Jacqueline J. Meulman

Nonlinear Principal Components Analysis: Introduction and Application.

The authors provide a didactic treatment of nonlinear (categorical) principal components analysis (PCA). This method is the nonlinear equivalent of standard PCA and reduces the observed variables to a number of uncorrelated principal components. The most important advantages of nonlinear over linear PCA are that it incorporates nominal and ordinal variables and that it can handle and discover nonlinear relationships between variables. Also, nonlinear PCA can deal with variables at their appropriate measurement level; for example, it can treat Likert-type scales ordinally instead of numerically. Every observed value of a variable can be referred to as a category. While performing PCA, nonlinear PCA converts every category to a numeric value, in accordance with the variables analysis level, using optimal quantification. The authors discuss how optimal quantification is carried out, what analysis levels are, which decisions have to be made when applying nonlinear PCA, and how the results can be interpreted. The strengths and limitations of the method are discussed. An example applying nonlinear PCA to empirical data using the program CATPCA (J. J. Meulman, W. J. Heiser, & SPSS, 2004) is provided.

The Integration of Multidimensional Scaling and Multivariate Analysis with Optimal Transformations.

The recent history of multidimensional data analysis suggests two distinct traditions that have developed along quite different lines. In multidimensional scaling (MDS), the available data typically describe the relationships among a set of objects in terms of similarity/dissimilarity (or (pseudo-)distances). In multivariate analysis (MVA), data usually result from observation on a collection of variables over a common set of objects. This paper starts from a very general multidimensional scaling task, defined on distances between objects derived from one or more sets of multivariate data. Particular special cases of the general problem, following familiar notions from MVA, will be discussed that encompass a variety of analysis techniques, including the possible use of optimal variable transformation. Throughout, it will be noted how certain data analysis approaches are equivalent to familiar MVA solutions when particular problem specifications are combined with particular distance approximations.

The Role of Parents and Peers in the Leisure Activities of Young Adolescents

In the previous chapter, we studied the organized and unorganized leisure patterns of contemporary children and teenagers. We established that ten to twelve year-old children commonly have different leisure interests than teenagers aged fourteen and fifteen and that girls generally prefer other leisure activities than boys. Most juveniles do not engage in these leisure activities on their own. The large majority spends an important part of their leisure time with peers. Many unorganized activities are thought up and undertaken together with peers, such as playing football in the streets, playing hide-and-seek, or just chat with local children or a group of friends from school. However, while also being engaged in organized activities at the sporting club or during music lessons, children meet and friendships are formed. In addition to the time that is spent with peers, many juveniles also spend an important part of their leisure time with their parents or siblings and engage, together with them, in all kinds of activities. In the leisure pursuits of contemporary children and teenagers, not only peer contacts, but also contacts with parents and other members of the family play an important part. The question of whether, during the transition from childhood into adolescence, alterations occur in the role parents and peers play in the leisure activities of juveniles has not been studied extensively so far. Hence, this question forms the central theme of the subsequent chapter.

Two Purposes for Matrix Factorization: A Historical Appraisal

Matrix factorization in numerical linear algebra (NLA) typically serves the purpose of restating some given problem in such a way that it can be solved more readily; for example, one major application is in the solution of a linear system of equations. In contrast, within applied statistics/psychometrics (AS/P), a much more common use for matrix factorization is in presenting, possibly spatially, the structure that may be inherent in a given data matrix obtained on a collection of objects observed over a set of variables. The actual components of a factorization are now of prime importance and not just as a mechanism for solving another problem. We review some connections between NLA and AS/P and their respective concerns with matrix factorization and the subsequent rank reduction of a matrix. We note in particular that several results available for many decades in AS/P were more recently (re)discovered in the NLA literature. Two other distinctions between NLA and AS/P are also discussed briefly: how a generalized singular value decomposition might be defined, and the differing uses for the (newer) methods of optimization based on cyclic or iterative projections.

The Structural Representation of Proximity Matrices with MATLAB

Preface Part I. (Multi- and Unidimensional) City-Block Scaling: 1. Linear unidimensional scaling 2. Linear multidimensional scaling 3. Circular scaling 4. LUS for two-mode proximity data Part II. The Representation of Proximity Matrices by Tree Structures: 5. Ultrametrics for symmetric proximity data 6. Additive trees for symmetric proximity data 7. Fitting multiple tree structures to a symmetric sroximity matrix 8. Ultrametrics and additive trees for two-mode (rectangular) proximity data Part III. The Representation of Proximity Matrices by Structures Dependent on Order (Only): 9. Anti-Robinson matrices for symmetric proximity data 10. Circular anti-Robinson matrices for symmetric proximity data 11. Anti-Robinson matrices for two-mode proximity data Appendix Bibliography Indices.

Characterization of Rheumatoid Arthritis Subtypes Using Symptom Profiles, Clinical Chemistry and Metabolomics Measurements

Objective The aim is to characterize subgroups or phenotypes of rheumatoid arthritis (RA) patients using a systems biology approach. The discovery of subtypes of rheumatoid arthritis patients is an essential research area for the improvement of response to therapy and the development of personalized medicine strategies. Methods In this study, 39 RA patients are phenotyped using clinical chemistry measurements, urine and plasma metabolomics analysis and symptom profiles. In addition, a Chinese medicine expert classified each RA patient as a Cold or Heat type according to Chinese medicine theory. Multivariate data analysis techniques are employed to detect and validate biochemical and symptom relationships with the classification. Results The questionnaire items ‘Red joints’, ‘Swollen joints’, ‘Warm joints’ suggest differences in the level of inflammation between the groups although c-reactive protein (CRP) and rheumatoid factor (RHF) levels were equal. Multivariate analysis of the urine metabolomics data revealed that the levels of 11 acylcarnitines were lower in the Cold RA than in the Heat RA patients, suggesting differences in muscle breakdown. Additionally, higher dehydroepiandrosterone sulfate (DHEAS) levels in Heat patients compared to Cold patients were found suggesting that the Cold RA group has a more suppressed hypothalamic-pituitary-adrenal (HPA) axis function. Conclusion Significant and relevant biochemical differences are found between Cold and Heat RA patients. Differences in immune function, HPA axis involvement and muscle breakdown point towards opportunities to tailor disease management strategies to each of the subgroups RA patient.

Stability of Nonlinear Principal Components Analysis: An Empirical Study Using the Balanced Bootstrap

Principal components analysis (PCA) is used to explore the structure of data sets containing linearly related numeric variables. Alternatively, nonlinear PCA can handle possibly nonlinearly related numeric as well as nonnumeric variables. For linear PCA, the stability of its solution can be established under the assumption of multivariate normality. For nonlinear PCA, however, standard options for establishing stability are not provided. The authors use the nonparametric bootstrap procedure to assess the stability of nonlinear PCA results, applied to empirical data. They use confidence intervals for the variable transformations and confidence ellipses for the eigenvalues, the component loadings, and the person scores. They discuss the balanced version of the bootstrap, bias estimation, and Procrustes rotation. To provide a benchmark, the same bootstrap procedure is applied to linear PCA on the same data. On the basis of the results, the authors advise using at least 1,000 bootstrap samples, using Procrustes rotation on the bootstrap results, examining the bootstrap distributions along with the confidence regions, and merging categories with small marginal frequencies to reduce the variance of the bootstrap results.

Constrained multidimensional scaling, including confirmation

Constrained and confirmatory multidimensional scaling (MDS) are not equivalent. Constraints refer to the translation of either theoretical or data analytical objectives into computational specifications. Confirma tion refers to a study of the balance between system atic and random variation in the data for modeling of the systematic part. Among the topics discussed from this perspective are the role of substantive theory in MDS studies, the type of constraints currently envis aged, and the relationships with other data analysis methods. This paper points out the possibility of using either sampling models or resampling schemes to study the stability of MDS solutions. Parallel to Akaikes (1974) information criterion for choosing one out of many models for the same data, a general sta bility criterion is proposed and illustrated, based on the ratio of within to total spread of configurations is sued from resampling.

Analyzing rectangular tables by joint and constrained multidimensional scaling

Abstract This paper selectively discusses Multidimensional Scaling methods, addressing both theory and applications. Most of the discussion is focussed upon the analysis of rectangular tables. The term joint MDS is introduced for an analysis which gives a meaningful joint representation of the set of row and the set of column objects of such a table so that the dissimilarities either within sets or between sets are optimally approximated. From this perspective, it is possible to give an unified account of the relationships between classical Multidimensional Scaling, Principal Components analysis and Correspondence Analysis. By imposing constraints the general model can be structurally changed in a number of ways. For example, it can be made discrete; that is, in principle it subsumes subset models and quadratic assignment as special cases. It can also be made more flexible in the sense that external information may be used in order to tune the analysis to specific hypotheses. Finally, an elegant joint MDS model called Multidimensional Unfolding is discussed. As a technique, Unfolding unfortunately suffers from being extremely sensitive to ill-conditioned data. A new way of coping with this problem is indicated and an application is given in which the Bootstrap technique succesfully establishes the stability of the results.

A Special Jackknife for Multidimensional Scaling

In this paper we develop a version of the Jackknife which seems especially suited for Multidimensional Scaling. It deletes one stimulus at a time, and combines the resulting solutions by a least squares matching method. The results can be used for stability analysis, and for purposes of cross validation.