36,809 research outputs found
Rate-optimal Bayesian intensity smoothing for inhomogeneous Poisson processes
We apply nonparametric Bayesian methods to study the problem of estimating
the intensity function of an inhomogeneous Poisson process. We exhibit a prior
on intensities which both leads to a computationally feasible method and enjoys
desirable theoretical optimality properties. The prior we use is based on
B-spline expansions with free knots, adapted from well-established methods used
in regression, for instance. We illustrate its practical use in the Poisson
process setting by analyzing count data coming from a call centre.
Theoretically we derive a new general theorem on contraction rates for
posteriors in the setting of intensity function estimation. Practical choices
that have to be made in the construction of our concrete prior, such as
choosing the priors on the number and the locations of the spline knots, are
based on these theoretical findings. The results assert that when properly
constructed, our approach yields a rate-optimal procedure that automatically
adapts to the regularity of the unknown intensity function
Nonparametric regression in exponential families
Most results in nonparametric regression theory are developed only for the
case of additive noise. In such a setting many smoothing techniques including
wavelet thresholding methods have been developed and shown to be highly
adaptive. In this paper we consider nonparametric regression in exponential
families with the main focus on the natural exponential families with a
quadratic variance function, which include, for example, Poisson regression,
binomial regression and gamma regression. We propose a unified approach of
using a mean-matching variance stabilizing transformation to turn the
relatively complicated problem of nonparametric regression in exponential
families into a standard homoscedastic Gaussian regression problem. Then in
principle any good nonparametric Gaussian regression procedure can be applied
to the transformed data. To illustrate our general methodology, in this paper
we use wavelet block thresholding to construct the final estimators of the
regression function. The procedures are easily implementable. Both theoretical
and numerical properties of the estimators are investigated. The estimators are
shown to enjoy a high degree of adaptivity and spatial adaptivity with
near-optimal asymptotic performance over a wide range of Besov spaces. The
estimators also perform well numerically.Comment: Published in at http://dx.doi.org/10.1214/09-AOS762 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Fast and scalable non-parametric Bayesian inference for Poisson point processes
We study the problem of non-parametric Bayesian estimation of the intensity
function of a Poisson point process. The observations are independent
realisations of a Poisson point process on the interval . We propose two
related approaches. In both approaches we model the intensity function as
piecewise constant on bins forming a partition of the interval . In
the first approach the coefficients of the intensity function are assigned
independent gamma priors, leading to a closed form posterior distribution. On
the theoretical side, we prove that as the posterior
asymptotically concentrates around the "true", data-generating intensity
function at an optimal rate for -H\"older regular intensity functions (). In the second approach we employ a gamma Markov chain prior on the
coefficients of the intensity function. The posterior distribution is no longer
available in closed form, but inference can be performed using a
straightforward version of the Gibbs sampler. Both approaches scale well with
sample size, but the second is much less sensitive to the choice of .
Practical performance of our methods is first demonstrated via synthetic data
examples. We compare our second method with other existing approaches on the UK
coal mining disasters data. Furthermore, we apply it to the US mass shootings
data and Donald Trump's Twitter data.Comment: 45 pages, 22 figure
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