# Branching processes

August 18, 2014 — February 7, 2020

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

`tscount`

one may find`campy`

,`ecoli`

,`ehec`

,`influenza`

and`measles`

.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

*The Annals of Probability*.

*The Annals of Probability*.

*Annales de l’Institut Henri Poincare (B) Probability and Statistics*.

*Communications in Statistics - Theory and Methods*.

*Journal of Time Series Analysis*.

*International Journal of Mathematics and Mathematical Sciences*.

*Statistica Neerlandica*.

*Notices of the AMS*.

*Lévy Processes and Stochastic Calculus*. Cambridge Studies in Advanced Mathematics 116.

*Applied Epidemiology Using R*.

*Sankhyā: The Indian Journal of Statistics, Series A (1961-2002)*.

*Classical and Modern Branching Processes*. The IMA Volumes in Mathematics and Its Applications 84.

*Journal of Machine Learning Research*.

*The European Physical Journal B*.

*Quantitative Finance*.

*Stochastic Processes and Their Applications*, A Special Issue on the Occasion of the 2013 International Year of Statistics,.

*arXiv:1412.7096 [q-Fin, Stat]*.

*Quantitative Finance*.

*IEEE Transactions on Information Theory*.

*Stochastic Geometry*. Lecture Notes in Mathematics 1892.

*International Statistical Review / Revue Internationale de Statistique*.

*Journal of Financial Econometrics*.

*Electron. Comm. Probab*.

*Advances in Applied Probability*.

*Probability in the Engineering and Informational Sciences*.

*Bernoulli*.

*Journal of Econometrics*.

*Communications in Partial Differential Equations*.

*Journal of Econometrics*.

*Journal of Applied Probability*.

*arXiv:1304.3741 [Math]*.

*Physical Review E*.

*Journal of Applied Probability*.

*The Annals of Probability*.

*Probability Surveys*.

*Journal of Computational and Graphical Statistics*.

*Theory of Probability & Its Applications*.

*Management Science*.

*Classical and Modern Branching Processes*. The IMA Volumes in Mathematics and Its Applications 84.

*Wiley StatsRef: Statistics Reference Online*.

*Communications in Statistics - Theory and Methods*.

*Lagrangian Probability Distributions*.

*Statistics*.

*SIAM Journal on Applied Mathematics*.

*Communications in Statistics*.

*Communications in Statistics - Theory and Methods*.

*American Journal of Mathematical and Management Sciences*.

*Physical Review E*.

*Markov Processes and Related Fields*.

*Biometrika*.

*Journal of Theoretical Probability*.

*An introduction to the theory of point processes*.

*An Introduction to the Theory of Point Processes*. Probability and Its Applications.

*Classical and Modern Branching Processes*. The IMA Volumes in Mathematics and Its Applications 84.

*Séminaire de Probabilités XXXIV*. Lecture Notes in Mathematics 1729.

*Physical Review E*.

*The Annals of Applied Probability*.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

*Stochastic Processes and Their Applications*.

*Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining*. KDD ’15.

*Journal of Applied Probability*.

*The Annals of Probability*.

*Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations*. University Lecture Series, v. 34.

*Neural Computation*.

*Journal of Time Series Analysis*.

*Journal of Applied Probability*.

*An Introduction to Superprocesses*. University Lecture Series, v. 20.

*Probability and Real Trees*. Lecture Notes in Mathematics 1920.

*Expositiones Mathematicae*.

*Advances in Applied Probability*.

*Journal of International Money and Finance*, Understanding International Commodity Price Fluctuations,.

*Communications in Nonlinear Science and Numerical Simulation*.

*Statistics*.

*Journal of Time Series Analysis*.

*Proceedings of the 23rd International Conference on Machine Learning*. ICML ’06.

*Random Struct. Algorithms*.

*Statistical Inference for Branching Processes*. Wiley Series in Probability and Mathematical Statistics.

*Branching Processes: Variation, Growth, and Extinction of Populations*.

*TEST*.

*Psychometrika*.

*Bernoulli*.

*The European Physical Journal B*.

*Physical Review E*.

*Biometrika*.

*Journal of Applied Probability*.

*Selected Works of C.C. Heyde*. Selected Works in Probability and Statistics.

*The Annals of Probability*.

*Communications in Statistics - Theory and Methods*.

*Physical Review E*.

*Séminaire de Probabilités XXXI*. Lecture Notes in Mathematics 1655.

*The Annals of Statistics*.

*Arkiv För Matematik*.

*Classical and Modern Branching Processes*. The IMA Volumes in Mathematics and Its Applications 84.

*Probability Surveys*.

*Proceedings of the Physical Society. Section A*.

*Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences*.

*Regression models for time series analysis*.

*Theoretical Statistics*. Springer Texts in Statistics.

*Acta Mathematica*.

*arXiv:1602.02803 [Math]*.

*Biometrika*.

*Advances in Applied Probability*.

*Neural Computation*.

*Bull. Amer. Math. Soc*.

*Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete*.

*arXiv:0902.4520 [Stat]*.

*Journal of Time Series Analysis*.

*arXiv:1507.02822 [Math, q-Fin, Stat]*.

*Probability Surveys*.

*The Annals of Probability*.

*Probability and Statistical Physics in Two and More Dimensions*.

*Physical Review E*.

*Stochastics and Dynamics*.

*Preprint*.

*Journal of the Australian Mathematical Society (Series A)*.

*Measure-Valued Branching Markov Processes*. Probability and Its Applications.

*arXiv:1202.3223 [Math]*.

*The Annals of Probability*.

*Journal of Probability and Statistical Science*.

*Bernoulli*.

*The Annals of Probability*.

*Science*.

*Advances in Applied Probability*.

*Advances in Applied Probability*.

*Handbook of Statistics*. Stochastic Processes: Modelling and Simulation.

*Stochastic Processes and Their Applications*.

*Proceedings of the Physical Society. Section A*.

*Proceedings of the Physical Society. Section A*.

*Proceedings of the 25th ACM International Conference on Information and Knowledge Management*. CIKM ’16.

*Journal of the American Statistical Association*.

*Communications in Statistics - Theory and Methods*.

*Aequationes Mathematicae*.

*Stochastic Processes and Their Applications*.

*Naval Research Logistics Quarterly*.

*Sankhyā: The Indian Journal of Statistics, Series A (1961-2002)*.

*Lévy Processes*.

*Journal of Applied Probability*.

*Annals of the Institute of Statistical Mathematics*.

*Journal of the American Statistical Association*.

*Pure and Applied Geophysics*.

*Journal of the Royal Statistical Society, Series B*.

*Probability, Statistics, and Stochastic Processes*.

*Annals of Mathematics*.

*The Annals of Mathematical Statistics*.

*Scandinavian Journal of Statistics*.

*Annals of the Institute of Statistical Mathematics*.

*Journal of the Australian Mathematical Society*.

*Australian Journal of Statistics*.

*arXiv:1606.01619 [Math]*.

*Journal of Applied Probability*.

*Journal of Physics D: Applied Physics*.

*Proceedings of the 2014 Tenth International Conference on Signal-Image Technology and Internet-Based Systems*. SITIS ’14.

*Statistica Neerlandica*.

*Proceedings of the Indian Academy of Sciences-Section A*.

*Gaussian Processes for Machine Learning*. Adaptive Computation and Machine Learning.

*The Journal of Mathematical Neuroscience*.

*Bulletin of the Belgian Mathematical Society - Simon Stevin*.

*The Annals of Statistics*.

*World Wide Web 2017, International Conference on*. WWW ’17.

*Pure and Applied Geophysics*.

*arXiv:0808.1828 [Physics, q-Fin]*.

*The European Physical Journal B*.

*arXiv:1101.1611 [Cond-Mat, Physics:physics]*.

*arXiv:1101.5564 [Cond-Mat, Physics:physics]*.

*Neuroscience*.

*Mathematical Notes of the Academy of Sciences of the USSR*.

*Biostatistics*.

*Journal of Applied Probability*.

*Statistics & Probability Letters*.

*Physical Review Letters*.

*Extreme Events in Nature and Society*. The Frontiers Collection.

*Physical Review Letters*.

*Physica A: Statistical Mechanics and Its Applications*.

*The Application of Econophysics*.

*Physical Review E*.

*The Annals of Probability*.

*Non-Linear Time Series*.

*An Introduction to Sparse Stochastic Processes*.

*Stochastic Processes and Their Applications*.

*Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete*.

*Journal of the American Statistical Association*.

*Journal of Mathematics of Kyoto University*.

*The Annals of Mathematical Statistics*.

*Advances in Statistical Analysis*.

*Communications in Statistics - Theory and Methods*.

*The Annals of Statistics*.

*Probability Theory and Related Fields*.

*The European Physical Journal B*.

*Proceedings of The 30th International Conference on Machine Learning*.

*Biometrika*.

*Biometrics*.

*Journal of the Optical Society of America A*.

*Statistics & Probability Letters*.

*Journal of Statistical Planning and Inference*.

*Proceedings of the 30th International Conference on Machine Learning (ICML-13)*.