508 research outputs found
State estimation for temporal point processes
This paper is concerned with combined inference for point processes on the
real line observed in a broken interval. For such processes, the classic
history-based approach cannot be used. Instead, we adapt tools from sequential
spatial point processes. For a range of models, the marginal and conditional
distributions are derived. We discuss likelihood based inference as well as
parameter estimation using the method of moments, conduct a simulation study
for the important special case of renewal processes and analyse a data set
collected by Diggle and Hawtin
Non-parametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets
We propose new summary statistics to quantify the association between the
components in coverage-reweighted moment stationary multivariate random sets
and measures. They are defined in terms of the coverage-reweighted cumulant
densities and extend classic functional statistics for stationary random closed
sets. We study the relations between these statistics and evaluate them
explicitly for a range of models. Unbiased estimators are given for all
statistics and applied to simulated examples.Comment: Added examples in version
A spectral mean for point sampled closed curves
We propose a spectral mean for closed curves described by sample points on
its boundary subject to mis-alignment and noise. First, we ignore mis-alignment
and derive maximum likelihood estimators of the model and noise parameters in
the Fourier domain. We estimate the unknown curve by back-transformation and
derive the distribution of the integrated squared error. Then, we model
mis-alignment by means of a shifted parametric diffeomorphism and minimise a
suitable objective function simultaneously over the unknown curve and the
mis-alignment parameters. Finally, the method is illustrated on simulated data
as well as on photographs of Lake Tana taken by astronauts during a Shuttle
mission
A J-function for inhomogeneous spatio-temporal point processes
We propose a new summary statistic for inhomogeneous intensity-reweighted
moment stationary spatio-temporal point processes. The statistic is defined
through the n-point correlation functions of the point process and it
generalises the J-function when stationarity is assumed. We show that our
statistic can be represented in terms of the generating functional and that it
is related to the inhomogeneous K-function. We further discuss its explicit
form under some specific model assumptions and derive a ratio-unbiased
estimator. We finally illustrate the use of our statistic on simulated data
Summary statistics for inhomogeneous marked point processes
We propose new summary statistics for intensity-reweighted moment stationary
marked point processes with particular emphasis on discrete marks. The new
statistics are based on the n-point correlation functions and reduce to cross
J- and D-functions when stationarity holds. We explore the relationships
between the various functions and discuss their explicit forms under specific
model assumptions. We derive ratio-unbiased minus sampling estimators for our
statistics and illustrate their use on a data set of wildfires
A non-homogeneous Semi-Markov model for Interval Censoring
Previous approaches to modelling interval-censored data have often relied on
assumptions of homogeneity in the sense that the censoring mechanism, the
underlying distribution of occurrence times, or both, are assumed to be
time-invariant. In this work, we introduce a model which allows for
non-homogeneous behaviour in both cases. In particular, we outline a censoring
mechanism based on semi-Markov processes in which interval generation is
assumed to be time-dependent and we propose a Markov point process model for
the underlying occurrence time distribution. We prove the existence of this
process and derive the conditional distribution of the occurrence times given
the intervals. We provide a framework within which the process can be
accurately modelled, and subsequently compare our model to homogeneous
approaches by way of a parametric example.Comment: 21 pages, 4 figure
Competing Claims in Public Space: The Construction of Frames in Different Relational Contexts
Contains fulltext :
151928.pdf (publisher's version ) (Open Access
XGBoostPP: Tree-based Estimation of Point Process Intensity Functions
We propose a novel tree-based ensemble method, named XGBoostPP, to
nonparametrically estimate the intensity of a point process as a function of
covariates. It extends the use of gradient-boosted regression trees (Chen &
Guestrin, 2016) to the point process literature via two carefully designed loss
functions. The first loss is based on the Poisson likelihood, working for
general point processes. The second loss is based on the weighted Poisson
likelihood, where spatially dependent weights are introduced to further improve
the estimation efficiency for clustered processes. An efficient greedy search
algorithm is developed for model estimation, and the effectiveness of the
proposed method is demonstrated through extensive simulation studies and two
real data analyses. In particular, we report that XGBoostPP achieves superior
performance to existing approaches when the dimension of the covariate space is
high, revealing the advantages of tree-based ensemble methods in estimating
complex intensity functions.Comment: 21 pages, 3 figure
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