Point processes

August 1, 2016 — February 18, 2019

Figure 1

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.

1 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.

2 Spatial point processes

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

See spatial point processes.

3 References

Aalen, Odd O. 1978. Nonparametric Inference for a Family of Counting Processes.” The Annals of Statistics.
Aalen, Odd O. 1989. A Linear Regression Model for the Analysis of Life Times.” Statistics in Medicine.
Achab, Bacry, Gaïffas, et al. 2017. Uncovering Causality from Multivariate Hawkes Integrated Cumulants.” In PMLR.
Adams, Murray, and MacKay. 2009. Tractable Nonparametric Bayesian Inference in Poisson Processes with Gaussian Process Intensities.” In.
Adelfio, and Schoenberg. 2009. Point Process Diagnostics Based on Weighted Second-Order Statistics and Their Asymptotic Properties.” Annals of the Institute of Statistical Mathematics.
Andersen, Borgan, Gill, et al. 1997. Statistical models based on counting processes. Springer series in statistics.
Anselin, Cohen, Cook, et al. 2000. Spatial Analyses of Crime.”
Arora, Ge, Ma, et al. 2015. Simple, Efficient, and Neural Algorithms for Sparse Coding.” In Proceedings of The 28th Conference on Learning Theory.
Arribas-Gil, and Müller. 2014. Pairwise Dynamic Time Warping for Event Data.” Computational Statistics & Data Analysis.
Azizpour, Giesecke, and others. 2008. Self-Exciting Corporate Defaults: Contagion Vs. Frailty.”
Bacry, Bompaire, Gaïffas, et al. 2020. Sparse and Low-Rank Multivariate Hawkes Processes.” Journal of Machine Learning Research.
Bacry, and Muzy. 2014. Hawkes Model for Price and Trades High-Frequency Dynamics.” Quantitative Finance.
———. 2016. First- and Second-Order Statistics Characterization of Hawkes Processes and Non-Parametric Estimation.” IEEE Transactions on Information Theory.
Baddeley, Adrian. 2007. Spatial Point Processes and Their Applications.” In Stochastic Geometry. Lecture Notes in Mathematics 1892.
Baddeley, Adrian, Gregori, Mateu, et al. 2006. Case Studies in Spatial Point Process Modeling.
Baddeley, Adrian, and Møller. 1989. Nearest-Neighbour Markov Point Processes and Random Sets.” International Statistical Review / Revue Internationale de Statistique.
Baddeley, Adrian, Møller, and Pakes. 2008. Properties of Residuals for Spatial Point Processes.” Annals of the Institute of Statistical Mathematics.
Baddeley, Adrian J, Møller, and Waagepetersen. 2000. Non- and Semi-Parametric Estimation of Interaction in Inhomogeneous Point Patterns.” Statistica Neerlandica.
Baddeley, Adrian, and Turner. 2000. Practical Maximum Pseudolikelihood for Spatial Point Patterns.” Australian & New Zealand Journal of Statistics.
———. 2006. Modelling Spatial Point Patterns in R.” In Case Studies in Spatial Point Process Modeling. Lecture Notes in Statistics 185.
Baddeley, A., Turner, Møller, et al. 2005. Residual Analysis for Spatial Point Processes (with Discussion).” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Baddeley, A. J., and Van Lieshout. 1995. Area-Interaction Point Processes.” Annals of the Institute of Statistical Mathematics.
Baddeley, A. J., Van Lieshout, and Møller. 1996. Markov Properties of Cluster Processes.” Advances in Applied Probability.
Bai, Chen, and Chen. 2015. Semiparametric Estimation of a Self-Exciting Regression Model with an Appication in Recurrent Event Data Analysis.” Statistica Sinica.
Barbieri, Quirk, Frank, et al. 2001. Construction and Analysis of Non-Poisson Stimulus-Response Models of Neural Spiking Activity.” Journal of Neuroscience Methods.
Barbour, A. D. n.d. Stein’s Method and Poisson Process Convergence.” Journal of Applied Probability.
Barbour, A.D., and Brown. 1992. Stein’s Method and Point Process Approximation.” Stochastic Processes and Their Applications.
Barron, and Cover. 1991. Minimum Complexity Density Estimation.” IEEE Transactions on Information Theory.
Basawa. 1980. Statistical Inference for Stochastic Processes.
Bashtannyk, and Hyndman. 2001. Bandwidth Selection for Kernel Conditional Density Estimation.” Computational Statistics & Data Analysis.
Bauwens, and Hautsch. 2006. Stochastic Conditional Intensity Processes.” Journal of Financial Econometrics.
Benichoux, Vincent, and 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.
Berman, and Diggle. 1989. Estimating Weighted Integrals of the Second-Order Intensity of a Spatial Point Process.” Journal of the Royal Statistical Society. Series B (Methodological).
Berman, and Turner. 1992. Approximating Point Process Likelihoods with GLIM.” Journal of the Royal Statistical Society. Series C (Applied Statistics).
Besag. 1977. Efficiency of Pseudolikelihood Estimation for Simple Gaussian Fields.” Biometrika.
Bowsher. 2007. Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models.” Journal of Econometrics.
Brémaud, Pierre. 1972. “A Martingale Approach to Point Processes.”
Brémaud, P., and Massoulié. 2002. Power Spectra of General Shot Noises and Hawkes Point Processes with a Random Excitation.” Advances in Applied Probability.
Brémaud, Pierre, Massoulié, and Ridolfi. 2005. Power Spectra of Random Spike Fields and Related Processes.” Advances in Applied Probability.
Brix, and Kendall. 2002. Simulation of Cluster Point Processes Without Edge Effects.” Advances in Applied Probability.
Brown, Cai, and Zhou. 2010. Nonparametric Regression in Exponential Families.” The Annals of Statistics.
Buckley, Eagleson, and Silverman. 1988. The Estimation of Residual Variance in Nonparametric Regression.” Biometrika.
Chang, Yi-Ping. 2001. Estimation of Parameters for Nonhomogeneous Poisson Process: Software Reliability with Change-Point Model.” Communications in Statistics - Simulation and Computation.
Chang, C., and Schoenberg. 2008. “Testing Separability in Multi-Dimensional Point Processes with Covariates.” Annals of the Institute of Statistical Mathematics.
Chaudhuri. 1991. Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation.” The Annals of Statistics.
Cheng, and Wicks. 2014. Event Detection Using Twitter: A Spatio-Temporal Approach.” PLoS ONE.
Chen, Feng, and Hall. 2013. Inference for a Nonstationary Self-Exciting Point Process with an Application in Ultra-High Frequency Financial Data Modeling.” Journal of Applied Probability.
Chen, Feng, Huggins, Yip, et al. 2008. Local Polynomial Estimation of Poisson Intensities in the Presence of Reporting Delays.” Journal of the Royal Statistical Society: Series C (Applied Statistics).
Chen, Feng, and Stindl. 2017. Direct Likelihood Evaluation for the Renewal Hawkes Process.” Journal of Computational and Graphical Statistics.
Chen, Louis H. Y., and Xia. 2011. Poisson Process Approximation for Dependent Superposition of Point Processes.” Bernoulli.
Chen, Feng, Yip, and Lam. 2011. On the Local Polynomial Estimators of the Counting Process Intensity Function and Its Derivatives.” Scandinavian Journal of Statistics.
Chilinski, and Silva. 2020. Neural Likelihoods via Cumulative Distribution Functions.” arXiv:1811.00974 [Cs, Stat].
Claeskens, Krivobokova, and Opsomer. 2009. Asymptotic Properties of Penalized Spline Estimators.” Biometrika.
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).
Cox, Dennis D., and O’Sullivan. 1990. Asymptotic Analysis of Penalized Likelihood and Related Estimators.” The Annals of Statistics.
Crisan, and Míguez. 2014. Particle-Kernel Estimation of the Filter Density in State-Space Models.” Bernoulli.
Cronie, and van Lieshout. 2016. Bandwidth Selection for Kernel Estimators of the Spatial Intensity Function.” arXiv:1611.10221 [Stat].
Cucala. 2008. Intensity Estimation for Spatial Point Processes Observed with Noise.” Scandinavian Journal of Statistics.
Cui, and Lund. 2009. A New Look at Time Series of Counts.” Biometrika.
Cunningham, Shenoy, and Sahani. 2008. Fast Gaussian Process Methods for Point Process Intensity Estimation.” In Proceedings of the 25th International Conference on Machine Learning. ICML ’08.
Dahlhaus, and Eichler. 2003. Causality and Graphical Models in Time Series Analysis.” Oxford Statistical Science Series.
Dahlhaus, and Polonik. 2009. Empirical Spectral Processes for Locally Stationary Time Series.” Bernoulli.
Daley, and Vere-Jones. 2003. An introduction to the theory of point processes.
———. 2008. An Introduction to the Theory of Point Processes. Probability and Its Applications.
Daneshmand, Gomez-Rodriguez, Song, et al. 2014. Estimating Diffusion Network Structures: Recovery Conditions, Sample Complexity & Soft-Thresholding Algorithm.” In ICML.
Das, Duffie, Kapadia, et al. 2007. Common Failings: How Corporate Defaults Are Correlated.” The Journal of Finance.
Diaconis, and Freedman. 1984. Asymptotics of Graphical Projection Pursuit.” The Annals of Statistics.
Díaz-Avalos, Juan, and Mateu. 2012. Similarity Measures of Conditional Intensity Functions to Test Separability in Multidimensional Point Processes.” Stochastic Environmental Research and Risk Assessment.
Diggle, Peter J. 1979. On Parameter Estimation and Goodness-of-Fit Testing for Spatial Point Patterns.” Biometrics.
Diggle, Peter. 1985. A Kernel Method for Smoothing Point Process Data.” Journal of the Royal Statistical Society. Series C (Applied Statistics).
Drovandi, Pettitt, and McCutchan. 2016. Exact and Approximate Bayesian Inference for Low Integer-Valued Time Series Models with Intractable Likelihoods.” Bayesian Analysis.
Eden, Frank, Barbieri, et al. 2004. Dynamic Analysis of Neural Encoding by Point Process Adaptive Filtering.” Neural Computation.
Eichler, Dahlhaus, and Dueck. 2016. Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions.” Journal of Time Series Analysis.
Ellis. 1991. Density Estimation for Point Processes.” Stochastic Processes and Their Applications.
Embrechts, Liniger, and Lin. 2011. Multivariate Hawkes Processes: An Application to Financial Data.” Journal of Applied Probability.
Fan, and Li. 2001. Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties.” Journal of the American Statistical Association.
Fatalov. 2012. Integral Functionals for the Exponential of the Wiener Process and the Brownian Bridge: Exact Asymptotics and Legendre Functions.” Mathematical Notes.
Feigin. 1976. Maximum Likelihood Estimation for Continuous-Time Stochastic Processes.” Advances in Applied Probability.
Filimonov, and 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.
Flaxman, Teh, and Sejdinovic. 2016. Poisson Intensity Estimation with Reproducing Kernels.” arXiv:1610.08623 [Stat].
Gaïffas, and Guilloux. 2012. High-Dimensional Additive Hazards Models and the Lasso.” Electronic Journal of Statistics.
Geyer, and Møller. 1994. Simulation Procedures and Likelihood Inference for Spatial Point Processes.” Scandinavian Journal of Statistics.
Giesecke, K., Kakavand, and Mousavi. 2008. Simulating Point Processes by Intensity Projection.” In Simulation Conference, 2008. WSC 2008. Winter.
———. 2011. Exact Simulation of Point Processes with Stochastic Intensities.” Operations Research.
Giesecke, Kay, and Schwenkler. 2011. Filtered Likelihood for Point Processes.” SSRN Scholarly Paper ID 1898344.
Goulard, Särkkä, and Grabarnik. 1996. Parameter Estimation for Marked Gibbs Point Processes Through the Maximum Pseudo-Likelihood Method.” Scandinavian Journal of Statistics.
Green. 1987. Penalized Likelihood for General Semi-Parametric Regression Models.” International Statistical Review / Revue Internationale de Statistique.
Guan. 2008a. Variance Estimation for Statistics Computed from Inhomogeneous Spatial Point Processes.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
———. 2008b. A Goodness-of-Fit Test for Inhomogeneous Spatial Poisson Processes.” Biometrika.
Guan, and Sherman. 2007. On Least Squares Fitting for Stationary Spatial Point Processes.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Gui, and Li. 2005. Penalized Cox Regression Analysis in the High-Dimensional and Low-Sample Size Settings, with Applications to Microarray Gene Expression Data.” Bioinformatics.
Häggström, van Lieshout, and Møller. 1999. Characterization Results and Markov Chain Monte Carlo Algorithms Including Exact Simulation for Some Spatial Point Processes.” Bernoulli.
Hansen. 2010. Penalized Maximum Likelihood Estimation for Generalized Linear Point Processes.” arXiv:1003.0848 [Math, Stat].
Hansen, Reynaud-Bouret, and Rivoirard. 2015. Lasso and Probabilistic Inequalities for Multivariate Point Processes.” Bernoulli.
Hardiman, and Bouchaud. 2014. Branching-Ratio Approximation for the Self-Exciting Hawkes Process.” Physical Review E.
Harte. 2010. PtProcess: An R Package for Modelling Marked Point Processes Indexed by Time.” Journal of Statistical Software.
Haslinger, Pipa, and Brown. 2010. Discrete Time Rescaling Theorem: Determining Goodness of Fit for Discrete Time Statistical Models of Neural Spiking.” Neural Computation.
Hawe, Kleinsteuber, and Diepold. 2013. Analysis Operator Learning and Its Application to Image Reconstruction.” IEEE Transactions on Image Processing.
Hawkes. 1971a. Point Spectra of Some Mutually Exciting Point Processes.” Journal of the Royal Statistical Society. Series B (Methodological).
———. 1971b. Spectra of Some Self-Exciting and Mutually Exciting Point Processes.” Biometrika.
Helmers, and Mangku. 1999. “Statistical Estimation of Poisson Intensity Functions.” ANN. INST. STAT. MATH.
Hosmer. 2011. Applied Survival Analysis: Regression Modeling of Time-To-Event Data.
Huang, and Ogata. 1999. Improvements of the Maximum Pseudo-Likelihood Estimators in Various Spatial Statistical Models.” Journal of Computational and Graphical Statistics.
Hurvich, Simonoff, and Tsai. 1998. Smoothing Parameter Selection in Nonparametric Regression Using an Improved Akaike Information Criterion.” Journal of the Royal Statistical Society. Series B (Statistical Methodology).
Iribarren, and Moro. 2011. Branching Dynamics of Viral Information Spreading.” Physical Review E.
Jensen, and Künsch. 1994. On Asymptotic Normality of Pseudo Likelihood Estimates for Pairwise Interaction Processes.” Annals of the Institute of Statistical Mathematics.
Jensen, and Møller. 1991. Pseudolikelihood for Exponential Family Models of Spatial Point Processes.” The Annals of Applied Probability.
Jovanović, Hertz, and Rotter. 2015. Cumulants of Hawkes Point Processes.” Physical Review E.
Juban, Fugon, and Kariniotakis. 2007. Probabilistic Short-Term Wind Power Forecasting Based on Kernel Density Estimators.” In.
Karr. 1986. Point Processes and Their Statistical Inference.
Kass, Amari, Arai, et al. 2018. Computational Neuroscience: Mathematical and Statistical Perspectives.” Annual Review of Statistics and Its Application.
Koenker, and Hallock. 2001. Quantile Regression.” The Journal of Economic Perspectives.
Koenker, and Machado. 1999. Goodness of Fit and Related Inference Processes for Quantile Regression.” Journal of the American Statistical Association.
Koenker, and Mizera. 2006. Density Estimation by Total Variation Regularization.” Advances in Statistical Modeling and Inference.
Konishi, and Kitagawa. 1996. Generalised Information Criteria in Model Selection.” Biometrika.
Kroese, and Botev. 2013. Spatial Process Generation.” arXiv:1308.0399 [Stat].
Kroll. 2016. Concentration Inequalities for Poisson Point Processes with Application to Adaptive Intensity Estimation.” arXiv:1612.07901 [Math, Stat].
Kvitkovičová, and Panaretos. 2011. Asymptotic Inference for Partially Observed Branching Processes.” Advances in Applied Probability.
Kwieciński, and Szekli. 1996. Some Monotonicity and Dependence Properties of Self-Exciting Point Processes.” The Annals of Applied Probability.
Lewis, Mohler, Brantingham, et al. 2012. Self-Exciting Point Process Models of Civilian Deaths in Iraq.” Security Journal.
Lindsey. 1995. Fitting Parametric Counting Processes by Using Log-Linear Models.” Journal of the Royal Statistical Society. Series C (Applied Statistics).
Mairal, Bach, Ponce, et al. 2009. Online Dictionary Learning for Sparse Coding.” In Proceedings of the 26th Annual International Conference on Machine Learning. ICML ’09.
Marcus, Marblestone, and Dean. 2014. The atoms of neural computation.” Science.
Martin, Jasra, and McCoy. 2013. Inference for a Class of Partially Observed Point Process Models.” Annals of the Institute of Statistical Mathematics.
Marzen, and Crutchfield. 2020. Inference, Prediction, and Entropy-Rate Estimation of Continuous-Time, Discrete-Event Processes.” arXiv:2005.03750 [Cond-Mat, Physics:nlin, Stat].
Matsumoto, and Yor. 2005. Exponential Functionals of Brownian Motion, I: Probability Laws at Fixed Time.” Probability Surveys.
McCullagh, and Møller. 2006. The Permanental Process.” Advances in Applied Probability.
Micchelli, and 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.
Mishra, Rizoiu, and Xie. 2016. Feature Driven and Point Process Approaches for Popularity Prediction.” In Proceedings of the 25th ACM International Conference on Information and Knowledge Management. CIKM ’16.
Mohler, Short, Brantingham, et al. 2011. Self-Exciting Point Process Modeling of Crime.” Journal of the American Statistical Association.
Møller, and Berthelsen. 2012. Transforming Spatial Point Processes into Poisson Processes Using Random Superposition.” Advances in Applied Probability.
Møller, and Rasmussen. 2006. Approximate Simulation of Hawkes Processes.” Methodology and Computing in Applied Probability.
Møller, and Waagepetersen. 2003. Statistical Inference and Simulation for Spatial Point Processes.
———. 2007. Modern Statistics for Spatial Point Processes.” Scandinavian Journal of Statistics.
Møller, and Waagepetersen. 2017. Some Recent Developments in Statistics for Spatial Point Patterns.” Annual Review of Statistics and Its Application.
Morimoto. 1963. Markov Processes and the H-Theorem.” Journal of the Physical Society of Japan.
Neustifter, Rathbun, and Shiffman. 2012. Mixed-Poisson Point Process with Partially-Observed Covariates: Ecological Momentary Assessment of Smoking.” Journal of Applied Statistics.
Oakes. 1975. The Markovian Self-Exciting Process.” Journal of Applied Probability.
Ogata, Yoshiko. 1978. The Asymptotic Behaviour of Maximum Likelihood Estimators for Stationary Point Processes.” Annals of the Institute of Statistical Mathematics.
Ogata, Y. 1981. On Lewis’ Simulation Method for Point Processes.” IEEE Transactions on Information Theory.
Ogata, Yosihiko. 1988. Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes.” Journal of the American Statistical Association.
Ogata, Y. 1999. Seismicity Analysis Through Point-Process Modeling: A Review.” Pure and Applied Geophysics.
Ogata, Yosihiko, and Akaike. 1982. On Linear Intensity Models for Mixed Doubly Stochastic Poisson and Self-Exciting Point Processes.” Journal of the Royal Statistical Society, Series B.
Ogata, Yosihiko, Matsu’ura, and Katsura. 1993. Fast Likelihood Computation of Epidemic Type Aftershock-Sequence Model.” Geophysical Research Letters.
Olshausen, and Field. 1996. Natural image statistics and efficient coding.” Network (Bristol, England).
Omi, Ueda, and Aihara. 2020. Fully Neural Network Based Model for General Temporal Point Processes.” arXiv:1905.09690 [Cs, Stat].
Ozaki. 1979. Maximum Likelihood Estimation of Hawkes’ Self-Exciting Point Processes.” Annals of the Institute of Statistical Mathematics.
Panaretos, and Zemel. 2016. Separation of Amplitude and Phase Variation in Point Processes.” The Annals of Statistics.
Paninski. 2004. Maximum Likelihood Estimation of Cascade Point-Process Neural Encoding Models.” Network: Computation in Neural Systems.
Pnevmatikakis. 2017. Compressed Sensing and Optimal Denoising of Monotone Signals.” In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).
Pouget-Abadie, and Horel. 2015. Inferring Graphs from Cascades: A Sparse Recovery Framework.” In Proceedings of The 32nd International Conference on Machine Learning.
Puri, and Tuan. 1986. Maximum Likelihood Estimation for Stationary Point Processes.” Proceedings of the National Academy of Sciences of the United States of America.
Rasmussen, Jakob G. 2011. Temporal Point Processes the Conditional Intensity Function.”
Rasmussen, Jakob Gulddahl. 2013. Bayesian Inference for Hawkes Processes.” Methodology and Computing in Applied Probability.
Rasmussen, Jakob Gulddahl, Møller, Aukema, et al. 2006. Bayesian Inference for Multivariate Point Processes Observed at Sparsely Distributed Times.”
Ravanbakhsh, Schneider, and Poczos. 2016. Deep Learning with Sets and Point Clouds.” In arXiv:1611.04500 [Cs, Stat].
Reynaud-Bouret. 2003. Adaptive Estimation of the Intensity of Inhomogeneous Poisson Processes via Concentration Inequalities.” Probability Theory and Related Fields.
Reynaud-Bouret, Rivoirard, Grammont, et al. 2014. Goodness-of-Fit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis.” The Journal of Mathematical Neuroscience.
Reynaud-Bouret, and Roy. 2007. “Some Non Asymptotic Tail Estimates for Hawkes Processes.” Bulletin of the Belgian Mathematical Society - Simon Stevin.
Reynaud-Bouret, and Schbath. 2010. Adaptive Estimation for Hawkes Processes; Application to Genome Analysis.” The Annals of Statistics.
Riabiz, Ardeshiri, and Godsill. 2016. A Central Limit Theorem with Application to Inference in α-Stable Regression Models.” In.
Ridolfi. 2005. Power Spectra of Random Spikes and Related Complex Signals.”
Ripley, and Kelly. 1977. Markov Point Processes.” Journal of the London Mathematical Society.
Rizoiu, Xie, Sanner, et al. 2017. Expecting to Be HIP: Hawkes Intensity Processes for Social Media Popularity.” In World Wide Web 2017, International Conference on. WWW ’17.
Rosser, and Cheng. 2014. A Self-Exciting Point Process Model for Predictive Policing: Implementation and Evaluation.”
Rubin. 1972. Regular Point Processes and Their Detection.” IEEE Transactions on Information Theory.
Saichev, and Sornette. 2011. Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.5564 [Cond-Mat, Physics:physics].
Schelldorfer, Meier, and Bühlmann. 2014. GLMMLasso: An Algorithm for High-Dimensional Generalized Linear Mixed Models Using ℓ1-Penalization.” Journal of Computational and Graphical Statistics.
Schoenberg, Frederic. 1999. Transforming Spatial Point Processes into Poisson Processes.” Stochastic Processes and Their Applications.
Schoenberg, Frederic Paik. 2002. On Rescaled Poisson Processes and the Brownian Bridge.” Annals of the Institute of Statistical Mathematics.
———. 2004. Testing Separability in Spatial-Temporal Marked Point Processes.” Biometrics.
———. 2005. Consistent Parametric Estimation of the Intensity of a Spatial–Temporal Point Process.” Journal of Statistical Planning and Inference.
Sevast’yanov. 1968. Renewal Equations and Moments of Branching Processes.” Mathematical Notes of the Academy of Sciences of the USSR.
Silverman. 1982. On the Estimation of a Probability Density Function by the Maximum Penalized Likelihood Method.” The Annals of Statistics.
———. 1984. Spline Smoothing: The Equivalent Variable Kernel Method.” The Annals of Statistics.
Simon, Friedman, Hastie, et al. 2011. Regularization Paths for Cox’s Proportional Hazards Model via Coordinate Descent.” Journal of Statistical Software.
Simpson, Illian, Lindgren, et al. 2011. Going Off Grid: Computationally Efficient Inference for Log-Gaussian Cox Processes.” arXiv:1111.0641 [Math, Stat].
Smith, and Brown. 2003. Estimating a State-Space Model from Point Process Observations.” Neural Computation.
Städler, and Mukherjee. 2013. Penalized Estimation in High-Dimensional Hidden Markov Models with State-Specific Graphical Models.” The Annals of Applied Statistics.
Stefanski, and Carroll. 1990. Deconvolving Kernel Density Estimators.” Statistics.
Thrampoulidis, Abbasi, and Hassibi. 2015. LASSO with Non-Linear Measurements Is Equivalent to One With Linear Measurements.” In Advances in Neural Information Processing Systems 28.
Tria, Loreto, Servedio, et al. 2013. The Dynamics of Correlated Novelties.” arXiv:1310.1953 [Physics].
Turlach. 1993. Bandwidth Selection in Kernel Density Estimation: A Review.”
Vacarescu. 2011. Filtering and Parameter Estimation for Partially Observed Point Processes.
van de Geer. 1995. Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes.” The Annals of Statistics.
van de Geer, Bühlmann, Ritov, et al. 2014. On Asymptotically Optimal Confidence Regions and Tests for High-Dimensional Models.” The Annals of Statistics.
Lieshout, Marie-Colette NM van. 2000. Markov Point Processes and Their Applications.
Lieshout, Marie-Colette N. M. van. 2011. On Estimation of the Intensity Function of a Point Process.” Methodology and Computing in Applied Probability.
Veen, and Schoenberg. 2008. Estimation of Space–Time Branching Process Models in Seismology Using an EM–Type Algorithm.” Journal of the American Statistical Association.
Vere-Jones, and Schoenberg. 2004. Rescaling Marked Point Processes.” Australian & New Zealand Journal of Statistics.
Wheatley. 2013. “Quantifying Endogeneity in Market Prices with Point Processes: Methods & Applications.” Masters Thesis.
Willett, and Nowak. 2007. Multiscale Poisson Intensity and Density Estimation.” IEEE Transactions on Information Theory.
Witten, Tibshirani, and Hastie. 2009. A Penalized Matrix Decomposition, with Applications to Sparse Principal Components and Canonical Correlation Analysis.” Biostatistics.
Wörmann, Hawe, and Kleinsteuber. 2013. Analysis Based Blind Compressive Sensing.” IEEE Signal Processing Letters.
Wu, Müller, and Zhang. 2013. Functional Data Analysis for Point Processes with Rare Events.” Statistica Sinica.
Zarezade, Upadhyay, Rabiee, et al. 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. WSDM ’17.
Zhang, Rui, Walder, and Rizoiu. 2020. Variational Inference for Sparse Gaussian Process Modulated Hawkes Process.” In Proceedings of the AAAI Conference on Artificial Intelligence.
Zhang, Cun-Hui, and Zhang. 2014. Confidence Intervals for Low Dimensional Parameters in High Dimensional Linear Models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Zhou, Zha, and Song. 2013. Learning Triggering Kernels for Multi-Dimensional Hawkes Processes.” In Proceedings of the 30th International Conference on Machine Learning (ICML-13).