Basic distance-derived statistics

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Basic distance-derived statistics

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With point data a range of core global statistics can be computed. Commonly produced measures of central tendency and spread include: identification of the location of the center of the point set (typically arithmetic mean, but optionally harmonic and geometric mean centers); identification of the median center (with or without weights and constraints); and standard distance, weighted standard distance and standard deviational ellipses. Each of these measures has been described earlier in Sections 4.2.5, Centroids and centers, and 4.5.3, Directional analysis of point datasets. A further measure, provided within some packages, is a table of all pairs of inter-point distances. This matrix is typically symmetric and completely specifies the arrangement of points — i.e. given this distance matrix the point set can be re-constructed, subject only to simple Euclidean transformations (translation, rotation and reflection)

Note that here we are describing Euclidean distance matrices — many other forms of distance matrix are possible and useful, although in general will not conform to the invariance characteristics of a Euclidean distance matrix. Examples including distance matrices that are computed using alternative metrics, such as squared Euclidean or Manhattan distance; non-symmetric distances matrices (e.g. based on transport infrastructure, routing and timetables); and proximity-based measures, such as similarity of genetic make-up or biomass. In cases where a matching distance matrix and proximity matrix are both available, Mantel tests (see Section 5.4.5, Proximity matrix comparisons) may be used to investigate the correlation between the two sets of measurements.