More Like this

1. Clustering Then Estimation of Spatio-Temporal Self-Exciting Processes

   https://doi.org/10.1287/ijoc.2022.0351  

   Zhang, Haoting; Zhan, Donglin; Anderson, James; Righter, Rhonda; Zheng, Zeyu (September 2024, INFORMS Journal on Computing)

   We propose a new estimation procedure for general spatio-temporal point processes that include a self-exciting feature. Estimating spatio-temporal self-exciting point processes with observed data is challenging, partly because of the difficulty in computing and optimizing the likelihood function. To circumvent this challenge, we employ a Poisson cluster representation for spatio-temporal self-exciting point processes to simplify the likelihood function and develop a new estimation procedure called “clustering-then-estimation” (CTE), which integrates clustering algorithms with likelihood-based estimation methods. Compared with the widely used expectation-maximization (EM) method, our approach separates the cluster structure inference of the data from the model selection. This has the benefit of reducing the risk of model misspecification. Our approach is computationally more efficient because it does not need to recursively solve optimization problems, which would be needed for EM. We also present asymptotic statistical results for our approach as theoretical support. Experimental results on several synthetic and real data sets illustrate the effectiveness of the proposed CTE procedure. 

   more » « less
2. Invariant Galton–Watson branching process for earthquake occurrence

   https://doi.org/10.1093/gji/ggac204  

   Kovchegov, Yevgeniy; Zaliapin, Ilya; Ben-Zion, Yehuda (June 2022, Geophysical Journal International)

   SUMMARY We propose a theoretical modelling framework for earthquake occurrence and clustering based on a family of invariant Galton–Watson (IGW) stochastic branching processes. The IGW process is a rigorously defined approximation to imprecisely observed or incorrectly estimated earthquake clusters modelled by Galton–Watson branching processes, including the Epidemic Type Aftershock Sequence (ETAS) model. The theory of IGW processes yields explicit distributions for multiple cluster attributes, including magnitude-dependent and magnitude-independent offspring number, cluster size and cluster combinatorial depth. Analysis of the observed seismicity in southern California demonstrates that the IGW model provides a close fit to the observed earthquake clusters. The estimated IGW parameters and derived statistics are robust with respect to the catalogue lower cut-off magnitude. The proposed model facilitates analyses of multiple quantities of seismicity based on self-similar tree attributes, and may be used to assess the proximity of seismicity to criticality. 

   more » « less
3. A Dirichlet Mixture Model of Hawkes Processes for Event Sequence Clustering

   Xu, Hongteng; Zha, Hongyuan (January 2017, Advances in neural information processing systems)

   How to cluster event sequences generated via different point processes is an interesting and important problem in statistical machine learning. To solve this problem, we propose and discuss an effective model-based clustering method based on a novel Dirichlet mixture model of a special but significant type of point processes — Hawkes process. The proposed model generates the event sequences with different clusters from the Hawkes processes with different parameters, and uses a Dirichlet distribution as the prior distribution of the clusters. We prove the identifiability of our mixture model and propose an effective variational Bayesian inference algorithm to learn our model. An adaptive inner iteration allocation strategy is designed to accelerate the convergence of our algorithm. Moreover, we investigate the sample complexity and the computational complexity of our learning algorithm in depth. Experiments on both synthetic and real-world data show that the clustering method based on our model can learn structural triggering patterns hidden in asynchronous event sequences robustly and achieve superior performance on clustering purity and consistency compared to existing methods. 

   more » « less
4. Scalable inference for space‐time Gaussian Cox processes

   https://doi.org/10.1111/jtsa.12457  

   Shirota, Shinichiro; Banerjee, Sudipto (April 2019, Journal of Time Series Analysis)

   The log‐Gaussian Cox process is a flexible and popular stochastic process for modeling point patterns exhibiting spatial and space‐time dependence. Model fitting requires approximation of stochastic integrals which is implemented through discretization over the domain of interest. With fine scale discretization, inference based on Markov chain Monte Carlo is computationally burdensome because of the cost of matrix decompositions and storage, such as the Cholesky, for high dimensional covariance matrices associated with latent Gaussian variables. This article addresses these computational bottlenecks by combining two recent developments: (i) a data augmentation strategy that has been proposed for space‐time Gaussian Cox processes that is based on exact Bayesian inference and does not require fine grid approximations for infinite dimensional integrals, and (ii) a recently developed family of sparsity‐inducing Gaussian processes, called nearest‐neighbor Gaussian processes, to avoid expensive matrix computations. Our inference is delivered within the fully model‐based Bayesian paradigm and does not sacrifice the richness of traditional log‐Gaussian Cox processes. We apply our method to crime event data in San Francisco and investigate the recovery of the intensity surface. 

   more » « less
5. Within-cluster resampling for multilevel models under informative cluster size

   https://doi.org/10.1093/biomet/asz035  

   Lee, D.; Kim, J. K.; Skinner, C. J. (July 2019, Biometrika)

   Summary A within-cluster resampling method is proposed for fitting a multilevel model in the presence of informative cluster size. Our method is based on the idea of removing the information in the cluster sizes by drawing bootstrap samples which contain a fixed number of observations from each cluster. We then estimate the parameters by maximizing an average, over the bootstrap samples, of a suitable composite loglikelihood. The consistency of the proposed estimator is shown and does not require that the correct model for cluster size is specified. We give an estimator of the covariance matrix of the proposed estimator, and a test for the noninformativeness of the cluster sizes. A simulation study shows, as in Neuhaus & McCulloch (2011), that the standard maximum likelihood estimator exhibits little bias for some regression coefficients. However, for those parameters which exhibit nonnegligible bias, the proposed method is successful in correcting for this bias. 

   more » « less