75,656 research outputs found
Colouring and breaking sticks: random distributions and heterogeneous clustering
We begin by reviewing some probabilistic results about the Dirichlet Process
and its close relatives, focussing on their implications for statistical
modelling and analysis. We then introduce a class of simple mixture models in
which clusters are of different `colours', with statistical characteristics
that are constant within colours, but different between colours. Thus cluster
identities are exchangeable only within colours. The basic form of our model is
a variant on the familiar Dirichlet process, and we find that much of the
standard modelling and computational machinery associated with the Dirichlet
process may be readily adapted to our generalisation. The methodology is
illustrated with an application to the partially-parametric clustering of gene
expression profiles.Comment: 26 pages, 3 figures. Chapter 13 of "Probability and Mathematical
Genetics: Papers in Honour of Sir John Kingman" (Editors N.H. Bingham and
C.M. Goldie), Cambridge University Press, 201
A new family of Markov branching trees: the alpha-gamma model
We introduce a simple tree growth process that gives rise to a new
two-parameter family of discrete fragmentation trees that extends Ford's alpha
model to multifurcating trees and includes the trees obtained by uniform
sampling from Duquesne and Le Gall's stable continuum random tree. We call
these new trees the alpha-gamma trees. In this paper, we obtain their splitting
rules, dislocation measures both in ranked order and in sized-biased order, and
we study their limiting behaviour.Comment: 23 pages, 1 figur
Entwined Paths, Difference Equations and the Dirac Equation
Entwined space-time paths are bound pairs of trajectories which are traversed
in opposite directions with respect to macroscopic time. In this paper we show
that ensembles of entwined paths on a discrete space-time lattice are simply
described by coupled difference equations which are discrete versions of the
Dirac equation. There is no analytic continuation, explicit or forced, involved
in this description. The entwined paths are `self-quantizing'. We also show
that simple classical stochastic processes that generate the difference
equations as ensemble averages are stable numerically and converge at a rate
governed by the details of the stochastic process. This result establishes the
Dirac equation in one dimension as a phenomenological equation describing an
underlying classical stochastic process in the same sense that the Diffusion
and Telegraph equations are phenomenological descriptions of stochastic
processes.Comment: 15 pages, 5 figures Replacement 11/02 contains minor editorial
change
Single camera pose estimation using Bayesian filtering and Kinect motion priors
Traditional approaches to upper body pose estimation using monocular vision
rely on complex body models and a large variety of geometric constraints. We
argue that this is not ideal and somewhat inelegant as it results in large
processing burdens, and instead attempt to incorporate these constraints
through priors obtained directly from training data. A prior distribution
covering the probability of a human pose occurring is used to incorporate
likely human poses. This distribution is obtained offline, by fitting a
Gaussian mixture model to a large dataset of recorded human body poses, tracked
using a Kinect sensor. We combine this prior information with a random walk
transition model to obtain an upper body model, suitable for use within a
recursive Bayesian filtering framework. Our model can be viewed as a mixture of
discrete Ornstein-Uhlenbeck processes, in that states behave as random walks,
but drift towards a set of typically observed poses. This model is combined
with measurements of the human head and hand positions, using recursive
Bayesian estimation to incorporate temporal information. Measurements are
obtained using face detection and a simple skin colour hand detector, trained
using the detected face. The suggested model is designed with analytical
tractability in mind and we show that the pose tracking can be
Rao-Blackwellised using the mixture Kalman filter, allowing for computational
efficiency while still incorporating bio-mechanical properties of the upper
body. In addition, the use of the proposed upper body model allows reliable
three-dimensional pose estimates to be obtained indirectly for a number of
joints that are often difficult to detect using traditional object recognition
strategies. Comparisons with Kinect sensor results and the state of the art in
2D pose estimation highlight the efficacy of the proposed approach.Comment: 25 pages, Technical report, related to Burke and Lasenby, AMDO 2014
conference paper. Code sample: https://github.com/mgb45/SignerBodyPose Video:
https://www.youtube.com/watch?v=dJMTSo7-uF
Metric Construction, Stopping Times and Path Coupling
In this paper we examine the importance of the choice of metric in path
coupling, and the relationship of this to \emph{stopping time analysis}. We
give strong evidence that stopping time analysis is no more powerful than
standard path coupling. In particular, we prove a stronger theorem for path
coupling with stopping times, using a metric which allows us to restrict
analysis to standard one-step path coupling. This approach provides insight for
the design of non-standard metrics giving improvements in the analysis of
specific problems.
We give illustrative applications to hypergraph independent sets and SAT
instances, hypergraph colourings and colourings of bipartite graphs.Comment: 21 pages, revised version includes statement and proof of general
stopping times theorem (section 2.2), and additonal remarks in section
Deterministic counting of graph colourings using sequences of subgraphs
In this paper we propose a deterministic algorithm for approximately counting
the -colourings of sparse random graphs . In particular, our
algorithm computes in polynomial time a approximation of
the logarithm of the number of -colourings of for with high probability over the graph instances.
Our algorithm is related to the algorithms of A. Bandyopadhyay et al. in SODA
'06, and A. Montanari et al. in SODA '06, i.e. it uses {\em spatial correlation
decay} to compute {\em deterministically} marginals of {\em Gibbs
distribution}. We develop a scheme whose accuracy depends on {\em
non-reconstruction} of the colourings of , rather than {\em
uniqueness} that are required in previous works. This leaves open the
possibility for our schema to be sufficiently accurate even for .
The set up for establishing correlation decay is as follows: Given
, we alter the graph structure in some specific region of
the graph by deleting edges between vertices of . Then we show that
the effect of this change on the marginals of Gibbs distribution, diminishes as
we move away from . Our approach is novel and suggests a new context
for the study of deterministic counting algorithms
A three-dimensional lattice gas model for amphiphilic fluid dynamics
We describe a three-dimensional hydrodynamic lattice-gas model of amphiphilic
fluids. This model of the non-equilibrium properties of oil-water-surfactant
systems, which is a non-trivial extension of an earlier two-dimensional
realisation due to Boghosian, Coveney and Emerton [Boghosian, Coveney, and
Emerton 1996, Proc. Roy. Soc. A 452, 1221-1250], can be studied effectively
only when it is implemented using high-performance computing and visualisation
techniques. We describe essential aspects of the model's theoretical basis and
computer implementation, and report on the phenomenological properties of the
model which confirm that it correctly captures binary oil-water and
surfactant-water behaviour, as well as the complex phase behaviour of ternary
amphiphilic fluids.Comment: 34 pages, 13 figures, high resolution figures available on reques
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