# Wilfrid Stephen KendallThe University of Warwick · Department of Statistics

Wilfrid Stephen Kendall

DSc

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202

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Introduction

Additional affiliations

October 1994 - present

October 1991 - September 1994

October 1988 - September 1991

## Publications

Publications (202)

We study optimal Markovian couplings of Markov processes, where the optimality is understood in terms of minimization of concave transport costs between the time-marginal distributions of the coupled processes. We provide explicit constructions of such optimal couplings for one-dimensional finite-activity L\'evy processes (continuous-time random wa...

Surface metrology is the area of engineering concerned with the study of geometric variation in surfaces. This paper explores the potential for modern techniques from spatial statistics to act as generative models for geometric variation in 3D-printed stainless steel. The complex macro-scale geometries of 3D-printed components pose a challenge that...

Surface metrology is the area of engineering concerned with the study of geometric variation in surfaces. This paper explores the potential for modern techniques from spatial statistics to be used to characterise geometric variation in 3D-printed stainless steel. The complex macro-scale geometries of 3D-printed components pose a challenge that is n...

The emergence of additive manufacture (AM) for metallic material enables components of near arbitrary complexity to be produced. This has potential to disrupt traditional engineering approaches. However, metallic AM components exhibit greater levels of variation in their geometric and mechanical properties compared to standard components, which is...

For sufficiently smooth targets of product form it is known that the variance of a single coordinate of the proposal in RWM (Random walk Metropolis) and MALA (Metropolis adjusted Langevin algorithm) should optimally scale as $n^{-1}$ and as $n^{-\frac{1}{3}}$ with dimension \(n\), and that the acceptance rates should be tuned to $0.234$ and $0.574$...

This paper reports a study of scale-invariant Rayleigh Random Flights ("RRF'') in random environments given by planar Scale-Invariant Random Spatial Networks ("SIRS") based on speed-marked Poisson line processes. RRF can be viewed as producing "randomly broken $\Pi$-geodesics" on the SIRSN; the aim of the study is to shed some light on whether a (n...

Classical coupling constructions arrange for copies of the same Markov process started at two different initial states to become equal as soon as possible. In this paper, we consider an alternative coupling framework in which one seeks to arrange for two different Markov (or other stochastic) processes to remain equal for as long as possible, when...

Maximal couplings are (probabilistic) couplings of Markov processes such that the tail prob-abilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before m...

Consider a separable Banach space W supporting a non-trivial Gaussian measure µ. The following is an immediate consequence of the theory of Gaussian measure on Banach spaces: there exist (almost surely) successful couplings of two W-valued Brownian motions B and B begun at starting points B(0) and B(0) if and only if the difference B(0) − B(0) of t...

We show how to build an immersion coupling of a two-dimensional Brownian motion $(W_1, W_2)$ along with $\binom{n}{2} + n= \tfrac12n(n+1)$ integrals of the form $\int W_1^iW_2^j \circ dW_2$, where $j=1,\ldots,n$ and $i=0, \ldots, n-j$ for some fixed $n$. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusi...

Classical coupling constructions arrange for copies of the \emph{same} Markov process started at two \emph{different} initial states to become equal as soon as possible. In this paper, we consider an alternative coupling framework in which one seeks to arrange for two \emph{different} Markov processes to remain equal for as long as possible, when s...

Consider an improper Poisson line process, marked by positive speeds so as to
satisfy a scale-invariance property (actually, scale-equivariance). The line
process can be characterized by its intensity measure, which belongs to a
one-parameter family if scale and Euclidean invariance are required. This paper
investigates a proposal by Aldous, namely...

This is a case study concerning the rate at which probabilistic coupling
occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov
diffusion (Brownian motion together with its time integral, or, slightly more
generally, together with a finite number of iterated time integrals). In this
case there can be no Markovian maximal coupl...

This paper develops the use of Dirichlet forms to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis-Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Diric...

BACKGROUND: In computed tomography (CT), the spot geometry is one of the main sources of error in CT images. Since X-rays do not arise from a point source, artefacts are produced. In particular there is a penumbra effect, leading to poorly defined edges within a reconstructed volume. Penumbra models can be simulated given a fixed spot geometry and...

In this paper we describe a perfect simulation algorithm for the stable M/G/
c
queue. Sigman (2011) showed how to build a dominated coupling-from-the-past algorithm for perfect simulation of the super-stable M/G/
c
queue operating under first-come-first-served discipline. Sigman's method used a dominating process provided by the corresponding M/G/1...

This paper answers a question of Emery (2009) by constructing an explicit
coupling of two copies of the BenesKaratzasRishel (1991) diffusion (BKR
diffusion), neither of which starts at the origin, and whose natural
filtrations agree. The paper commences with a brief survey of probabilistic
coupling, defining a immersed coupling (the natural filtrat...

The purpose of this chapter is to exemplify construction of selected coupling-from-the-past algorithms, using simple examples and discussing code which can be run in the statistical scripting language R. The simple examples are: symmetric random walk with two reflecting boundaries, a very basic continuous state-space Markov chain, the Ising model w...

The use of barycentres in data analysis is illustrated, using as example a
dataset of hurricane trajectories.

Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial/final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination/source. S...

In this paper, we consider pursuit-evasion and probabilistic consequences of
some geometric notions for bounded and suitably regular domains in Euclidean
space that are CAT(kappa) for some kappa > 0. These geometric notions are
useful for analyzing the related problems of (a) existence/nonexistence of
successful evasion strategies for the Man in Li...

The theory of random lines has a celebrated history, reaching back 300 years into the past to the work of Buffon, and forming a major part of the field of stochastic geometry. Recently it has found application in the derivation of surprising non-stochastic results concerning effective planar networks (Aldous and Kendall, 2008). The following is an...

Perfect simulation refers to the art of converting suitable Markov Chain Monte Carlo algorithms into algorithms which return exact draws from the target distribution, instead of long-time approximations. The theoretical concepts underlying perfect simulation have a long history, but they were first drawn together to form a practical simulation tech...

An extensive update to a classic text
Stochastic geometry and spatial statistics play a fundamental role in many modern branches of physics, materials sciences, engineering, biology and environmental sciences. They offer successful models for the description of random two- and three-dimensional micro and macro structures and statistical methods fo...

We consider the problem of estimating neural activity from measurements of
the magnetic fields recorded by magnetoencephalography. We exploit the temporal
structure of the problem and model the neural current as a collection of
evolving current dipoles, which appear and disappear, but whose locations are
constant throughout their lifetime. This ful...

Two random processes X and Y on a metric space are said to be epsilon-shy
coupled if there is positive probability of them staying at least a positive
distance epsilon apart from each other forever. Interest in the literature
centres on nonexistence results subject to topological and geometric
conditions; motivation arises from the desire to gain a...

A commonly used metric for comparing the resilience of key predistribution schemes is fail s , which measures the proportion of network connections which are 'broken' by an adversary which has compromised s nodes. In 'Random key predistribution schemes for sensor networks', Chan, Perrig and Song present a formula for measuring the resilience in a c...

This paper introduces a new approach to analyzing spatial point data
clustered along or around a system of curves or "fibres." Such data arise in
catalogues of galaxy locations, recorded locations of earthquakes, aerial
images of minefields and pore patterns on fingerprints. Finding the underlying
curvilinear structure of these point-pattern data s...

The stationary isotropic Poisson line process was used to derive upper bounds
on mean excess network geodesic length in Aldous and Kendall [Adv. in Appl.
Probab. 40 (2008) 1-21]. The current paper presents a study of the geometry and
fluctuations of near-geodesics in the network generated by the line process.
The notion of a "Poissonian city" is in...

We prove weak laws of large numbers and central limit theorems of Lindeberg
type for empirical centres of mass (empirical Fr\'echet means) of independent
non-identically distributed random variables taking values in Riemannian
manifolds. In order to prove these theorems we describe and prove a simple kind
of Lindeberg-Feller central approximation t...

Focussing on the work of Sir John Kingman, one of the world's leading researchers in probability and mathematical genetics, this book touches on the important areas of these subjects in the last 50 years. Leading authorities give a unique insight into a wide range of currently topical problems. Papers in probability concentrate on combinatorial and...

We exhibit some explicit co-adapted couplings for n-dimensional Brownian motion and all its Levy stochastic areas. In the two-dimensional case we show how to derive exact asymptotics for the coupling time under various mixed coupling strategies, using Dufresne's formula for the distribution of exponential functionals of Brownian motion. This yields...

There are many variations on what one may regard as statistical shape, depending on the application in mind. The focus of this chapter is the statistical analysis of the shapes determined by finite sequences of points in a Euclidean space. We shall draw together a range of ideas from statistical shape theory, including distributions, diffusions, es...

Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, their probability theory, and the challenging problems raised by their statistical analysis. It has grown rapidly in response to challenges in all kinds...

The aims of APTS are to equip first-year PhD students in statistical sciences in the
UK with clear “mental maps” of some important areas in the discipline, and to help
them gain confidence and resources (analytical and computing tools, literature
entry-points) so that they are enabled to find out more for themselves. The intention
is thereby to add...

Benjamini, Burdzy and Chen (2007) introduced the notion of a shy coupling: a
coupling of a Markov process such that, for suitable starting points, there is
a positive chance of the two component processes of the coupling staying a
positive distance away from each other for all time. Among other results, they
showed no shy couplings could exist for...

This paper discusses the uses of computer algebra within statistics and probability. A distinction is drawn between the use of computer algebra packages to support investigations, by performing calculations, ankl their use to implement structure; to build in elements of a theory (such as stochastic calculus or the Taylor string theory of Barndorff...

In designing a network to link n points in a square of area n , we might be guided by the following two desiderata. First, the total network length should not be much greater than the length of the shortest network connecting all points. Second, the average route length (taken over source-destination pairs) should not be much greater than the avera...

We describe a method for inferring vascular (tree-like) structures from 2D and 3D imagery. A Bayesian formulation is used
to make effective use of prior knowledge of likely tree structures with the observed being modelled locally with intensity
profiles as being Gaussian. The local feature models are estimated by combination of a multiresolution, w...

Describes asymptotics of maximum lateral deviation (size and location), and a simple stochastic growth model which leads to a bound on the variance.

The principal aim of stochastic geometry is the mathematical analysis of random geometric structures. Fundamental examples of such structures are point processes of geometric objects, random tessellations of space into convex or non-convex regions, random systems of non-overlapping balls (or more general convex bodies), or excursion and level sets...

The paper ‘Modern statistics for spatial point processes’ by Jesper Møller and Rasmus P. Waagepetersen is based on a special invited lecture given by the authors at the 21st Nordic Conference on Mathematical Statistics, held at Rebild, Denmark, in June 2006. At the conference, Antti Penttinen and Eva B. Vedel Jensen were invited to discuss the pape...

Minimal area regions are constructed for Brownian paths and perturbed Brownian paths. While the theoretical optimal region
cannot be obtained in closed form, we provide practical confidence regions based on numerical approximations and local time
arguments. These regions are used to provide informal convergence assessments for both Monte Carlo and...

Pairwise interaction point processes with strong interaction are usually difficult to sample. We discuss how Besag lattice processes can be used in a simulated tem- pering MCMC scheme to help with the simulation of such processes. We show how the N-fold way algorithm can be used to sample the lattice processes efficiently and introduce the N-fold w...

We present a novel approach to examining local anisotropy in planar point processes. Our method is based on a kernel Principal Component Analysis and produces a tensor field that describes local orientation. The approach is illustrated on an example examining pore patterns in ink fingerprints.

This paper generalizes the work of Kendall [Electron. Comm. Probab. 9 (2004)
140--151], which showed that perfect simulation, in the form of dominated
coupling from the past, is always possible (although not necessarily practical)
for geometrically ergodic Markov chains. Here, we consider the more general
situation of positive recurrent chains and...

It is shown how to construct a successful co-adapted coupling of two copies of an n-dimensional Brownian motion (B<sub>1</sub>, …, B<sub>n</sub>) while simultaneously coupling all corresponding copies of Lévy stochastic areas ∫B<sub>i</sub> dB<sub>j</sub>−∫B<sub>j</sub> dB<sub>i</sub>. It is conjectured that successful co-adapted couplings still ex...

This article begins by presenting basic common features, and by listing and giving references for current commercial computer algebra packages. There follow two simple introductory examples of the use of computer algebra in statistical contexts, and then a discussion of some typical constraints and features of computer algebra systems: it is import...

When considering a Monte Carlo estimation procedure, the path produced by successive partial estimates is often used as a guide for informal convergence diagnostics. However the confidence region associated with that path cannot be derived simplistically from the confidence interval for the estimate itself. An asymptotically correct approach can be...

This note extends the work of Foss and Tweedie (1998), who showed that availability of the classic Coupling from the Past (CFTP) algorithm of Propp and Wilson (1996) is essentially equivalent to uniform ergodicity for a Markov chain (see also Hobert and Robert 2004). In this note we show that all geometrically ergodic chains possess dominated CFTP...

The Kolmogorov (1934)diffusion is the two-dimensional diffusion gen- erated by real Brownian motionB and its time integral R B dt. In this paper we construct successful co-adapted couplings for iterated Kolmogorov dif- fusions defined by adding iterated time integrals R R B ds dt, . . . as further components to the original Kolmogorov diffusion. A...

This research report describes Mathematica note-books available on the web at http://www.warwick.ac.uk/statsdept/ Staff/WSK/. They range from a reference notebook which displays the source code of Itovsn3, through a simple introductory notebook, to notebooks which describe examples relating to problems in math-ematical finance, coupling theory, and...

We study percolation and Ising models defined on generalizations of quad-trees used in multiresolution image analysis. These can be viewed as trees for which each mother vertex has
2
d
daughter vertices, and for which daughter vertices are linked together in d-dimensional Euclidean configurations. Retention probabilities and interaction strengths d...

We study percolation and Ising models defined on generalizations of quad-trees used in multiresolution image analysis. These can be viewed as trees for which each mother vertex has 2<sup>d</sup> daughter vertices, and for which daughter vertices are linked together in d-dimensional Euclidean configurations. Retention probabilities and interaction s...

IntroductionThe detection of artery and vein structure is of interest in both medical diagnosis and surgicalplanning. Here we examine vascular structure in the eye which is important for example todiagnose abnormal vessel growth due to diabetic retinopathy. The results presented here focuson retinal blood vessels; however all methods can be applied...

We describe a method for inferring tree-like vascular structures from 2D imagery. A Markov Chain Monte Carlo (MCMC) algorithm is employed to produce approximate samples from the posterior distribution given local feature estimates, derived from likelihood maximisation for a Gaussian intensity profile. A multiresolution scheme, in which coarse scale...

The theory of general state-space Markov chains can be strongly related to the case of discrete state-space by use of the notion of small sets and associated minorization conditions. The general theory shows that small sets exist for all Markov chains on state-spaces with countably generated [sigma]-algebras, though the minorization provided by the...

The usual direct method of simulation for cluster processes requires the generation of the parent point process over a region larger than the actual observation window, since we have to allow for all possible parents giving rise to observed daughter points, and some of these parents may fall outwith the observation window. When there is no a priori...

We describe a method for inferring tree-like vascular structures from 2D imagery. A Markov chain Monte Carlo (MCMC) algorithm is employed to sample from the posterior distribution given local feature estimates, derived from likelihood maximisation for a Gaussian intensity profile. A multiresolution scheme, in which coarse scale estimates are used t...

We describe a method for inferring vascular (tree-like) structures from 2D and 3D imagery. A Bayesian formulation is used to make eective use of prior knowledge of likely tree structures with the observed being modelled locally with intensity pro les as being Gaussian. The local feature models are estimated by combination of a multiresolution, wind...

Cluster point processes are important models for many biological data sets, and their simulation is often required for purposes of model exploration and statistical inference. The usual direct method of simulation requires the generation of the parent point process over a region larger than the actual observation window, since one has to allow for...

This paper is a preliminary report on the results of an investigation into the diffusion of Euclidean shape, using computer algebra to reduce complicated intermediate calculations to an informative final form. The computer algebra takes the form of an extension to the symbolic Ito calculus described in W.S. Kendall (1988). A substantially more deta...

This is the text of a talk I gave at Temasek Junior College during the ICFS in March 2000. The purpose of the talk was to communicate something of the excitement of modern mathematical science to a general audience of school students. I have edited it lightly, for example to clarify a reference to a visiting pop star, but otherwise have left the te...

Consider a polynomial-valued stochastic process
, where the coefficients
are real-valued or complex-valued semimartingales. The time-evolving zeros can be viewed as stochastic processes in their own right: the ?root-processes?of P The root-processes are complex-valued semimartingales except when they collide with each other or explode to infinity,...

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatia...

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatia...

In this paper we investigate the application of perfect simulation, in particular Coupling from The Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatia...

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero...

This document describes a C implementation of a perfect algorithm for a Metropolis -Hastings simulation of a point process, based on a decomposition into types relating to dierent square sub-regions, and a dominating process which is composed of independent discrete-time M=M=1 queues, one for each square sub-region, each in equilibrium. Mathematica...

Simulation plays an important role in stochastic geometry and related fields, because all but the simplest random set models tend to be intractable to analysis. Many simulation algorithms deliver (approximate) samples of such random set models, for example by simulating the equilibrium distribution of a Markov chain such as a spatial birth-and-deat...

This document describes a C implementation of an algorithm for a perfect simulation of a Boolean model conditioned to cover a set of points S = fx 1 ; x 2 ; :::; x k g in a bounded window W . This is done by using perfect simulation for a set of correlated Poisson random variables conditioned all to be non-zero. Mathematical details are to be found...

In this paper we present a perfect simulation method for obtaining perfect samples from collections of correlated Poisson random variables conditioned to be positive. We show how to use this method to produce a perfect sample from a Boolean model conditioned to cover a set of points: in [11] this special case was treated in a more complicated way....

This document describes version 2.18 of an actively developing implementation of Itovsn3 using the special object-oriented and mathematicalprogramming features of AXIOM code. It supersedes Version 1, which was a direct translation from (a simplied form of) the Mathematica version of [9], which itself was derived from the REDUCE version described in...

In this paper we study the notion of an efficient coupling of Markov processes. Informally, an efficient coupling is one which couples at the maximum possible exponential rate, as given by the spectral gap. This notion is of interest not only for its own sake, but also of growing importance arising from the recent advent of methods of "perfect simu...

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain , a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear co...

Symbolic Ito calculus refers both to the implementation of Ito calculus in a computer algebra package and to its application. This article 1 reports on progress in the implementation of Ito calculus in the powerful and innovative computer algebra package AXIOM, in the context of a decade of previous implementations and applications. It is shown how...

This essay is an introduction to the statistical theory of shape: a branch of statistics which arose independently from two source problems, one in archaeology, one concerning the biology of growth. After a brief history including a summary of the two source problems, the geometry and probability of statistical shape is described for two different...

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Xi be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear co...