The nonconvergence of factals: A nonmetric common factor analysis

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.In this paper it is shown that this algorithm is based upon an erroneous assumption, namely that the least squares loss function (which is in this case a nonscale free loss function) can be transformed into a scalefree loss function. A consequence of this is that monotonical convergence of the algorithm can not be guaranteed.     

A monotonically convergent algorithm for orthogonal congruence rotation

Brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix. This method employs an algorithm for which convergence from every starting point is not guaranteed. In the present paper, an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point. Specifically, it is proven that the function value converges monotonically, and that the difference between subsequent iterates converges to zero. In addition to the better convergence properties, another advantage of the present algorithm over Brokken's one is that it is easier to program. The algorithms are compared on 80 simulated data sets, and it turned out that the new algorithm performed well in all cases, whereas Brokken's algorithm failed in almost half the cases. The derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. The authors are obliged to Willem J. Heiser and Jos M. F. ten Berge for useful comments on an earlier version of this paper.     

Regularized Generalized Canonical Correlation Analysis

Regularized generalized canonical correlation analysis (RGCCA) is a generalization of regularized canonical correlation analysis to three or more sets of variables. It constitutes a general framework for many multi-block data analysis methods. It combines the power of multi-block data analysis methods (maximization of well identified criteria) and the flexibility of PLS path modeling (the researcher decides which blocks are connected and which are not). Searching for a fixed point of the stationary equations related to RGCCA, a new monotonically convergent algorithm, very similar to the PLS algorithm proposed by Herman Wold, is obtained. Finally, a practical example is discussed.     

A monotonically convergent algorithm for factals

Takane, Young, and de Leeuw proposed a procedure called FACTALS for the analysis of variables of mixed measurement levels (numerical, ordinal, or nominal). Mooijaart pointed out that their algorithm does not necessarily converge, and Nevels proposed a new algorithm for the case of nominal variables. In the present paper it is shown that Nevels' procedure is incorrect, and a new procedure for handling nominal variables is proposed. In addition, a procedure for handling ordinal variables is proposed. Using these results, a monotonically convergent algorithm is constructed for FACTALS of any mixture of variables.The authors are obliged to Jos ten Berge for stimulating comments on an earlier version of this paper. The research of H. A. L. Kiers has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. The research of Y. Takane has been supported by the Natural Sciences and Engineering Research Council of Canada, grant number A6394, and by the McGill-IBM Cooperative Grant.     

The analysis of proximities: Multidimensional scaling with an unknown distance function. II

The first in the present series of two papers described a computer program for multidimensional scaling on the basis of essentially nonmetric data. This second paper reports the results of two kinds of test applications of that program. The first application is to artificial data generated by monotonically transforming the interpoint distances in a known spatial configuration. The purpose is to show that the recovery of the original metric configuration does not depend upon the particular transformation used. The second application is to measures of interstimulus similarity and confusability obtained from some actual psychological experiments.     

Mathematical formulation of multivariate euclidean models for discrimination methods

Multivariate models for the triangular and duo-trio methods are described in this paper. In both cases, the mathematical formulation of Euclidean models for these methods is derived and evaluated for the bivariate case using numerical quadrature. Theoretical results are compared with those obtained using Monte Carlo simulation which is validated by comparison with previously published theoretical results for univariate models of these methods. This work is discussed in light of its importance to the development of a new theory for multidimensional scaling in which the traditional assumption can be eliminated that proximity measures and perceptual distances are monotonically related.     

Robust canonical discriminant analysis

A method for robust canonical discriminant analysis via two robust objective loss functions is discussed. These functions are useful to reduce the influence of outliers in the data. Majorization is used at several stages of the minimization procedure to obtain a monotonically convergent algorithm. An advantage of the proposed method is that it allows for optimal scaling of the variables. In a simulation study it is shown that under the presence of outliers the robust functions outperform the ordinary least squares function, both when the underlying structure is linear in the variables as when it is nonlinear. Furthermore, the method is illustrated with empirical data.The research of the first author was supported by the Netherlands Organization of Scientific Research (NWO grant 560-267-029).     

An efficient algorithm for joint correspondence analysis

Joint correspondence analysis is a technique for constructing reduced-dimensional representations of pairwise relationships among categorical variables. The technique was proposed by Greenacre as an alternative to multiple correspondence analysis. Joint correspondence analysis differs from multiple correspondence analysis in that it focuses solely on between-variable relationships. Greenacre described one alternating least-squares algorithm for conducting joint correspondence analysis. Another alternating least-squares algorithm is described in this article. The algorithm is guaranteed to converge, and does so in fewer iterations than does the algorithm proposed by Greenacre. A modification of the algorithm for handling Heywood cases is described. The algorithm is illustrated on two data sets.     

Experimental and theoretical results for some models of random-dot-pattern discrimination

Summary The popular, but low-power, two-stimulus signal-detection paradigms are extended to generate response-criterion-dependent psychometric functions. Maximumlikelihood estimation within a sufficiently powerful experimental design is described, and theoretical conditions for the resulting psychometric functions to be monotonically increasing are presented. These results are applied to an experiment on the detection of spatial-density differences in briefly presented random-dot patterns. The results confirm, extend, and complement current notions put forth by H. B. Barlow and his co-workers on mechanisms of internal spatial-density representation and the efficiency of intermediate levels of visual processing.     

APPARENT WEIGHT DIFFERENCES AS A SOURCE OF AFTER-EFFECTS

Two experiments are reported. In the first, the influence of stimulus size on apparent weight, i.e., the size-weight illusion, was scaled. The relationship between heaviness and size could be described by a simple power function. In the second experiment it was demonstrated that the illusory weight differences give rise to aftereffects. The size of the after-effect increased monotonically with the amount of apparent weight difference.     

Majorization as a tool for optimizing a class of matrix functions

The problem of minimizing a general matrix, trace function, possibly subject to certain constraints, is approached by means of majorizing this function by one having a simple quadratic shape and whose minimum is easily found. It is shown that the parameter set that minimizes the majorizing function also decreases the matrix trace function, which in turn provides a monotonically convergent algorithm for minimizing the matrix trace function iteratively. Three algorithms based on majorization for solving certain least squares problems are shown to be special cases. In addition, by means of several examples, it is noted how algorithms may be provided for a wide class of statistical optimization tasks for which no satisfactory algorithms seem available.The Netherlands organization for scientific research(NWO) is gratefully acknowledged for funding this project. This research was conducted while the author was supported by a PSYCHON-grant (560-267-011) from this organization. The author is obliged to Jos ten Berge, Willem Heiser, and Wim Krijnen for helpful comments on an earlier version of this paper.     

Memory hazard functions: a vehicle for theory development and test

A framework is developed to rigorously test an entire class of memory retention functions by examining hazard properties. Evidence is provided that the memory hazard function is not monotonically decreasing. Yet most of the proposals for retention functions, which have emerged from the psychological literature, imply that memory hazard is monotonically decreasing over the entire temporal range. Furthermore, the few remaining proposals, that do not have monotonically decreasing hazard, have difficulty fitting data over both short-term and long-term intervals. A new 2-trace hazard model is developed that successfully circumvents these difficulties. This new model is used to account for the size of memory span and the time course of proactive and retroactive interference effects. The model can fit the retention characteristics of H. M., the famous amnesic patient, as well as normal experimental participants. The model is also used to account for the time course of the misinformation effect.     

Mental size scaling examined

Observers made same-different shape judgments of stimuli that were identical in shape and size different in shape but not in size, or differed in relative size along a number of steps for both same- and different- shaped forms. "Same" judgment RTs increased monotonically with increases in the magnitude of the relative size difference. In contrast, "different" judgment RTs were unaffected by changes in relative size. A second experiment in which stimulus presentation was successive rather than simultaneous yielded essentially the same results. Consideration was given to a dual- process model in which a time-consuming analog process normalizes stimuli that are size discrepant prior to a comparison stage that is operative for those structures responsible for "same" decisions but not for "different" judgments. Some evidence that seems inconsistent with a normalization process which operates prior to contact with memory was discussed, and an alternative explanation in which the early detection of a size difference causes changes in decision criteria was suggested.     

Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis

Multidimensional scaling is the problem of representingn objects geometrically byn points, so that the interpoint distances correspond in some sense to experimental dissimilarities between objects. In just what sense distances and dissimilarities should correspond has been left rather vague in most approaches, thus leaving these approaches logically incomplete. Our fundamental hypothesis is that dissimilarities and distances are monotonically related. We define a quantitative, intuitively satisfying measure of goodness of fit to this hypothesis. Our technique of multidimensional scaling is to compute that configuration of points which optimizes the goodness of fit. A practical computer program for doing the calculations is described in a companion paper.     

Predicting the shape of distance functions in curve tracing: Evidence for a zoom lens operator

Jolicoeur, Ullman, and MacKay (1986) showed that the time to confirm that two dots are on the same curve increases monotonically, but nonlinearly, as the curve distance between the two dots increases. These displays contained two curves and two dots. On same trials, the two dots were on the same curve (target curve), while the other curve served as a foil (distractor curve). The monotonically increasing effects of curve distance on response times for same trials suggested that the intervening curve segment was traced. In the present investigation of the source of the nonlinearity in these distance functions, it was hypothesized that differences in the distractor curves may have allowed a curve tracing operator with zoom lens properties to widen its receptive field while tracing parts of certain target curves. The wider receptive field may have allowed faster tracing over certain segments, owing to a reduced number of shifts required by the operator to scan the curve. The consequence of training certain segments of the curve more quickly than other segments of the curve would be a nonlinear effect of distance. A new set of stimuli was created for testing this hypothesis directly. Fairly linear distance effects were found for stimuli that contained a distractor curve that constrained the breadth of the postulated curve tracing operator, whereas stimuli that contained a distractor curve that could allow for a larger receptive field yielded nonlinear distance functions. The results are compared with the predictions of three quantitative models: pixel-by-pixel tracing; Jolicoeur, Ullman, and MacKay's (1991) bipartite operator; and a new zoom lens model, analogous to the zoom lens model of visual attention. The results were fit best by the latter model, in which tracing is accomplished by tracking the curve with a variably sized local operator.     

Direction of gaze and emergence of speech in the second year

Orienting to the right member of a pair of identical pictures increased monotonically with age in a longitudinal sample of 14 infants seen monthly from 13 to 22 months. The magnitude of the correlation between this orientation bias and a measure of language development also rose with age reaching a peak at 20 months and then declining. It was suggested that the acceleration of vocabulary and maturational changes in the central nervous system that occur between 16 and 22 months are associated with a special excitatory state in the temporal cortex of the left hemisphere.     

Functional Differences Between Upright and Rotated Images

In Experiment I subjects imaged an alphanumeric character either upright or upside-down, and triggered a test display character. Their task was to decide as quickly as possible whether the test character was NORMAL or MIRRORED. On 72% of the trials the test was at the orientation imaged. Reaction time (RT) was then about 200 ms longer in the upside-down image condition. This difference reduced with practice. On the remaining trials the orientation of the test character differed from that of the prepared image. For upright images RT increased monotonically with the angular difference in orientation between test and image. For upside-down images RT did not increase monotonically with angular difference as there was a wide dip around the upright. Further experiments suggested that upside-down images can be rotated, but at considerably slower rates than upright ones, and that the apparent rates of rotation for upside-down images are dependent upon the width of the sector tested. These results indicate that visual short-term memory (STM) and long-term memory (LTM) are distinct; that the process of mental rotation does not operate directly upon LTM; and that functionally, upright and rotated images may differ in important ways.     

Global optimality of the successive Maxbet algorithm

The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.The authors are obliged to Henk Kiers for critical comments on a previous draft.