arXiv subjects

Robert J. Adler

Publications and source records attributed to Robert J. Adler.

30 records · Page 2Linked to original sources

arXiv2014

On the existence of paths between points in high level excursion sets of Gaussian random fields

The structure of Gaussian random fields over high levels is a well researched and well understood area, particularly if the field is smooth. However, the question as to whether or not two or more points which lie in an excursion set belong to the same connected component has constantly eluded analysis. We study this problem from the point of view of large deviations, finding the asymptotic probabilities that two such points are connected by a path laying within the excursion set, and so belong to the same component. In addition, we obtain a characterization and descriptions of the most likely paths, given that one exists.

math.PR↗

arXiv2013

Rotation and scale space random fields and the Gaussian kinematic formula

We provide a new approach, along with extensions, to results in two important papers of Worsley, Siegmund and coworkers closely tied to the statistical analysis of fMRI (functional magnetic resonance imaging) brain data. These papers studied approximations for the exceedence probabilities of scale and rotation space random fields, the latter playing an important role in the statistical analysis of fMRI data. The techniques used there came either from the Euler characteristic heuristic or via tube formulae, and to a large extent were carefully attuned to the specific examples of the paper. This paper treats the same problem, but via calculations based on the so-called Gaussian kinematic formula. This allows for extensions of the Worsley-Siegmund results to a wide class of non-Gaussian cases. In addition, it allows one to obtain results for rotation space random fields in any dimension via reasonably straightforward Riemannian geometric calculations. Previously only the two-dimensional case could be covered, and then only via computer algebra. By adopting this more structured approach to this particular problem, a solution path for other, related problems becomes clearer.

math.PR↗

arXiv2013

High level excursion set geometry for non-Gaussian infinitely divisible random fields

We consider smooth, infinitely divisible random fields $(X(t),t\in M)$, $M\subset {\mathbb{R}}^d$, with regularly varying Levy measure, and are interested in the geometric characteristics of the excursion sets \[A_u=\{t\in M:X(t)>u\}\] over high levels u. For a large class of such random fields, we compute the $u\to\infty$ asymptotic joint distribution of the numbers of critical points, of various types, of X in $A_u$, conditional on $A_u$ being nonempty. This allows us, for example, to obtain the asymptotic conditional distribution of the Euler characteristic of the excursion set. In a significant departure from the Gaussian situation, the high level excursion sets for these random fields can have quite a complicated geometry. Whereas in the Gaussian case nonempty excursion sets are, with high probability, roughly ellipsoidal, in the more general infinitely divisible setting almost any shape is possible.

math.PR↗

arXiv2013

Crackle: The Persistent Homology of Noise

We study the homology of simplicial complexes built via deterministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either become trivial as the number $n$ of vertices grows, or can contain more and more complex structures. The different behaviours are consequences of different underlying distributions for the generation of vertices, and we consider three illustrative examples, when the vertices are sampled from Gaussian, exponential, and power-law distributions in $\R^d$. We also discuss consequences of our results for manifold learning with noisy data, describing the topological phenomena that arise in this scenario as `crackle', in analogy to audio crackle in temporal signal analysis.

math.PR↗

arXiv2012

Efficient Monte Carlo for high excursions of Gaussian random fields

Our focus is on the design and analysis of efficient Monte Carlo methods for computing tail probabilities for the suprema of Gaussian random fields, along with conditional expectations of functionals of the fields given the existence of excursions above high levels, b. Naïve Monte Carlo takes an exponential, in b, computational cost to estimate these probabilities and conditional expectations for a prescribed relative accuracy. In contrast, our Monte Carlo procedures achieve, at worst, polynomial complexity in b, assuming only that the mean and covariance functions are Hölder continuous. We also explain how to fine tune the construction of our procedures in the presence of additional regularity, such as homogeneity and smoothness, in order to further improve the efficiency.

math.PR↗

arXiv2012

Multiple testing of local maxima for detection of peaks in 1D

A topological multiple testing scheme for one-dimensional domains is proposed where, rather than testing every spatial or temporal location for the presence of a signal, tests are performed only at the local maxima of the smoothed observed sequence. Assuming unimodal true peaks with finite support and Gaussian stationary ergodic noise, it is shown that the algorithm with Bonferroni or Benjamini--Hochberg correction provides asymptotic strong control of the family wise error rate and false discovery rate, and is power consistent, as the search space and the signal strength get large, where the search space may grow exponentially faster than the signal strength. Simulations show that error levels are maintained for nonasymptotic conditions, and that power is maximized when the smoothing kernel is close in shape and bandwidth to the signal peaks, akin to the matched filter theorem in signal processing. The methods are illustrated in an analysis of electrical recordings of neuronal cell activity.

math.ST↗

arXiv2010

Peak Detection as Multiple Testing

This paper considers the problem of detecting equal-shaped non-overlapping unimodal peaks in the presence of Gaussian ergodic stationary noise, where the number, location and heights of the peaks are unknown. A multiple testing approach is proposed in which, after kernel smoothing, the presence of a peak is tested at each observed local maximum. The procedure provides strong control of the family wise error rate and the false discovery rate asymptotically as both the signal-to-noise ratio (SNR) and the search space get large, where the search space may grow exponentially as a function of SNR. Simulations assuming a Gaussian peak shape and a Gaussian autocorrelation function show that desired error levels are achieved for relatively low SNR and are robust to partial peak overlap. Simulations also show that detection power is maximized when the smoothing bandwidth is close to the bandwidth of the signal peaks, akin to the well-known matched filter theorem in signal processing. The procedure is illustrated in an analysis of electrical recordings of neuronal cell activity.

stat.ME↗

arXiv2010

Persistent Homology for Random Fields and Complexes

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the algebraic topology of the excursion sets of random fields.

math.PR↗

arXiv2009

Gaussian processes, kinematic formulae and Poincaré's limit

We consider vector valued, unit variance Gaussian processes defined over stratified manifolds and the geometry of their excursion sets. In particular, we develop an explicit formula for the expectation of all the Lipschitz--Killing curvatures of these sets. Whereas our motivation is primarily probabilistic, with statistical applications in the background, this formula has also an interpretation as a version of the classic kinematic fundamental formula of integral geometry. All of these aspects are developed in the paper. Particularly novel is the method of proof, which is based on a an approximation to the canonical Gaussian process on the $n$-sphere. The $n\to\infty$ limit, which gives the final result, is handled via recent extensions of the classic Poincaré limit theorem.

math.DG↗

arXiv2008

Global geometry under isotropic Brownian flows

We consider global geometric properties of a codimension one manifold embedded in Euclidean space, as it evolves under an isotropic and volume preserving Brownian flow of diffeomorphisms. In particular, we obtain expressions describing the expected rate of growth of the Lipschitz-Killing curvatures, or intrinsic volumes, of the manifold under the flow. These results shed new light on some of the intriguing growth properties of flows from a global perspective, rather than the local perspective, on which there is a much larger literature.

math.PR↗

arXiv2007

Excursion sets of stable random fields

Studying the geometry generated by Gaussian and Gaussian- related random fields via their excursion sets is now a well developed and well understood subject. The purely non-Gaussian scenario has, however, not been studied at all. In this paper we look at three classes of stable random fields, and obtain asymptotic formulae for the mean values of various geometric characteristics of their excursion sets over high levels. While the formulae are asymptotic, they contain enough information to show that not only do stable random fields exhibit geometric behaviour very different from that of Gaussian fields, but they also differ significantly among themselves.

math.PR↗

arXiv2005

Validity of the expected Euler characteristic heuristic

We study the accuracy of the expected Euler characteristic approximation to the distribution of the maximum of a smooth, centered, unit variance Gaussian process f. Using a point process representation of the error, valid for arbitrary smooth processes, we show that the error is in general exponentially smaller than any of the terms in the approximation. We also give a lower bound on this exponential rate of decay in terms of the maximal variance of a family of Gaussian processes f^x, derived from the original process f.

math.PR↗