#### 5.7 Displaying similarity matrices

To illustrate how Multi Dimensional Scaling (MDS) works, we have selected 5 out of the 23 clean Michelsberg sites. For each possible combination of these 5 sites the Phi value has been calculated and displayed in the so-called dissimilarity matrix.

 52B-168 52B-192 52B-223 52E-150 52E-188 52B-168 .000 52B-192 .387 .000 52B-223 .565 .520 .000 52E-150 .468 .555 .471 .000 52E-188 .645 .548 .745 .687 .000
Table 13 Dissimilarity matrix of the 5 selected Michelsberg sites

This matrix shows that 52B-168 and 52B-192 are most similar (lowest dissimilarity) and 52B-223 and 52E-188 differ most (highest dissimilarity). Yet it is difficult to understand the 'position' of the sites in relation to each other, on the basis of such a matrix. A simple presentation method is a relation diagram, where the strength of the resemblance among sites is visualized by the number of 'elastic bands'.

Fig. 43 Visualizing a dissimilarity matrix

less detail

Multi Dimensional Scaling obtains the same results as such a relation diagram in a mathematical way: it converts the matrix into a picture, where the distances between all sites match the various differences as much as possible. Very similar sites are close together and strongly dissimilar sites are far apart. The picture may be 1-dimensional, 2-dimensional, 3-dimensional or n-dimensional, as desired. The less dimensions, the harder it is for the MDS-procedure to reflect all mutual distances accurately. The picture is then, as it were, under a lot of strain. Displays in a higher dimension have less strain (S), but are much harder to visualize. A 1-D diagram yields in general an inaccurate picture with too much strain. When the strain is not too high, we prefer a 2-dimensional display. This is a simple picture, easy to explain. A 3-D diagram has even less strain, but is also harder to use. For the 2-D MDS-diagram of the 5 selected Michelsberg sites the value of S is very low (0.007). Yet we can clearly see how the MDS-procedure has so to speak stretched the distance between 52B-168 and 52B-192, in order to express the position of all sites in the best possible way.

Fig. 44 Multi Dimensional Scaling diagram of the 5 selected Michelsberg sites