doi:10.1016/S0167-8655(99)00135-X 
Copyright © 2000 Elsevier Science B.V. All rights reserved.
Multidimensional scaling of interval-valued dissimilarity data
T. Denœux 
, 
and M. Masson
Université de Technologie de Compiègne, UMR CNRS 6599 Heudiasyc, BP 20529, F-60205, Compiègne Cedex, France
Received 9 April 1999;
Revised 3 September 1999.
Available online 23 December 1999.
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Abstract
Multidimensional scaling is a well-known technique for representing measurements of dissimilarity among objects as points in a p-dimensional space. In this paper, this method is extended to the case where dissimilarities are only known to lie within certain intervals. Each object is then no longer represented as point, but as a region of 
, in such a way that the minimum and maximum distances between two regions approximate the lower and upper bounds of the dissimilarity interval between the two objects. Experiments with real data demonstrate the ability of this method to represent both the structure and the precision of dissimilarity measurements.
Author Keywords: Author Keywords: Multidimensional scaling; Interval-valued data; Exploratory data analysis; Data visualization
Fig. 1. Minimum and maximum distances between two spheric regions.
Fig. 2. Minimum and maximum distances between two boxes.
Fig. 3. Two-dimensional metric configuration for the fats and oil data (hypersphere model). Each circle represents one of the eight objects. The lower and upper Euclidean distances between any two circles approximate the distances between the corresponding objects.
Fig. 4. Real distances (x-axis) vs reconstructed distances (y-axis) for the fats and oil data using the hypersphere model (
: δij− vs dij−; ×: δij+ vs dij+). The closeness of the points to the first diagonal reflects the good quality of the model.
Fig. 5. Two-dimensional metric configuration for the fats and oil data (hyperbox model). Each rectangle represents one of the eight objects. The lower and upper Euclidean distances between any two rectangles approximate the distances between the corresponding objects.
Fig. 6. Real distances (x-axis) vs reconstructed distances (y-axis) for the fats and oil data using the hyperbox model (: δij− vs dij−; ×: δij+ vs dij+).
Fig. 7. Two-dimensional non-metric configuration for the vowel data (hypersphere model).
Fig. 8. Shepard diagram for the vowel data using the hypersphere model (: lower distances; ×: upper distances; –: isotonic regression function f).
Fig. 9. Two-dimensional non-metric configuration for the vowel data (hyperbox model).
Fig. 10. Shepard diagram for the vowel data using the hyperbox model (: lower distances; ×: upper distances; –: isotonic regression function f).
Table 1. Fats and oils data
Table 2. Fats and oils data: interval-valued distances from Table 1
Table 3. Vowel data: words used in recording the vowels
Table 4. Vowel data: interval-valued distance matrix
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