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Appendix
Published in Arup Bose, Koushik Saha, Random Circulant Matrices, 2018
Poisson point process: The simplest example of a point process is a Poisson point process. Let λ(x),x∈ℝd be a locally integrable positive function so that Λ(B):=∫Bλ(x)dx<∞foranyboundedregionB.
P
Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
Poisson distribution a probability distribution widely used in system modeling. A non-negative integer-valued random variable X is Poissondistributed if -aa k /k!, where a is a positive parameter sometimes called the intensity of the distribution. For example, the number of `purely' random events occurring over a time interval of t often follows a Poisson distribution with parameter a proportional to t. Poisson process a random point process denoting the occurrence of a sequence of events at discrete points in time. The time difference between the different events is a random variable. For a given time interval of length T , the number of events (points in the process) is a random variable with Poisson distribution given by the folk -T lowing probability law P[N = k] = (T )k!e , where N is the number of events that occur in the interval of length T , and is the expected number of occurrences of events per unit time. The time interval between any two events is a random variable with exponential distribution given by F( ) = 1 - e- , > 0. Poisson's equation a partial differential equation that expresses the relation between the scalar potential V and the charge density at any point . Mathematically described by 2 V = , where 2 is the Laplacian, is the forcing function, V is the equation's solution, and is the dielectric constant of the medium. polariton term originally used to designate the polarization field particles analogous to photons. It is currently used to denote the coupling of the electromagnetic field with the polar excitations in solids.
Point processes
Published in Peter Guttorp, Stochastic Modeling of Scientific Data, 2018
The third way of describing a point process is by using the complete intensity function () λ(t|Nt)=limh→01hP(N(t,t+h]=1|Nt)
Sequential Change-Point Detection for Mutually Exciting Point Processes
Published in Technometrics, 2023
Haoyun Wang, Liyan Xie, Yao Xie, Alex Cuozzo, Simon Mak
Point processes are widely used for modeling discrete events data, which consists of a series of event times and additional associated information. Recently, a class of mutually-exciting nonhomogeneous point processes called Hawkes processes (Hawkes 1971) has gained much popularity in the statistics and machine learning literature. The intensity function of the Hawkes process consists of a deterministic part and a stochastic part, which captures the triggering or inhibiting effects of past events on future events. For example, each earthquake is usually followed by a sequence of aftershock activities and the occurrence rate of aftershocks can be represented in the stochastic part of the intensity function (Ogata 1988). Hawkes processes provide a flexible model for capturing spatio-temporal correlations, and have been successfully applied in a wide range of domains including seismology (Ogata 1988, 1998), criminology (Mohler et al. 2011), epidemiology (Rizoiu et al. 2018), social networks (Yang and Zha 2013), finance (Hawkes 2018), and neural activity (Reynaud-Bouret et al. 2013).
Analysis and description of crimes in Mexico city using point pattern analysis within networks
Published in Annals of GIS, 2023
Iulian Teodor Vlad, Carlos Diaz, Pablo Juan, Somnath Chaudhuri
In this respect, point processes are a powerful statistical tool used to analyse random events that occur at point locations, such as crimes, earthquakes and forest fires among many others (Diggle, 1979; Daley and Vere-Jones, 2003; Baddeley, Møller, and Waagepetersen, 2000; Baddeley, 2006; Daley and Vere-Jones, 2008; Baddeley and Turner, 2005; Baddeley, Rubak, and Turner, 2016; Gonzalez et al., 2016).
Maintenance planning estimations and policies optimization for single-unit systems using Hawkes processes
Published in Quality Technology & Quantitative Management, 2023
Lirong Cui, Fengming Kang, Jingyuan Shen
Failures of the repairable single-unit systems can be modelled by many ways, in which the use of point processes is one of the most popular methods. Among them, the renewal process and Poisson process are usually employed, for example, see, Ross (1996). In our paper, a Hawkes process is used to model the failures of the repairable single-unit system, which is rarely used in reliability literature. Hawkes process was first introduced by A.G. Hawkes (1971a), A. G. Hawkes (1971b) based on the background of earthquakes, since then there has been great developments on its theory and applications. Hawkes processes are self-exciting point processes, i.e., the occurred events increase the probabilities of the future event occurrences. Hawkes process is an extension of Poisson process, and all related events are dependent of each other. Hawkes processes or self-exciting point processes have been used in reliability, especially in the software reliability modeling such as Chen and Singpurwalla (1997) and Ledoux (2008). The key idea on applications of self-exciting processes in software reliability modeling is to treat a failure process of the software as a counting process described by the self-exciting process, which can capture the dependence among failures on history. Other works with Hawkes processes in reliability, most of them, are related to shock processes, for example, see, Ross (2014), Gouno and Guérineau (2015), Cui et al. (2018), Li et al. (2018), J. Verheugd et al. (2020), and Ertekin et al. (2015) proposed a reactive point process for analysis of power failures in the underground electrical system of New York. Doyen et al. (2017) studied the possibility of establishing equivalent models with a geometric reduction of age or intensity, in which many possible intensity functions including Hawkes process intensity function are proposed for describing maintenance behaviors. Cui et al. (2019) proposed a partial self-exciting Hawkes process for some possible applications in reliability. Cui et al. (2020) presented a new method on moments of Hawkes processes, which extends the famous Dynkin’s formula. Cui and Wu (2021) provided the moments for Hawkes processes with Gamma decay kernel functions. Cui and Shen (2021) discussed an extension of Hawkes processes with ephemeral nearest effects.