compound poisson
Recently Published Documents


TOTAL DOCUMENTS

871
(FIVE YEARS 107)

H-INDEX

40
(FIVE YEARS 5)

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2820
Author(s):  
Emanuele Dolera ◽  
Stefano Favaro

The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set {1,…,n}, with n∈N, which is indexed by real parameters α and θ such that either α∈[0,1) and θ>−α, or α<0 and θ=−mα for some m∈N. For α=0, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either α∈(0,1), or α<0. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large n asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s α diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of α-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case.


2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Mattia Prosperi ◽  
Simone Marini ◽  
Christina Boucher

Abstract Background Identification of motifs and quantification of their occurrences are important for the study of genetic diseases, gene evolution, transcription sites, and other biological mechanisms. Exact formulae for estimating count distributions of motifs under Markovian assumptions have high computational complexity and are impractical to be used on large motif sets. Approximated formulae, e.g. based on compound Poisson, are faster, but reliable p value calculation remains challenging. Here, we introduce ‘motif_prob’, a fast implementation of an exact formula for motif count distribution through progressive approximation with arbitrary precision. Our implementation speeds up the exact calculation, usually impractical, making it feasible and posit to substitute currently employed heuristics. Results We implement motif_prob in both Perl and C+ + languages, using an efficient error-bound iterative process for the exact formula, providing comparison with state-of-the-art tools (e.g. MoSDi) in terms of precision, run time benchmarks, along with a real-world use case on bacterial motif characterization. Our software is able to process a million of motifs (13–31 bases) over genome lengths of 5 million bases within the minute on a regular laptop, and the run times for both the Perl and C+ + code are several orders of magnitude smaller (50–1000× faster) than MoSDi, even when using their fast compound Poisson approximation (60–120× faster). In the real-world use cases, we first show the consistency of motif_prob with MoSDi, and then how the p-value quantification is crucial for enrichment quantification when bacteria have different GC content, using motifs found in antimicrobial resistance genes. The software and the code sources are available under the MIT license at https://github.com/DataIntellSystLab/motif_prob. Conclusions The motif_prob software is a multi-platform and efficient open source solution for calculating exact frequency distributions of motifs. It can be integrated with motif discovery/characterization tools for quantifying enrichment and deviation from expected frequency ranges with exact p values, without loss in data processing efficiency.


Author(s):  
Maryam Rahmati ◽  
Parisa Rezanejad Asl ◽  
Javad Mikaeli ◽  
Hojjat Zeraati ◽  
Aliakbar Rasekhi

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenguang Yu ◽  
Peng Guo ◽  
Qi Wang ◽  
Guofeng Guan ◽  
Yujuan Huang ◽  
...  

AbstractIn this paper, we model the insurance company’s surplus by a compound Poisson risk model, where the surplus process can only be observed at random observation times. It is assumed that the insurer observes its surplus level periodically to decide on dividend payments and capital injection at the interobservation time having an $\operatorname{Erlang}(n)$ Erlang ( n ) distribution. If the observed surplus level is greater than zero but less than injection line $b_{1} > 0$ b 1 > 0 , the shareholders should immediately inject a certain amount of capital to bring the surplus level back to the injection line $b_{1}$ b 1 . If the observed surplus level is larger than dividend line $b_{2}$ b 2 ($b_{2} > b_{1}$ b 2 > b 1 ), any excess of the surplus over $b_{2}$ b 2 is immediately paid out as dividends to the shareholders of the company. Ruin is declared when the observed surplus level is negative. We derive the explicit expressions of the Gerber–Shiu function, the expected discounted capital injection, and the expected discounted dividend payments. Numerical illustrations are also given to analyze the effect of random observation times on actuarial quantities.


Sign in / Sign up

Export Citation Format

Share Document