XLSTAT - Multidimensional Scaling (MDS)

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Multidimensional Scaling principle

Multidimensional Scaling (MDS) is used to go from a proximity matrix (similarity or dissimilarity) between a series of N objects to the coordinates of these same objects in a p-dimensional space. p is generally fixed at 2 or 3 so that the objects may be visualized easily.

For example, with MDS, it is possible to reconstitute the position of towns on a map very precisely from the distances in kilometers (the dissimilarity in this case being the Euclidean distance) between the towns, modulo a rotation and a symmetrical transformation. Practically, MDS is often used in psychometry (perception analysis) and marketing (distances between products obtained from consumer classifications) but there are applications in a large number of domains.

Type of Multidimensional Scaling

There are two types of MDS depending on the nature of the dissimilarity observed:

Metric MDS	Non metric MDS
Absolute MDS Ratio MDS Interval MDS Polynomial MDS	Ordinal (2 options) Raw Stress Normalized Stress Kruskal's stress (2 options)

Note: for a given number of dimensions, the weaker the stress, the better the quality of the representation. Furthermore, the higher the number of dimensions, the weaker the stress.

Multidimensional Scaling algorithm in XLSTAT

XLSTAT uses the SMACOF (Scaling by MAjorizing a COnvex Function) algorithm which minimizes the "Normalized Stress" (de Leeuw, 1977).