A Non-metric Procedure for Transforming Dissimilarities to Euclidean Distances Useful in Numerical Taxonomy and Ecology
✍ Scribed by L. P. Lefkovitch
- Publisher
- John Wiley and Sons
- Year
- 1989
- Tongue
- English
- Weight
- 995 KB
- Volume
- 31
- Category
- Article
- ISSN
- 0323-3847
No coin nor oath required. For personal study only.
✦ Synopsis
Engineering and Statistical Research Centre Agriculture Canada Contribution No. 1-727 from the Engineering and Statistical Reeearch Centre Summu y
Because (a) in agglomerative cluster analysis, i t is the smallest dissimilarities which are used, (b) in numerical ecology, little or no information is given about the resemblance between two localities if there are very few (or no) species in common, (c) for 8 given dissimilarity, the number of ways two objects can differ in the states shown by their attributes is small for small dissimilarities, and disproportionately larger for larger dissimilarities, (d) the major axes in principal coordinates analpis are debrmined by the largest dissimilarities, which from (843) are the least informative, and (e) the 'horseshoe' pattern often shown by two-dimensional scaling8 of dissimilarities, it is concluded that the largest dissimilarities should be replaced by values determined from the smallest.
The proposal is to retain the dissimilarities corresponding with the edges in the relative neighbourhood graph, and to replace the remaining by the shortest p a t h on the dissimilarity-weighted graph. The new dissimilarities, which are linearly related with the retained subset, are shown to be Euclidean, and usually the effective dimensionality is reduced in comparison with the original set. Theinternal disposition of distinct subsets of objects is little affected, but any distinct subsets tend to be further separated.
By contrast, if the smallest dissimilarities represent random differences, while the larger subset represents those which are systematic, i t is the latter which should be retained. The notion of the relative external graph is introduced, and the procedures required to obtain this, and to replace the deleted empirical smallest dissimilarities briefly described. However, this resulte in increase in the effective dimension, changes in the internal disposition of distinct subsets, and reduction in their separation.-Two numerical examples, based on empirical. data, illustrate some of the consequences of the transformations.