3.2 Mapping

### 3.2.2 Non-metric multidimensional scaling

If the metric nature of transformation is abandoned, non-metric multidimensional scaling [Kruskal and Wish, 1981][Wilkinson et al., 1992][Cox et al., 1994] or non-linear dimensionality reduction [Li et al., 1995] can be defined as a mapping which only has to obey the monotonicity constraint. Popular methods for doing this transformation, which differ mainly on the cost function[5] to be minimized, are Kruskal's approach [Kruskal and Wish, 1981], the Guttman approach [Cox et al., 1994] and Sammon's mapping (non-linear mapping) [Kohonen, 1995][Li et al., 1995].
Although these methods could lead to a potential solution to our mapping problem[6], a fairly new method, the self-organizing map algorithm, has proved to outperform them [Li et al., 1995].

[5] The cost function tries to estimate how well a configuration satisfies the requirements. It is a prerequisite for creating an optimization model

[6] Keeping in mind that abandoning the metric nature of the transformation will not be satisfactory for our particular problem, and thus a way to recover it will have to be sought.

Cyberspace geography visualization - 15 October 1995

Luc Girardin, The Graduate Institute of International Studies