Hawkes processes

December 22, 2019 — December 22, 2019

branching
count data
functional analysis
point processes
probability
social graph
statistics
time series
virality
Figure 1

At the intersection of point processes and branching processes can be found the Hawkes process.

The classic is the univariate, temporal, linear Hawkes process.git Recall the log likelihood of a generic point process, with occurrence times \(\{t_i\}.\)

\[ \begin{aligned} L_\theta(t_{1:N}) &:= -\int_0^T\lambda^*_\theta(t)dt + \int_0^T\log \lambda^*_\theta(t) dN_t\\ &= -\int_0^T\lambda^*_\theta(t)dt + \sum_{j} \log \lambda^*_\theta(t_j) \end{aligned} \]

\(\lambda^*(t)\) is shorthand for \(\lambda^*(t|\mathcal{F}_t)\), and we call this the intensity. This term is what distinguishes various point processes. For the Hawkes process is in particular we have

\[ \lambda^*(t) = \mu + \int_{-\infty}^t \eta\phi(t-s)dNs. \] where \(\phi_\kappa(t)\) is the influence kernel with parameter \(\kappa\), \(\eta\) the branching ratio, and

0.1 Time-inhomogeneous extension

Partial notes to an extension that I have looked at. 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.

\[ \psi_{\nu}(t):= \frac{\mathbb{I}_{0< t \leq \nu}}{\nu} \]

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

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

1 Tools

2 References

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