Topological Detection of Phenomenological Bifurcations with Unreliable Kernel Densities
arxiv(2024)
摘要
Phenomenological (P-type) bifurcations are qualitative changes in stochastic
dynamical systems whereby the stationary probability density function (PDF)
changes its topology. The current state of the art for detecting these
bifurcations requires reliable kernel density estimates computed from an
ensemble of system realizations. However, in several real world signals such as
Big Data, only a single system realization is available – making it impossible
to estimate a reliable kernel density. This study presents an approach for
detecting P-type bifurcations using unreliable density estimates. The approach
creates an ensemble of objects from Topological Data Analysis (TDA) called
persistence diagrams from the system's sole realization and statistically
analyzes the resulting set. We compare several methods for replicating the
original persistence diagram including Gibbs point process modelling, Pairwise
Interaction Point Modelling, and subsampling. We show that for the purpose of
predicting a bifurcation, the simple method of subsampling exceeds the other
two methods of point process modelling in performance.
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