Hawkes processes

An intersection of point processes and branching processes is the Hawkes process. The classic is the univariate linear Hawkes process. For now we’ll assume it to be indexed by time.

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

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.

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

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

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

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