Topology has proven to be a useful tool in the current quest for ”insights on the data”, since it characterises objects through their connectivity structure, in an easy and interpretable way. More specifically, the new, but growing, field of TDA (Topological Data Analysis) deals with Persistent Homology, a multiscale version of Homology Groups summarized by the Persistence Diagram and its functional representations (Persistence Landscapes, Silhouettes etc). All of these objects, how- ever, are designed and work only for static point clouds. We define a new topological summary, the Landscape Surface, that takes into account the changes in the topology of a dynamical point cloud such as a (possibly very high dimensional) time series. We prove its continuity and its stability and, finally, we sketch a simple example.

Topological summaries for Time-Varying Data / Padellini, Tullia; Brutti, Pierpaolo. - ELETTRONICO. - (2017), pp. 747-752. (Intervento presentato al convegno SIS 2017. Statistics and Data Science: new challenges, new generations tenutosi a Florence).

Topological summaries for Time-Varying Data

PADELLINI, TULLIA
;
Pierpaolo Brutti
2017

Abstract

Topology has proven to be a useful tool in the current quest for ”insights on the data”, since it characterises objects through their connectivity structure, in an easy and interpretable way. More specifically, the new, but growing, field of TDA (Topological Data Analysis) deals with Persistent Homology, a multiscale version of Homology Groups summarized by the Persistence Diagram and its functional representations (Persistence Landscapes, Silhouettes etc). All of these objects, how- ever, are designed and work only for static point clouds. We define a new topological summary, the Landscape Surface, that takes into account the changes in the topology of a dynamical point cloud such as a (possibly very high dimensional) time series. We prove its continuity and its stability and, finally, we sketch a simple example.
2017
SIS 2017. Statistics and Data Science: new challenges, new generations
persistent homology; time series; topological Inference
04 Pubblicazione in atti di convegno::04b Atto di convegno in volume
Topological summaries for Time-Varying Data / Padellini, Tullia; Brutti, Pierpaolo. - ELETTRONICO. - (2017), pp. 747-752. (Intervento presentato al convegno SIS 2017. Statistics and Data Science: new challenges, new generations tenutosi a Florence).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1138161
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