# Branching processes and their statistics

A diverse class of stochastic models that I am mildly obsessed with.

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.

I will assume linear intensity models except where otherwise indicated.

## To learn

• Basic handling of process 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. Amazingly this works in continuous state spaces too.
• Theory of inference for the continuous state case; (Is there any? Perhaps in the Lévy process literature?)
• connection to the non-stochastic epidemiological SIR DE models (What to they do about noise?)
• Connection to stable processes and Lévy processes.

## Discrete index, discrete state: The Galton-Watson process and friends

There are many standard expositions. Two good ones:

This section got long enough to break out separately. See my notes on long-memory Galton-Watson process.

## Continuous index, discrete state: the Hawkes Process

If you have a integer-valued state space, but a continuous time index, and linear intensities, then this is a Hawkes Point Process, the cluster point process. Consider also change of time.

See my masters thesis.

As an example, here is a time-based one, the Hawkes process.

We start with the log likelihood of a generic point process, with occurrence times $$\{t_i\}.$$ \begin{align*} 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{align*}

I recall the the intensity for the Hawkes process in in particular:

$\lambda^*(t) = \mu + \int_{-\infty}^t \eta\phi(t-s)dNs.$ where $$\phi_\kappa(t)$$ is the influence kernel with parameter $$\kappa$$, $$\eta$$ the branching ratio, and the star $$\lambda^*(t)$$ is shorthand for $$\lambda^*(t|\mathcal{F}_t)$$.

### Time-inhomogeneous extension

My non-parametric of choice here will be of convolution kernel type. That is, I will introduce an additional convolution kernel $$\psi$$, and functions of the form

$\mu(t) = \mu + \sum_{1 \leq j \leq p}\omega_i\psi_{\nu_j}(t-t_j)$

for some set of kernel bandwidths $$\{\nu_j\}_{1 \leq j \leq p}$$, kernel weights $$\{\omega_{\nu_j}\}_{1 \leq j \leq p}$$, kernel locations $$\{\tau_j\}_{1 \leq j \leq p}$$.

There are many kernels available. We start with the top hat kernel, the piecewise-constant function.

$\psi_{\nu}(t):= \frac{\mathbb{I}_{0\lt t \leq \nu}}{\nu}$

giving the following background intensity

$\mu(t) = \mu + \sum_{1\leq j\le p}\omega_j\frac{\mathbb{I}_{(0, \nu_j]}(t-\tau_j)}{\nu_j}.$

I augment the parameter vector to include the kernel weights $$\theta':=( \mu,\eta,\kappa, \boldsymbol\omega).$$ We could also try to infer kernel locations and bandwidths.

The hypothesized generating model now has conditional intensity process

$\lambda_{\theta'}(t|\mathcal{F}_t) = \mu + \sum_{j=2}^n \omega_j \mathbb{I}_{[\tau_{j-1},\tau_j)}(t) + \eta \sum_{t_i\lt t}\phi_\kappa(t-t_i).$

## Continuous index, continuous state: The CSBP type of Lévy process

Aldous gives an expo on these. Super trendy at the moment: It turns out that growing trees is connected in a deep but purportedly simple way to “glueing together” excursions of random processes, oh, and a bunch of trippy fractals and random trees and stuff.

Lee and Hopcraft (LeHJ08) also found an analogous result for discrete state branching processes.

### 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 Weiner process (as a Lévy process) there is a well-understood estimation theory, with twiddly flourishes on top even. For CSBPs I am not aware of any specific examples except specific constrained cases, notably, the discrete state case, which rather defeats the purpose. Can you do more with general nonparametric Lévy measure inference in this continuous case? How much more? Over98 seems to be one of the few refs. Surely the finance folks are onto this?

## Discrete index, continuous state

Umm. Is this well-defined? I suppose so. Can’t find any literature references though. It surely has a fancy name. “Marked Galton-Watson Process”? Some kind of compound Poisson, I imagine.

## Special issues for multivariate branching processes

If you are looking at cross-excitation between variables then I have some additional matter at contagion processes.

## Superprocesses

Measure-valued state or something? Are these even branching processes, or did they just seem to be so because I ran into them at the end of a branching process seminar? Can’t recall, must investigate later.

Dynk04, Dynk91 and Ethe00 were recommended to me for this.

## Classic data sets

• Tomás Aragón’s free online epidemiology textbook (Arag12) lists, among others,

• In the R package tscount one may find campy, ecoli, ehec, influenza and measles.

• Gapminder hosts some disease datasets including tubercolosis, 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.

## Implementations

IHSEP is Feng Chen’s software to continuous index, discrete state branching processes.

Spatstat is for spatial point processes.