English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Pro

Presents the zaia usage as premium feature

ZAIA

Presents AI Discovery search usage as a premium feature

AI Discovery

Presents AI Discovery explain and summary usage as a premium feature

Content Analysis

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Presents the pdf analysis as premium feature

PDF Analysis

1500 Top-up

3000 Top-up

Zendy Basic

2500 Top-up

Zendy Unlimited

Presents the access of premium content as premium feature

Premium Content

In topological data analysis, persistent homology characterizes robust topological features in data and it has a summary representation, called a persistence diagram. Statistical research for persistence diagrams have been actively developed, and the persistence weighted kernel shows several advantages over other statistical methods for persistence diagrams. If data is drawn from some probability distribution, the corresponding persistence diagram have randomness. Then, the expectation of the persistence diagram by the persistence weighted kernel is well-defined. In this paper, we study relationships between a probability distribution and the persistence weighted kernel in the viewpoint of (1) the strong law of large numbers and the central limit theorem, (2) a confidence interval to estimate the expectation of the persistence weighted kernel numerically, and (3) the stability theorem to ensure the continuity of the map from a probability distribution to the expectation. In numerical experiments, we demonstrate our method gives an interesting counterexample to a common view in topological data analysis.

On the expectation of a persistence diagram by the persistence weighted kernel