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In this chapter we shall consider several non-Euclidean distance measures that are popular in the environmental sciences: the Bray-Curtis dissimilarity, the L1 distance (also called the city-block or Manhattan distance) and the Jaccard index for presence-absence data.
The concept of distance between two samples or between two variables is fundamental in multivariate analysis – almost everything we do has a relation with this measure. If we talk about a single variable we take this concept for granted.
Measures of statistical distance are widely used in techniques such as clustering and classification, when we wish to identify objects that are in some sense similar to each other. The choice of distance between data points is an important one, and there are a large number of measures to choose from.
There are metrics that are useful in data analysis, i.e. given data we want to measure some notion of distance between (the distribution of) subsets of the data and use this in some way. There is of course overlap between these contexts and so there are distances that are useful in both, but they are motivated by slightly di erent considerations.
We discuss an approximation framework for model-based inference using statistical distances. Emphasis is placed on identifying in what sense and which statistical distances can be interpreted as loss functions and used for model assessment.
The distance between two sample units depends on which other sample units are included in the data matrix. Since this is an inverse weighting, common species are downplayed and rare species are weighted more strongly.
This document is also available in PDF: Distance.pdf. This is a working document which will be (hopefully frequently) updated with materials con-cerning the discrepancies between two distributions/parameters or the diversities of a set of distri-butions/parameters.
