On the law of the iterated logarithm and strong invariance principles in stochastic geometry

We study the law of the iterated logarithm (Khinchin (1924), Kolmogorov (1929)) and related strong invariance principles in stochastic geometry. As potential applications, we think of well-known functionals such as functionals defined on the $k$-nearest neighbors graph and important functionals in topological data analysis such as the Euler characteristic and persistent Betti numbers

On the stability of the filtration functions for weakly dependent data with applications to structural break detection

In this paper, we study the stability of commonly used filtration functions in topological data analysis under small pertubations of the underlying nonrandom point cloud. Relying on these stability results, we then develop a test procedure to detect and determine structural breaks in a sequence of topological data objects obtained from weakly dependent data. The proposed method applies for instance to statistics of persistence diagrams of $\mathbb{R}^d$-valued Bernoulli shift systems under the \v{C}ech or Vietoris-Rips filtration

Two-sample tests for relevant differences in persistence diagrams

We study two-sample tests for relevant differences in persistence diagrams obtained from $L^p$-$m$-approximable data $(\mathcal{X}_t)_t$ and $(\mathcal{Y}_t)_t$. To this end, we compare variance estimates w.r.t.\ the Wasserstein metrics on the space of persistence diagrams. In detail, we consider two test procedures. The first compares the Fr{\'e}chet variances of the two samples based on estimators for the Fr{\'e}chet mean of the observed persistence diagrams $PD(\mathcal{X}_i)$ ($1\le i\le m$), resp., $PD(\mathcal{Y}_j)$ ($1\le j\le n$) of a given feature dimension. We use classical functional central limit theorems to establish consistency of the testing procedure. The second procedure relies on a comparison of the so-called independent copy variances of the respective samples. Technically, this leads to functional central limit theorems for U-statistics built on $L^p$-$m$-approximable sample data

Functional central limit theorems for persistent Betti numbers on cylindrical networks

We study functional central limit theorems for persistent Betti numbers obtained from networks defined on a Poisson point process. The limit is formed in large volumes of cylindrical shape stretching only in one dimension. The results cover a directed sublevel-filtration for stabilizing networks and the Cech and Vietoris-Rips complex on the random geometric graph. The presented functional central limit theorems open the door to a variety of statistical applications in topological data analysis and we consider goodness-of-fit tests in a simulation study

On the asymptotic normality of persistent Betti numbers

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process $(r,s) \mapsto n^{-1/2} (\beta^{r,s}_q ( \mathcal{K}(n^{1/d} S_n))-\mathbb{E}[\beta^{r,s}_q ( \mathcal{K}( n^{1/d} S_n))])$. So far, pointwise limit theorems have been established in different set-ups. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime, see Yogeshwaran et al. [2017] and Hiraoka et al. [2018]. In this contribution, we derive a strong stabilizing property (in the spirit of Penrose and Yukich [2001] of persistent Betti numbers and generalize the existing results on the asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that the multivariate asymptotic normality holds for all pairs $(r,s)$, $0\le r\le s<\infty$, and that it is not affected by percolation effects in the underlying random geometric graph

Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics

The present contribution investigates multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in practice for the construction of confidence intervals, motivating the use of the bootstrap in this capacity. However, the standard nonparametric bootstrap does not directly provide for asymptotically valid confidence intervals in some situations. A smoothed bootstrap procedure, instead, is shown to give consistent estimation in these settings. The present work relates to other general results in the area of stabilizing statistics, including central limit theorems for functionals of Poisson and Binomial processes in the critical regime. Specific statistics considered include the persistent Betti numbers of \v{C}ech and Vietoris-Rips complexes over point sets in $\mathbb R^d$, along with Euler characteristics, and the total edge length of the $k$-nearest neighbor graph. Special emphasis is made throughout to weakening the necessary conditions needed to establish bootstrap consistency. In particular, the assumption of a continuous underlying density is not required. A simulation study is provided to assess the performance of the smoothed bootstrap for finite sample sizes, and the method is further applied to the cosmic web dataset from the Sloan Digital Sky Survey (SDSS). Source code is available at github.com/btroycraft/stabilizing_statistics_bootstrap.Comment: 59 pages, 3 figures. Restructured paper with alternate problem settings moved to appendix. Rewrote data analysis and simulations study sections to be more comprehensive, moved each to the end of the pape

The Right to a Fair Trial in the Context of Counter-Terrorism: The use and suppression of sensitive information in Australia and the United Kingdom

In the recent fight against terrorism Western liberal democracies have significantly expanded pre-emptive measures, such as inchoate and preparatory offences or control orders. As these measures rely increasingly on the use of sensitive information, their application poses a dilemma. On the one hand, sensitive information may be necessary as evidence in an open court to justify the coercive measure or demonstrate the innocence of the suspect. On the other hand, states are reluctant to disclose such information where there is a risk to national security, preferring either to supress the information or to use it in secret. Such practices, however, may seriously violate the principle of fairness - and its attached individual right to a fair trial - a principle sitting not only at the core of the criminal justice system, but also forming part of the rule of law and democracy itself. The thesis poses the questions of what limitations are acceptable to the right to a fair trial, and what safeguards are necessary when states allow the suppression or use of sensitive information in criminal and related proceedings. The thesis is therefore concerned with finding an appropriate judicial methodology for addressing the dilemma in court. It argues that without a proper process (often generally described as balancing), minimum standards of fairness are more likely to be lowered due to security pressures. Principles, however, which emphasise the right to a fair trial and require justifications for any limitation in the interest of national security are capable of retaining higher standards. Hence the thesis suggests that while what is fair must be decided in the particular circumstances, what needs to be taken into consideration in order to achieve fairness can be defined. By comparing the case law from Australia and the United Kingdom, the thesis then offers an in-depths analyses of various degrees of balancing and principles when dealing with sensitive information, as well as the dynamics and interaction that accompany the two approaches between the branches of government. The two countries are particularly suitable for such an enquiry as they share a legal heritage, but have diverged increasingly over the last decades in how to protect human rights. While the thesis generally favours a principled approach as now predominantly applied in the UK, it does not simply propose a legal transplant for Australia, which so far has rejected any legislation including principles. Rather the comparison points out the reasons why Australian judges behave differently and challenges the Australian Parliament to amend the relevant legislation in accordance with its own values in order to retain high standards of fair trial protection in proceedings dealing with sensitive information