We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success ...

Abstract The aim of this paper is twofold. First, a new functional representation accelerometer data distributional nature introduced to build complete individualized profile each subject’s physical activity levels. Second, we extend two nonparametric regression models, kernel smoothing and ridge regression, handle survey obtain reliable conclusions about the influence activity. advantages prop...

Bayesian Data Analysis. Bayesian inference is too narrow; Bayesian statistics is too broad. Bayes is a good brand name; Statistics using conditional. Bayesian Data Analysis: Straightline fitting. Stephen F. Gull. Cavendish Laboratory,. Madingley Road,. Cambridge CB3 OHE, U.K Abstract. A Bayesian Overview. Bayesian data analysis. John K. Kruschke. . Bayesian methods have garnered huge interest i...

We define a new topological summary for data that we call the persistence landscape. In contrast to the standard topological summaries, the barcode and the persistence diagram, it is easy to combine with statistical analysis, and its associated computations are much faster. This summary obeys a Strong Law of Large Numbers and a Central Limit Theorem. Under certain finiteness conditions, this al...

We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation t...

We present a new method for analyzing imagebased motion data with topological methods for flow fields. A video sensor captures motion of subjects and yields discrete samples of velocity vectors in the image domain. We show how to construct a smooth, continuous, globally defined bidirectional flow field which evaluates determined vectors at given samples. We then apply a topological analysis of ...

simplicial complex: Given a set X, an abstract simplicial complex C with vertex set X is a set of finite subsets of X, the simplices, such that the elements of X belong to C and if σ ∈ C and τ ⊂ σ then τ ∈ C. Homology: Intuitively, homology (with coefficient in a field) associates to any topological space X, a family of vector spaces, the so-called homology groups Hk(X), k = 0, 1, . . ., each o...

Topological data analysis, such as persistent homology has shown beneficial properties for machine learning in many tasks. Topological representations, such as the persistence diagram (PD), however, have a complex structure (multiset of intervals) which makes it difficult to combine with typical machine learning workflows. We present novel compact fixed-size vectorial representations of PDs bas...