In this paper we revisit and extend some classical results on persistent homology. We start by extending the notion of merge trees to all continuous functions on some general topological spaces. We revisit the concept of homological dimension, previously introduced by other authors and show that the suprema in the definitions of these concepts is attained generically in the sense of Baire. We then generalize the Wasserstein stability theorem to irregular settings, giving explicit bounds on the constants in the theorem and sharp bounds on its regime of validity. Finally, we use this generalized Wasserstein stability theorem to show a stochastic stability theorem for persistence diagrams.
This paper investigates the propreties of the persistence diagrams stemming from almost surely continuous random processes on $[0,t]$. We focus our study on two variables which together characterize the barcode : the number of points of the persistence diagram inside a rectangle $]−\infty,x]\times[x+\varepsilon,\infty[$, $N^{x,x+\varepsilon}$ and the number of bars of length $\geq \varepsilon$, $N^\varepsilon$. For processes with the strong Markov property, we show both of these variables admit a moment generating function and in particular moments of every order. Switching our attention to semimartingales, we show the asymptotic behaviour of $N^\varepsilon$ and $N^{x,x+\varepsilon}$ as $\varepsilon \to 0$ and of $N^\varepsilon$ as $\varepsilon \to \infty$. Finally, we study the repercussions of the classical stability theorem of barcodes and illustrate our results with some examples, most notably Brownian motion and empirical functions converging to the Brownian bridge.
We describe the topology of superlevel sets of ($\alpha$-stable) Lévy processes $X$ by introducing so-called stochastic $\zeta$-functions, which are defined in terms of the widely used $\text{Pers}_p$-functional in the theory of persistence modules. The latter share many of the properties commonly attributed to $\zeta$-functions in analytic number theory, among others, we show that for $\alpha$-stable processes, these (tail) $\zeta$-functions always admit a meromorphic extension to the entire complex plane with a single pole at $\alpha$, of known residue and that the analytic properties of these $\zeta$-functions are related to the asymptotic expansion of a dual variable, which counts the number of variations of $X$ of size $\geq \varepsilon$. Finally, using these results, we devise a new statistical parameter test using the topology of these superlevel sets.
Euler and Betti curves of stochastic processes defined on a $d$-dimensional compact Riemannian manifold which are almost surely in a Sobolev space $W^{n,s}(X,\mathbb{R})$ (with $d<n$) are stable under perturbations of the distributions of said processes in a Wasserstein metric. Moreover, Wasserstein stability is shown to hold for all $p>\frac{d}{n}$ for persistence diagrams stemming from functions in $W^{n,s}(X,\mathbb{R})$.
In this paper, we attempt to reproduce the results obtained by Sovacool
et al. in their recent paper the differences in carbon emissions
reduction between countries pursuing renewable electricity versus nuclear
power. We have found several
aws in the statistical analysis
performed theirein, notably the lack of time series considerations and
lack of consideration of statistical artifacts which arise from the nature
of the data.
The data used in this paper is the same as that used in Sovacool et al.'s analysis and is publically available here, as is our analysis.
We provide a response to Sovacool et al.'s response to our paper "On Sovacool's et al. study on the differences in
carbon emissions reduction between countries
pursuing renewable electricity versus nuclear power".