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As we are dealing with increasingly complex data, our need for characterising them through a few, interpretable features has grown considerably. Topology has proven to be a useful tool in this quest for "insights on the data", since it characterises objects through their connectivity structure, i.e. connected components, loops and voids. More specifically, the new, but growing, field of TDA (Topological Data Analysis) deals “Persistent Homologyâ€, a multiscale version of Homology Groups, which allows us to give a measure of importance to each topological feature and to deal with noise.

Topological inference is typically carried out using the Persistence Diagram, which is a scatter plot of the generators of persistent homology groups in the data, and its functional representations (Persistence Landscapes, Silhouettes etc). All of these objects, however, are designed and work only for static point clouds. Building on existing tools from TDA, we define a new topological summary, the “Landscape Surfaceâ€, that takes into account the changes in the topology of a dynamical point cloud such as a (possibly multivariate) time serie. We prove its continuity and its stability and, finally, we show its use in some applications.