# Multidimensional Scaling (MDS)

Multidimensional Scaling (MDS) is part of:

• ### System configuration

• Windows:
• Versions: 9x/Me/NT/2000/XP/Vista/Win 7/Win 8
• Excel: 97 and later
• Processor: 32 or 64 bits
• Hard disk: 150 Mb
• Mac OS X:
• OS: OS X
• Excel: X, 2004 and 2011
• Hard disk: 150Mb.

## Benefits

• Easy and user-friendly
• Data and results shared seamlessly
• Modular
• Didactic
• Affordable
• Accessible - Available in many languages
• Automatable and customizable

### 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).