An Application of Particle Filter in Point Processes

We consider a stochastic motion in a bounded planar region and a point process of events on a trajectory of the motion. The aim is to use the particle filter for the estimation of the conditional intensity of the point process and simultaneously for the reconstruction of the trajectory which is considered unknown. Using a simulated dataset numerical results are obtained and presented. Introduction Consider a stochastic motion in a bounded planar region A within a time interval [0, T ]. Let us assume that the equation of the motion Yt, t ∈ [0, T ] is in the form dYt = b(Yt, t)dt+ σ(Yt, t)dWt, (1) whereWt is a two dimensional Wiener process. Discussion on a constrained motion in a bounded region is given e.g. in Brillinger (2003). During the motion discrete events occur which form a point process N on the trajectory of the motion, see Fig.1. The aim is to estimate the conditional intensity λ∗ (for the exact definition see Daley, Vere-Jones (2008), ch.14) λ(t)dt ≈ E[N(dt)|Ht−] (2) of the point process, where Ht− denotes the history of the process to the time t, and to reconstruct the unknown trajectory given the observed events in space and time. For solving the first part of the task, i.e. the estimation of the conditional intensity, a filter was suggested in Eden et al. (2004) using Gaussian approximation of the posterior distribution. Later this Gaussian assumption was relaxed in Ergun et al. (2007) using the particle filter (Doucet et al. (2001)). We use this technique for the solution of both parts of the task, i.e. while Ergun et al. (2007) and Eden et al. (2004) deal with a known trajectory, here the trajectory is considered as not observed. A simulation study is provided to achieve numerical results. Filtering Problem Consider a point process of events in the time interval (0, T ], T > 0. Divide this interval into M subintervals ((k − 1)∆, k∆], k = 1, . . . ,M of equal length ∆ = TM−1. In practise for a given realization of the point process we choose ∆ so small that there is at most one event in each subinterval. Let ∆Nk be the indicator of the occurence of an event in the k−th interval, i.e. ∆Nk = 1 if there is an event in ((k − 1)∆, k∆] and it is equal to zero otherwise. Let us denote N1:k = [∆N1, . . . ,∆Nk]. Further let θ ∈ R be the random state parameter of the process of events and for each k = 1, . . . ,M denote θ1:k = (θ1, . . . , θk) the sequence of state parameters θj corresponding to j-th subinterval. The model which we consider for our data is based on the state equation for the parameter in the form θk = Fθk−1 + ηk, (3) where F is a fixed d× d evolution matrix and η = {ηk} M k=1 is a zero mean Gaussian noise with covariance matrix Qk. The solution of the filtering problem is the evaluation of conditional expectation E[g(θk)|N1:k] of any summary statistic g of the state parameter θk, k = 1, . . .M . 108 WDS'09 Proceedings of Contributed Papers, Part I, 108–112, 2009. ISBN 978-80-7378-101-9 © MATFYZPRESS