Principal component analysis of persistent homology rank functions with case studies of spatial point patterns, sphere packing and colloids Article Swipe
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·
· 2016
· Open Access
·
· DOI: https://doi.org/10.1016/j.physd.2016.03.007
· OA: W801015201
Persistent homology, while ostensibly measuring changes in topology, captures\nmultiscale geometrical information. It is a natural tool for the analysis of\npoint patterns. In this paper we explore the statistical power of the\n(persistent homology) rank functions. For a point pattern $X$ we construct a\nfiltration of spaces by taking the union of balls of radius $a$ centered on\npoints in $X$, $X_a = \\cup_{x\\in X}B(x,a)$. The rank function\n${\\beta}_k(X):{\\{(a,b)\\in \\mathbb{R}^2: a\\leq b\\}} \\to \\mathbb{R}$ is then\ndefined by ${\\beta}_k(X)(a,b) = rank ( \\iota_*:H_k(X_a) \\to H_k(X_b))$ where\n$\\iota_*$ is the induced map on homology from the inclusion map on spaces. We\nconsider the rank functions as lying in a Hilbert space and show that under\nreasonable conditions the rank functions from multiple simulations or\nexperiments will lie in an affine subspace. This enables us to perform\nfunctional principal component analysis which we apply to experimental data\nfrom colloids at different effective temperatures and of sphere packings with\ndifferent volume fractions. We also investigate the potential of rank functions\nin providing a test of complete spatial randomness of 2D point patterns using\nthe distances to an empirically computed mean rank function of binomial point\npatterns in the unit square.\n
