Fast multi-resolution segmentation for nonstationary Hawkes process using cumulants
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Fast multi-resolution segmentation for nonstationary Hawkes process using cumulants · Zhidong Li1 · Xuhui Fan2 · Yang Wang1 · Arcot Sowmya2 · Fang Chen1
Feng Zhou1
Received: 30 April 2020 / Accepted: 8 May 2020 © Springer Nature Switzerland AG 2020
Abstract The stationarity is assumed in the vanilla Hawkes process, which reduces the model complexity but introduces a strong assumption. In this paper, we propose a fast multi-resolution segmentation algorithm to capture the time-varying characteristics of the nonstationary Hawkes process. The proposed algorithm is based on the first- and second-order cumulants. Except for the computation efficiency, the algorithm can provide a hierarchical view of the segmentation at different resolutions. We extensively investigate the impact of hyperparameters on the performance of this algorithm. To ease the choice of hyperparameter, we propose a refined Gaussian process-based segmentation algorithm, which is proved to be a robust method. The proposed algorithm is applied to a real vehicle collision dataset, and the outcome shows some interesting hierarchical dynamic time-varying characteristics. Keywords Hawkes process · Nonstationary · Segmentation · Cumulants
1 Introduction The point process data is a common data type in real applications. To model this kind of point process data, various statistical models have been proposed to disclose its underlying temporal dynamics, such as homogeneous Poisson process [28], inhomogeneous Poisson process [30] and Hawkes process [11]. In this paper, we focus on the Hawkes process. Hawkes process is widely used to model the self-exciting phenomenon which can be observed in many fields, like crime [16], ecosystem [10], transportation [7] and TV programs [17]. An important way to characterize a temporal point process is through the definition of a conditional intensity. The specific Hawkes process conditional intensity is:
t
λ(t) = μ +
φ(t − s)dN(s) = μ +
0
φ(t − ti ),
(1)
ti 0 is the baseline intensity which is constant, {ti } are the timestamps of events before time t, N(t) is the
B
Feng Zhou [email protected]
1
University of Technology Sydney, Ultimo, Australia
2
University of New South Wales, Sydney, Australia
corresponding counting process and φ(·) > 0 is the triggering kernel. The summation of triggering kernels explains the nature of self-excitation, which is the occurrence of events in the past will intensify the intensity of events occurring in the future. It is straightforward to see that the conditional intensity of Hawkes process is unchanged over timeshifting because μ is a constant and φ(·) only depends on τ = t − ti , not on t, which means the stationarity [11,25]. The assumption of stationarity leads to reduced model complexity and easy inference. However, the point process data generated in many real applications has nonstationary properties, which means its first-, second- and higher-order cumulants (moments) are changing over time. Applying the vanilla Hawkes process directly to the non
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