916 research outputs found
Gaussian processes, kinematic formulae and Poincar\'e'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 -sphere. The
limit, which gives the final result, is handled via recent
extensions of the classic Poincar\'e limit theorem.Comment: Published in at http://dx.doi.org/10.1214/08-AOP439 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Distortion and Excluded Middles
Why is there so much distortion in ordinary, political, social, and ethical argument? Since we have a pervasive interest in reasoning well and corresponding abilities, the extent of distortion invites explanation. The leading candidates are the need to economize, widespread, fallacious heuristics or assumptions, and self-defensive biases. I argue that these are not sufficient. An additional force is the intellectual pressure generated by acceptance of norms of conversation and argument, which exclude ‘middles’ of, prominently, neither accept (believe) nor reject (disbelieve). I conjecture that the distortion we find is due to intellectual and normative pressures generated by our commitment to these excluded-middle norms and if, or when, their force is lessened, there is likely to be less distortion
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.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1055 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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.Comment: 35 pages, 1 figur
High level excursion set geometry for non-Gaussian infinitely divisible random fields
We consider smooth, infinitely divisible random fields ,
, with regularly varying Levy measure, and are
interested in the geometric characteristics of the excursion sets over high levels u. For a large class of such random fields, we
compute the asymptotic joint distribution of the numbers of
critical points, of various types, of X in , conditional on 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.Comment: Published in at http://dx.doi.org/10.1214/11-AOP738 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Jointly reconstructing ground motion and resistivity for ERT-based slope stability monitoring
Electrical resistivity tomography (ERT) is increasingly being used to investigate unstable slopes and monitor the hydrogeological processes within. But movement of electrodes or incorrect placement of electrodes with respect to an assumed model can introduce significant resistivity artefacts into the reconstruction. In this work, we demonstrate a joint resistivity and electrode movement reconstruction algorithm within an iterative Gauss–Newton framework. We apply this to ERT monitoring data from an active slow-moving landslide in the UK. Results show fewer resistivity artefacts and suggest that electrode movement and resistivity can be reconstructed at the same time under certain conditions. A new 2.5-D formulation for the electrode position Jacobian is developed and is shown to give accurate numerical solutions when compared to the adjoint method on 3-D models. On large finite element meshes, the calculation time of the newly developed approach was also proven to be orders of magnitude faster than the 3-D adjoint method and addressed modelling errors in the 2-D perturbation and adjoint electrode position Jacobian
A Framework for Generalising the Newton Method and Other Iterative Methods from Euclidean Space to Manifolds
The Newton iteration is a popular method for minimising a cost function on
Euclidean space. Various generalisations to cost functions defined on manifolds
appear in the literature. In each case, the convergence rate of the generalised
Newton iteration needed establishing from first principles. The present paper
presents a framework for generalising iterative methods from Euclidean space to
manifolds that ensures local convergence rates are preserved. It applies to any
(memoryless) iterative method computing a coordinate independent property of a
function (such as a zero or a local minimum). All possible Newton methods on
manifolds are believed to come under this framework. Changes of coordinates,
and not any Riemannian structure, are shown to play a natural role in lifting
the Newton method to a manifold. The framework also gives new insight into the
design of Newton methods in general.Comment: 36 page
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