21,944 research outputs found
Deconvolution of point processes
The superposition of two independent point processes can be described by
multiplication of their probability generating functionals (p.g.fl.s). The
inverse operation, which can be viewed as a deconvolution, is defined by
dividing the superposed process by one of its constituent p.g.fl.s. The
deconvolved process is computed using the higher-order chain rule for Gateaux
differentials. The higher-order quotient rule for Gateaux differentials is
first established and then applied to point processes
Learning Determinantal Point Processes
Determinantal point processes (DPPs), which arise in random matrix theory and
quantum physics, are natural models for subset selection problems where
diversity is preferred. Among many remarkable properties, DPPs offer tractable
algorithms for exact inference, including computing marginal probabilities and
sampling; however, an important open question has been how to learn a DPP from
labeled training data. In this paper we propose a natural feature-based
parameterization of conditional DPPs, and show how it leads to a convex and
efficient learning formulation. We analyze the relationship between our model
and binary Markov random fields with repulsive potentials, which are
qualitatively similar but computationally intractable. Finally, we apply our
approach to the task of extractive summarization, where the goal is to choose a
small subset of sentences conveying the most important information from a set
of documents. In this task there is a fundamental tradeoff between sentences
that are highly relevant to the collection as a whole, and sentences that are
diverse and not repetitive. Our parameterization allows us to naturally balance
these two characteristics. We evaluate our system on data from the DUC 2003/04
multi-document summarization task, achieving state-of-the-art results
Stable marked point processes
In many contexts such as queuing theory, spatial statistics, geostatistics
and meteorology, data are observed at irregular spatial positions. One model of
this situation involves considering the observation points as generated by a
Poisson process. Under this assumption, we study the limit behavior of the
partial sums of the marked point process , where X(t) is a
stationary random field and the points t_i are generated from an independent
Poisson random measure on . We define the sample
mean and sample variance statistics and determine their joint asymptotic
behavior in a heavy-tailed setting, thus extending some finite variance results
of Karr [Adv. in Appl. Probab. 18 (1986) 406--422]. New results on subsampling
in the context of a marked point process are also presented, with the
application of forming a confidence interval for the unknown mean under an
unknown degree of heavy tails.Comment: Published at http://dx.doi.org/10.1214/009053606000001163 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Cumulants of Hawkes point processes
We derive explicit, closed-form expressions for the cumulant densities of a
multivariate, self-exciting Hawkes point process, generalizing a result of
Hawkes in his earlier work on the covariance density and Bartlett spectrum of
such processes. To do this, we represent the Hawkes process in terms of a
Poisson cluster process and show how the cumulant density formulas can be
derived by enumerating all possible "family trees", representing complex
interactions between point events. We also consider the problem of computing
the integrated cumulants, characterizing the average measure of correlated
activity between events of different types, and derive the relevant equations.Comment: 11 pages, 4 figure
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