# 3WaySym-Scal: three-way symbolic multidimensional scaling

Abstract: 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.

## Summary (2 min read)

### 1 Introduction

- Classical multidimensional scaling (MDS) models the dissimilarities among a set of objects as distances between points in a low dimensional space.
- Then, rather than using an average value of dissimilarity for each object pair one would wish to retain the information contained in the interval or histogram of dissimilarities obtained for each pair of objects.
- Both formulations are in line with the hyperbox approach.
- The hypersphere interpretation would be to state that the car is centered around a top speed of 180 km/h and a fuel consumption of 9 liters per 100 km and give a radius.
- All of the methods described above for MDS of symbolic data treat the two-way one-mode case.

### 2 MDS of Interval Dissimilarities

- The authors now review briefly the case of two-way one-mode MDS of interval dissimilarities.
- This objective is achieved by representing the objects by rectangles and approximate the upper bound of the dissimilarity by the maximum distance between the rectangles and the lower bound by the minimum distance between the rectangles.
- 2 )1/2 . (2) This definition implies that rotation of the axes changes the distances between the hyperboxes because they are always parallel to the rotated axes.
- This sensitivity for rotation can be seen as an asset because it makes a solution rotational unique, which is not true for ordinary MDS.
- For more details on iterative majorization and its use in three-way MDS, see, for example, De Leeuw and Heiser (1980) and Borg and Groenen (2005).

### 3 Two-Mode Three-Way MDS of Interval Data

- The I-Scal algorithm developed by Groenen et al. (2006) can be extended quite easily to two-mode three-way interval data.
- Let X and R denote here the centers and spreads of the hyperboxes in the common space.
- Then, the weighted Euclidean model restrictions imply that the hyperboxes for the individual replication ` are modelled as X` = XV` (4) R` = RV`, (5) where V` is a p×p diagonal matrix with dimension weights for replication `.
- The 3WaySym-Scal algorithm defined later updates X and R for fixed V` followed by updating V` for fixed X and R both using the majorizing function at the right of (8).

### 4 Synthesized Musical Instruments

- To illustrate their method, the authors consider an empirical data set where the entries in each of two dissimilarity matrices are an interval of values.
- On each occasion the expert listened to each pair of sounds and indicated a range of dissimilarity for each pair on a calibrated slider scale going from very similar to very different.
- The authors also present the results obtained analyzing the data from occasion one and occasion two separately using the I-Scal algorithm that is two separate two-way analyses in Figure 6.
- The results for the second occasion analyzed alone reflect the physical space the best, and the solution from the first occasion alone shows the most deviations from the physical space: 8, 3, 6 are too far to the left, 3 is too low, 7 is too far to the left, and 1 is too far to the right.

### 5 Discussion and Conclusions

- The authors have presented an MDS technique for symbolic data that deals with threeway two-mode fuzzy dissimilarities consisting of a interval of values observed for each pair of objects, for each source.
- By representing the objects as hypercubes, the authors are able to convey information contained when the dissimilarity between the objects or for any object pair needs to be expressed as a interval of values not a single value, and when one has data from more than one source.
- The 3WaySym-Scal algorithm for MDS of interval dissimilarities is based on iterative majorization, and the I-Scal algorithm created to deal with the case when dissimilarities are two-way, one-mode data and are given by a range or interval of values.
- The present model can be extended along at least two lines.
- First, one could allow for individual rotations of the common space.

Did you find this useful? Give us your feedback

...read more

##### References

^{1}

6,444 citations

^{1}

4,126 citations

1,093 citations

573 citations