Point processes


Another intermittent obsession, tentatively placemarked. Discrete-state random fields/processes with a continuous index. In general I also assume they are non-lattice and simple, which terms I will define if I need them.

The most interesting class for me are the branching processes.

I’ve just spent 6 months thinking about nothing else, so I won’t write much here.

There are comprehensive introductions. (Daley and Vere-Jones 2003, 2008; Møller and Waagepetersen 2003)

A curious thing is that much point process estimation theory concerns estimating statistics from a single realisation of the point process. But in fact you may have many point process realisations. This is not news per se, just a new emphasis.

Temporal point processes

Sometimes including spatiotemporal point processes, depending on mood.

In these, one has an arrow of time which simplifies things because you know that you “only need to consider the past of a process to understand its future”, which potentially simplifies many calculations about the conditional intensity processes; We consider only interactions from the past to the future, rather than some kind of mutual interaction.

In particular, for nice processes you can do fairly cheap likelihood calculations to estimate process parameters etc.

Using the regular point process representation of the probability density of the occurrences, we have the following joint log likelihood for all the occurrences

\[\begin{aligned} L_\theta(t_{1:N}) &:= -\int_0^T\lambda^*_\theta(t)dt + \int_0^T\log \lambda^*_\theta(t) dN_t\\ &= -\int_0^T\lambda^*_\theta(t)dt + \sum_{j} \log \lambda^*_\theta(t_j) \end{aligned}\]

I do a lot of this, for example, over at the branching processes notebook, and I have no use at the moment for other types of process, so I won’t say much about other cases for the moment.

See also change of time.

Spatial point processes

Processes without an arrow of time arise naturally, say as processes where you observe only snaphosts of the dynamics, or where whatever dynamics that gave rise to the process being too slow to be considered as anything but static (forests).

See spatial point processes.

Aalen, Odd. 1978. “Nonparametric Inference for a Family of Counting Processes.” The Annals of Statistics 6 (4): 701–26. https://doi.org/10.1214/aos/1176344247.

Aalen, Odd O. 1989. “A Linear Regression Model for the Analysis of Life Times.” Statistics in Medicine 8 (8): 907–25. https://doi.org/10.1002/sim.4780080803.

Achab, Massil, Emmanuel Bacry, Stéphane Gaïffas, Iacopo Mastromatteo, and Jean-Francois Muzy. 2017. “Uncovering Causality from Multivariate Hawkes Integrated Cumulants.” In PMLR. http://arxiv.org/abs/1607.06333.

Adams, Ryan Prescott, Iain Murray, and David J. C. MacKay. 2009. “Tractable Nonparametric Bayesian Inference in Poisson Processes with Gaussian Process Intensities.” In, 1–8. ACM Press. https://doi.org/10.1145/1553374.1553376.

Adelfio, Giada, and Frederic Paik Schoenberg. 2009. “Point Process Diagnostics Based on Weighted Second-Order Statistics and Their Asymptotic Properties.” Annals of the Institute of Statistical Mathematics 61 (4): 929–48. https://doi.org/10.1007/s10463-008-0177-1.

Andersen, Per Kragh, Ornulf Borgan, Richard D. Gill, and Niels Keiding. 1997. Statistical Models Based on Counting Processes. Corr. 2. print. Springer Series in Statistics. New York, NY: Springer.

Arora, Sanjeev, Rong Ge, Tengyu Ma, and Ankur Moitra. 2015. “Simple, Efficient, and Neural Algorithms for Sparse Coding.” In Proceedings of the 28th Conference on Learning Theory, 40:113–49. Paris, France: PMLR. http://proceedings.mlr.press/v40/Arora15.html.

Arribas-Gil, Ana, and Hans-Georg Müller. 2014. “Pairwise Dynamic Time Warping for Event Data.” Computational Statistics & Data Analysis 69 (January): 255–68. https://doi.org/10.1016/j.csda.2013.08.011.

Azizpour, Shariar, Kay Giesecke, and others. 2008. “Self-Exciting Corporate Defaults: Contagion Vs. Frailty.” Stanford University working paper series. http://web.stanford.edu/dept/MSandE/cgi-bin/people/faculty/giesecke/pdfs/selfexciting.pdf.

Bacry, Emmanuel, Martin Bompaire, Stéphane Gaïffas, and Jean-Francois Muzy. 2020. “Sparse and Low-Rank Multivariate Hawkes Processes.” Journal of Machine Learning Research 21 (50): 1–32. http://jmlr.org/papers/v21/15-114.html.

Bacry, Emmanuel, and Jean-François Muzy. 2014. “Hawkes Model for Price and Trades High-Frequency Dynamics.” Quantitative Finance 14 (7): 1147–66. https://doi.org/10.1080/14697688.2014.897000.

———. 2016. “First- and Second-Order Statistics Characterization of Hawkes Processes and Non-Parametric Estimation.” IEEE Transactions on Information Theory 62 (4, 4): 2184–2202. https://doi.org/10.1109/TIT.2016.2533397.

Baddeley, Adrian. 2007. “Spatial Point Processes and Their Applications.” In Stochastic Geometry, edited by Wolfgang Weil, 1–75. Lecture Notes in Mathematics 1892. Springer Berlin Heidelberg. http://ahvaz.ist.unomaha.edu/azad/temp/sac/07-baddeley-point-process-poisson-coverage-sensor-simulation.pdf.

Baddeley, Adrian, Pablo Gregori, Jorge Mateu, Radu Stoica, and Dietrich Stoyan. 2006. Case Studies in Spatial Point Process Modeling. Vol. 185. Springer. http://link.springer.com/content/pdf/10.1007/0-387-31144-0.pdf.

Baddeley, Adrian J, Jesper Møller, and Rasmus Plenge Waagepetersen. 2000. “Non- and Semi-Parametric Estimation of Interaction in Inhomogeneous Point Patterns.” Statistica Neerlandica 54 (3): 329–50. https://doi.org/10.1111/1467-9574.00144.

Baddeley, Adrian, and Jesper Møller. 1989. “Nearest-Neighbour Markov Point Processes and Random Sets.” International Statistical Review / Revue Internationale de Statistique 57 (2): 89–121. https://doi.org/10.2307/1403381.

Baddeley, Adrian, Jesper Møller, and Anthony G. Pakes. 2008. “Properties of Residuals for Spatial Point Processes.” Annals of the Institute of Statistical Mathematics 60 (3): 627–49. http://link.springer.com/article/10.1007/s10463-007-0116-6.

Baddeley, Adrian, and Rolf Turner. 2000. “Practical Maximum Pseudolikelihood for Spatial Point Patterns.” Australian & New Zealand Journal of Statistics 42 (3): 283–322. https://doi.org/10.1111/1467-842X.00128.

———. 2006. “Modelling Spatial Point Patterns in R.” In Case Studies in Spatial Point Process Modeling, edited by Adrian Baddeley, Pablo Gregori, Jorge Mateu, Radu Stoica, and Dietrich Stoyan, 23–74. Lecture Notes in Statistics 185. Springer New York. http://link.springer.com/chapter/10.1007/0-387-31144-0_2.

Baddeley, A. J., and Marie-Colette NM Van Lieshout. 1995. “Area-Interaction Point Processes.” Annals of the Institute of Statistical Mathematics 47 (4): 601–19. https://doi.org/10.1007/BF01856536.

Baddeley, A. J., Marie-Colette NM Van Lieshout, and J. Møller. 1996. “Markov Properties of Cluster Processes.” Advances in Applied Probability 28 (2): 346–55. https://doi.org/10.2307/1428060.

Baddeley, A., R. Turner, J. Møller, and M. Hazelton. 2005. “Residual Analysis for Spatial Point Processes (with Discussion).” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67 (5): 617–66. https://doi.org/10.1111/j.1467-9868.2005.00519.x.

Bai, Fangfang, Feng Chen, and Kani Chen. 2015. “Semiparametric Estimation of a Self-Exciting Regression Model with an Appication in Recurrent Event Data Analysis.” Statistica Sinica. https://doi.org/10.5705/ss.2013.217.

Barbieri, Riccardo, Michael C Quirk, Loren M Frank, Matthew A Wilson, and Emery N Brown. 2001. “Construction and Analysis of Non-Poisson Stimulus-Response Models of Neural Spiking Activity.” Journal of Neuroscience Methods 105 (1): 25–37. https://doi.org/10.1016/S0165-0270(00)00344-7.

Barron, A. R., and T. M. Cover. 1991. “Minimum Complexity Density Estimation.” IEEE Transactions on Information Theory 37 (4): 1034–54. https://doi.org/10.1109/18.86996.

Basawa, Ishwar. 1980. Statistical Inference for Stochastic Processes. Academic Press.

Bashtannyk, David M., and Rob J. Hyndman. 2001. “Bandwidth Selection for Kernel Conditional Density Estimation.” Computational Statistics & Data Analysis 36 (3): 279–98. https://doi.org/10.1016/S0167-9473(00)00046-3.

Bauwens, Luc, and Nikolaus Hautsch. 2006. “Stochastic Conditional Intensity Processes.” Journal of Financial Econometrics 4 (3): 450–93. https://doi.org/10.1093/jjfinec/nbj013.

Benichoux, Alexis, Emmanuel Vincent, and Rémi Gribonval. 2013. “A Fundamental Pitfall in Blind Deconvolution with Sparse and Shift-Invariant Priors.” In ICASSP-38th International Conference on Acoustics, Speech, and Signal Processing-2013. https://hal.inria.fr/hal-00800770/.

Berman, Mark, and Peter Diggle. 1989. “Estimating Weighted Integrals of the Second-Order Intensity of a Spatial Point Process.” Journal of the Royal Statistical Society. Series B (Methodological) 51 (1): 81–92. https://publications.csiro.au/rpr/pub?list=BRO&pid=procite:d5b7ecd7-435c-4dab-9063-f1cf2fbdf4cb.

Berman, Mark, and T. Rolf Turner. 1992. “Approximating Point Process Likelihoods with GLIM.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 41 (1): 31–38. https://doi.org/10.2307/2347614.

Besag, Julian. 1977. “Efficiency of Pseudolikelihood Estimation for Simple Gaussian Fields.” Biometrika 64 (3): 616–18. https://doi.org/10.2307/2345341.

Bowsher, Clive G. 2007. “Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models.” Journal of Econometrics 141 (2): 876–912. https://doi.org/10.1016/j.jeconom.2006.11.007.

Brémaud, Pierre. 1972. “A Martingale Approach to Point Processes.” University of California, Berkeley.

Brémaud, Pierre, Laurent Massoulié, and Andrea Ridolfi. 2005. “Power Spectra of Random Spike Fields and Related Processes.” Advances in Applied Probability 37 (4): 1116–46. https://doi.org/10.1239/aap/1134587756.

Brémaud, P., and L. Massoulié. 2002. “Power Spectra of General Shot Noises and Hawkes Point Processes with a Random Excitation.” Advances in Applied Probability 34 (1): 205–22. https://doi.org/10.1239/aap/1019160957.

Brix, Anders, and Wilfrid S. Kendall. 2002. “Simulation of Cluster Point Processes Without Edge Effects.” Advances in Applied Probability 34 (2): 267–80. https://doi.org/10.1239/aap/1025131217.

Brown, Lawrence D., T. Tony Cai, and Harrison H. Zhou. 2010. “Nonparametric Regression in Exponential Families.” The Annals of Statistics 38 (4): 2005–46. https://doi.org/10.1214/09-AOS762.

Buckley, M. J., G. K. Eagleson, and B. W. Silverman. 1988. “The Estimation of Residual Variance in Nonparametric Regression.” Biometrika 75 (2): 189–99. https://doi.org/10.1093/biomet/75.2.189.

Chang, C., and F. P. Schoenberg. 2008. “Testing Separability in Multi-Dimensional Point Processes with Covariates.” Annals of the Institute of Statistical Mathematics.

Chang, Yi-Ping. 2001. “Estimation of Parameters for Nonhomogeneous Poisson Process: Software Reliability with Change-Point Model.” Communications in Statistics - Simulation and Computation 30 (3): 623–35. https://doi.org/10.1081/SAC-100105083.

Chaudhuri, Probal. 1991. “Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation.” The Annals of Statistics 19 (2): 760–77. https://doi.org/10.1214/aos/1176348119.

Chen, Feng, and Peter Hall. 2013. “Inference for a Nonstationary Self-Exciting Point Process with an Application in Ultra-High Frequency Financial Data Modeling.” Journal of Applied Probability 50 (4): 1006–24. https://doi.org/10.1239/jap/1389370096.

Chen, Feng, Richard M. Huggins, Paul S. F. Yip, and K. F. Lam. 2008. “Local Polynomial Estimation of Poisson Intensities in the Presence of Reporting Delays.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 57 (4): 447–59. https://doi.org/10.1111/j.1467-9876.2008.00624.x.

Chen, Feng, and Tom Stindl. 2017. “Direct Likelihood Evaluation for the Renewal Hawkes Process.” Journal of Computational and Graphical Statistics 27 (1): 1–13. https://doi.org/10.1080/10618600.2017.1341324.

Chen, Feng, Paul S. F. Yip, and K. F. Lam. 2011. “On the Local Polynomial Estimators of the Counting Process Intensity Function and Its Derivatives.” Scandinavian Journal of Statistics 38 (4): 631–49. https://doi.org/10.1111/j.1467-9469.2011.00733.x.

Chen, Louis H. Y., and Aihua Xia. 2011. “Poisson Process Approximation for Dependent Superposition of Point Processes.” Bernoulli 17 (2): 530–44. https://doi.org/10.3150/10-BEJ290.

Cheng, Tao, and Thomas Wicks. 2014. “Event Detection Using Twitter: A Spatio-Temporal Approach.” PLoS ONE 9 (6): e97807. https://doi.org/10.1371/journal.pone.0097807.

Claeskens, Gerda, Tatyana Krivobokova, and Jean D. Opsomer. 2009. “Asymptotic Properties of Penalized Spline Estimators.” Biometrika 96 (3): 529–44. https://doi.org/10.1093/biomet/asp035.

Cox, Dennis D., and Finbarr O’Sullivan. 1990. “Asymptotic Analysis of Penalized Likelihood and Related Estimators.” The Annals of Statistics 18 (4): 1676–95. https://doi.org/10.1214/aos/1176347872.

Cox, D. R. 1965. “On the Estimation of the Intensity Function of a Stationary Point Process.” Journal of the Royal Statistical Society: Series B (Methodological) 27 (2): 332–37. https://doi.org/10.1111/j.2517-6161.1965.tb01500.x.

Crisan, Dan, and Joaquín Míguez. 2014. “Particle-Kernel Estimation of the Filter Density in State-Space Models.” Bernoulli 20 (4): 1879–1929. https://doi.org/10.3150/13-BEJ545.

Cronie, O., and M. N. M. van Lieshout. 2016. “Bandwidth Selection for Kernel Estimators of the Spatial Intensity Function.” November 30, 2016. http://arxiv.org/abs/1611.10221.

Cucala, Lionel. 2008. “Intensity Estimation for Spatial Point Processes Observed with Noise.” Scandinavian Journal of Statistics 35 (2): 322–34. https://doi.org/10.1111/j.1467-9469.2007.00583.x.

Cui, Yunwei, and Robert Lund. 2009. “A New Look at Time Series of Counts.” Biometrika 96 (4): 781–92. https://doi.org/10.1093/biomet/asp057.

Cunningham, John P., Krishna V. Shenoy, and Maneesh Sahani. 2008. “Fast Gaussian Process Methods for Point Process Intensity Estimation.” In Proceedings of the 25th International Conference on Machine Learning, 192–99. ICML ’08. New York, NY, USA: ACM Press. https://doi.org/10.1145/1390156.1390181.

Dahlhaus, Rainer, and Michael Eichler. 2003. “Causality and Graphical Models in Time Series Analysis.” Oxford Statistical Science Series, 115–37. http://galton.uchicago.edu/~eichler/hsss.pdf.

Dahlhaus, Rainer, and Wolfgang Polonik. 2009. “Empirical Spectral Processes for Locally Stationary Time Series.” Bernoulli 15 (1): 1–39. https://doi.org/10.3150/08-BEJ137.

Daley, Daryl J., and David Vere-Jones. 2003. An Introduction to the Theory of Point Processes. 2nd ed. Vol. 1. Elementary theory and methods. New York: Springer. http://ebooks.springerlink.com/UrlApi.aspx?action=summary&v=1&bookid=108085.

———. 2008. An Introduction to the Theory of Point Processes. 2nd ed. Vol. 2. General theory and structure. Probability and Its Applications. New York: Springer. http://link.springer.com/chapter/10.1007/978-0-387-49835-5_7.

Daneshmand, Hadi, Manuel Gomez-Rodriguez, Le Song, and Bernhard Schölkopf. 2014. “Estimating Diffusion Network Structures: Recovery Conditions, Sample Complexity & Soft-Thresholding Algorithm.” In ICML. http://arxiv.org/abs/1405.2936.

Das, Sanjiv R., Darrell Duffie, Nikunj Kapadia, and Leandro Saita. 2007. “Common Failings: How Corporate Defaults Are Correlated.” The Journal of Finance 62 (1): 93–117. https://doi.org/10.1111/j.1540-6261.2007.01202.x.

Diaconis, Persi, and David Freedman. 1984. “Asymptotics of Graphical Projection Pursuit.” The Annals of Statistics 12 (3): 793–815. http://www.jstor.org/stable/2240961.

Diggle, Peter. 1985. “A Kernel Method for Smoothing Point Process Data.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 34 (2): 138–47. https://doi.org/10.2307/2347366.

Diggle, Peter J. 1979. “On Parameter Estimation and Goodness-of-Fit Testing for Spatial Point Patterns.” Biometrics 35 (1): 87–101. https://doi.org/10.2307/2529938.

Díaz-Avalos, Carlos, P. Juan, and J. Mateu. 2012. “Similarity Measures of Conditional Intensity Functions to Test Separability in Multidimensional Point Processes.” Stochastic Environmental Research and Risk Assessment 27 (5): 1193–1205. https://doi.org/10.1007/s00477-012-0654-1.

Drovandi, Christopher C., Anthony N. Pettitt, and Roy A. McCutchan. 2016. “Exact and Approximate Bayesian Inference for Low Integer-Valued Time Series Models with Intractable Likelihoods.” Bayesian Analysis 11 (2): 325–52. https://doi.org/10.1214/15-BA950.

Eden, U, L Frank, R Barbieri, V Solo, and E Brown. 2004. “Dynamic Analysis of Neural Encoding by Point Process Adaptive Filtering.” Neural Computation 16 (5): 971–98. https://doi.org/10.1162/089976604773135069.

Eichler, Michael, Rainer Dahlhaus, and Johannes Dueck. 2016. “Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions.” Journal of Time Series Analysis, January, n/a–n/a. https://doi.org/10.1111/jtsa.12213.

Ellis, Steven P. 1991. “Density Estimation for Point Processes.” Stochastic Processes and Their Applications 39 (2): 345–58. https://doi.org/10.1016/0304-4149(91)90087-S.

Embrechts, Paul, Thomas Liniger, and Lu Lin. 2011. “Multivariate Hawkes Processes: An Application to Financial Data.” Journal of Applied Probability 48A (August): 367–78. https://doi.org/10.1239/jap/1318940477.

Fan, Jianqing, and Runze Li. 2001. “Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties.” Journal of the American Statistical Association 96 (456): 1348–60. https://doi.org/10.1198/016214501753382273.

Fatalov, V. R. 2012. “Integral Functionals for the Exponential of the Wiener Process and the Brownian Bridge: Exact Asymptotics and Legendre Functions.” Mathematical Notes 92 (1-2): 79–98. https://doi.org/10.1134/S0001434612070103.

Feigin, Paul David. 1976. “Maximum Likelihood Estimation for Continuous-Time Stochastic Processes.” Advances in Applied Probability 8 (4): 712–36. https://doi.org/10.2307/1425931.

Filimonov, Vladimir, and Didier Sornette. 2013. “Apparent Criticality and Calibration Issues in the Hawkes Self-Excited Point Process Model: Application to High-Frequency Financial Data.” SSRN Scholarly Paper ID 2371284. Rochester, NY: Social Science Research Network. http://arxiv.org/abs/1308.6756.

Flaxman, Seth, Yee Whye Teh, and Dino Sejdinovic. 2016. “Poisson Intensity Estimation with Reproducing Kernels.” October 27, 2016. http://arxiv.org/abs/1610.08623.

Gaïffas, Stéphane, and Agathe Guilloux. 2012. “High-Dimensional Additive Hazards Models and the Lasso.” Electronic Journal of Statistics 6: 522–46. https://doi.org/10.1214/12-EJS681.

Geer, Sara van de. 1995. “Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes.” The Annals of Statistics 23 (5): 1779–1801. https://doi.org/10.1214/aos/1176324323.

Geer, Sara van de, Peter Bühlmann, Ya’acov Ritov, and Ruben Dezeure. 2014. “On Asymptotically Optimal Confidence Regions and Tests for High-Dimensional Models.” The Annals of Statistics 42 (3): 1166–1202. https://doi.org/10.1214/14-AOS1221.

Geyer, Charles J., and Jesper Møller. 1994. “Simulation Procedures and Likelihood Inference for Spatial Point Processes.” Scandinavian Journal of Statistics, 359–73. http://www.jstor.org/stable/4616323.

Giesecke, Kay, and Gustavo Schwenkler. 2011. “Filtered Likelihood for Point Processes.” SSRN Scholarly Paper ID 1898344. Rochester, NY: Social Science Research Network. http://www.stanford.edu/~gschwenk/pdf/filtered_rev3.pdf.

Giesecke, K., H. Kakavand, and M. Mousavi. 2008. “Simulating Point Processes by Intensity Projection.” In Simulation Conference, 2008. WSC 2008. Winter, 560–68. https://doi.org/10.1109/WSC.2008.4736114.

———. 2011. “Exact Simulation of Point Processes with Stochastic Intensities.” Operations Research 59 (5): 1233–45. https://doi.org/10.1287/opre.1110.0962.

Goulard, Michel, Aila Särkkä, and Pavel Grabarnik. 1996. “Parameter Estimation for Marked Gibbs Point Processes Through the Maximum Pseudo-Likelihood Method.” Scandinavian Journal of Statistics, 365–79. http://www.jstor.org/stable/4616410.

Green, Peter J. 1987. “Penalized Likelihood for General Semi-Parametric Regression Models.” International Statistical Review / Revue Internationale de Statistique 55 (3): 245–59. https://doi.org/10.2307/1403404.

Guan, Yongtao. 2008a. “Variance Estimation for Statistics Computed from Inhomogeneous Spatial Point Processes.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 70 (1): 175–90. http://moya.bus.miami.edu/~yguan/Papers/Papers%20accepted/JRSSB_07.pdf.

———. 2008b. “A Goodness-of-Fit Test for Inhomogeneous Spatial Poisson Processes.” Biometrika 95 (4): 831–45. https://doi.org/10.1093/biomet/asn045.

Guan, Yongtao, and Michael Sherman. 2007. “On Least Squares Fitting for Stationary Spatial Point Processes.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69 (1): 31–49. https://doi.org/10.1111/j.1467-9868.2007.00575.x.

Gui, Jiang, and Hongzhe Li. 2005. “Penalized Cox Regression Analysis in the High-Dimensional and Low-Sample Size Settings, with Applications to Microarray Gene Expression Data.” Bioinformatics 21 (13): 3001–8. https://doi.org/10.1093/bioinformatics/bti422.

Hansen, Niels Richard. 2010. “Penalized Maximum Likelihood Estimation for Generalized Linear Point Processes.” March 3, 2010. http://arxiv.org/abs/1003.0848.

Hansen, Niels Richard, Patricia Reynaud-Bouret, and Vincent Rivoirard. 2015. “Lasso and Probabilistic Inequalities for Multivariate Point Processes.” Bernoulli 21 (1): 83–143. https://doi.org/10.3150/13-BEJ562.

Hardiman, Stephen J., and Jean-Philippe Bouchaud. 2014. “Branching-Ratio Approximation for the Self-Exciting Hawkes Process.” Physical Review E 90 (6): 062807. https://doi.org/10.1103/PhysRevE.90.062807.

Harte, David. 2010. “PtProcess: An R Package for Modelling Marked Point Processes Indexed by Time.” Journal of Statistical Software 35 (8): 1–32. http://www.jstatsoft.org/v35/i08.

Haslinger, Robert, Gordon Pipa, and Emery Brown. 2010. “Discrete Time Rescaling Theorem: Determining Goodness of Fit for Discrete Time Statistical Models of Neural Spiking.” Neural Computation 22 (10): 2477–2506. https://doi.org/10.1162/NECO_a_00015.

Hawe, S., M. Kleinsteuber, and K. Diepold. 2013. “Analysis Operator Learning and Its Application to Image Reconstruction.” IEEE Transactions on Image Processing 22 (6): 2138–50. https://doi.org/10.1109/TIP.2013.2246175.

Hawkes, Alan G. 1971a. “Point Spectra of Some Mutually Exciting Point Processes.” Journal of the Royal Statistical Society. Series B (Methodological) 33 (3): 438–43. https://www.researchgate.net/profile/Alan_Hawkes2/publication/266241162_Point_spectra_of_some_mutually_exciting_point_process/links/56bb0ffa08ae0a6bc955f936.pdf.

———. 1971b. “Spectra of Some Self-Exciting and Mutually Exciting Point Processes.” Biometrika 58 (1): 83–90. https://doi.org/10.1093/biomet/58.1.83.

Häggström, Olle, Marie-Colette N. M. van Lieshout, and Jesper Møller. 1999. “Characterization Results and Markov Chain Monte Carlo Algorithms Including Exact Simulation for Some Spatial Point Processes.” Bernoulli 5 (4): 641–58. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.1330.

Helmers, Roelof, and I. Wayan Mangku. 1999. “Statistical Estimation of Poisson Intensity Functions.” ANN. INST. STAT. MATH 51: 265–80.

Hosmer, David W. 2011. Applied Survival Analysis: Regression Modeling of Time-to-Event Data. Wiley-Interscience.

Huang, Fuchun, and Yosihiko Ogata. 1999. “Improvements of the Maximum Pseudo-Likelihood Estimators in Various Spatial Statistical Models.” Journal of Computational and Graphical Statistics 8 (3): 510–30. https://doi.org/10.1080/10618600.1999.10474829.

Hurvich, Clifford M., Jeffrey S. Simonoff, and Chih-Ling Tsai. 1998. “Smoothing Parameter Selection in Nonparametric Regression Using an Improved Akaike Information Criterion.” Journal of the Royal Statistical Society. Series B (Statistical Methodology) 60 (2): 271–93. http://www.jstor.org/stable/2985940.

Iribarren, José Luis, and Esteban Moro. 2011. “Branching Dynamics of Viral Information Spreading.” Physical Review E 84 (4): 046116. https://doi.org/10.1103/PhysRevE.84.046116.

Jensen, Jens Ledet, and Hans R. Künsch. 1994. “On Asymptotic Normality of Pseudo Likelihood Estimates for Pairwise Interaction Processes.” Annals of the Institute of Statistical Mathematics 46 (3): 475–86. https://doi.org/10.1007/BF00773511.

Jensen, Jens Ledet, and Jesper Møller. 1991. “Pseudolikelihood for Exponential Family Models of Spatial Point Processes.” The Annals of Applied Probability 1 (3): 445–61. https://doi.org/10.1214/aoap/1177005877.

Jovanović, Stojan, John Hertz, and Stefan Rotter. 2015. “Cumulants of Hawkes Point Processes.” Physical Review E 91 (4): 042802. https://doi.org/10.1103/PhysRevE.91.042802.

Juban, Jérémie, Lionel Fugon, and Georges Kariniotakis. 2007. “Probabilistic Short-Term Wind Power Forecasting Based on Kernel Density Estimators.” In. https://hal-mines-paristech.archives-ouvertes.fr/hal-00526011/document.

Karr, Alan F. 1986. Point Processes and Their Statistical Inference. New York: Marcel Dekker Inc. http://books.google.com?id=Gdgx6RPp40EC.

Kass, Robert E., Shun-Ichi Amari, Kensuke Arai, Emery N. Brown, Casey O. Diekman, Markus Diesmann, Brent Doiron, et al. 2018. “Computational Neuroscience: Mathematical and Statistical Perspectives.” Annual Review of Statistics and Its Application 5 (1): 183–214. https://doi.org/10.1146/annurev-statistics-041715-033733.

Koenker, Roger, and Kevin F. Hallock. 2001. “Quantile Regression.” The Journal of Economic Perspectives 15 (4): 143–56. http://ajbuckeconbikesail.net/Econ616/Quantile/Koenker-Hallock.pdf.

Koenker, Roger, and José A. F. Machado. 1999. “Goodness of Fit and Related Inference Processes for Quantile Regression.” Journal of the American Statistical Association 94 (448): 1296–1310. https://doi.org/10.1080/01621459.1999.10473882.

Koenker, Roger, and Ivan Mizera. 2006. “Density Estimation by Total Variation Regularization.” Advances in Statistical Modeling and Inference, 613–34. http://ysidro.econ.uiuc.edu/~roger/research/densiles/Doksum.pdf.

Konishi, Sadanori, and Genshiro Kitagawa. 1996. “Generalised Information Criteria in Model Selection.” Biometrika 83 (4): 875–90. https://doi.org/10.1093/biomet/83.4.875.

Kroese, Dirk P., and Zdravko I. Botev. 2013. “Spatial Process Generation.” August 1, 2013. http://arxiv.org/abs/1308.0399.

Kroll, Martin. 2016. “Concentration Inequalities for Poisson Point Processes with Application to Adaptive Intensity Estimation.” December 23, 2016. http://arxiv.org/abs/1612.07901.

Kvitkovičová, Andrea, and Victor M. Panaretos. 2011. “Asymptotic Inference for Partially Observed Branching Processes.” Advances in Applied Probability 43 (4): 1166–90. https://doi.org/10.1239/aap/1324045703.

Kwieciński, Andrzej, and Ryszard Szekli. 1996. “Some Monotonicity and Dependence Properties of Self-Exciting Point Processes.” The Annals of Applied Probability 6 (4): 1211–31. http://www.jstor.org/stable/2245152.

Lewis, Erik, George Mohler, P. Jeffrey Brantingham, and Andrea L. Bertozzi. 2012. “Self-Exciting Point Process Models of Civilian Deaths in Iraq.” Security Journal 25 (3): 244–64. https://doi.org/10.1057/sj.2011.21.

Lieshout, Marie-Colette NM Van. 2000. Markov Point Processes and Their Applications. London: Imperial College Press.

Lieshout, Marie-Colette N. M. van. 2011. “On Estimation of the Intensity Function of a Point Process.” Methodology and Computing in Applied Probability 14 (3): 567–78. https://doi.org/10.1007/s11009-011-9244-9.

Lindsey, J. K. 1995. “Fitting Parametric Counting Processes by Using Log-Linear Models.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 44 (2): 201–12. https://doi.org/10.2307/2986345.

Mairal, Julien, Francis Bach, Jean Ponce, and Guillermo Sapiro. 2009. “Online Dictionary Learning for Sparse Coding.” In Proceedings of the 26th Annual International Conference on Machine Learning, 689–96. ICML ’09. New York, NY, USA: ACM. https://doi.org/10.1145/1553374.1553463.

Marcus, Gary, Adam Marblestone, and Thomas Dean. 2014. “The Atoms of Neural Computation.” Science 346 (6209): 551–52. https://doi.org/10.1126/science.1261661.

Martin, James S., Ajay Jasra, and Emma McCoy. 2013. “Inference for a Class of Partially Observed Point Process Models.” Annals of the Institute of Statistical Mathematics 65 (3): 413–37. https://doi.org/10.1007/s10463-012-0375-8.

Marzen, S. E., and J. P. Crutchfield. 2020. “Inference, Prediction, and Entropy-Rate Estimation of Continuous-Time, Discrete-Event Processes.” May 7, 2020. http://arxiv.org/abs/2005.03750.

Matsumoto, Hiroyuki, and Marc Yor. 2005. “Exponential Functionals of Brownian Motion, II: Some Related Diffusion Processes.” Probability Surveys 2: 348–84. https://doi.org/10.1214/154957805100000168.

McCullagh, Peter, and Jesper Møller. 2006. “The Permanental Process.” Advances in Applied Probability 38 (4): 873–88. https://doi.org/10.1017/S0001867800001361.

Micchelli, Charles A., and Peder Olsen. 2000. “Penalized Maximum-Likelihood Estimation, the Baum–Welch Algorithm, Diagonal Balancing of Symmetric Matrices and Applications to Training Acoustic Data.” Journal of Computational and Applied Mathematics 119 (1–2): 301–31. https://doi.org/10.1016/S0377-0427(00)00385-X.

Mishra, Swapnil, Marian-Andrei Rizoiu, and Lexing Xie. 2016. “Feature Driven and Point Process Approaches for Popularity Prediction.” In Proceedings of the 25th ACM International Conference on Information and Knowledge Management, 1069–78. CIKM ’16. New York, NY, USA: ACM. https://doi.org/10.1145/2983323.2983812.

Mohler, G. O., M. B. Short, P. J. Brantingham, F. P. Schoenberg, and G. E. Tita. 2011. “Self-Exciting Point Process Modeling of Crime.” Journal of the American Statistical Association 106 (493): 100–108. https://doi.org/10.1198/jasa.2011.ap09546.

Morimoto, Tetsuzo. 1963. “Markov Processes and the H-Theorem.” Journal of the Physical Society of Japan 18 (3): 328–31. https://doi.org/10.1143/JPSJ.18.328.

Møller, Jesper, and Kasper K. Berthelsen. 2012. “Transforming Spatial Point Processes into Poisson Processes Using Random Superposition.” Advances in Applied Probability 44 (1): 42–62. https://doi.org/10.1239/aap/1331216644.

Møller, Jesper, and Jakob G. Rasmussen. 2006. “Approximate Simulation of Hawkes Processes.” Methodology and Computing in Applied Probability 8 (1): 53–64. https://doi.org/10.1007/s11009-006-7288-z.

Møller, Jesper, and Rasmus Waagepetersen. 2017. “Some Recent Developments in Statistics for Spatial Point Patterns.” Annual Review of Statistics and Its Application 4 (1): 317–42. https://doi.org/10.1146/annurev-statistics-060116-054055.

Møller, Jesper, and Rasmus Plenge Waagepetersen. 2007. “Modern Statistics for Spatial Point Processes.” Scandinavian Journal of Statistics 34 (4): 643–84. https://doi.org/10.1111/j.1467-9469.2007.00569.x.

———. 2003. Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC. https://doi.org/10.1201/9780203496930.

Neustifter, Benjamin, Stephen L. Rathbun, and Saul Shiffman. 2012. “Mixed-Poisson Point Process with Partially-Observed Covariates: Ecological Momentary Assessment of Smoking.” Journal of Applied Statistics 39 (4): 883–99. https://doi.org/10.1080/02664763.2011.626848.

Oakes, David. 1975. “The Markovian Self-Exciting Process.” Journal of Applied Probability 12 (1): 69. https://doi.org/10.2307/3212408.

Ogata, Y. 1981. “On Lewis’ Simulation Method for Point Processes.” IEEE Transactions on Information Theory 27 (1): 23–31. https://doi.org/10.1109/TIT.1981.1056305.

———. 1999. “Seismicity Analysis Through Point-Process Modeling: A Review.” Pure and Applied Geophysics 155 (2-4): 471–507. https://doi.org/10.1007/s000240050275.

Ogata, Yoshiko. 1978. “The Asymptotic Behaviour of Maximum Likelihood Estimators for Stationary Point Processes.” Annals of the Institute of Statistical Mathematics 30 (1): 243–61. https://doi.org/10.1007/BF02480216.

Ogata, Yosihiko. 1988. “Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes.” Journal of the American Statistical Association 83 (401): 9–27. https://doi.org/10.1080/01621459.1988.10478560.

Ogata, Yosihiko, and Hirotugu Akaike. 1982. “On Linear Intensity Models for Mixed Doubly Stochastic Poisson and Self-Exciting Point Processes.” Journal of the Royal Statistical Society, Series B 44: 269–74. https://doi.org/10.1007/978-1-4612-1694-0_20.

Ogata, Yosihiko, Ritsuko S. Matsu’ura, and Koichi Katsura. 1993. “Fast Likelihood Computation of Epidemic Type Aftershock-Sequence Model.” Geophysical Research Letters 20 (19): 2143–6. https://doi.org/10.1029/93GL02142.

Olshausen, B. A., and D. J. Field. 1996. “Natural Image Statistics and Efficient Coding.” Network (Bristol, England) 7 (2): 333–39. https://doi.org/10.1088/0954-898X/7/2/014.

Ozaki, T. 1979. “Maximum Likelihood Estimation of Hawkes’ Self-Exciting Point Processes.” Annals of the Institute of Statistical Mathematics 31 (1): 145–55. https://doi.org/10.1007/BF02480272.

Panaretos, Victor M., and Yoav Zemel. 2016. “Separation of Amplitude and Phase Variation in Point Processes.” The Annals of Statistics 44 (2): 771–812. https://doi.org/10.1214/15-AOS1387.

Paninski, Liam. 2004. “Maximum Likelihood Estimation of Cascade Point-Process Neural Encoding Models.” Network: Computation in Neural Systems 15 (4): 243–62. https://doi.org/10.1088/0954-898X/15/4/002.

Pnevmatikakis, Eftychios A. 2017. “Compressed Sensing and Optimal Denoising of Monotone Signals.” In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 4740–4. https://doi.org/10.1109/ICASSP.2017.7953056.

Pouget-Abadie, Jean, and Thibaut Horel. 2015. “Inferring Graphs from Cascades: A Sparse Recovery Framework.” In Proceedings of the 32nd International Conference on Machine Learning. http://arxiv.org/abs/1505.05663.

Puri, Madan L., and Pham D. Tuan. 1986. “Maximum Likelihood Estimation for Stationary Point Processes.” Proceedings of the National Academy of Sciences of the United States of America 83 (3): 541–45. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC322899/.

Rasmussen, Jakob G. 2011. “Temporal Point Processes the Conditional Intensity Function.” http://people.math.aau.dk/~jgr/teaching/punktproc11/tpp.pdf.

Rasmussen, Jakob Gulddahl. 2013. “Bayesian Inference for Hawkes Processes.” Methodology and Computing in Applied Probability 15 (3): 623–42. https://doi.org/10.1007/s11009-011-9272-5.

Rasmussen, Jakob Gulddahl, Jesper Møller, B. H. Aukema, K. F. Raffa, and J. Zhu. 2006. “Bayesian Inference for Multivariate Point Processes Observed at Sparsely Distributed Times.” Department of Mathematical Sciences, Aalborg University. http://vbn.aau.dk/ws/files/4751438/R-2006-24.pdf.

Ravanbakhsh, Siamak, Jeff Schneider, and Barnabas Poczos. 2016. “Deep Learning with Sets and Point Clouds.” In. http://arxiv.org/abs/1611.04500.

Reynaud-Bouret, Patricia. 2003. “Adaptive Estimation of the Intensity of Inhomogeneous Poisson Processes via Concentration Inequalities.” Probability Theory and Related Fields 126 (1). https://doi.org/10.1007/s00440-003-0259-1.

Reynaud-Bouret, Patricia, Vincent Rivoirard, Franck Grammont, and Christine Tuleau-Malot. 2014. “Goodness-of-Fit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis.” The Journal of Mathematical Neuroscience 4 (1): 3. https://doi.org/10.1186/2190-8567-4-3.

Reynaud-Bouret, Patricia, and Emmanuel Roy. 2007. “Some Non Asymptotic Tail Estimates for Hawkes Processes.” Bulletin of the Belgian Mathematical Society - Simon Stevin 13 (5): 883–96.

Reynaud-Bouret, Patricia, and Sophie Schbath. 2010. “Adaptive Estimation for Hawkes Processes; Application to Genome Analysis.” The Annals of Statistics 38 (5): 2781–2822. https://doi.org/10.1214/10-AOS806.

Riabiz, Marina, Tohid Ardeshiri, and Simon Godsill. 2016. “A Central Limit Theorem with Application to Inference in α-Stable Regression Models.” In, 70–82. http://jmlr.org/proceedings/papers/v55/riabiz16.html.

Ridolfi, Andrea. 2005. “Power Spectra of Random Spikes and Related Complex Signals.” Institut de systèmes de communication SECTION DES SYSTÈMES DE COMMUNICATION POUR L’OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES PAR laurea di dottore in ingegneria elettronica, Politecnico di Milano. http://icwww.epfl.ch/~ridolfi/research/thesis_andrea.pdf.

Ripley, B. D., and F. P. Kelly. 1977. “Markov Point Processes.” Journal of the London Mathematical Society s2-15 (1): 188–92. https://doi.org/10.1112/jlms/s2-15.1.188.

Rizoiu, Marian-Andrei, Lexing Xie, Scott Sanner, Manuel Cebrian, Honglin Yu, and Pascal Van Hentenryck. 2017. “Expecting to Be HIP: Hawkes Intensity Processes for Social Media Popularity.” In World Wide Web 2017, International Conference on, 1–9. WWW ’17. Perth, Australia: International World Wide Web Conferences Steering Committee. https://doi.org/10.1145/3038912.3052650.

Rosser, G., and T. Cheng. 2014. “A Self-Exciting Point Process Model for Predictive Policing: Implementation and Evaluation.” http://leeds.gisruk.org/abstracts/GISRUK2015_submission_33.pdf.

Rubin, Izhak. 1972. “Regular Point Processes and Their Detection.” IEEE Transactions on Information Theory 18 (5): 547–57. https://doi.org/10.1109/TIT.1972.1054897.

Saichev, A., and D. Sornette. 2011. “Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes.” January 28, 2011. http://arxiv.org/abs/1101.5564.

Schelldorfer, Jürg, Lukas Meier, and Peter Bühlmann. 2014. “GLMMLasso: An Algorithm for High-Dimensional Generalized Linear Mixed Models Using ℓ1-Penalization.” Journal of Computational and Graphical Statistics 23 (2): 460–77. https://doi.org/10.1080/10618600.2013.773239.

Schoenberg, Frederic. 1999. “Transforming Spatial Point Processes into Poisson Processes.” Stochastic Processes and Their Applications 81 (2): 155–64. https://doi.org/10.1016/S0304-4149(98)00098-2.

Schoenberg, Frederic Paik. 2002. “On Rescaled Poisson Processes and the Brownian Bridge.” Annals of the Institute of Statistical Mathematics 54 (2): 445–57. https://doi.org/10.1023/A:1022494523519.

———. 2004. “Testing Separability in Spatial-Temporal Marked Point Processes.” Biometrics 60 (2): 471–81. http://www.stat.ucla.edu/~frederic/papers/sep133.pdf.

———. 2005. “Consistent Parametric Estimation of the Intensity of a Spatial–Temporal Point Process.” Journal of Statistical Planning and Inference 128 (1): 79–93. https://doi.org/10.1016/j.jspi.2003.09.027.

Sevast’yanov, B. A. 1968. “Renewal Equations and Moments of Branching Processes.” Mathematical Notes of the Academy of Sciences of the USSR 3 (1): 3–10. https://doi.org/10.1007/BF01386956.

Silverman, B. W. 1982. “On the Estimation of a Probability Density Function by the Maximum Penalized Likelihood Method.” The Annals of Statistics 10 (3): 795–810. https://doi.org/10.1214/aos/1176345872.

———. 1984. “Spline Smoothing: The Equivalent Variable Kernel Method.” The Annals of Statistics 12 (3): 898–916. https://doi.org/10.1214/aos/1176346710.

Simon, Noah, Jerome Friedman, Trevor Hastie, and Rob Tibshirani. 2011. “Regularization Paths for Cox’s Proportional Hazards Model via Coordinate Descent.” Journal of Statistical Software 39 (5). http://www.jstatsoft.org/v39/i05/paper.

Simpson, Daniel, Janine Illian, Finn Lindgren, Sigrunn Sørbye, and Håvard Rue. 2011. “Going Off Grid: Computationally Efficient Inference for Log-Gaussian Cox Processes.” November 1, 2011. http://arxiv.org/abs/1111.0641.

Smith, A, and E Brown. 2003. “Estimating a State-Space Model from Point Process Observations.” Neural Computation 15 (5): 965–91. https://doi.org/10.1162/089976603765202622.

Städler, Nicolas, and Sach Mukherjee. 2013. “Penalized Estimation in High-Dimensional Hidden Markov Models with State-Specific Graphical Models.” The Annals of Applied Statistics 7 (4): 2157–79. https://doi.org/10.1214/13-AOAS662.

Stefanski, Leonard A., and Raymond J. Carroll. 1990. “Deconvolving Kernel Density Estimators.” Statistics 21 (2): 169–84. https://doi.org/10.1080/02331889008802238.

Thrampoulidis, Chrtistos, Ehsan Abbasi, and Babak Hassibi. 2015. “LASSO with Non-Linear Measurements Is Equivalent to One with Linear Measurements.” In Advances in Neural Information Processing Systems 28, edited by C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, R. Garnett, and R. Garnett, 3402–10. Curran Associates, Inc. http://papers.nips.cc/paper/5739-lasso-with-non-linear-measurements-is-equivalent-to-one-with-linear-measurements.pdf.

Tria, F., V. Loreto, V. D. P. Servedio, and S. H. Strogatz. 2013. “The Dynamics of Correlated Novelties.” October 7, 2013. https://doi.org/10.1038/srep05890.

Turlach, Berwin A. 1993. “Bandwidth Selection in Kernel Density Estimation: A Review.” http://www.stat.washington.edu/courses/stat527/s13/readings/Turlach.pdf.

Vacarescu. 2011. Filtering and Parameter Estimation for Partially Observed Point Processes.

Veen, Alejandro, and Frederic P Schoenberg. 2008. “Estimation of Space–Time Branching Process Models in Seismology Using an EM–Type Algorithm.” Journal of the American Statistical Association 103 (482): 614–24. https://doi.org/10.1198/016214508000000148.

Vere-Jones, David, and Frederic Paik Schoenberg. 2004. “Rescaling Marked Point Processes.” Australian & New Zealand Journal of Statistics 46 (1): 133–43. https://doi.org/10.1111/j.1467-842X.2004.00319.x.

Wheatley, Spencer. 2013. “Quantifying Endogeneity in Market Prices with Point Processes: Methods & Applications.” Masters Thesis. ETH Zürich: Masters Thesis.

Willett, R. M., and R. D. Nowak. 2007. “Multiscale Poisson Intensity and Density Estimation.” IEEE Transactions on Information Theory 53 (9): 3171–87. https://doi.org/10.1109/TIT.2007.903139.

Witten, Daniela M., Robert Tibshirani, and Trevor Hastie. 2009. “A Penalized Matrix Decomposition, with Applications to Sparse Principal Components and Canonical Correlation Analysis.” Biostatistics, January, kxp008. https://doi.org/10.1093/biostatistics/kxp008.

Wörmann, Julian, Simon Hawe, and Martin Kleinsteuber. 2013. “Analysis Based Blind Compressive Sensing.” IEEE Signal Processing Letters 20 (5): 491–94. https://doi.org/10.1109/LSP.2013.2252900.

Wu, Shuang, Hans-Georg Müller, and Zhen Zhang. 2013. “Functional Data Analysis for Point Processes with Rare Events.” Statistica Sinica 23 (1): 1–23. http://www3.stat.sinica.edu.tw/sstest/oldpdf/A23n11.pdf.

Zarezade, Ali, Utkarsh Upadhyay, Hamid R. Rabiee, and Manuel Gomez-Rodriguez. 2017. “RedQueen: An Online Algorithm for Smart Broadcasting in Social Networks.” In Proceedings of the Tenth ACM International Conference on Web Search and Data Mining, 51–60. WSDM ’17. New York, NY, USA: ACM Press. https://doi.org/10.1145/3018661.3018684.

Zhang, Cun-Hui, and Stephanie S. Zhang. 2014. “Confidence Intervals for Low Dimensional Parameters in High Dimensional Linear Models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76 (1): 217–42. https://doi.org/10.1111/rssb.12026.

Zhou, Ke, Hongyuan Zha, and Le Song. 2013. “Learning Triggering Kernels for Multi-Dimensional Hawkes Processes.” In Proceedings of the 30th International Conference on Machine Learning (ICML-13), 1301–9. http://machinelearning.wustl.edu/mlpapers/papers/icml2013_zhou13.