arXiv subjects

Wilfrid Stephen Kendall

Publications and source records attributed to Wilfrid Stephen Kendall.

3 recordsLinked to original sources

arXiv2020

Rayleigh Random Flights on the Poisson line SIRSN

We study scale-invariant Rayleigh Random Flights ("RRF") in random environments given by planar Scale-Invariant Random Spatial Networks ("SIRSN") based on speed-marked Poisson line processes. A natural one-parameter family of such RRF (with scale-invariant dynamics) can be viewed as producing "randomly-broken local geodesics" on the SIRSN; we aim to shed some light on a conjecture that a (non-broken) geodesic on such a SIRSN will never come to a complete stop en route. (If true, then all such geodesics can be represented as doubly-infinite sequences of sequentially connected line segments. This would justify a natural procedure for computing geodesics.) The family of these RRF ("SIRSNRRF"), is introduced via a novel axiomatic theory of abstract scattering representations for Markov chains (itself of independent interest). Palm conditioning (specifically the Mecke-Slivnyak theorem for Palm probabilities of Poisson point processes) and ideas from the ergodic theory of random walks in random environments are used to show that at a critical value of the parameter the speed of the scale-invariant SIRSNRRF neither diverges to infinity nor tends to zero, thus supporting the conjecture.

math.PR↗

arXiv2020

Counterexamples for optimal scaling of Metropolis-Hastings chains with rough target densities

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 $0.574$. We establish counterexamples to demonstrate that smoothness assumptions of the order of $\mathcal{C}^1(\mathbb{R})$ for RWM and $\mathcal{C}^3(\mathbb{R})$ for MALA are indeed required if these scaling rates are to hold. The counterexamples identify classes of marginal targets for which these guidelines are violated, obtained by perturbing a standard Normal density (at the level of the potential for RWM and the second derivative of the potential for MALA) using roughness generated by a path of fractional Brownian motion with Hurst exponent $H$. For such targets there is strong evidence that RWM and MALA proposal variances should optimally be scaled as $n^{-\frac{1}{H}}$ and as $n^{-\frac{1}{2+H}}$ and will then obey anomalous acceptance rate guidelines. Useful heuristics resulting from this theory are discussed. The paper develops a framework capable of tackling optimal scaling results for quite general Metropolis-Hastings algorithms (possibly depending on a random environment).

math.PR↗

arXiv2017

Perches, Post-holes and Grids

The "Planning in the Early Medieval Landscape" project (PEML) , funded by the Leverhulme Trust, has organized and collated a substantial quantity of images, and has used this as evidence to support the hypothesis that Anglo-Saxon building construction was based on grid-like planning structures based on fixed modules or quanta of measurement. We report on the development of some statistical contributions to the debate concerning this hypothesis. In practice the PEML images correspond to data arising in a wide variety of different forms. It does not seem feasible to produce a single automatic method which can be applied uniformly to all such images; even the initial chore of cleaning up an image (removing extraneous material such as legends and physical features which do not bear on the planning hypothesis) typically presents a separate and demanding challenge for each different image. Moreover care must be taken, even in the relatively straightforward cases of clearly defined ground-plans (for example for large ecclesiastical buildings of the period), to consider exactly what measurements might be relevant. We report on pilot statistical analyses concerning three different situations. These establish not only the presence of underlying structure (which indeed is often visually obvious), but also provide an account of the numerical evidence supporting the deduction that such structure is present. We contend that statistical methodology thus contributes to the larger historical debate and provides useful input to the wide and varied range of evidence that has to be debated.

stat.AP↗