Generalized Galton-Watson processes

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

There are many standard expositions. Two good ones:

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

  • Steven Lalley’s intro.

Working through some generalisations of the Galton-Watson process as an INAR process. That is, this is something like the Galton-Watson process, but

Consider

  • van Harn & Steutel’s work on “F-stable branching processes.” Also bounded influence kernel?

  • Lee, Hopcraft, Jakeman and Williams on discrete stable processes. Discrete state, continuous time - How do these differ from the usual Hawkes processes, if at all?

Long Memory Galton-Watson

For my own edification and amusement I would like to walk through the construction of a particular analogue of the continuous time Hawkes point process on a discrete index set.

Specifically, a non-Markovian generalisation of the Galton-Watson process which still operates in quantised time, but has interesting, possibly-unbounded influence kernels, like the Hawkes process.

I denote a realisation of the process \(\{N_t\}_{t\in\mathbb{N}}\). and the associated non-negative increment process \(\{X_t\}\equiv\{N_t-N_{t-1}\}\) and a conditional non-negative pseudo-intensity process \(\lambda_t\equiv g(\{N_s\}_{s \lt t})\), adapted to the whole history \(\{N_s\}_{s \lt t}\). By “pseudo-intensity” I mean that the innovation law \(X_t\sim\mathcal{L}_t\) is parameterised (solely, for now) by some scalar-valued process \(\lambda_t(\mathcal{F}(X_t))\). That is, \(\{X_t\}|\{N_s\}_{s \lt t}\sim \mathcal{L}(\lambda_t)\). For the moment I will take this be be Poisson. To complete the analogy with the Hawkes process I choose the dependence on the past values of the process linear with influence kernel \(\phi\): This is also close to clustering, and indeed there are lots of papers noticing the connection.

\[ \lambda_t\equiv \phi * X \]

Then a linear conditional intensity process \(\lambda_t\) would be

\[ \lambda_t := \mu + \eta\sum_{0 \leq s <t} \phi(s-t-1)N_s \]

The \(-1\) in \(\phi(s-t-1)\) is to make sure our influence kernel is defined on \(\mathbb{N}_0\), which is convenient for typical count distribution functions.

If the kernel has bounded support such that

\[ s>p\Rightarrow\phi(s)=0 \]

then we have an autoregressive count process of order p. More on that in a moment.

What influence kernel shape will we use?

Geometric distributions are natural, although it doesn’t have to be strictly monotonic, or even unimodal. Poisson or negative binomial would also work. We could in general give any arbitrary probability mass function as influence kernel, or use a nonparametric form.

\[ \phi_\text{Exp}(i) = \sum_{0 \leq k <K} b_ke^{a_ki} \]

for some \(\{a_k, b_k\}\).

If we expect to be using sparsifying lasso penalties for such a kernel we probably want to decompose the kernel in a way that minimises correlation between mixture components to improve our odds of correctly identifying dependency at different scales. If we constrain our distributions to be positive the only way to do this is for them to be completely orthogonal is to have disjoint support.

Intermediately, we could choose a Poisson mixture

\[ \phi_\text{Pois}(i) = \sum_{0 \leq k <K} \frac{a_k^i}{i!} e^{-a_k} \]

There is a subtlety here with regard to the filtration - do we set up the kernel strictly to regard triggering events at previous timesteps? If so, no problem. If we want to allow same-day triggering, we might allow the exogenous events to also contribute to the kernel, in which case we might have to estimate an extra influence parameter, or find some principled way to include it in the kernel weights.

🏗 unconditional distribution using, e.g. generator fns.

Autoregressive characterisation

Steutel and van Harn characterised this process in 1979 - see (Steutel and van Harn 1979) (Wait - is this strictly true, that we can make this go with a thinning operator? Many related definitions here, muddying the waters)

We need their binomial thinning operator \(\odot\), which is defined for some count RV \(X\) by

\[ \alpha\odot X = \sum_{i=1}^X N_i \]

for \(N_i\) independent \(\operatorname{Bernoulli}(\alpha)\) RVs.

In terms of generating functions,

\(G_{\alpha\odot X}(s)=G_{X}(1-\alpha+\alpha s)\)

There are many generalisation of this operator - see (Weiß 2008) for an overview.

Anyway, you can use this thinning operator to construct an autoregressive time series model driven by thinned versions of its history.

(Maybe it would be simpler to use Fokkianos’ GLM characterisation? I think they are equivalent or nearly equivalent in ths case - certainly with stable distributions they are.)

Estimation of parameters

Well studied for finite-order GINAR(p) processes.

Influence kernels

Hardiman et al propose multiple-scale exponential kernels to simultaneously estimate decays and branching ratios Bacry et al 2012 have a related nonparametric method based on estimating the kernel in the spectral domain. Convergence properties are unclear.

We are also free to use a sum-of-exponentials kernel, possibly calculating the branching ratio from that alone, and some measure of tail-heaviness from that.

Possibly Smooth-lasso (penalises component CHANGE)

Endo-exo models

Note that we can still recover the endo-exo model with this by simply calculating the projected ratio between exogenous and endogenous events. It would be interesting to derive the properties of this as a single parameter of interest.

Al-Osh, M. A., and A. A. Alzaid. 1987. “First-Order Integer-Valued Autoregressive (INAR(1)) Process.” Journal of Time Series Analysis 8 (3): 261–75. https://doi.org/10.1111/j.1467-9892.1987.tb00438.x.

Al-Osh, Mohamed A., and Emad-Eldin A. A. Aly. 1992. “First Order Autoregressive Time Series with Negative Binomial and Geometric Marginals.” Communications in Statistics - Theory and Methods 21 (9): 2483–92. https://doi.org/10.1080/03610929208830925.

Aly, Emad-Eldin A. A., and Nadjib Bouzar. 2005. “Stationary Solutions for Integer-Valued Autoregressive Processes.” International Journal of Mathematics and Mathematical Sciences 2005 (1): 1–18. https://doi.org/10.1155/IJMMS.2005.1.

Alzaid, A., and M. Al-Osh. 1988. “First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties.” Statistica Neerlandica 42 (1): 53–61. https://doi.org/10.1111/j.1467-9574.1988.tb01521.x.

Aragón, Tomás J. 2012. Applied Epidemiology Using R. MedEpi Publishing. http://www. medepi. net/epir/index. html. Calendar Time. Accessed. http://www.medepi.net/docs/EpidemiologyUsingR.pdf.

Barndorff-Nielsen, O. E., and M. Sørensen. 1994. “A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes.” International Statistical Review / Revue Internationale de Statistique 62 (1): 133–65. https://doi.org/10.2307/1403550.

Bhat, B. R., and S. R. Adke. 1981. “Maximum Likelihood Estimation for Branching Processes with Immigration.” Advances in Applied Probability 13 (3): 498–509. https://doi.org/10.2307/1426782.

Bhattacharjee, M. C. 1987. “The Time to Extinction of Branching Processes and Log-Convexity: I.” Probability in the Engineering and Informational Sciences 1 (03): 265–78. https://doi.org/10.1017/S0269964800000048.

Bibby, Bo Martin, and Michael Sørensen. 1995. “Martingale Estimation Functions for Discretely Observed Diffusion Processes.” Bernoulli 1 (1/2): 17–39. https://doi.org/10.2307/3318679.

Böckenholt, Ulf. 1998. “Mixed INAR(1) Poisson Regression Models: Analyzing Heterogeneity and Serial Dependencies in Longitudinal Count Data.” Journal of Econometrics 89 (1–2): 317–38. https://doi.org/10.1016/S0304-4076(98)00069-4.

Cui, Yunwei, and Robert Lund. 2009. “A New Look at Time Series of Counts.” Biometrika 96 (4). Oxford Academic: 781–92. https://doi.org/10.1093/biomet/asp057.

Drost, Feike C., Ramon van den Akker, and Bas J. M. 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) 71 (2): 467–85. https://doi.org/10.1111/j.1467-9868.2008.00687.x.

Dwass, Meyer. 1969. “The Total Progeny in a Branching Process and a Related Random Walk.” Journal of Applied Probability 6 (3): 682–86. https://doi.org/10.2307/3212112.

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–n/a. https://doi.org/10.1111/jtsa.12213.

Fokianos, Konstantinos. 2011. “Some Recent Progress in Count Time Series.” Statistics 45 (1): 49–58. https://doi.org/10.1080/02331888.2010.541250.

Freeland, R. K., and B. P. M. McCabe. 2004. “Analysis of Low Count Time Series Data by Poisson Autoregression.” Journal of Time Series Analysis 25 (5): 701–22. https://doi.org/10.1111/j.1467-9892.2004.01885.x.

Gehler, Peter V., Alex D. Holub, and Max Welling. 2006. “The Rate Adapting Poisson Model for Information Retrieval and Object Recognition.” In Proceedings of the 23rd International Conference on Machine Learning, 337–44. ICML ’06. New York, NY, USA: ACM. https://doi.org/10.1145/1143844.1143887.

Geiger, Jochen, and Lars Kauffmann. 2004. “The Shape of Large Galton-Watson Trees with Possibly Infinite Variance.” Random Struct. Algorithms 25 (3): 311–35. https://doi.org/10.1002/rsa.20021.

Hall, Andreia, Manuel Scotto, and João Cruz. 2009. “Extremes of Integer-Valued Moving Average Sequences.” TEST 19 (2): 359–74. https://doi.org/10.1007/s11749-009-0158-6.

Harn, K. van, and F. W. Steutel. 1993. “Stability Equations for Processes with Stationary Independent Increments Using Branching Processes and Poisson Mixtures.” Stochastic Processes and Their Applications 45 (2): 209–30. https://doi.org/10.1016/0304-4149(93)90070-K.

Harn, K. van, F. W. Steutel, and W. Vervaat. 1982. “Self-Decomposable Discrete Distributions and Branching Processes.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 61 (1): 97–118. https://doi.org/10.1007/BF00537228.

Hawkes, Alan G., and David Oakes. 1974. “A Cluster Process Representation of a Self-Exciting Process.” Journal of Applied Probability 11 (3): 493. https://doi.org/10.2307/3212693.

Kedem, Benjamin, and Konstantinos Fokianos. 2002. Regression Models for Time Series Analysis. Chichester; Hoboken, NJ: John Wiley & Sons. http://site.ebrary.com/id/10274729.

Kratz, Peter, and Etienne Pardoux. 2016. “Large Deviations for Infectious Diseases Models,” February. http://arxiv.org/abs/1602.02803.

Kraus, Andrea, and Victor M. Panaretos. 2014. “Frequentist Estimation of an Epidemic’s Spreading Potential When Observations Are Scarce.” Biometrika 101 (1): 141–54. https://doi.org/10.1093/biomet/ast049.

Kvitkovičová, Andrea, and Victor M. Panaretos. 2011. “Asymptotic Inference for Partially Observed Branching Processes.” Advances in Applied Probability 43 (4): 1166–90. https://doi.org/10.1239/aap/1324045703.

Laredo, Catherine, Olivier David, and Aurélie Garnier. 2009. “Inference for Partially Observed Multitype Branching Processes and Ecological Applications,” February. http://arxiv.org/abs/0902.4520.

Latour, Alain. 1998. “Existence and Stochastic Structure of a Non-Negative Integer-Valued Autoregressive Process.” Journal of Time Series Analysis 19 (4): 439–55. https://doi.org/10.1111/1467-9892.00102.

Lee, W. H., K. I. Hopcraft, and E. Jakeman. 2008. “Continuous and Discrete Stable Processes.” Physical Review E 77 (1): 011109. https://doi.org/10.1103/PhysRevE.77.011109.

McKenzie, Ed. 1986. “Autoregressive Moving-Average Processes with Negative-Binomial and Geometric Marginal Distributions.” Advances in Applied Probability 18 (3): 679–705. https://doi.org/10.2307/1427183.

———. 1988. “Some ARMA Models for Dependent Sequences of Poisson Counts.” Advances in Applied Probability 20 (4): 822–35. https://doi.org/10.2307/1427362.

McKenzie, Eddie. 2003. “Discrete Variate Time Series.” In Handbook of Statistics, edited by c Raoand and d Shanbhag, 21:573–606. Stochastic Processes: Modelling and Simulation. Elsevier. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.485.8657.

Monteiro, Magda, Manuel G. Scotto, and Isabel Pereira. 2012. “Integer-Valued Self-Exciting Threshold Autoregressive Processes.” Communications in Statistics - Theory and Methods 41 (15): 2717–37. https://doi.org/10.1080/03610926.2011.556292.

Nanthi, K., and M.T. Wasan. 1984. “Branching Processes.” Stochastic Processes and Their Applications 18 (2): 189. https://doi.org/10.1016/0304-4149(84)90183-2.

Pardoux, Etienne, and Brice Samegni-Kepgnou. 2016. “Large Deviation Principle for Poisson Driven SDEs in Epidemic Models,” June. http://arxiv.org/abs/1606.01619.

———. 2017. “Large Deviation Principle for Epidemic Models.” Journal of Applied Probability 54 (3): 905–20. https://doi.org/10.1017/jpr.2017.41.

Sandkühler, J., and A. A. Eblen-Zajjur. 1994. “Identification and Characterization of Rhythmic Nociceptive and Non-Nociceptive Spinal Dorsal Horn Neurons in the Rat.” Neuroscience 61 (4): 991–1006. https://doi.org/10.1016/0306-4522(94)90419-7.

Soltani, A. R., A. Shirvani, and F. Alqallaf. 2009. “A Class of Discrete Distributions Induced by Stable Laws.” Statistics & Probability Letters 79 (14): 1608–14. https://doi.org/10.1016/j.spl.2009.03.030.

Steutel, F. W., and K. van Harn. 1979. “Discrete Analogues of Self-Decomposability and Stability.” The Annals of Probability 7 (5): 893–99. https://doi.org/10.1214/aop/1176994950.

Turkman, Kamil Feridun, Manuel González Scotto, and Patrícia de Zea Bermudez. 2014. “Models for Integer-Valued Time Series.” In Non-Linear Time Series, 199–244. Springer International Publishing.

Wei, C. Z., and J. Winnicki. 1990. “Estimation of the Means in the Branching Process with Immigration.” The Annals of Statistics 18 (4): 1757–73. https://doi.org/10.1214/aos/1176347876.

Weiß, Christian H. 2008. “Thinning Operations for Modeling Time Series of Counts—a Survey.” Advances in Statistical Analysis 92 (3): 319–41. https://doi.org/10.1007/s10182-008-0072-3.

———. 2009. “A New Class of Autoregressive Models for Time Series of Binomial Counts.” Communications in Statistics - Theory and Methods 38 (4): 447–60. https://doi.org/10.1080/03610920802233937.

Winnicki, J. 1991. “Estimation of the Variances in the Branching Process with Immigration.” Probability Theory and Related Fields 88 (1): 77–106. https://doi.org/10.1007/BF01193583.

Zeger, Scott L. 1988. “A Regression Model for Time Series of Counts.” Biometrika 75 (4): 621–29. https://doi.org/10.1093/biomet/75.4.621.

Zeger, Scott L., and Bahjat Qaqish. 1988. “Markov Regression Models for Time Series: A Quasi-Likelihood Approach.” Biometrics 44 (4): 1019–31. https://doi.org/10.2307/2531732.

Zheng, Haitao, and Ishwar V. Basawa. 2008. “First-Order Observation-Driven Integer-Valued Autoregressive Processes.” Statistics & Probability Letters 78 (1): 1–9. https://doi.org/10.1016/j.spl.2007.04.017.

Zheng, Haitao, Ishwar V. Basawa, and Somnath Datta. 2007. “First-Order Random Coefficient Integer-Valued Autoregressive Processes.” Journal of Statistical Planning and Inference 137 (1): 212–29. https://doi.org/10.1016/j.jspi.2005.12.003.