arXiv subjects

Katharine Turner

Publications and source records attributed to Katharine Turner.

29 records · Page 2Linked to original sources

arXiv2018

Rips filtrations for quasi-metric spaces and asymmetric functions with stability results

The Rips filtration over a finite metric space and its corresponding persistent homology are prominent methods in TDA to summarise the shape of data. Crucial to their use is the bottleneck stability result. A generalisation of the Rips filtration to any symmetric function $f:X\times X\to \mathbb{R}$ was defined by Chazal, De Silva and Oudot, and they showed it was stable with respect to the correspondence distortion distance. Allowing asymmetry, we consider four different persistence modules. The first method is through symmetrisation. For $a\in [0,1]$ we can construct a symmetric function $sym_a(f)(x,y)=a \min \{d(x,y), d(y,x)\}+ (1-a)\max \{d(x,y), d(y,x)\}$. We can then follow the apply the standard theory for symmetric functions and get stability as a corollary. The second method is to construct a filtration of ordered tuple complexes where tuple $(x_0, x_2, \ldots x_p)\in Rips^{dir}(X)_t$ if $d(x_i, x_j)\leq t$ for all $i\leq j$. These two methods have the same persistent homology as the standard Rips filtration when applied to a metric space, or more generally to a symmetric function. We consider two constructions using a filtration of directed graphs. We have directed graphs $\{D(X)_t\}$, where directed edges $x\to y$ are included in $D(X)_t$ whenever $\max\{f(x,y),f(x,x),f(y,y)\}\leq t$. From this we construct a preorder where $x\leq y$ if there is a path from $x$ to $y$ in $D(X)_t$. We build persistence modules using the strongly connected components of the graphs $D(X)_t$, which are the equivalence classes of the associated preorders. We consider persistence modules using a generalisation of poset topology to preorders. The Gromov-Hausdorff distance can be extended as a correspondence distortion distance to set-function pairs. We prove that all these new constructions enjoy the same stability results.

math.AT↗

arXiv2017

Fitting Markovian binary trees using global and individual demographic data

We consider a class of branching processes called Markovian binary trees, in which the individuals lifetime and reproduction epochs are modeled using a transient Markovian arrival process (TMAP). We estimate the parameters of the TMAP based on population data containing information on age-specific fertility and mortality rates. Depending on the degree of detail of the available data, a weighted non-linear regression method or a maximum likelihood method is applied. We discuss the optimal choice of the number of phases in the TMAP, and we provide confidence intervals for the model outputs. The results are illustrated using real data on human and bird populations.

stat.AP↗

arXiv2016

Hypothesis Testing for Topological Data Analysis

Persistent homology is a vital tool for topological data analysis. Previous work has developed some statistical estimators for characteristics of collections of persistence diagrams. However, tools that provide statistical inference for observations that are persistence diagrams are limited. Specifically, there is a need for tests that can assess the strength of evidence against a claim that two samples arise from the same population or process. We propose the use of randomization-style null hypothesis significance tests (NHST) for these situations. The test is based on a loss function that comprises pairwise distances between the elements of each sample and all the elements in the other sample. We use this method to analyze a range of simulated and experimental data. Through these examples we experimentally explore the power of the p-values. Our results show that the randomization-style NHST based on pairwise distances can distinguish between samples from different processes, which suggests that its use for hypothesis tests upon persistence diagrams is reasonable. We demonstrate its application on a real dataset of fMRI data of patients with ADHD.

stat.AP↗

arXiv2016

Topological analysis of the connectome of digital reconstructions of neural microcircuits

A recent publication provides the network graph for a neocortical microcircuit comprising 8 million connections between 31,000 neurons (H. Markram, et al., Reconstruction and simulation of neocortical microcircuitry, Cell, 163 (2015) no. 2, 456-492). Since traditional graph-theoretical methods may not be sufficient to understand the immense complexity of such a biological network, we explored whether methods from algebraic topology could provide a new perspective on its structural and functional organization. Structural topological analysis revealed that directed graphs representing connectivity among neurons in the microcircuit deviated significantly from different varieties of randomized graph. In particular, the directed graphs contained in the order of $10^7$ simplices Ð groups of neurons with all-to-all directed connectivity. Some of these simplices contained up to 8 neurons, making them the most extreme neuronal clustering motif ever reported. Functional topological analysis of simulated neuronal activity in the microcircuit revealed novel spatio-temporal metrics that provide an effective classification of functional responses to qualitatively different stimuli. This study represents the first algebraic topological analysis of structural connectomics and connectomics-based spatio-temporal activity in a biologically realistic neural microcircuit. The methods used in the study show promise for more general applications in network science.

q-bio.NC↗

arXiv2015

Principal Component Analysis of Persistent Homology Rank Functions with case studies of Spatial Point Patterns, Sphere Packing and Colloids

Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the (persistent homology) rank functions. For a point pattern $X$ we construct a filtration of spaces by taking the union of balls of radius $a$ centered on points in $X$, $X_a = \cup_{x\in X}B(x,a)$. The rank function $β_k(X):{\{(a,b)\in \mathbb{R}^2: a\leq b\}} \to \mathbb{R}$ is then defined by $β_k(X)(a,b) = rank ( ι_*:H_k(X_a) \to H_k(X_b))$ where $ι_*$ is the induced map on homology from the inclusion map on spaces. We consider the rank functions as lying in a Hilbert space and show that under reasonable conditions the rank functions from multiple simulations or experiments will lie in an affine subspace. This enables us to perform functional principal component analysis which we apply to experimental data from colloids at different effective temperatures and of sphere packings with different volume fractions. We also investigate the potential of rank functions in providing a test of complete spatial randomness of 2D point patterns using the distances to an empirically computed mean rank function of binomial point patterns in the unit square.

math.ST↗

arXiv2014

Probabilistic Fréchet Means for Time Varying Persistence Diagrams

In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In 2011, Mileyko and his collaborators made the first study of the properties of the Fréchet mean in $(\mathcal{D}_p,W_p)$, the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fréchet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fréchet mean definition to the realm of vineyards. We fix this problem by altering the original definition of Fréchet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each $N$ a map $(\mathcal{D}_p)^N \to \mathbb{P}(\mathcal{D}_p)$. We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on time-varying persistence diagrams, better known as vineyards.

math.PR↗

arXiv2014

Persistent Homology Transform for Modeling Shapes and Surfaces

In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. This statistic is a collection of persistence diagrams - multiscale topological summaries used extensively in topological data analysis. We use the PHT to represent shapes and execute operations such as computing distances between shapes or classifying shapes. We prove the map from the space of simplicial complexes in $\mathbb{R}^3$ into the space spanned by this statistic is injective. This implies that the statistic is a sufficient statistic for probability densities on the space of piecewise linear shapes. We also show that a variant of this statistic, the Euler Characteristic Transform (ECT), admits a simple exponential family formulation which is of use in providing likelihood based inference for shapes and surfaces. We illustrate the utility of this statistic on simulated and real data.

math.ST↗

arXiv2013

Cone fields and topological sampling in manifolds with bounded curvature

Often noisy point clouds are given as an approximation of a particular compact set of interest. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of μ-critical points in an annular region. Since an offset of a set deformation retracts to the set itself provided that there are no critical points of the distance function nearby, we can use this theorem to show when the offset of a point cloud is homotopy equivalent to the set it is sampled from. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature. In the process, we prove stability theorems for μ-critical points when the ambient space is a manifold.

cs.CG↗

arXiv2013

Fréchet Means for Distributions of Persistence diagrams

Given a distribution $ρ$ on persistence diagrams and observations $X_1,...X_n \stackrel{iid}{\sim} ρ$ we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams $X_1,...X_n$. If the underlying measure $ρ$ is a combination of Dirac masses $ρ= \frac{1}{m} \sum_{i=1}^m δ_{Z_i}$ then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from $ρ$. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.

math.ST↗

arXiv2012

Harmonic tori in de Sitter spaces $S^{2n}_1$

We show that all superconformal harmonic immersions from genus one surfaces into de Sitter spaces $ S ^ {2n}_1 $ with globally defined harmonic sequence are of finite-type and hence result merely from solving a pair of ordinary differential equations. As an application, we prove that all Willmore tori in $ S ^ 3 $ without umbilic points can be constructed in this simple way.

math.DG↗

arXiv2011

Toda frames, harmonic maps and extended Dynkin diagrams

We prove that all immersions of a genus one surface into G/T possessing a Toda frame can be constructed by integrating a pair of commuting vector fields on a finite dimensional Lie algebra. Here G is any simple real Lie group (not necessarily compact), T is a Cartan subgroup and the k-symmetric space structure on G/T is induced from the Coxeter automorphism. We provide necessary and sufficient conditions for the existence of a Toda frame for a harmonic map into G/T and describe those G/T to which the theory applies in terms of involutions of extended Dynkin diagrams.

math.DG↗