On measures of dissimilarity between point patterns: Classification based on prototypes and multidimensional scaling

Abstract

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Metrics and barycenters for point pattern data

“…From a more abstract point of view, if we think of a point pattern as an element of a metric space (N, τ ), where the metric τ reflects the concept of distance in an appropriate problem-related way, there is a number of standard methods which can be applied, including multidimensional scaling, disriminant and cluster analysis techniques. This is a stance already taken in Schuhmacher (2014), Section 1.4, and Mateu et al (2015). In the metric space (N, τ ) we can furthermore define a Fréchet mean of order q ≥ 1; that is, for data ξ 1 , .…”

“…Among others Schoenberg and Tranbarger (2008), Diez et al (2012) and Mateu et al (2015) have treated Fréchet means of order 1 (medians) for the spike time metric under the name of prototypes. However, computations in 2-d and higher were only possible for very small data sets due to a prohibitive computational cost of O(n 6 ) for the distance between two point patterns with n points each.…”

Statistical analysis of complex and spatially dependent data: A review of Object Oriented Spatial Statistics

“…For the sake of completeness in the present overview on spatial object data, we finally mention the viewpoint of those who interpret spatial patterns as object data (Mateu et al, 2015). Here, data are a discrete collection of (possibly linked) spatial points, such as point patterns, trees, graphs or networks.…”

“…For instance, one may want to analyze profiles of neuronal activation (i.e., random linked points) measured on different patients in response to a stimulus, in order to identify healthy/unhealty conditions. Additional examples of these types of data are available in environmental studies: Mateu et al (2015) deal with plant communities with high biodiversity, aiming at the classification of plant species according to their spatial structure and dislocation. Unlike traditional analyses of point patterns, in the object oriented approach the attention is posed on revealing similarities and correlations between entire spatial objects, rather than characterizing the interactions between single points within a given object.…”

Statistical tests for comparisons of spatial and spatio-temporal point patterns

“…The use of more complex functional descriptors (including marks, for example) in the ANOVA design also constitutes an open field of research. Finally, the definition of the factorial analysis statistics but using complex functional distances between patterns (Mateu et al, 2015) or between functions, could be a challenging field of research.…”