs significantly better. We also apply CA to a long-standing question regarding the authorship of the Dutch national anthem Wilhelmus and provide further support that it can be attributed to the author Datheen, among several contenders.

Calculating weighted scores from a multiple correspondence analysis solution

Calculating weighted scores from a multiple correspondence analysis solution

Retrieving a contingency table from a correspondence analysis solution

Correspondence analysis (CA) is a dimension reduction technique for categorical data. In particular, CA is typically applied to a contingency matrix in order to visualize the relationships within and between the categories of the two variables as represented by its rows and columns. The CA solution can be obtained by considering a singular value decomposition of the so-called matrix of standardized residuals. Inverse correspondence analysis considers the problem of retrieving the data underlying a given low-dimensional CA solution. Using the specific structure of the CA solutions as well as the characteristics of the original data we formulate the inverse CA problem in an integer linear programming context. Considering various conditions involving the dimensions of the original data matrix, the number of observations, the precision and dimensionality of the CA solution, we show that by solving the integer linear programs, the original data can be retrieved.

Exploratory Visual Inspection of Category Associations and Correlation Estimation in Multidimensional Subspaces

In this paper, we aimed to estimate associations among categories in a multi-way contingency table. To simplify estimation and interpretation of results, we stacked multiple variables to form a two-way stacked table and analyzed it using the biplot in correspondence analysis (CA) paradigm. The correspondence analysis biplot allowed visual inspection of category associations in a twodimensional plane, and the CA solution numerically estimated the category relationships. We utilized parallel analysis and identified two statistically meaningful dimensions with which a plane was constructed. In the plane, we examined metric space mapping, which was converted into correlations, between school districts and categories of school-relevant variables. The results showed differential correlation patterns among school districts and this correlational information may be useful for stake holders or policy makers to pinpoint possible causes of low school performance and school-relevant behaviors.

Using Correspondence Analysis in Multiple Case Studies

In qualitative research of multiple case studies, Miles and Huberman proposed to summarize the separate cases in a so-called meta-matrix that consists of cases by variables. Yin discusses cross-case synthesis to study this matrix. We propose correspondence analysis (CA) as a useful tool to study this matrix. CA is a quantitative method that yields a graphical display of the rows and of the columns of a matrix. The rows and the columns receive coordinates that can be interpreted as quantifications, hence the cases can be compared using these quantifications. Using an example from qualitative educational research into teaching philosophy, we illustrate both methods and their complementarity. We discuss special features of the application of CA to case study research, such as flexible ways of coding the data, and the stability of the CA solution when the number of cases is much smaller than the number of variables.

Meal Mapping

A new methodology is introduced that allows the design of meal solutions (such as chilled and frozen ready meals, menu choices in catering and food service) based on empirical assessments of fit between meal centres and side components. The necessary input data are collected by means of a consumer survey. Survey participants are asked to indicate, for a list of relevant meal centres, which side components they typically combine with the meal centres. In the first step of the statistical analysis, a parametric model of centre-side combination is estimated. In a second step, the predicted probabilities are subjected to multiple correspondence analysis and mapped into low-dimensional space. In a third step, the principal coordinates representing meal centres and side components in the correspondence analysis solution are subjected to cluster analysis to identify distinct groups of compatible centre-side combinations. The methodology is demonstrated in the context of a pan-European study of meals with fresh and minimally processed pork products as meal centres. The best-fitting solution suggested five meal segments: a “from the grill” segment (skewers or sausages served with condiments and beer or spirits), a “Sunday roast” segment (shoulder, gammon roast, collar roast or medallions served with potatoes and sauce, combined with beans, carrots or cabbage), a “Bolognese” segment (minced meat with pasta), a “lean cuisine” segment (tenderloin or small cuts combined served with rice, combined with tomatoes, sweet peppers or lettuce plus wine) and a miscellaneous segment of rarely consumed products.

The contributions of rare objects in correspondence analysis

Correspondence analysis, when used to understand relationships in a table of counts (for example, abundance data in ecology), has been criticized as being too sensitive to objects (for example, species) that occur with very low frequency or in very few samples. Here I show that this criticism is generally unfounded. This is demonstrated in several data sets by calculating the actual contributions of rare objects to the results of correspondence analysis and canonical correspondence analysis, both to the determination of the ordination axes and to the chi-square distance. It is a fact that rare objects are often positioned as outliers in correspondence analysis ordinations, which gives the impression that they are highly influential, but their low weight offsets their distant positions and reduces their effect on the results. An alternative scaling of the correspondence analysis solution, the contribution biplot, is proposed as a way of displaying the results in order to avoid the problem of outlying and low contributing rare objects. In this new scaling of the biplot (or triplot in canonical correspondence analysis), species points have coordinates that are directly related to their contributions to the solution.

Oblique rotation in correspondence analysis: A step forward in the search for the simplest interpretation

Correspondence analysis (CA) is a popular method that can be used to analyse relationships between categorical variables. It is closely related to several popular multivariate analysis methods such as canonical correlation analysis and principal component analysis. Like principal component analysis, CA solutions can be rotated orthogonally as well as obliquely into a simple structure without affecting the total amount of explained inertia. However, some specific aspects of CA prevent standard rotation procedures from being applied in a straightforward fashion. In particular, the role played by weights assigned to points and dimensions and the duality of CA solutions are unique to CA. For orthogonal simple structure rotation, procedures recently have been proposed. In this paper, we construct oblique rotation methods for CA that take into account these specific difficulties. We illustrate the benefits of our oblique rotation procedure by means of two illustrative examples.

CAR: AMATLABPackage to Compute Correspondence Analysis with Rotations

Correspondence analysis (CA) is a popular method that can be used to analyse relationships between categorical variables. Like principal component analysis, CA solutions can be rotated both orthogonally and obliquely to simple structure without affecting the total amount of explained inertia. We describe a <b>MATLAB</b> package for computing CA. The package includes orthogonal and oblique rotation of axes. It is designed not only for advanced users of <b>MATLAB</b> but also for beginners. Analysis can be done using a user-friendly interface, or by using command lines. We illustrate the use of <b>CAR</b> with one example.

Seriation by constrained correspondence analysis: A simulation study

One of the many areas in which correspondence analysis (CA) is an effective method, concerns seriation problems. For example, CA is a well-known technique for the seriation of archaeological assemblages. A problem with the CA seriation solution, however, is that only a relative ordering of the assemblages is obtained. To improve the usual CA solution, a constrained CA approach that incorporates additional information in the form of equality and inequality constraints concerning the time points of the assemblages may be considered. Using such constraints, explicit dates can be assigned to the seriation solution. The set of constraints that can be used in CA by introducing interval constraints is extended. That is, constraints that put the CA solution within a specific time frame. Moreover, the quality of the constrained CA solution is studied in a simulation study. In particular, by means of the simulation study we are able to assess how well ordinary, and constrained CA can recover the true time order. Furthermore, for the constrained approach, it is shown that the true dates are retrieved satisfactory. The simulation study is set up in such a way that it mimics the data of a series of ceramic assemblages consisting of the locally produced tableware from Sagalassos (SW Turkey). It is found that the dating of the assemblages on the basis of constraints appears to work quite well.

Diagnostics for regression dependence in tables re-ordered by the dominant correspondence analysis solution

Correspondence analysis is an exploratory technique for analyzing the interaction in a contingency table. Tables with meaningful orders of the rows and columns may be analyzed using a model-based correspondence analysis that incorporates order constraints. However, if there exists a permutation of the rows and columns of the contingency table so that the rows are regression dependent on the columns and, vice versa, the columns are regression dependent on the rows, then both implied orders are reflected in the first dimension of the unconstrained correspondence analysis [Schriever, B.F., 1983. Scaling of order dependent categorical variables with correspondence analysis. International Statistical Review 51, 225–238]. Thus, using unconstrained correspondence analysis, we may still find that the data fit an ordinal stochastic model. Fit measures are formulated that may be used to verify whether the re-ordered contingency table is regression dependent in either the rows or columns. Using several data examples, it is shown that the fit indices may complement the usual geometric interpretation of the unconstrained correspondence analysis solution in low-dimensional space.

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.

Inverse correspondence analysis

In correspondence analysis (CA), rows and columns of a data matrix are depicted as points in low-dimensional space. The row and column profiles are approximated by minimizing the so-called weighted chi-squared distance between the original profiles and their approximations, see for example, [Theory and applications of correspondence analysis, Academic Press, New York, 1984]. In this paper, we will study the inverse CA problem, that is, the possibilities for retrieving one or more data matrices from a low-dimensional CA solution. We will show that there exists a nonempty closed and bounded polyhedron of such matrices. We also present two algorithms to find the vertices of the polyhedron: an exact algorithm that finds all vertices and a heuristic approach for larger sized problems that will find some of the vertices. A proof that the maximum of the Pearson chi-squared statistic is attained at one of the vertices is given. In addition, it is discussed how extra equality constraints on some elements of the data matrix can be imposed on the inverse CA problem. As a special case, we present a method for imposing integer restrictions on the data matrix as well. The approach to inverse CA followed here is similar to the one employed by De Leeuw and Groenen [J. Classification 14 (1997) 3] in their inverse multidimensional scaling problem.

Optimal Representation of Supplementary Variables in Biplots from Principal Component Analysis and Correspondence Analysis

AbstractThis paper treats the topic of representing supplementary variables in biplots obtained by principal component analysis (PCA) and correspondence analysis (CA). We follow a geometrical approach where we minimize errors that are obtained when the scores of the PCA or CA solution are projected onto a vector that represents a supplementary variable. This paper shows that optimal directions for supplementary variables can be found by solving a regression problem, and justifies that earlier formulae from Gabriel are optimal in the least squares sense. We derive new results regarding the geometrical properties, goodness of fit statistics and the interpretation of supplementary variables. It is shown that supplementary variables can be represented by plotting their correlation coefficients with the axes of the biplot only when the proper type of scaling is used. We discuss supplementary variables in an ecological context and give illustrations with data from an environmental monitoring survey.

Annotation: correspondence analysis.

Correspondence analysis is a useful technique in the investigation of relationships between categorical variables. By displaying these relationships graphically, correspondence analysis solutions may allow investigators to obtain insights into the structure of their data more readily than would be obtained by simply studying tables of numerical values and examining associated chi-squared statistics. Some experience of using correspondence analysis solutions in practice is generally necessary to appreciate the method's usefulness fully. An interesting application of the method in a longitudinal study of cognitive development is given in Lautrey and Cibois (1992), and some suggestions as to how best to combine correspondence analysis with a more formal method such as log-linear modelling are given in Van der Heyden and de Leeuw (1985).

Correspondence analysis and optimal structural representations

Many well-known measures for the comparison of distinct partitions of the same set ofn objects are based on the structure of class overlap presented in the form of a contingency table (e.g., Pearson's chi-square statistic, Rand's measure, or Goodman-Kruskal'sτ b ), but they all can be rephrased through the use of a simple cross-product index defined between the corresponding entries from twon ×n proximity matrices that provide particular a priori (numerical) codings of the within- and between-class relationships for each of the partitions. We consider the task of optimally constructing the proximity matrices characterizing the partitions (under suitable restriction) so as to maximize the cross-product measure, or equivalently, the Pearson correlation between their entries. The major result presented states that within the broad classes of matrices that are either symmetric, skew-symmetric, or completely arbitrary, optimal representations are already derivable from what is given by a simple one-dimensional correspondence analysis solution. Besides severely limiting the type of structures that might be of interest to consider for representing the proximity matrices, this result also implies that correspondence analysis beyond one dimension must always be justified from logical bases other than the optimization of a single correlational relationship between the matrices representing the two partitions.

Reply to Greenacre's Commentary on the Carroll-Green-Schaffer Scaling of Two-Way Correspondence Analysis Solutions

Reply to Greenacre's Commentary on the Carroll-Green-Schaffer Scaling of Two-Way Correspondence Analysis Solutions

Reply to Greenacre's Commentary on the Carroll-Green-Schaffer Scaling of Two-Way Correspondence Analysis Solutions

The authors consider Greenacre's criticisms of the CGS scaling of two-way correspondence analysis. They suggest that Greenacre's indictment of this scaling approach also implies an indictment of multiple correspondence analysis interpretations—interpretations that have been made by him (and by other contributors to correspondence analysis) prior to the publication of their articles.