Nonparametric Estimation of Probability Density Functions of Random Persistence Diagrams
Nonparametric Estimation of Probability Density Functions of Random Persistence Diagrams
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发表时间：
2018-03
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通讯作者：
V. Maroulas;Joshua L. Mike;Christopher Oballe
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作者：
V. Maroulas;Joshua L. Mike;Christopher Oballe
我们引入了一种非参数的方法来估计随机持久性图的全局概率密度函数。准确地说，核密度函数的中心在一个给定的持久性图和一个给定的带宽被构造。我们的方法封装的拓扑特征的数量，并认为在一个稳定的方式在对角线附近的功能的出现或消失。特别是，我们的内核结构单独跟踪长持久性特征，同时将对角线附近的特征视为一个集体单元。选择将短持久性特征描述为一组减少了计算时间，同时保持了准确性。事实上，我们证明了相关的核密度估计收敛到真实的分布作为持久性图的数量增加，相应地缩小带宽。我们还建立了收敛的平均绝对偏差估计，根据瓶颈度量定义。最后，给出了典型数据集的核密度估计的例子。
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 reduces computation time 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.