The probability generating functional

Westcott, Mark

doi:10.1017/s1446788700011095

Cited by 99 publications

(30 citation statements)

References 20 publications

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“…limit operations, the latter requiring the space to be small enough (consistent with uniqueness of determination of the distribution of the random entity involved) so as to allow the maximum flexibility in manipulation. Westcott (1972) principally used for G [. ] not U but the space V _ V(X) …”

Section: And Type Of Reportmentioning

confidence: 99%

On the Probability Generating Functional for Point Processes.

Daley¹,

Vere-Jones²

1984

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Section: And Type Of Reportmentioning

confidence: 99%

On the Probability Generating Functional for Point Processes.

Daley¹,

Vere-Jones²

1984

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“…In order to realize this expected value is important to remember the fidi of the point process which allows us to rewrite the p.g.fl [51] as…”

Section: Probability Generating Functionalmentioning

confidence: 99%

“…where the final representation is obtained thanks to the combinatorics interpretation of a Janossy measure [51] applied to the fidi. This representation can then be extended to joint point processes, where a new process Υ is introduced, with similar characteristics to Φ but on space Y ∈ R dy (in this application it can considered as the measurement space and it has a mapping to the state space).…”

Section: Probability Generating Functionalmentioning

confidence: 99%

Predicting multiple target tracking performance for applications on video sequences

Tapiero

Medeiros

2017

Machine Vision and Applications

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This dissertation presents a framework to predict the performance of multiple target tracking (MTT) techniques. The framework is based on the mathematical descriptors of point processes, the probability generating functional (p.g.fl). It is shown that conceptually the p.g.fls of MTT techniques can be interpreted as a transform that can be marginalized to an expression that encodes all the information regarding the likelihood model as well as the underlying assumptions present in a given tracking technique. In order to use this approach for tracker performance prediction in video sequences, a framework that combines video quality assessment concepts and the marginalized transform is introduced. The multiple hypothesis tracker (MHT), Joint Probabilistic Data Association (JPDA), Markov Chain Monte Carlo (MCMC) data association, and the Probability Hypothesis Density filter (PHD) are used as a test cases. We introduce their transforms and perform a numerical comparison to predict their performance under identical conditions. We also introduce the concepts that present the base for estimation in general and for applications in computer vision.ii ACKNOWLEDGEMENTS

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“…As mentioned again by Diggle and Milne (), they are stationary by construction. However, the only stationary mixed Poisson processes that are ergodic are those for which the mixing distribution is concentrated at a single point, thus giving an (ordinary) Poisson process (e.g., Westcott, , p. 464). It follows that nontrivial processes of this type can be orderly but never ergodic.…”

Section: Introductionmentioning

confidence: 99%

Modeling spatial overdispersion via the generalized Waring point process

Zografi

Xekalaki

2019

Scandinavian J Statistics

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Modeling spatial overdispersion requires point process models with finite‐dimensional distributions that are overdisperse relative to the Poisson distribution. Fitting such models usually heavily relies on the properties of stationarity, ergodicity, and orderliness. In addition, although processes based on negative binomial finite‐dimensional distributions have been widely considered, they typically fail to simultaneously satisfy the three required properties for fitting. Indeed, it has been conjectured by Diggle and Milne that no negative binomial model can satisfy all three properties. In light of this, we change perspective and construct a new process based on a different overdisperse count model, namely, the generalized Waring (GW) distribution. While comparably tractable and flexible to negative binomial processes, the GW process is shown to possess all required properties and additionally span the negative binomial and Poisson processes as limiting cases. In this sense, the GW process provides an approximate resolution to the conundrum highlighted by Diggle and Milne.

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The probability generating functional

Cited by 99 publications

References 20 publications

On the Probability Generating Functional for Point Processes.

On the Probability Generating Functional for Point Processes.

Predicting multiple target tracking performance for applications on video sequences

Modeling spatial overdispersion via the generalized Waring point process

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