### Functional Data Analysis Using a Topological Summary Statistic: The Smooth Euler Characteristic Transform

We introduce a novel statistic, the smooth Euler characteristic transform (SECT), which is designed to integrate shape information into regression models by representing shapes and surfaces as a collection of curves. Its construction is based on theory from topological data analysis (TDA). Due to its well-defined inner product structure, the SECT can be used in a wider range of functional and nonparametric modeling approaches than other previously proposed topological summary statistics. We provide mathematical properties of this statistic, notably, its injectivity, which is an implication for statistical sufficiency.

We illustrate the utility of the SECT in a radiomics context by showing that the topological quantification of tumors, assayed by magnetic resonance imaging (MRI), are better predictors of clinical outcomes in patients with glioblastoma multiforme (GBM). We show that topological features of tumors captured by the SECT alone explain more of the variance in patient survival than gene expression, volumetric features, and morphometric features.