# Cluster models

### a.k.a. cascade distributions, Galton-Watson models

Usefulness: 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

A type of count model for a Markov stochastic pure-birth branching process in discrete time. They can be generalised into non-Markov processes or into, say, or miscellaneous other types of branching process.

This needs a better intro, but the Galton-Watson process is the one here.

There are many standard expositions. Two good ones:

• Gesine Reinert’s Introduction to Branching Processes: Parts 1 and 2.

• Steven Lalley’s intro.

The distribution of subcritical processes are sometimes tedious to calculate, although we can get a nice form for the generating function for a geometric offspring distribution.

Set $$\frac{1}{\lambda+1}=p$$ and $$q=1-p$$. We write $$G^n\equiv G\cdot G\cdot \dots \cdot G\cdot G$$ for the $$n$$-fold composition of $$G$$. Then the (non-critical) geometric offspring distribution branching process obeys the identity

$1-G^n(s;\lambda) = \frac{\lambda^n(\lambda-1)(1-s)}{\lambda(\lambda^n-1)(1-s)+\lambda-1}$

This can get us a formula for the first two factorial moments, and hence the mean and variance.

More generally the machinery of Lagrangian distributions is all we need.

Maybe I should use (???) to get the moments? Dominic Yeo has a great explanation as always.

🚧 🚧 🚧

# Refs

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Bowman, K. O., and L. R. Shenton. 1989. “The Distribution of a Moment Estimator for a Parameter of the Generalized Poision Distribution.” Communications in Partial Differential Equations 14 (4): 867–93. https://doi.org/10.1080/03605308908820611.

Burridge, James. 2013. “Cascade Sizes in a Branching Process with Gamma Distributed Generations,” April. http://arxiv.org/abs/1304.3741.

Consul, P. C. 2014. “Lagrange and Related Probability Distributions.” In Wiley StatsRef: Statistics Reference Online. John Wiley & Sons, Ltd. http://onlinelibrary.wiley.com/doi/10.1002/9781118445112.stat01038/abstract.

Consul, P. C., and Felix Famoye. 2006. Lagrangian Probability Distributions. Boston: Birkhäuser. https://books.google.com/books/about/Lagrangian_Probability_Distributions.html?id=UGbkIwT63qsC.

———. 1992. “Generalized Poisson Regression Model.” Communications in Statistics - Theory and Methods 21 (1): 89–109. https://doi.org/10.1080/03610929208830766.

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———. 1984. “Maximum Likelihood Estimation for the Generalized Poisson Distribution.” Communications in Statistics - Theory and Methods 13 (12): 1533–47. https://doi.org/10.1080/03610928408828776.

Consul, P., and L. Shenton. 1972. “Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions.” SIAM Journal on Applied Mathematics 23 (2): 239–48. https://doi.org/10.1137/0123026.

Houdré, Christian. 2002. “Remarks on Deviation Inequalities for Functions of Infinitely Divisible Random Vectors.” The Annals of Probability 30 (3): 1223–37. https://doi.org/10.1214/aop/1029867126.

Imoto, Tomoaki. 2016. “Properties of Lagrangian Distributions.” Communications in Statistics - Theory and Methods 45 (3): 712–21. https://doi.org/10.1080/03610926.2013.835414.

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———. 1951. “Investigation into the Higher Moments of a Nucleon Cascade.” Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences 54: 245–62. http://www.jstor.org/stable/20488536.

Li, S, F Famoye, and C Lee. 2010. “On the Generalized Lagrangian Probability Distributions.” Journal of Probability and Statistical Science 8 (1): 113–23.

Messel, H. 1952. “The Solution of the Fluctuation Problem in Nucleon Cascade Theory: Homogeneous Nuclear Matter.” Proceedings of the Physical Society. Section A 65 (7): 465. https://doi.org/10.1088/0370-1298/65/7/301.

Messel, H., and R. B. Potts. 1952. “Note on the Fluctuation Problem in Cascade Theory.” Proceedings of the Physical Society. Section A 65 (10): 854. https://doi.org/10.1088/0370-1298/65/10/310.

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.

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Neyman, Jerzy. 1965. “Certain Chance Mechanisms Involving Discrete Distributions.” Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 27 (2/4): 249–58. http://www.jstor.org/stable/25049369.

Otter, Richard. 1948. “The Number of Trees.” Annals of Mathematics 49 (3): 583–99. https://doi.org/10.2307/1969046.

———. 1949. “The Multiplicative Process.” The Annals of Mathematical Statistics 20 (2): 206–24. https://doi.org/10.1214/aoms/1177730031.

Pardoux, Etienne, and Brice Samegni-Kepgnou. 2017. “Large Deviation Principle for Epidemic Models.” Journal of Applied Probability 54 (3): 905–20. https://doi.org/10.1017/jpr.2017.41.

Pazsit, I. 1987. “Note on the Calculation of the Variance in Linear Collision Cascades.” Journal of Physics D: Applied Physics 20 (2): 151. https://doi.org/10.1088/0022-3727/20/2/001.

Ramakrishnan, Alladi, and S. K. Srinivasan. 1956. “A New Approach to the Cascade Theory.” In Proceedings of the Indian Academy of Sciences-Section A, 44:263–73. Springer. http://link.springer.com/article/10.1007/BF03046051#page-1.

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.

Shoukri, M. M., and P. C. Consul. 1987. “Some Chance Mechanisms Generating the Generalized Poisson Probability Models.” In Biostatistics, edited by Ian B. MacNeill, Gary J. Umphrey, Allan Donner, and V. Krishna Jandhyala, 259–68. Dordrecht: Springer Netherlands. http://link.springer.com/10.1007/978-94-009-4794-8_15.

Sibuya, Masaaki, Norihiko Miyawaki, and Ushio Sumita. 1994. “Aspects of Lagrangian Probability Distributions.” Journal of Applied Probability 31: 185–97. https://doi.org/10.2307/3214956.