A new metric between distributions of point processes

Schuhmacher, Dominic; Xia, Aihua

doi:10.1239/aap/1222868180

Cited by 38 publications

(37 citation statements)

References 20 publications

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“…Some alternatives are discussed in Mateu et al (2015). Another option, that will prove to be more appropriate for our applications in Section 4, is the optimal matching distance introduced by Schuhmacher and Xia (2008). Letting x = {x 1 , .…”

Section: Continuous Time Observationsmentioning

confidence: 99%

Spatial Birth–Death–Move Processes: Basic Properties and Estimation of their Intensity Functions

Lavancier

Guével

2021

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Many spatiotemporal data record the time of birth and death of individuals, along with their spatial trajectories during their lifetime, whether through continuous‐time observations or discrete‐time observations. Natural applications include epidemiology, individual‐based modelling in ecology, spatiotemporal dynamics observed in bioimaging and computer vision. The aim of this article is to estimate in this context the birth and death intensity functions that depend in full generality on the current spatial configuration of all alive individuals. While the temporal evolution of the population size is a simple birth–death process, observing the lifetime and trajectories of all individuals calls for a new paradigm. To formalise this framework, we introduce spatial birth–death–move processes, where the birth and death dynamics depends on the current spatial configuration of the population and where individuals can move during their lifetime according to a continuous Markov process with possible interactions. We consider non‐parametric kernel estimators of their birth and death intensity functions. The setting is original because each observation in time belongs to a non‐vectorial, infinite dimensional space and the dependence between observations is barely tractable. We prove the consistency of the estimators in the presence of continuous‐time and discrete‐time observations, under fairly simple conditions. We moreover discuss how we can take advantage in practice of structural assumptions made on the intensity functions and we explain how data‐driven bandwidth selection can be conducted, despite the unknown (and sometimes undefined) second order moments of the estimators. We finally apply our statistical method to the analysis of the spatiotemporal dynamics of proteins involved in exocytosis in cells, providing new insights on this complex mechanism.

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Section: Continuous Time Observationsmentioning

confidence: 99%

Spatial Birth–Death–Move Processes: Basic Properties and Estimation of their Intensity Functions

Lavancier

Guével

2021

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…For s = 0, cost c (1) is never higher than c (0) so one can always choose â = 1. It is interesting to note that for sufficiently low sensing costs, according to (14), we only take action if the probability of existence is not too low or too high. That is, we measure the target if we are not very confident whether the target exists or not, and we therefore gain valuable information by measuring it.…”

Section: Analysis I: One Potential Targetmentioning

confidence: 99%

“…Therefore, metric-driven sensor management, in which the cost function is related to a metric, provides a clear interpretation of what the objective is, and also measurable results in terms of performance evaluation. In this respect, the optimal subpattern assignment metric (OSPA) is a widely-used metric [14], [15]. A sensor management algorithm based on the OSPA metric was proposed in [16].…”

Section: Introductionmentioning

confidence: 99%

An analysis on metric-driven multi-target sensor management: GOSPA versus OSPA

García-Fernández¹,

Hernandez²,

Maskell³

2021

Preprint

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This paper presents an analysis on sensor management using a cost function based on a multi-target metric, in particular, the optimal subpattern-assignment (OSPA) metric, the unnormalised OSPA (UOSPA) metric and the generalised OSPA (GOSPA) metric (α = 2). We consider the problem of managing an array of sensors, where each sensor is able to observe a region of the surveillance area, not covered by other sensors, with a given sensing cost. We look at the case in which there are far-away, independent potential targets, at maximum one per sensor region. In this set-up, the optimal action using GOSPA is taken for each sensor independently, as we may expect. On the contrary, as a consequence of the spooky effect at a distance in optimal OSPA/UOSPA estimation, the optimal actions for different sensors using OSPA and UOSPA are entangled.

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“…It holds d 2 ¤ d 2 and d 2 metricizes weak convergence (see Proposition 2.3. (ii) and (iii) in [26]). We consider the Bernoulli process µ n with intensity measure ν n and the Poisson point process µ with intensity measure ν as given in Definition 4.18.…”

Section: The Vague Topologymentioning

confidence: 93%

“…We start by considering the Wasserstein metrics d 2 and d 2 . Since the definitions of d 2 and d 2 are a bit lengthy and we do not actually utilise the definitions in our arguments we will only provide a reference where the metrics are defined (see [26]). It holds d 2 ¤ d 2 and d 2 metricizes weak convergence (see Proposition 2.3.…”

Section: The Vague Topologymentioning

confidence: 99%

Convergence of the Genealogy of the Spatial Cannings Model

Heuer¹

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In this thesis we consider the genealogy of a spatial Cannings model. This is a population model in which individuals are distributed over a countable set of sites G. The reproduction of individuals at each site is panmictic (exchangeable) and preserves the local population size. The offspring then migrate to other sites in G, also in an exchangeable manner. We consider the spatial coalescent introduced by sampling n individuals at present time and tracking their ancestral lines back in time. The resulting process is the spatial Cannings coalescent. Our main result shows, that an appropriately time-rescaled spatial Cannings coalescent converges to a spatial Ξ-coalescent in the large population limit.The key feature of our result is that the spatial structure is preserved into the limit as opposed to a fast migration limit. The influence of the migration on the local population size can yield a time-inhomogeneous limit and, in case of sites with a small population size, our limiting process may not have a strongly continuous semigroup. I would like to thank my friends and colleagues Alexander Hartmann and Fabian Telschow, who made for great targets to bounce mathematical ideas and questions off and also for their input regarding some formal as well as formatting questions concerning this thesis. Last, but not least, I would like to thank my parents Jürgen and Gabriela Heuer for their great support over the years. The Backward Model 92. The Wright-Fisher model in which we choose pν I k;x k;x,i q iI k;x to be a vector of i.i.d. Poisson distributed random variables conditioned on their sum being equal |I k;x |. An alternative way of describing this distribution would be that the vector is multinomially distributed. More precisely we consider an urn containing one ball for each color i I k;x . Now we draw |I k;x | times with replacement from the urn and set νI k;x k;x,i to be the total number of draws of color i.

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A new metric between distributions of point processes

Cited by 38 publications

References 20 publications

Spatial Birth–Death–Move Processes: Basic Properties and Estimation of their Intensity Functions

Spatial Birth–Death–Move Processes: Basic Properties and Estimation of their Intensity Functions

An analysis on metric-driven multi-target sensor management: GOSPA versus OSPA

Convergence of the Genealogy of the Spatial Cannings Model

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