348 research outputs found
Rank Reduction of Correlation Matrices by Majorization
A novel algorithm is developed for the problem of finding a low-rank correlation matrix nearest to a given correlation matrix. The algorithm is based on majorization and, therefore, it is globally convergent. The algorithm 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. Mainly, such an application concerns interest rates.rank, correlation matrix, majorization, lognormal price processes
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.Nowadays there is an increasing tendency of visualizing data in favor of
tables. An important advantage is that results can be derived more
easily and interpretation is more direct. For example, the correlations of
returns of stock market indices can be mapped by multidimensional
scaling, separating closely related markets from less related markets. In
this inaugural address, dynamic visualization is added as a new
element to such maps. We discuss applications of this idea in
interactively constructing a map, modeling changing correlations over
time of stock exchanges by dynamic maps, and interactive construction
of a map of the Dutch political parties. Combining visualization with
interaction and dynamics provides an easier way to gain insight in
complex data than using static visualization
Symbolic Multidimensional Scaling
__Abstract__
Multidimensional scaling (MDS) is a technique that visualizes dissimilarities between pairs of objects as distances between points in a low dimensional space. In symbolic MDS, a dissimilarity is not just a value but can represent an interval or even a histogram. Here, we present an overview of developments for symbolic MDS. We discuss how interval dissimilarities they can be represented by (concentric) circles or rectangles, how replications can be represented by a three-way MDS version, and show how nested intervals of distances can be obtained for representing histogram dissimilarities. The various models are illustrated by empirical examples
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
Majorization algorithms for inspecting circles, ellipses, squares, rectangles, and rhombi
In several disciplines, as diverse as shape analysis, location
theory, quality control, archaeology, and psychometrics, it can be
of interest to fit a circle through a set of points. We use the
result that it suffices to locate a center for which the variance
of the distances from the center to a set of given points is
minimal. In this paper, we propose a new algorithm based on
iterative majorization to locate the center. This algorithm is
guaranteed to yield a series nonincreasing variances until a
stationary point is obtained. In all practical cases, the
stationary point turns out to be a local minimum. Numerical
experiments show that the majorizing algorithm is stable and fast.
In addition, we extend the method to fit other shapes, such as a
square, an ellipse, a rectangle, and a rhombus by making use of
the class of distances and dimension weighting. In addition,
we allow for rotations for shapes that might be rotated in the
plane. We illustrate how this extended algorithm can be used as a
tool for shape recognition
A new model for visualizing interactions in analysis of variance
In analysis of variance, there is usually little attention for interpreting
the terms of the effects themselves, especially for interaction
effects. One of the reasons is that the number of interaction-effect
terms increases rapidly with the number of predictor variables and
the number of categories. In this paper, we propose a new model,
called the interaction decomposition model, that allows to visualize
the interactions. We argue that with the help of the visualization, the
interaction-effect terms are much easier to interpret. We apply our
method to predict holiday spending1 using seven categorical predictor
variables
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
The Past, Present, and Future of Multidimensional Scaling
Multidimensional scaling (MDS) has established itself as a standard tool for statisticians and applied researchers. Its success is due to its simple and easily interpretable representation of potentially complex structural data. These data are typically embedded into a 2-dimensional map, where the objects of interest (items, attributes, stimuli, respondents, etc.) correspond to points such that those that are near to each other are empirically similar, and those that are far apart are different. In this paper, we pay tribute to several important developers of MDS and give a subjective overview of milestones in MDS developments. We also discuss the present situation of MDS and give a brief outlook on its future
More on Multidimensional Scaling and Unfolding in R: smacof Version 2
The smacof package offers a comprehensive implementation of multidimensional scaling (MDS) techniques in R. Since its first publication (De Leeuw and Mair 2009b) the functionality of the package has been enhanced, and several additional methods, features and utilities were added. Major updates include a complete re-implementation of multidimensional unfolding allowing for monotone dissimilarity transformations, including row-conditional, circular, and external unfolding. Additionally, the constrained MDS implementation was extended in terms of optimal scaling of the external variables. Further package additions include various tools and functions for goodness-of-fit assessment, unidimensional scaling, gravity MDS, asymmetric MDS, Procrustes, and MDS biplots. All these new package functionalities are illustrated using a variety of real-life applications
Multidimensional Scaling with Regional Restrictions for Facet Theory: An Application to Levi's Political Protest Data
Multidimensional scaling (MDS) is often used for the analysis of correlation matrices of items generated by a facet theory design. The emphasis of the analysis is on regional hypotheses on the location of the items in the MDS solution. An important regional hypothesis is the axial constraint where the items from different levels of a facet are assumed to be located in different parallel slices. The simplest approach is to do an MDS and draw the parallel lines separating the slices as good as possible by hand. Alternatively, Borg and Shye (1995) propose to automate the second step. Borg and Groenen (1997, 2005) proposed a simultaneous approach for ordered facets when the number of MDS dimensions equals the number of facets. In this paper, we propose a new algorithm that estimates an MDS solution subject to axial constraints without the restriction that the number of facets equals the number of dimensions. The algorithm is based on constrained iterative majorization of De Leeuw and Heiser (1980) with special constraints. This algorithm is applied to Levi’s (1983) data on political protests
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