Point process intensities and statistical estimation thereof

On understanding and estimating the intensity function of inhomogeneous Poisson 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.
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.
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. 2016. “First- and Second-Order Statistics Characterization of Hawkes Processes and Non-Parametric Estimation.” IEEE Transactions on Information Theory 62 (4): 2184–2202. https://doi.org/10.1109/TIT.2016.2533397.
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.
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.
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.
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.
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.
Chilinski, Pawel, and Ricardo Silva. 2020. “Neural Likelihoods via Cumulative Distribution Functions.” arXiv:1811.00974 [cs, Stat], June. http://arxiv.org/abs/1811.00974.
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, 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.
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.
Crisan, Dan, and Joaquín Míguez. 2014. “Particle-kernel estimation of the filter density in state-space models.” Bernoulli 20 (4): 1879–929. 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.” arXiv:1611.10221 [stat], November. http://arxiv.org/abs/1611.10221.
Cronie, Ottmar, Mehdi Moradi, and Christophe A. N. Biscio. 2021. “Statistical Learning and Cross-Validation for Point Processes.” arXiv:2103.01356 [math, Stat], March. http://arxiv.org/abs/2103.01356.
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.
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.
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.
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–. 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.
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.
Flaxman, Seth, Yee Whye Teh, and Dino Sejdinovic. 2016. “Poisson Intensity Estimation with Reproducing Kernels.” arXiv:1610.08623 [stat], October. 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, 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.
Gelfand, Alan, and Sudipto Banerjee. 2010. “Multivariate Spatial Process Models.” In Handbook of Spatial Statistics, edited by Alan Gelfand, Peter Diggle, Montserrat Fuentes, and Peter Guttorp, 20103158:495–515. CRC Press. https://doi.org/10.1201/9781420072884-c28.
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. 2008. “A Goodness-of-Fit Test for Inhomogeneous Spatial Poisson Processes.” Biometrika 95 (4): 831–45. https://doi.org/10.1093/biomet/asn045.
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.” arXiv:1003.0848 [math, Stat], March. 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.
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.
Helmers, Roelof, and I. Wayan Mangku. 1999. “Statistical Estimation of Poisson Intensity Functions.” ANN. INST. STAT. MATH 51: 265–80.
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.
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.
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.
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.
Kroll, Martin. 2016. “Concentration Inequalities for Poisson Point Processes with Application to Adaptive Intensity Estimation.” arXiv:1612.07901 [math, Stat], December. http://arxiv.org/abs/1612.07901.
Lee, Young, Kar Wai Lim, and Cheng Soon Ong. 2016. “Hawkes Processes with Stochastic Excitations.” In ICML. http://arxiv.org/abs/1609.06831.
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.
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.
Marzen, S. E., and J. P. Crutchfield. 2020. “Inference, Prediction, and Entropy-Rate Estimation of Continuous-Time, Discrete-Event Processes.” arXiv:2005.03750 [cond-Mat, Physics:nlin, Stat], May. http://arxiv.org/abs/2005.03750.
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.
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.
Ng, Tin Lok James, and Andrew Zammit-Mangion. 2020. “Non-Homogeneous Poisson Process Intensity Modeling and Estimation Using Measure Transport.” arXiv:2007.00248 [stat], July. http://arxiv.org/abs/2007.00248.
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.
Omi, Takahiro, Naonori Ueda, and Kazuyuki Aihara. 2020. “Fully Neural Network Based Model for General Temporal Point Processes.” arXiv:1905.09690 [cs, Stat], January. http://arxiv.org/abs/1905.09690.
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.
Pnevmatikakis, Eftychios A. 2017. “Compressed Sensing and Optimal Denoising of Monotone Signals.” In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 4740–44. https://doi.org/10.1109/ICASSP.2017.7953056.
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/.
Ravanbakhsh, Siamak, Jeff Schneider, and Barnabas Poczos. 2016. “Deep Learning with Sets and Point Clouds.” In arXiv:1611.04500 [cs, Stat]. 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.
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.
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 Paik. 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.
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.
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.
Soen, Alexander, Alexander Mathews, Daniel Grixti-Cheng, and Lexing Xie. 2021. “UNIPoint: Universally Approximating Point Processes Intensities.” arXiv:2007.14082 [cs, Stat], March. http://arxiv.org/abs/2007.14082.
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.
Turlach, Berwin A. 1993. “Bandwidth Selection in Kernel Density Estimation: A Review.” http://www.stat.washington.edu/courses/stat527/s13/readings/Turlach.pdf.
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.
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.
Zhang, Rui, Christian Walder, and Marian-Andrei Rizoiu. 2020. “Variational Inference for Sparse Gaussian Process Modulated Hawkes Process.” In Proceedings of the AAAI Conference on Artificial Intelligence, 34:6803–10. https://doi.org/10.1609/aaai.v34i04.6160.

No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.