Branching processes
August 18, 2014 — February 7, 2020
branching
count data
functional analysis
Lévy processes
point processes
probability
statistics
time series
A diverse class of stochastic models that I am mildly obsessed with, where over some index set (usually time, space, or both) there are distributed births of some kind, and we count the total population.
In particular, I am interested in “pure birth” branching processes, where each event leads to certain numbers of offspring with a certain probability. These correspond to certain types of “cluster” and “self-excitation” processes.
These come in Markov and non-Markov flavours, depending on, loosely, whether the notional particles in the system have a memoryless life cycle or not.
1 To learn
- Basic handling of processes defined on a multidimensional index set, i.e. space-time processes and branching random fields. (“cluster processes”) Maybe I’ve done that over at spatial point processes by now?
- The various connections to trees, and hence the connection to networks.
- Connection to stable processes and Lévy processes.
2 We do not care about time
3 Discrete index, discrete state, Markov: The Galton-Watson process
This section got long enough to break out separately. See my notes on long-memory Galton-Watson process.
4 Continuous index, discrete state: the Hawkes Process
If you have an integer-valued state space, but a continuous time index, and linear intensities, then this is a Hawkes point process, the cluster point process. See my masters thesis, or my Hawkes process notes.
5 Continuous index, continuous state
Aldous gives an expo on the Continuous State Branching process. I do not know much about these. Perhaps I could know more if I read Z. Li (2011), which introduces CSBPs as a special case of Measure-valued branching processes, and also connects them with superprocesses (Etheridge 2000; Dynkin 2004, 1991) were recommended to me for the latter.
5.1 Parameter estimation
I’m curious about this, and Lévy process inference in general. It’s interesting because such processes are always incompletely sampled; What’s the best you can do with finitely many samples from a continuous branching process? For the simple case of the Wiener process (as a Lévy process) there is a well-understood estimation theory, with twiddly flourishes on top. For CSBPs I am not aware of any general methods. (Overbeck 1998) seems to be one of the few refs and is rather constrained. Surely the finance folks are onto this?
6 Discrete index, continuous state
Popular in physics as a contagion model. See Burridge (2013a);Burridge (2013b) for some handy relations between these models, Gamma processes, martingales and limits of negative binomial distributions via renewal theory and Kendall’s identity.
7 Special issues for multivariate branching processes
If you are looking at cross-excitation between variables then I have some additional matter at contagion processes.
8 Classic data sets
Data sets which might be explored for their branching process nature tend towards the epidemiological.
Tomás Aragón’s free online epidemiology textbook (Aragón 2012) lists, among others,
In the R package
tscountone may findcampy,ecoli,ehec,influenzaandmeasles.Gapminder hosts some disease datasets including tuberculosis, malaria, and diarrhoea.
If you include lifecycle questions in your model (death and birth rates) you might use population data sets. The UN department of Economic and Social Affairs has global population data sets.
9 Implementations
IHSEP is Feng Chen’s software for continuous index, discrete state branching processes.
Spatstat is for spatial point processes.
10 References
Aldous. 1991. “The Continuum Random Tree. I.” The Annals of Probability.
———. 1993. “The Continuum Random Tree III.” The Annals of Probability.
Aldous, and Pitman. 1998. “Tree-Valued Markov Chains Derived from Galton-Watson Processes.” Annales de l’Institut Henri Poincare (B) Probability and Statistics.
Al-Osh, Mohamed A., and Aly. 1992. “First Order Autoregressive Time Series with Negative Binomial and Geometric Marginals.” Communications in Statistics - Theory and Methods.
Al-Osh, M. A., and Alzaid. 1987. “First-Order Integer-Valued Autoregressive (INAR(1)) Process.” Journal of Time Series Analysis.
Aly, and Bouzar. 2005. “Stationary Solutions for Integer-Valued Autoregressive Processes.” International Journal of Mathematics and Mathematical Sciences.
Alzaid, and Al-Osh. 1988. “First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties.” Statistica Neerlandica.
Applebaum. 2004. “Lévy Processes — from Probability to Finance and Quantum Groups.” Notices of the AMS.
———. 2009. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 116.
Aragón. 2012. Applied Epidemiology Using R.
Athreya, and Keiding. 1977. “Estimation Theory for Continuous-Time Branching Processes.” Sankhyā: The Indian Journal of Statistics, Series A (1961-2002).
Athreya, and Vidyashankar. 1997. “Large Deviation Rates for Supercritical and Critical Branching Processes.” In Classical and Modern Branching Processes. The IMA Volumes in Mathematics and Its Applications 84.
Bacry, Emmanuel, Bompaire, Gaïffas, et al. 2020. “Sparse and Low-Rank Multivariate Hawkes Processes.” Journal of Machine Learning Research.
Bacry, E., Dayri, and Muzy. 2012. “Non-Parametric Kernel Estimation for Symmetric Hawkes Processes. Application to High Frequency Financial Data.” The European Physical Journal B.
Bacry, E., Delattre, Hoffmann, et al. 2013a. “Modelling Microstructure Noise with Mutually Exciting Point Processes.” Quantitative Finance.
———, et al. 2013b. “Some Limit Theorems for Hawkes Processes and Application to Financial Statistics.” Stochastic Processes and Their Applications, A Special Issue on the Occasion of the 2013 International Year of Statistics,.
Bacry, Emmanuel, Jaisson, and Muzy. 2014. “Estimation of Slowly Decreasing Hawkes Kernels: Application to High Frequency Order Book Modelling.” arXiv:1412.7096 [q-Fin, Stat].
Bacry, Emmanuel, 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. 2007. “Spatial Point Processes and Their Applications.” In Stochastic Geometry. Lecture Notes in Mathematics 1892.
Barndorff-Nielsen, and Sørensen. 1994. “A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes.” International Statistical Review / Revue Internationale de Statistique.
Bauwens, and Hautsch. 2006. “Stochastic Conditional Intensity Processes.” Journal of Financial Econometrics.
Bertoin, Yor, and others. 2001. “On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable.” Electron. Comm. Probab.
Bhat, and Adke. 1981. “Maximum Likelihood Estimation for Branching Processes with Immigration.” Advances in Applied Probability.
Bhattacharjee. 1987. “The Time to Extinction of Branching Processes and Log-Convexity: I.” Probability in the Engineering and Informational Sciences.
Bibby, and Sørensen. 1995. “Martingale Estimation Functions for Discretely Observed Diffusion Processes.” Bernoulli.
Böckenholt. 1998. “Mixed INAR(1) Poisson Regression Models: Analyzing Heterogeneity and Serial Dependencies in Longitudinal Count Data.” Journal of Econometrics.
Bowman, and Shenton. 1989. “The Distribution of a Moment Estimator for a Parameter of the Generalized Poision Distribution.” Communications in Partial Differential Equations.
Bowsher. 2007. “Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models.” Journal of Econometrics.
Brown, and Hewitt. 1975. “Inference for the Diffusion Branching Process.” Journal of Applied Probability.
Burridge. 2013a. “Cascade Sizes in a Branching Process with Gamma Distributed Generations.” arXiv:1304.3741 [Math].
———. 2013b. “Crossover Behavior in Driven Cascades.” Physical Review E.
Caballero, M. E., and Chaumont. 2006. “Conditioned Stable Lévy Processes and the Lamperti Representation.” Journal of Applied Probability.
Caballero, M. Emilia, Garmendia, and Bravo. 2013. “A Lamperti-Type Representation of Continuous-State Branching Processes with Immigration.” The Annals of Probability.
Caballero, Maria-Emilia, Lambert, and Bravo. 2009. “Proof(s) of the Lamperti Representation of Continuous-State Branching Processes.” Probability Surveys.
Chen, and Hall. 2016. “Nonparametric Estimation for Self-Exciting Point Processes—A Parsimonious Approach.” Journal of Computational and Graphical Statistics.
Chistyakov. 1964. “A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes.” Theory of Probability & Its Applications.
Çinlar. 1975. “Exceptional Paper—Markov Renewal Theory: A Survey.” Management Science.
Cohn. 1997. “Stochastic Monotonicity and Branching Processes.” In Classical and Modern Branching Processes. The IMA Volumes in Mathematics and Its Applications 84.
Consul, P. C. 2014. “Lagrange and Related Probability Distributions.” In Wiley StatsRef: Statistics Reference Online.
Consul, P. C., and Famoye. 1992. “Generalized Poisson Regression Model.” Communications in Statistics - Theory and Methods.
———. 2006. Lagrangian Probability Distributions.
Consul, P. C., and Felix. 1989. “Minimum Variance Unbiased Estimation for the Lagrange Power Series Distributions.” Statistics.
Consul, P., and Shenton. 1972. “Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions.” SIAM Journal on Applied Mathematics.
Consul, P. C., and Shenton. 1973. “Some Interesting Properties of Lagrangian Distributions.” Communications in Statistics.
Consul, P.C., and Shoukri. 1984. “Maximum Likelihood Estimation for the Generalized Poisson Distribution.” Communications in Statistics - Theory and Methods.
Consul, P.C., and Shoukri. 1988. “Some Chance Mechanisms Related to a Generalized Poisson Probability Model.” American Journal of Mathematical and Management Sciences.
Crane, Schweitzer, and Sornette. 2010. “Power Law Signature of Media Exposure in Human Response Waiting Time Distributions.” Physical Review E.
Crisan, Del Moral, and Lyons. 1999. “Discrete Filtering Using Branching and Interacting Particle Systems.” Markov Processes and Related Fields.
Cui, and Lund. 2009. “A New Look at Time Series of Counts.” Biometrika.
Curien, and Le Gall. 2013. “The Brownian Plane.” Journal of Theoretical Probability.
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.
Dekking, and Speer. 1997. “On the Shape of the Wavefront of Branching Random Walk.” In Classical and Modern Branching Processes. The IMA Volumes in Mathematics and Its Applications 84.
Del Moral, and Miclo. 2000. “Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering.” In Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics 1729.
Deschâtres, and Sornette. 2005. “Dynamics of Book Sales: Endogenous Versus Exogenous Shocks in Complex Networks.” Physical Review E.
Doney, and Kyprianou. 2006. “Overshoots and Undershoots of Lévy Processes.” The Annals of Applied Probability.
Drost, Akker, and Werker. 2009. “Efficient Estimation of Auto-Regression Parameters and Innovation Distributions for Semiparametric Integer-Valued AR(p) Models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Duembgen, and Podolskij. 2015. “High-Frequency Asymptotics for Path-Dependent Functionals of Itô Semimartingales.” Stochastic Processes and Their Applications.
Du, Farajtabar, Ahmed, et al. 2015. “Dirichlet-Hawkes Processes with Applications to Clustering Continuous-Time Document Streams.” In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’15.
Dwass. 1969. “The Total Progeny in a Branching Process and a Related Random Walk.” Journal of Applied Probability.
Dynkin. 1991. “Branching Particle Systems and Superprocesses.” The Annals of Probability.
———. 2004. Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations. University Lecture Series, v. 34.
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.
Embrechts, Liniger, and Lin. 2011. “Multivariate Hawkes Processes: An Application to Financial Data.” Journal of Applied Probability.
Etheridge. 2000. An Introduction to Superprocesses. University Lecture Series, v. 20.
Evans. 2008. Probability and Real Trees. Lecture Notes in Mathematics 1920.
Falkner, and Teschl. 2012. “On the Substitution Rule for Lebesgue–Stieltjes Integrals.” Expositiones Mathematicae.
Feigin. 1976. “Maximum Likelihood Estimation for Continuous-Time Stochastic Processes.” Advances in Applied Probability.
Filimonov, Bicchetti, Maystre, et al. 2014. “Quantification of the High Level of Endogeneity and of Structural Regime Shifts in Commodity Markets.” Journal of International Money and Finance, Understanding International Commodity Price Fluctuations,.
Filimonov, Wheatley, and Sornette. 2015. “Effective Measure of Endogeneity for the Autoregressive Conditional Duration Point Processes via Mapping to the Self-Excited Hawkes Process.” Communications in Nonlinear Science and Numerical Simulation.
Fokianos. 2011. “Some Recent Progress in Count Time Series.” Statistics.
Freeland, and McCabe. 2004. “Analysis of Low Count Time Series Data by Poisson Autoregression.” Journal of Time Series Analysis.
Gehler, Holub, and Welling. 2006. “The Rate Adapting Poisson Model for Information Retrieval and Object Recognition.” In Proceedings of the 23rd International Conference on Machine Learning. ICML ’06.
Geiger, and Kauffmann. 2004. “The Shape of Large Galton-Watson Trees with Possibly Infinite Variance.” Random Struct. Algorithms.
Godoy, Solo, Min, et al. 2016. “Local Likelihood Estimation of Time-Variant Hawkes Models.” In.
Guttorp. 1991. Statistical Inference for Branching Processes. Wiley Series in Probability and Mathematical Statistics.
Haccou, Jagers, and Vatutin. 2005. Branching Processes: Variation, Growth, and Extinction of Populations.
Hall, Scotto, and Cruz. 2009. “Extremes of Integer-Valued Moving Average Sequences.” TEST.
Halpin, and Boeck. 2013. “Modelling Dyadic Interaction with Hawkes Processes.” Psychometrika.
Hansen, Reynaud-Bouret, and Rivoirard. 2015. “Lasso and Probabilistic Inequalities for Multivariate Point Processes.” Bernoulli.
Hardiman, Bercot, and Bouchaud. 2013. “Critical Reflexivity in Financial Markets: A Hawkes Process Analysis.” The European Physical Journal B.
Hardiman, and Bouchaud. 2014. “Branching-Ratio Approximation for the Self-Exciting Hawkes Process.” Physical Review E.
Hawkes. 1971. “Spectra of Some Self-Exciting and Mutually Exciting Point Processes.” Biometrika.
Hawkes, and Oakes. 1974. “A Cluster Process Representation of a Self-Exciting Process.” Journal of Applied Probability.
Heyde, and Seneta. 2010. “Estimation Theory for Growth and Immigration Rates in a Multiplicative Process.” In Selected Works of C.C. Heyde. Selected Works in Probability and Statistics.
Houdré. 2002. “Remarks on Deviation Inequalities for Functions of Infinitely Divisible Random Vectors.” The Annals of Probability.
Imoto. 2016. “Properties of Lagrangian Distributions.” Communications in Statistics - Theory and Methods.
Iribarren, and Moro. 2011. “Branching Dynamics of Viral Information Spreading.” Physical Review E.
Jacod. 1997. “On Continuous Conditional Gaussian Martingales and Stable Convergence in Law.” In Séminaire de Probabilités XXXI. Lecture Notes in Mathematics 1655.
Jacod, Podolskij, and Vetter. 2010. “Limit Theorems for Moving Averages of Discretized Processes Plus Noise.” The Annals of Statistics.
Jagers. 1969. “Renewal Theory and the Almost Sure Convergence of Branching Processes.” Arkiv För Matematik.
———. 1997. “Towards Dependence in General Branching Processes.” In Classical and Modern Branching Processes. The IMA Volumes in Mathematics and Its Applications 84.
János Engländer. 2007. “Branching Diffusions, Superdiffusions and Random Media.” Probability Surveys.
Jánossy, and Messel. 1950. “Fluctuations of the Electron-Photon Cascade - Moments of the Distribution.” Proceedings of the Physical Society. Section A.
———. 1951. “Investigation into the Higher Moments of a Nucleon Cascade.” Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences.
Kedem, and Fokianos. 2002. Regression models for time series analysis.
Keener. 2009. “Curved Exponential Families.” In Theoretical Statistics. Springer Texts in Statistics.
Kesten. 1973. “Random Difference Equations and Renewal Theory for Products of Random Matrices.” Acta Mathematica.
Kratz, and Pardoux. 2016. “Large Deviations for Infectious Diseases Models.” arXiv:1602.02803 [Math].
Kraus, and Panaretos. 2014. “Frequentist Estimation of an Epidemic’s Spreading Potential When Observations Are Scarce.” Biometrika.
Kvitkovičová, and Panaretos. 2011. “Asymptotic Inference for Partially Observed Branching Processes.” Advances in Applied Probability.
Lakshmanan, Sadtler, Tyler-Kabara, et al. 2015. “Extracting Low-Dimensional Latent Structure from Time Series in the Presence of Delays.” Neural Computation.
Lamperti. 1967a. “Continuous-State Branching Processes.” Bull. Amer. Math. Soc.
———. 1967b. “The Limit of a Sequence of Branching Processes.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete.
Laredo, David, and Garnier. 2009. “Inference for Partially Observed Multitype Branching Processes and Ecological Applications.” arXiv:0902.4520 [Stat].
Latour. 1998. “Existence and Stochastic Structure of a Non-Negative Integer-Valued Autoregressive Process.” Journal of Time Series Analysis.
Laub, Taimre, and Pollett. 2015. “Hawkes Processes.” arXiv:1507.02822 [Math, q-Fin, Stat].
Le Gall. 2005. “Random Trees and Applications.” Probability Surveys.
———. 2013. “Uniqueness and Universality of the Brownian Map.” The Annals of Probability.
Le Gall, and Miermont. 2012. “Scaling Limits of Random Trees and Planar Maps.” Probability and Statistical Physics in Two and More Dimensions.
Lee, Hopcraft, and Jakeman. 2008. “Continuous and Discrete Stable Processes.” Physical Review E.
Levina, and Herrmann. 2013. “The Abelian Distribution.” Stochastics and Dynamics.
Lewis, and Mohler. 2011. “A Nonparametric EM Algorithm for Multiscale Hawkes Processes.” Preprint.
Li, Zeng-Hu. 2000. “Asymptotic Behaviour of Continuous Time and State Branching Processes.” Journal of the Australian Mathematical Society (Series A).
Li, Zenghu. 2011. Measure-Valued Branching Markov Processes. Probability and Its Applications.
———. 2012. “Continuous-State Branching Processes.” arXiv:1202.3223 [Math].
———. 2014. “Path-Valued Branching Processes and Nonlocal Branching Superprocesses.” The Annals of Probability.
Li, S, Famoye, and Lee. 2010. “On the Generalized Lagrangian Probability Distributions.” Journal of Probability and Statistical Science.
Li, Yingying, and Mykland. 2007. “Are Volatility Estimators Robust with Respect to Modeling Assumptions?” Bernoulli.
Liniger. 2009. “Multivariate Hawkes Processes.”
Lyons. 1990. “Random Walks and Percolation on Trees.” The Annals of Probability.
Marsan, and Lengliné. 2008. “Extending Earthquakes’ Reach Through Cascading.” Science.
McKenzie, Ed. 1986. “Autoregressive Moving-Average Processes with Negative-Binomial and Geometric Marginal Distributions.” Advances in Applied Probability.
———. 1988. “Some ARMA Models for Dependent Sequences of Poisson Counts.” Advances in Applied Probability.
McKenzie, Eddie. 2003. “Discrete Variate Time Series.” In Handbook of Statistics. Stochastic Processes: Modelling and Simulation.
Meiners. 2009. “Weighted Branching and a Pathwise Renewal Equation.” Stochastic Processes and Their Applications.
Messel. 1952. “The Solution of the Fluctuation Problem in Nucleon Cascade Theory: Homogeneous Nuclear Matter.” Proceedings of the Physical Society. Section A.
Messel, and Potts. 1952. “Note on the Fluctuation Problem in Cascade Theory.” Proceedings of the Physical Society. Section A.
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.
Monteiro, Scotto, and Pereira. 2012. “Integer-Valued Self-Exciting Threshold Autoregressive Processes.” Communications in Statistics - Theory and Methods.
Mutafchiev. 1995. “Local Limit Approximations for Lagrangian Distributions.” Aequationes Mathematicae.
Nanthi, and Wasan. 1984. “Branching Processes.” Stochastic Processes and Their Applications.
Neuts. 1978. “Renewal Processes of Phase Type.” Naval Research Logistics Quarterly.
Neyman. 1965. “Certain Chance Mechanisms Involving Discrete Distributions.” Sankhyā: The Indian Journal of Statistics, Series A (1961-2002).
Nolan. 2001. “Maximum Likelihood Estimation and Diagnostics for Stable Distributions.” In Lévy Processes.
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, 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.
Olofsson. 2005. Probability, Statistics, and Stochastic Processes.
Otter. 1948. “The Number of Trees.” Annals of Mathematics.
———. 1949. “The Multiplicative Process.” The Annals of Mathematical Statistics.
Overbeck. 1998. “Estimation for Continuous Branching Processes.” Scandinavian Journal of Statistics.
Ozaki. 1979. “Maximum Likelihood Estimation of Hawkes’ Self-Exciting Point Processes.” Annals of the Institute of Statistical Mathematics.
Pakes. 1971a. “On the Critical Galton-Watson Process with Immigration.” Journal of the Australian Mathematical Society.
———. 1971b. “On a Theorem of Quine and Seneta for the Galton-Watson Process With Immigration.” Australian Journal of Statistics.
Pardoux, and Samegni-Kepgnou. 2016. “Large Deviation Principle for Poisson Driven SDEs in Epidemic Models.” arXiv:1606.01619 [Math].
———. 2017. “Large Deviation Principle for Epidemic Models.” Journal of Applied Probability.
Pazsit. 1987. “Note on the Calculation of the Variance in Linear Collision Cascades.” Journal of Physics D: Applied Physics.
Pinto, and Chahed. 2014. “Modeling Multi-Topic Information Diffusion in Social Networks Using Latent Dirichlet Allocation and Hawkes Processes.” In Proceedings of the 2014 Tenth International Conference on Signal-Image Technology and Internet-Based Systems. SITIS ’14.
Podolskij, and Vetter. 2010. “Understanding Limit Theorems for Semimartingales: A Short Survey: Limit Theorems for Semimartingales.” Statistica Neerlandica.
Ramakrishnan, and Srinivasan. 1956. “A New Approach to the Cascade Theory.” In Proceedings of the Indian Academy of Sciences-Section A.
Rasmussen, and Williams. 2006. Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning.
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.
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.
Saichev, A., Helmstetter, and Sornette. 2005. “Power-Law Distributions of Offspring and Generation Numbers in Branching Models of Earthquake Triggering.” Pure and Applied Geophysics.
Saichev, A., Malevergne, and Sornette. 2008. “Theory of Zipf’s Law and of General Power Law Distributions with Gibrat’s Law of Proportional Growth.” arXiv:0808.1828 [Physics, q-Fin].
Saichev, A. I., and Sornette. 2010. “Generation-by-Generation Dissection of the Response Function in Long Memory Epidemic Processes.” The European Physical Journal B.
Saichev, A., and Sornette. 2011a. “Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.1611 [Cond-Mat, Physics:physics].
———. 2011b. “Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.5564 [Cond-Mat, Physics:physics].
Sandkühler, and Eblen-Zajjur. 1994. “Identification and Characterization of Rhythmic Nociceptive and Non-Nociceptive Spinal Dorsal Horn Neurons in the Rat.” Neuroscience.
Sevast’yanov. 1968. “Renewal Equations and Moments of Branching Processes.” Mathematical Notes of the Academy of Sciences of the USSR.
Shoukri, and Consul. 1987. “Some Chance Mechanisms Generating the Generalized Poisson Probability Models.” In Biostatistics.
Sibuya, Miyawaki, and Sumita. 1994. “Aspects of Lagrangian Probability Distributions.” Journal of Applied Probability.
Soltani, Shirvani, and Alqallaf. 2009. “A Class of Discrete Distributions Induced by Stable Laws.” Statistics & Probability Letters.
Sood, Mathieu, Shreim, et al. 2010. “Interacting Branching Process as a Simple Model of Innovation.” Physical Review Letters.
Sornette, Didier. 2006. “Endogenous Versus Exogenous Origins of Crises.” In Extreme Events in Nature and Society. The Frontiers Collection.
Sornette, Didier, Deschâtres, Gilbert, et al. 2004. “Endogenous Versus Exogenous Shocks in Complex Networks: An Empirical Test Using Book Sale Rankings.” Physical Review Letters.
Sornette, D, and Helmstetter. 2003. “Endogenous Versus Exogenous Shocks in Systems with Memory.” Physica A: Statistical Mechanics and Its Applications.
Sornette, D., Malevergne, and Muzy. 2004. “Volatility Fingerprints of Large Shocks: Endogenous Versus Exogenous.” In The Application of Econophysics.
Sornette, D., and Utkin. 2009. “Limits of Declustering Methods for Disentangling Exogenous from Endogenous Events in Time Series with Foreshocks, Main Shocks, and Aftershocks.” Physical Review E.
Steutel, and van Harn. 1979. “Discrete Analogues of Self-Decomposability and Stability.” The Annals of Probability.
Turkman, Scotto, and Bermudez. 2014. “Models for Integer-Valued Time Series.” In Non-Linear Time Series.
Unser, and Tafti. 2014. An Introduction to Sparse Stochastic Processes.
van Harn, and Steutel. 1993. “Stability Equations for Processes with Stationary Independent Increments Using Branching Processes and Poisson Mixtures.” Stochastic Processes and Their Applications.
van Harn, Steutel, and Vervaat. 1982. “Self-Decomposable Discrete Distributions and Branching Processes.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete.
Veen, and Schoenberg. 2008. “Estimation of Space–Time Branching Process Models in Seismology Using an EM–Type Algorithm.” Journal of the American Statistical Association.
Watanabe. 1968. “A Limit Theorem of Branching Processes and Continuous State Branching Processes.” Journal of Mathematics of Kyoto University.
Weiner. 1965. “An Integral Equation in Age Dependent Branching Processes.” The Annals of Mathematical Statistics.
Weiß. 2008. “Thinning Operations for Modeling Time Series of Counts—a Survey.” Advances in Statistical Analysis.
———. 2009. “A New Class of Autoregressive Models for Time Series of Binomial Counts.” Communications in Statistics - Theory and Methods.
Wei, and Winnicki. 1990. “Estimation of the Means in the Branching Process with Immigration.” The Annals of Statistics.
Wheatley. 2013. “Quantifying Endogeneity in Market Prices with Point Processes: Methods & Applications.” Masters Thesis.
Winnicki. 1991. “Estimation of the Variances in the Branching Process with Immigration.” Probability Theory and Related Fields.
Yaari, Nowak, Rakocy, et al. 2008. “Microscopic Study Reveals the Singular Origins of Growth.” The European Physical Journal B.
Yang, and Zha. 2013. “Mixture of Mutually Exciting Processes for Viral Diffusion.” In Proceedings of The 30th International Conference on Machine Learning.
Zeger. 1988. “A Regression Model for Time Series of Counts.” Biometrika.
Zeger, and Qaqish. 1988. “Markov Regression Models for Time Series: A Quasi-Likelihood Approach.” Biometrics.
Zhao, and Singer. 2013. “Fourier–Bessel Rotational Invariant Eigenimages.” Journal of the Optical Society of America A.
Zheng, and Basawa. 2008. “First-Order Observation-Driven Integer-Valued Autoregressive Processes.” Statistics & Probability Letters.
Zheng, Basawa, and Datta. 2007. “First-Order Random Coefficient Integer-Valued Autoregressive Processes.” Journal of Statistical Planning and Inference.
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).