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The Gifi system of analyzing categorical data through nonlinear varieties of classical multivariate analysis techniques is reviewed. The system is characterized by the optimal scaling of categorical variables which is implemented through alternating least squares algorithms. The main technique of homogeneity analysis is presented, along with its extensions and generalizations leading to nonmetric principal components analysis and canonical correlation analysis. A brief account of stability issues and areas of applications of the techniques is also given.

The symmetric difference between sets of qualitative elements (called features) forms the basis of a distance model that can be used as a general framework for fitting a particular class of graphs, which includes additive trees, hierarchical trees and circumplex structures. It is shown how to parametrize this fitting problem in terms of a lattice of subsets, and how inclusion relations between feature sets lead to additivity of distance along paths in a graph. An algorithm based on alternating least squares and on the recent method of cluster differences scaling is described, and illustrated for the general case.

Graphical representation of nonsymmetric relationships data has usually proceeded via separate displays for the symmetric and the skew-symmetric parts of a data matrix. DEDICOM avoids splitting the data into symmetric and skew-symmetric parts, but lacks a graphical representation of the results. Chino's GIPSCAL combines features of both models, but may have a poor goodness-of-fit compared to DEDICOM. We simplify and generalize Chino's method in such a way that it fits the data better. We develop an alternating least squares algorithm for the resulting method, called Generalized GIPSCAL, and adjust it to handle GIPSCAL as well. In addition, we show that Generalized GIPSCAL is a constrained variant of DEDICOM and derive necessary and sufficient conditions for equivalence of the two models. Because these conditions are rather mild, we expect that in many practical cases DEDICOM and Generalized GIPSCAL are (nearly) equivalent, and hence that the graphical representation from Generalized GIPSCAL can be used to display the DEDICOM results graphically. Such a representation is given for an illustration. Finally, we show Generalized GIPSCAL to be a generalization of another method for joint representation of the symmetric and skew-symmetric parts of a data matrix.

Recently, a number of methods have been proposed for the exploratory analysis of mixtures of qualitative and quantitative variables. In these methods for each variable an object by object similarity matrix is constructed, and these are consequently analyzed by means of three-way methods like INDSCAL, IDIOSCAL and TUCKALS-3. When the number of observation units (objects) is large, algorithms for INDSCAL, IDIOSCAL and TUCKALS-3 become. inefficient or even infeasible. The present paper offers variants of these algorithms that can handle large numbers of objects in case the similarity matrices are of rank much smaller than the number of objects, which is usually the case. In addition, it is shown that results of the three-way methods at hand are essentially based only on certain aggregate measures for the variables, like variances and covariances for numerical variables, and bivariate and marginal frequencies for nominal variables.

Approaches that allow one to rapidly understand tissue structure and functionality in situ remain to be developed. Such techniques are required in many instances, including where there is a need to remove with a high degree of confidence positive tumour margins during surgical excision. As biological tissue has little contrast, gold standard confirmation of surgical margins is conventionally undertaken by histopathological diagnosis of tissue architecture via optical microscopy. Vibrational spectroscopy techniques, when coupled to sophisticated computational analyses, are capable of constructing bio-molecular contrast images of unstained tissue. To assess the relative applicability of a range of candidate algorithms to distinguish the in situ bio-molecular structures of a complex tissue, the empty modelling approach of multivariate curve resolution-alternating least squares (MCR-ALS) was compared to hierarchical cluster analysis (HCA) or principal component analysis (PCA). Such chemometric analyses were applied to Raman images of benign (tumour-adjacent) endometrium, stage I and stage II endometrioid cancer. Re-constructed images from the in situ bio-molecular tissue architectures highlighted features associated with glandular epithelium, stroma, glandular lumen and myometrium. Of the tested chemometric analyses, MCR-ALS provided the best bio-molecular contrast images, superior to those derived following HCA or PCA, with clear and defined margins of histological features. Iteratively-resolved spectra identified wavenumbers responsible for the contrast image. Wavenumbers 1234 cm(-1) (Amide III), 1390 cm(-1) (CH(3) bend), 1675 cm(-1) (Amide I/lipid), 1275 cm(-1) (Amide III), 918 cm(-1) (proline) and 936 cm(-1) (proline, valine and proteins) were responsible for generating the majority of the contrast within MCR-ALS-generated images. Applications of sophisticated computational analyses coupled with vibrational spectroscopy techniques have the potential to lend novel functionality insights into bio-molecular structures in vivo.

A constrained version of Three-mode Factor Analysis model is considered in order to make its interpretation easier. The constraints are obtained by fixing some elements of the core to zero and requiring orthonormal factor loadings. An algorithm to solve the related minimization problem and an example of core constraints with theoretically interesting features, are given.

The qualitative regression is implemented according to two different approaches: optimization of a criterion in the sense of the least squares or the least absolute deviations. It has generated considerable amount of research, whether it is in analysis multicriterion or in conjoint analysis , its methods find wide range of applications. By placing us in the least squares approach, we consider the general problem as a canonical analysis under constraints, requiring during its implementation, the determination of the orthogonal projection of a vector on a convex polyhedron ; we propose MCCP algorithm which improve NNLS algorithm (of Lawson and Hanson, 1974).In the canonical analysis under the positivity constraints, we propose ACCE algorithm.One of the most important application of ACCE is the statistical analysis of preference data, used by marketers to determine consumers' preferred core benefits. We make a comparative study between least squares approach (UTA, Jacquet-Lagrèze and Siskos, 1982) and least squares approach (ACCE)., La régression qualitative est mise en oeuvre selon deux approches distinctes: optimisation d'un critère au sens des moindres écarts quadratiques ("moindres carrés") ou des moindres écarts absolus ("moindres écarts"). Elle a suscité ces dernières années de nombreux travaux car, que ce soit en analyse multicritère ou en analyse des mesures conjointes, ses méthodes trouvent un vaste champ d'application. En nous situant dans une approche moindres carrés, nous considérons le problème général comme une analyse canonique sous contraintes, requérant lors de sa mise en oeuvre, la détermination de la projection orthogonale d'un vecteur sur un polyèdre convexe. Ce dernier problème rentre dans le cadre de la régression multiple sous contraintes sur les coefficients et permet, en outre, d'apporter une réponse à la question de la protection de la régression, rencontrée lors du traitement de données en économie, géologie, physique, etc. Notre étude nous a amené à définir la notion d'approche de type ADOPT. Notion essentielle et féconde, elle est a la base des algorithmes MCCB et ACCE que nous proposons et qui résolvent les problèmes des contraintes, respectivement de bornes en régression et de positivité des facteurs en analyse canonique. L'application de la régression qualitative en analyse multicritère est abordée dans le problème de la désagrégation de la préférence globale. Nous y effectuons une étude comparative des approches moindres carrés (ACCE) et moindres écarts (UTA, Jacquet-Lagrèze et Siskos, 1982). Le programme informatique REGALS que nous avons mis au point, a permis le traitement des exemples qui illustrent les méthodes étudiées tout au long du document.

The qualitative regression is implemented according to two different approaches: optimization of a criterion in the sense of the least squares or the least absolute deviations. It has generated considerable amount of research, whether it is in analysis multicriterion or in conjoint analysis , its methods find wide range of applications. By placing us in the least squares approach, we consider the general problem as a canonical analysis under constraints, requiring during its implementation, the determination of the orthogonal projection of a vector on a convex polyhedron ; we propose MCCP algorithm which improve NNLS algorithm (of Lawson and Hanson, 1974).In the canonical analysis under the positivity constraints, we propose ACCE algorithm.One of the most important application of ACCE is the statistical analysis of preference data, used by marketers to determine consumers' preferred core benefits. We make a comparative study between least squares approach (UTA, Jacquet-Lagrèze and Siskos, 1982) and least squares approach (ACCE)., La régression qualitative est mise en oeuvre selon deux approches distinctes: optimisation d'un critère au sens des moindres écarts quadratiques ("moindres carrés") ou des moindres écarts absolus ("moindres écarts"). Elle a suscité ces dernières années de nombreux travaux car, que ce soit en analyse multicritère ou en analyse des mesures conjointes, ses méthodes trouvent un vaste champ d'application. En nous situant dans une approche moindres carrés, nous considérons le problème général comme une analyse canonique sous contraintes, requérant lors de sa mise en oeuvre, la détermination de la projection orthogonale d'un vecteur sur un polyèdre convexe. Ce dernier problème rentre dans le cadre de la régression multiple sous contraintes sur les coefficients et permet, en outre, d'apporter une réponse à la question de la protection de la régression, rencontrée lors du traitement de données en économie, géologie, physique, etc. Notre étude nous a amené à définir la notion d'approche de type ADOPT. Notion essentielle et féconde, elle est a la base des algorithmes MCCB et ACCE que nous proposons et qui résolvent les problèmes des contraintes, respectivement de bornes en régression et de positivité des facteurs en analyse canonique. L'application de la régression qualitative en analyse multicritère est abordée dans le problème de la désagrégation de la préférence globale. Nous y effectuons une étude comparative des approches moindres carrés (ACCE) et moindres écarts (UTA, Jacquet-Lagrèze et Siskos, 1982). Le programme informatique REGALS que nous avons mis au point, a permis le traitement des exemples qui illustrent les méthodes étudiées tout au long du document.

We discuss properties of some data-analytic methods which are intimately related to each other: alternating least squares (ALS), correspondence analysis and more recently Breiman and Friedman's ACE algorithm. The application of these methods to regression produces nonparametric estimators of nonlinear transformations, both of the response and the predictors. These procedures are among the most powerful tools for data analysis, but missing awareness of some artifacts could lead to inappropriate interpretations. We point out some anomalies as well as some curiosities in the mathematics of these methods, and we relate them to some areas in computer-aided tomography, projection pursuit regression and nonlinear devices in the theory of noise.

The DEDICOM model is a model for representing asymmetric relations among a set of objects by means of a set of coordinates for the objects on a limited number of dimensions. The present paper offers an alternating least squares algorithm for fitting the DEDICOM model. The model can be generalized to represent any number of sets of relations among the same set of objects. An algorithm for fitting this three-way DEDICOM model is provided as well. Based on the algorithm for the three-way DEDICOM model an algorithm is developed for fitting the IDIOSCAL model in the least squares sense.

In FACTALS an alternating least squares algorithm is utilized. Mooijaart (1984) has shown that this algorithm is based upon an erroneous assumption. This paper gives a proper solution for the loss function used in FACTALS.

We describe a convergent procedure for fitting the common factor analysis model to multivariate data whose variables may be nominal, ordinal or interval. Any mixture of measurement levels is permitted. There may be any pattern of missing data. As distinguished from previous work, the nonmetric relations (nominal or ordinal) are assumed on the raw observations (not on the correlations), and the model fitted is the common factor analysis model (not the principal components model) which isolates common from unique factor variation. The computational algorithm, based on the alternating least squares principle, is monotonically convergent and efficient. An illustrative example is presented.

Non-linear canonical correlation analysis is a method for canonical correlation analysis with optimal scaling features. The method fits many kinds of discrete data. The different parameters are solved for in an alternating least squares way and the corresponding program is called CANALS. An application of CANALS is discussed and also a study of the stability of the scaling results.

A method is developed to investigate the additive structure of data that (a) may be measured at the nominal, ordinal or cardinal levels, (b) may be obtained from either a discrete or continuous source, (c) may have known degrees of imprecision, or (d) may be obtained in unbalanced designs. The method also permits experimental variables to be measured at the ordinal level. It is shown that the method is convergent, and includes several previously proposed methods as special cases. Both Monte Carlo and empirical evaluations indicate that the method is robust.

Through external analysis of two-mode data one attempts to map the elements of one mode (e.g., attributes) as vectors in a fixed space of the elements of the other mode (e.g., stimuli). This type of analysis is extended to three-mode data, for instance, when the ratings are made by more individuals. It is described how alternating least squares algorithms for three-mode principal component analysis (PCA) are adapted to enable external analysis, and it is demonstrated that these techniques are useful for exploring differences in the individuals' mappings of the attribute vectors in the fixed stimulus space. Conditions are described under which individual differences may be ignored. External three-mode PCA is illustrated with data from a person perception experiment, designed after two studies by Rosenberg and his associates whose results were used as external information.

The author has submitted the data of a random survey on Moroccan workers in the Netherlands to two statistical analysis techniques: “Homals” (“Homogeneity analysis by means of alternating least squares”) and the canonical correlation technique, in order to test and evaluate a migrants integration model. The assumptions of the model are very well confirmed: social incorporation and cultural adaptation are very clearly correlated; the correlation is also very positive between the migrant’s incorporation and his subjective perception of it; finally the strongest correlation is observed between the importance of objective changes in the individual’s original culture and his degree of positive evaluation of new cultural values.

The authors discuss and illustrate the advantages and limitations of a family of new approaches to the analysis of metric and nonmetric data in marketing research. The general method, which is based on alternating least squares optimal scaling procedures, extends the analytical flexibility of the general linear model procedures (ANOVA, regression, canonical correlation, discriminant analysis, etc.) to situations in which the data (1) are measured at any mixture of the nominal, ordinal, or interval levels and (2) are derived from either a discrete or continuous distribution. The relationship of these procedures to traditional linear models and to other nonmetric approaches (such as multidimensional scaling and conjoint analysis) is reviewed.

FACTALS is a nonmetric common factor analysis model for multivariate data whose variables may be nominal, ordinal or interval. In FACTALS an Alternating Least Squares algorithm is utilized which is claimed to be monotonically convergent.

Homogeneity analysis, or multiple correspondence analysis, is usually applied tok separate variables. In this paper we apply it to sets of variables by using sums within sets. The resulting technique is called OVERALS. It uses the notion of optimal scaling, with transformations that can be multiple or single. The single transformations consist of three types: nominal, ordinal, and numerical. The corresponding OVERALS computer program minimizes a least squares loss function by using an alternating least squares algorithm. Many existing linear and nonlinear multivariate analysis techniques are shown to be special cases of OVERALS. An application to data from an epidemiological survey is presented.

The Gifi system of analyzing categorical data through nonlinear varieties of classical multivariate analysis techniques is reviewed. The system is characterized by the optimal scaling of categorical variables which is implemented through alternating least squares algorithms. The main technique of homogeneity analysis is presented, along with its extensions and generalizations leading to nonmetric principal components analysis and canonical correlation analysis. Several examples are used to illustrate the methods. A brief account of stability issues and areas of applications of the techniques is also given.

Sylvester matrix equations are ubiquitous in scientific computing. However, few solution techniques exist for their generalized multiterm version, as they recently arose in stochastic Galerkin finite element discretizations and isogeometric analysis. In this work, we consider preconditioning techniques for the iterative solution of generalized Sylvester equations. They consist in constructing low Kronecker rank approximations of either the operator itself or its inverse. In the first case, applying the preconditioning operator requires solving standard Sylvester equations, for which very efficient solution methods have already been proposed. In the second case, applying the preconditioning operator only requires computing matrix-matrix multiplications, which are also highly optimized on modern computer architectures. Moreover, low Kronecker rank approximate inverses can be easily combined with sparse approximate inverse techniques, thereby further speeding up their application with little or no damage to their preconditioning capability.

Failures in milk coagulation during cheese manufacturing can lead to decreased yield, anomalous behaviour of cheese during storage, significant impact on cheese quality and process wastes. This study proposes a Process Analytical Technology approach based on FT-NIR spectroscopy for milk renneting control during cheese manufacturing. Multivariate Curve Resolution optimized by Alternating Least Squares (MCR-ALS) was used for data analysis and development of Multivariate Statistical Process Control (MSPC) charts. Fifteen renneting batches were set up varying temperature (30, 35, 40 °C), milk pH (6.3, 6.5, 6.7), and fat content (0.1, 2.55, 5 g/100 mL). Three failure batches were also considered. The MCR-ALS models well described the coagulation processes (explained variance ≥99.93%; lack of fit

Projecte final de Màster Oficial realitzat en col.laboració amb Universitat de Barcelona. Departament d’Electrònica. English: In this work, a Multivariate Curve Resolution (MCR) with Alternating Least Squares (ALS) method is described and used to identify the concentrations of a two-component (ethanol and acetone) mixture analysed with an Ion Mobility Spectrometer. Results allow us to distinguish qualitatively both components at lower concentrations, whereas fail to detect ethanol at higher concentrations. The impossibility of detecting etanol at higher concentrations is caused by higher acetone’s proton affinity.

PRINCIPALS analysis (principal components analysis by alternating least squares optimal scaling) provides an approach for improving the reliability and convergent and discriminant validity of measures used in marketing research. Given a set of items designed to measure a theoretical construct or conceptual dimension, PRINCIPALS rescales the original response categories of each item to interval-level measurement and maximizes their unidimensional communality. PRINCIPALS is complementary to the traditional approaches for improving the measurement quality of scales used in marketing. The authors present the transformation in the general form, then illustrate it in two attitude research examples. The stability of the results is also examined. In both examples the reliability and convergent and discriminant validity of measures based on the tripartite attitude model are substantially improved after PRINCIPALS rescaling.

Background A set of silver coins from the collection of Déri Museum Debrecen (Hungary) was examined by X-ray fluorescent elemental analysis with the aim to assign the coins to different groups with the best possible precision based on the acquired chemical information and to build models, which arrange the coins according to their historical periods. Results Principal component analysis, linear discriminant analysis, partial least squares discriminant analysis, classification and regression trees and multivariate curve resolution with alternating least squares were applied to reveal dominant pattern in the data and classify the coins into several groups. We also identified those chemical components, which are present in small percentages, but are useful for the classification of the coins. With the coins divided into two groups according to adequate historical periods, we have obtained a correct classification (76-78%) based on the chemical compositions. Conclusions X-ray fluorescent elemental analysis together with multivariate data analysis methods is suitable to group medieval coins according to historical periods.