A short note on a method of seriation

Defays, Daniel

doi:10.1111/j.2044-8317.1978.tb00571.x

Cited by 49 publications

(32 citation statements)

References 2 publications

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“…For example, Defays [21] demonstrated that the task of finding a best-fitting unidimensional scale for given inter-object proximities can be solved solely by permuting the rows and columns of the data matrix such that a certain patterning among cell entries is satisfied. The desired numerical scale values can be immediately deduced from the reordered matrix.…”

Section: Discussionmentioning

confidence: 99%

Anti-Robinson Structures for Analyzing Three-Way Two-Mode Proximity Data

Köhn¹

2014

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Multiple discrete (non-spatial) and continuous (spatial) structures can be fitted to a proximity matrix to increase the information extracted about the relations among the row and column objects vis-à-vis a representation featuring only a single structure. However, using multiple discrete and continuous structures often leads to ambiguous results that make it difficult to determine the most faithful representation of the proximity matrix in question. We propose to resolve this dilemma by using a nonmetric analogue of spectral matrix decomposition, namely, the decomposition of the proximity matrix into a sum of equally-sized matrices, restricted only to display an order-constrained patterning, the anti-Robinson (AR) form. Each AR matrix captures a unique amount of the total variability of the original data. As our ultimate goal, we seek to extract a small number of matrices in AR form such that their sum allows for a parsimonious, but faithful reconstruction of the total variability among the original proximity entries. Subsequently, the AR matrices are treated as separate proximity matrices. Their specific patterning lends them immediately to the representation by a single (discrete non-spatial) ultrametric cluster dendrogram and a single (continuous spatial) unidimensional scale. Because both models refer to the same data base and involve the same number of parameters, estimated through least-squares, a direct comparison of their differential fit is legitimate. Thus, one can readily determine whether the amount of variability associated which each AR matrix is most faithfully represented by a discrete or a continuous structure, and which model provides in sum the most appropriate representation of the original proximity matrix. We propose an extension of the order-constrained anti-Robinson decomposition of square-symmetric proximity matrices to the analysis of individual differences of three-way data, with the third way representing individual data sources. An application to judgments of schematic face stimuli illustrates the method.

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Section: Discussionmentioning

confidence: 99%

Anti-Robinson Structures for Analyzing Three-Way Two-Mode Proximity Data

Köhn¹

2014

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“…This special case was already singled out by Guttman (1968). Defays (1978), Heiser (1981), and Hubert and Arabie (1986) show that one-dimensional scaling is essentially a combinatorial problem. Because either the result Fx(Y) = 0 or the fact of finite convergence stops all considerations having to do with rate of convergence, we assume from now onthatp > 1.…”

Section: Derivatives Of the Guttman Transformmentioning

confidence: 99%

Convergence of the majorization method for multidimensional scaling

Leeuw

1988

Journal of Classification

184

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“…Branch-and-bound algorithms have been developed for seriation of asymmetric matrices (DeCani, 1972;Flueck & Korsh, 1974), as well as least-squares unidimensional scaling (Delays, 1978). Relative to dynamic programming, branchand-bound algorithms frequently require less storage, especially when using depth-first solution strategies.…”

Section: Introductionmentioning

confidence: 99%