Multidimensional Scaling with Hyperbox Model for Percentile Dissimilarities

In symbolic data analysis, we can consider more complex dissimilarity data existing between objects. Dissimilarity between objects may be described in various ways, including using a single value, an interval, a histogram, and a distribution. Analysis of such data may be carried out using multidimensional scaling. For histogram-valued dissimilarity data, Groenen and Winsberg proposed the “Hist- Scal” algorithm, which focuses on the percentiles of histogram dissimilarities [3]. For the I-Scal algorithm [2], the solution is guaranteed to improved after every iteration. However, for the Hist-Scal algorithm, this improvement cannot be guaranteed since iterative majorization is used in combination with the weighted monotone regression. In this paper, we propose a new improved algorithm applicable to percentile dissimilarities, called “the hyperbox model Percen-Scal” algorithm. Since the algorithm is based on iterative majorization, an improvement in the solution is guaranteed after every iteration of the algorithm.We apply both the hyperbox model Percen-Scal and the Hist-Scal algorithms to artificial data and compare the results obtained from both methods. We then discuss the validity of the results produced using the hyperbox model Percen-Scal algorithm.