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  • Salón G101, CIMAT Guanajuato

Sesión conjunta del seminario de Estadística y las Sesiones ATD

13:00 - 14:00. Nonparametric Estimation of Probability Density Functions of Random Persistence Diagrams
Vasileios Maroulas, University of Tennessee at Knoxville 

Resumen: We introduce a nonparametric way to estimate the global probability density function for a random persistence diagram. Precisely, a kernel density function centered at a given persistence diagram and a given bandwidth is constructed. Our approach encapsulates the number of topological features and considers the appearance or disappearance of features near the diagonal in a stable fashion. In particular, the structure of our kernel individually tracks long persistence features, while considering features near the diagonal as a collective unit. The choice to describe short persistence features as a group saves computational time and reduces scaling while simultaneously retaining accuracy. Indeed, we prove that the associated kernel density estimate converges to the true distribution as the number of persistence diagrams increases and the bandwidth shrinks accordingly. We also establish the convergence of the mean absolute deviation estimate, defined according to the bottleneck metric. Lastly, examples of kernel density estimation are presented for typical underlying datasets.