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A methodology is developed for data analysis based on empirically constructedgeodesic metric spaces. For a probability distribution, the length along a pathbetween two points can be defined as the amount of probability mass accumulatedalong the path. The geodesic, then, is the shortest such path and defines ageodesic metric. Such metrics are transformed in a number of ways to produceparametrised families of geodesic metric spaces, empirical versions of whichallow computation of intrinsic means and associated measures of dispersion.These reveal properties of the data, based on geometry, such as those that aredifficult to see from the raw Euclidean distances. Examples of applicationinclude clustering and classification. For certain parameter ranges, the spacesbecome CAT(0) spaces and the intrinsic means are unique. In one case, a minimalspanning tree of a graph based on the data becomes CAT(0). In another, aso-called "metric cone" construction allows extension to CAT($k$) spaces. It isshown how to empirically tune the parameters of the metrics, making it possibleto apply them to a number of real cases.

Empirical geodesic graphs and CAT(k) metrics for data analysis