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

Author(s)
Hans-Friedrich Köhn

ABSTRACT

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.

KEYWORDS

Anti-Robinson Structure, Three-Way Data, Individual Differences, Ultrametric, Unidimensional Scale

Anti-Robinson Structure, Three-Way Data, Individual Differences, Ultrametric, Unidimensional Scale

Cite this paper

Köhn, H. (2014) Anti-Robinson Structures for Analyzing Three-Way Two-Mode Proximity Data.*Applied Mathematics*, **5**, 983-1003. doi: 10.4236/am.2014.56094.

Köhn, H. (2014) Anti-Robinson Structures for Analyzing Three-Way Two-Mode Proximity Data.

References

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[2] Carroll, J.D. and Arabie, P. (1980) Multidimensional Scaling. Annual Review of Psychology, 31, 607-649.

http://dx.doi.org/10.1146/annurev.ps.31.020180.003135

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http://dx.doi.org/10.1007/BF02296969

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http://dx.doi.org/10.1007/BF01890116

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http://dx.doi.org/10.1111/j.2044-8317.1995.tb01065.x

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http://dx.doi.org/10.2307/276978

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http://dx.doi.org/10.1007/BF02294996

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http://dx.doi.org/10.1007/s00357-001-0032-z

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http://dx.doi.org/10.3758/BF03210535

[21] Defays, D. (1978) A Short Note on a Method of Seriation. British Journal of Mathematical and Statistical Psychology, 31, 49-53.

http://dx.doi.org/10.1111/j.2044-8317.1978.tb00571.x

[1] Hubert, L.J., Arabie, P. and Meulman, J. (2006) The Structural Representation of Proximity Matrices with MATLAB. SIAM, Philadelphia.

[2] Carroll, J.D. and Arabie, P. (1980) Multidimensional Scaling. Annual Review of Psychology, 31, 607-649.

http://dx.doi.org/10.1146/annurev.ps.31.020180.003135

[3] Carroll, J.D. (1976) Spatial, Non-Spatial and Hybrid Models for Scaling. Psychometrika, 41, 439-463.

http://dx.doi.org/10.1007/BF02296969

[4] Carroll, J.D. and Pruzansky, S. (1980) Discrete and Hybrid Scaling Models. In: Lantermann, E. and Feger, H., Eds., Similarity and Choice, Huber, Bern, 108-139.

[5] Carroll, J.D., Clark, L.A. and DeSarbo, W.S. (1984) The Representation of Three-Way Proximities Data by Single and Multiple Tree Structure Models. Journal of Classification, 1, 25-74.

http://dx.doi.org/10.1007/BF01890116

[6] Hubert, L.J. and Arabie, P. (1995) Iterative Projection Strategies for the Least-Squares Fitting of Tree Structures to Proximity Data. British Journal of Mathematical and Statistical Psychology, 48, 281-317.

http://dx.doi.org/10.1111/j.2044-8317.1995.tb01065.x

[7] Robinson, W.S. (1951) A Method for Chronologically Ordering Archaeological Deposits. American Antiquity, 19, 293301.

http://dx.doi.org/10.2307/276978

[8] Hubert, L.J. and Arabie, P. (1994) The Analysis of Proximity Matrices through Sums of Matrices Having (Anti-)Robinson Forms. British Journal of Mathematical and Statistical Psychology, 47, 1-40.

http://dx.doi.org/10.1111/j.2044-8317.1994.tb01023.x

[9] Rendl, F. (2002) The Quadratic Assignment Problem. In: Drezner, Z. and Hamacher, H.W., Eds., Facility Location, Springer, Berlin, 439-457.

http://dx.doi.org/10.1007/978-3-642-56082-8_14

[10] Hubert, L.J., Arabie, P. and Meulman, J. (2001) Combinatorial Data Analysis: Optimization by Dynamic Programming. SIAM, Philadelphia.

http://dx.doi.org/10.1137/1.9780898718553

[11] Brusco, M.J. (2002) A Branch-and-Bound Algorithm for Fitting Anti-Robinson Structures to Symmetric Dissimilarities Matrices. Psychometrika, 67, 459-471.

http://dx.doi.org/10.1007/BF02294996

[12] Brusco, M.J. and Stahl, S. (2005) Branch-and-Bound Applications in Combinatorial Data Analysis. Springer, New York.

[13] Dykstra, R.L. (1983) An Algorithm for Restricted Least-Squares Regression. Journal of the American Statistical Association, 78, 837-842.

http://dx.doi.org/10.1080/01621459.1983.10477029

[14] Deutsch, F. (2001) Best Approximation in Inner Product Spaces. Springer, New York.

http://dx.doi.org/10.1007/978-1-4684-9298-9

[15] Boyle, J.P. and Dykstra, R.L. (1985) A Method for Finding Projections onto the Intersection of Convex Sets in Hilbert Spaces. In: Dykstra, R.L., Robertson, R. and Wright, F.T., Eds., Advances in Order Restricted Inference (Vol. 37), Lecture Notes in Statistics, Springer, Berlin, 28-47.

[16] Brusco, M.J. (2002) Integer Programming Methods for Seriation and Unidimensional Scaling of Proximity Matrices: A Review and Some Extensions. Journal of Classification, 19, 45-67.

http://dx.doi.org/10.1007/s00357-001-0032-z

[17] Hubert, L.J. and Arabie, P. (1986) Unidimensional Scaling and Combinatorial Optimization. In: de Leeuw, J., Meulman, J., Heiser, W. and Critchley, F., Eds., Multidimensional Data Analysis, DSWO Press, Leiden, 181-196.

[18] Hubert, L.J. and Arabie, P. (1988) Relying on Necessary Conditions for Optimization: Unidimensional Scaling and Some Extensions. In: Bock, H.H., Ed., Classification and Related Methods of Data Analysis, Elsevier, Amsterdam, 463-472.

[19] Barthélemy, J.P. and Guénoche, A. (1991) Tree and Proximity Representations. Wiley, Chichester.

[20] Tversky, A. and Krantz, D. (1969) Similarity of Schematic Faces: A Test of Interdimensional Additivity. Perception and Psychophysics, 5, 124-128.

http://dx.doi.org/10.3758/BF03210535

[21] Defays, D. (1978) A Short Note on a Method of Seriation. British Journal of Mathematical and Statistical Psychology, 31, 49-53.

http://dx.doi.org/10.1111/j.2044-8317.1978.tb00571.x