aleatory.processes.HawkesProcess#

class aleatory.processes.HawkesProcess(mu=1.0, alpha=1.0, beta=1.0, rng=None)[source]#

Hawkes process#

../_images/hawkes_draw.png

Notes#

A Hawkes process is a counting process \(N(t)\) whose conditional intensity process is given by

\[\lambda^{\ast}(t) = \mu + \sum_{T_i <t} \phi(t-T_i), \qquad t\geq 0,\]

where

  • \(\mu>0\) is the baseline intensity (rate of events in the absence of previous events) or background arrival rate

  • \(\phi: \mathbb{R}^{+}\rightarrow \mathbb{R}\), is the excitation function or triggering kernel, a non-negative function representing the influence of past events, and

  • \(T_i\), are the times of prior events.

In particular, we assume

\[\phi(t) = \alpha \exp(-\beta t)\]

i.e. an exponentially decaying excitation function.

Constructor, Methods, and Attributes#

__init__(mu=1.0, alpha=1.0, beta=1.0, rng=None)[source]#
Parameters:

  • mu (double) – the baseline intensity

  • alpha (double) – the \(\alpha >0\) in the excitation function above

  • beta (double) – the \(\beta >0\) in the excitation function above

  • beta – the \(\beta >0\) in the excitation function above

  • rng (numpy.random.Generator) – a custom random number generator

Methods

__init__([mu, alpha, beta, rng])

parameter double mu:

the baseline intensity

draw(N[, T, style, colormap, mode, title, ...])

plot(N[, T, title, suptitle])

sample([T])

simulate(N[, T])

Simulate paths/trajectories from the instanced stochastic process.

Attributes

alpha

intensity

mu

rng