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.

翻译：现象性（P型）分岔是随机动力系统中平稳概率密度函数拓扑结构发生质变的现象。当前检测此类分岔的先进技术需要从系统实现集合中计算可靠的核密度估计值。然而在诸如大数据等真实世界信号中，通常仅能获取单一系统实现，这使得可靠核密度估计无法实现。本研究提出一种利用不可靠密度估计检测P型分岔的方法。该方法从系统的唯一实现中构建拓扑数据分析领域的对象集合（称为持久性图），并对所得集合进行统计分析。我们比较了包括吉布斯点过程建模、成对交互点建模和子采样在内的多种原始持久性图复制方法。结果表明，在预测分岔的任务中，简单的子采样方法在性能上优于其他两种点过程建模方法。