A new model for visualizing interactions in analysis of variance

In analysis of variance, there is usually little attention for interpretingthe terms of the effects themselves, especially for interactioneffects. One of the reasons is that the number of interaction-effectterms increases rapidly with the number of predictor variables andthe number of categories. In this paper, we propose a new model,called the interaction decomposition model, that allows to visualizethe interactions. We argue that with the help of the visualization, theinteraction-effect terms are much easier to interpret. We apply ourmethod to predict holiday spending1 using seven categorical predictorvariables.

Generalized bi-additive modelling for categorical data

Generalized linear modelling (GLM) is a versatile technique, which may be viewed as a generalization of well-known techniques such as least squares regression, analysis of variance, loglinear modelling, and logistic regression. In may applications, low-order interaction (such as bivariate interaction) terms are included in the model. However, as the number of categorical variables increases, the total number of low-order interactions also increases dramatically. In this papaer, we propose to constrain bivariate interactions by a bi-additive model which allows a simple graphical representation in which each category of every variable is represented by a vector.

Rank reduction of correlation matrices by majorization

In this paper a novel method is developed for the problem of finding a low-rank correlation matrix nearest to a given correlation matrix. The method is based on majorization and therefore it is globally convergent. The method is computationally efficient, is straightforward to implement, and can handle arbitrary weights on the entries of the correlation matrix. A simulation study suggests that majorization compares favourably with competing approaches in terms of the quality of the solution within a fixed computational time. The problem of rank reduction of correlation matrices occurs when pricing a derivative dependent on a large number of assets, where the asset prices are modelled as correlated log-normal processes.correlation matrix;lognormal price processes;majorization;rank

3WaySym-Scal: three-way symbolic multidimensional scaling

Multidimensional scaling aims at reconstructing dissimilarities between pairs of objects by distances in a low dimensional space.However, in some cases the dissimilarity itself is not known, but the range, or a histogram of the dissimilarities is given. This type of data fall in the wider class of symbolic data (see Bock and Diday (2000)). We model three-way two-mode data consisting of an interval of dissimilarities for each object pair from each of K sources by a set of intervals of the distances defined as the minimum and maximum distance between two sets of embedded rectangles representing the objects. In this paper, we provide a new algorithm called 3WaySym-Scal using iterative majorization, that is based on an algorithm, I-Scal developed for the two-way case where the dissimilarities are given by a range of values ie an interval (see Groenen et al. (2006)).The advantage of iterative majorization is that each iteration is guaranteed to improve the solution until no improvement is possible. We present the results on an empirical data set on synthetic musical tones.2WaySym-Scal;interval data;multidimensional scaling;symbolic data analysis;three-way data

Dynamische Meerdimensionele Schaling: Statistiek op de Kaart

Er is een steeds sterkere tendens om onderzoeksgegevens te visualiseren in plaats van in tabellen weer te geven. Een belangrijk voordeel van visualisatie is dat de resultaten vaak direct duidelijk en eenvoudig te interpreteren zijn. Zo kan met behulp van meerdimensionele schaling de samenhang tussen rendementen van indexen van aandelenmarkten gerepresenteerd worden door een kaart waaruit blijkt welke beurzen nauw aan elkaar gerelateerd zijn en welke niet. In deze rede wordt dynamische visualisatie als nieuw element toegevoegd aan dit soort kaarten. Er worden toepassingen besproken van het interactief construeren van zo???n kaart, bewegende kaarten van verandering van samenhang tussen aandelenmarkten in de tijd, en interactieve constructie van een kaart van de politieke partijen in Nederland. De combinatie van visualisatie met interactie en dynamiek maakt het mogelijk om op eenvoudiger wijze inzicht te krijgen in gecompliceerde gegevens dan met statische visualisatie alleen.multidimensional scaling;visualization;interactive majorization;interactive MDS;statistics

Majorization algorithms for inspecting circles, ellipses, squares, rectangles, and rhombi

In several disciplines, as diverse as shape analysis, locationtheory, quality control, archaeology, and psychometrics, it can beof interest to fit a circle through a set of points. We use theresult that it suffices to locate a center for which the varianceof the distances from the center to a set of given points isminimal. In this paper, we propose a new algorithm based oniterative majorization to locate the center. This algorithm isguaranteed to yield a series nonincreasing variances until astationary point is obtained. In all practical cases, thestationary point turns out to be a local minimum. Numericalexperiments show that the majorizing algorithm is stable and fast.In addition, we extend the method to fit other shapes, such as asquare, an ellipse, a rectangle, and a rhombus by making use ofthe class of $l_p$ distances and dimension weighting. In addition,we allow for rotations for shapes that might be rotated in theplane. We illustrate how this extended algorithm can be used as atool for shape recognition.iterative majorization;location;optimization;shape analysis

Weighted Majorization Algorithms for Weighted Least Squares Decomposition Models

For many least-squares decomposition models efficient algorithms are well known. A more difficult problem arises in decomposition models where each residual is weighted by a nonnegative value. A special case is principal components analysis with missing data. Kiers (1997) discusses an algorithm for minimizing weighteddecomposition models by iterative majorization. In this paper, we for computing a solution. We will show that the algorithm by Kiers is a special case of our algorithm. Here, we will apply weighted majorization to weighted principal components analysis, robust Procrustes analysis, and logistic bi-additive models of which the two parameter logistic model in item response theory is a specialcase. Simulation studies show that weighted majorization is generally faster than the method by Kiers by a factor one to four and obtains the same or better quality solutions. For logistic bi-additive models, we propose a new iterative majorization algorithm called logistic majorization.iterative majorization;IRT;logistic bi-additive model;robust Procrustes analysis;weighted principal component analysis;two parameter logistic model

Nonlinear support vector machines through iterative majorization and I-splines

To minimize the primal support vector machine (SVM) problem, wepropose to use iterative majorization. To do so, we propose to use it-erative majorization. To allow for nonlinearity of the predictors, we use(non)monotone spline transformations. An advantage over the usual ker-nel approach in the dual problem is that the variables can be easily inter-preted. We illustrate this with an example from the literature.iterative majorization;support vector machines;I-Splines

SVM-Maj: a majorization approach to linear support vector machines with different hinge errors

Support vector machines (SVM) are becoming increasingly popular for the prediction of a binary dependent variable. SVMs perform very well with respect to competing techniques. Often, the solution of an SVM is obtained by switching to the dual. In this paper, we stick to the primal support vector machine (SVM) problem, study its effective aspects, and propose varieties of convex loss functions such as the standard for SVM with the absolute hinge error as well as the quadratic hinge and the Huber hinge errors. We present an iterative majorization algorithm that minimizes each of the adaptations. In addition, we show that many of the features of an SVM are also obtained by an optimal scaling approach to regression. We illustrate this with an example from the literature and do a comparison of different methods on several empirical data sets.iterative majorization;I-splines;absolute hinge error;huber hinge error;optimal scaling;quadratic hinge error;support vector machines

Solving and interpreting binary classification problems in marketing with SVMs

Marketing problems often involve inary classification of customers into ``buyers'' versus ``non-buyers'' or ``prefers brand A'' versus ``prefers brand B''. These cases require binary classification models such as logistic regression, linear, andquadratic discriminant analysis. A promising recent technique forthe binary classification problem is the Support Vector Machine(Vapnik (1995)), which has achieved outstanding results in areas ranging from Bioinformatics to Finance. In this paper, we compare the performance of the Support Vector Machine against standard binary classification techniques on a marketing data set and elaborate on the interpretation of the obtained results.