Geometric goodness of fit measure to detect patterns in data point clouds

We derived a geometric goodness-of-fit index, similar to $R^2$ using topological data analysis techniques. We build the Vietoris-Rips complex from the data-cloud projected onto each variable. Estimating the area of the complex and their domain, we create an index that measures the emptiness of the space with respect to the data. We made the analysis with an own package called TopSA (Topological Sensitivy Analysis).