Abstract
The impact from past to future is a vital feature in modelling time series data, which has been described by many point processes, e.g. the Hawkes process. In classical Hawkes process, the triggering kernel is assumed to be a deterministic function. However, the triggering kernel can vary with time due to the system uncertainty in real applications. To model this kind of variance, we propose a Hawkes process variant with stochastic triggering kernel, which incorporates the variation of triggering kernel over time. In this model, the triggering kernel is considered to be an independent multivariate Gaussian distribution. We derive and implement a tractable inference algorithm based on variational auto-encoder. Results from synthetic and real data experiments show that the underlying mean triggering kernel and variance band can be recovered, and using the stochastic triggering kernel is more accurate than the vanilla Hawkes process in capacity planning.
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Notes
- 1.
Customarily, \(Q(\cdot )\) is used for encoder’s distribution in VAE, but here to be consistent with the previous discussion \(P(\cdot )\) is used.
- 2.
Based on the Poisson process, the probability could be larger when intensity is low.
References
Adams, R.P., Murray, I., MacKay, D.J.: Tractable nonparametric Bayesian inference in Poisson processes with Gaussian process intensities. In: Proceedings of the 26th Annual International Conference on Machine Learning, pp. 9–16. ACM (2009)
Cox, D.R.: Some statistical methods connected with series of events. J. Roy. Stat. Soc. Ser. B (Methodol.) 17, 129–164 (1955)
Cunningham, J.P., Byron, M.Y., Shenoy, K.V., Sahani, M.: Inferring neural firing rates from spike trains using Gaussian processes. In: Advances in Neural Information Processing Systems, pp. 329–336 (2008)
Cunningham, J.P., Shenoy, K.V., Sahani, M.: Fast Gaussian process methods for point process intensity estimation. In: Proceedings of the 25th International Conference on Machine Learning, pp. 192–199. ACM (2008)
Dassios, A., Zhao, H., et al.: Exact simulation of Hawkes process with exponentially decaying intensity. Electron. Commun. Probab. 18, 1–13 (2013)
Doersch, C.: Tutorial on variational autoencoders. arXiv preprint arXiv:1606.05908 (2016)
Du, N., Dai, H., Trivedi, R., Upadhyay, U., Gomez-Rodriguez, M., Song, L.: Recurrent marked temporal point processes: embedding event history to vector. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1555–1564. ACM (2016)
Embrechts, P., Liniger, T., Lin, L.: Multivariate Hawkes processes: an application to financial data. J. Appl. Probab. 48(A), 367–378 (2011)
Gregory, P., Loredo, T.J.: A new method for the detection of a periodic signal of unknown shape and period. Astrophys. J. 398, 146–168 (1992)
Hawkes, A.G.: Spectra of some self-exciting and mutually exciting point processes. Biometrika 58(1), 83–90 (1971)
Kingma, D.P., Welling, M.: Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114 (2013)
Lee, Y., Lim, K.W., Ong, C.S.: Hawkes processes with stochastic excitations. In: International Conference on Machine Learning, pp. 79–88 (2016)
Menon, A.K., Lee, Y.: Predicting short-term public transport demand via inhomogeneous Poisson processes. In: Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, pp. 2207–2210. ACM (2017)
Mohler, G.O., Short, M.B., Brantingham, P.J., Schoenberg, F.P., Tita, G.E.: Self-exciting point process modeling of crime. J. Am. Stat. Assoc. 106(493), 100–108 (2011)
Møller, J., Syversveen, A.R., Waagepetersen, R.P.: Log gaussian cox processes. Scand. J. Stat. 25(3), 451–482 (1998)
Ogata, Y., Vere-Jones, D.: Inference for earthquake models: a self-correcting model. Stoch. Process. Appl. 17(2), 337–347 (1984)
Ogata, Y.: On lewis’ simulation method for point processes. IEEE Trans. Inf. Theory 27(1), 23–31 (1981)
Rodriguez, M.G., Balduzzi, D., Schölkopf, B.: Uncovering the temporal dynamics of diffusion networks. arXiv preprint arXiv:1105.0697 (2011)
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Zhou, F. et al. (2019). Hawkes Process with Stochastic Triggering Kernel. In: Yang, Q., Zhou, ZH., Gong, Z., Zhang, ML., Huang, SJ. (eds) Advances in Knowledge Discovery and Data Mining. PAKDD 2019. Lecture Notes in Computer Science(), vol 11439. Springer, Cham. https://doi.org/10.1007/978-3-030-16148-4_25
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DOI: https://doi.org/10.1007/978-3-030-16148-4_25
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