aleatory.processes.MixedPoissonProcess#

class aleatory.processes.MixedPoissonProcess(intensity, intensity_args=None, intensity_kwargs=None, rng=None)[source]#

Mixed Poisson Process#

../_images/mixed_poisson_draw.png

Notes#

In probability theory, a mixed Poisson process is a special point process that is a generalization of a Poisson process. Mixed Poisson processes are simple example for Cox processes.

A Mixed Poisson process (MPP) \(\{N(t), t\in [0,\infty)\}\) is a counting process with counting distribution of the form:

\[P(N(t)= n) = \int_0^{\infty} \frac{1}{n!} e^{-\lambda t} (\lambda t)^n d \Lambda(\lambda), \qquad n\in \mathbb{N},\]

where \(\Lambda\) is the structure distribution given by

\[\Lambda(\lambda) = P(\Lambda \leq \lambda)\]

with \(\Lambda(0)=0\). This type of distribution is known as mixed Poisson distribution which gives the name to the processes.

Constructor, Methods, and Attributes#

__init__(intensity, intensity_args=None, intensity_kwargs=None, rng=None)[source]#
Parameters:

  • intensity (callable) – a callable function which defines the structure distribution \(\Lambda\)

  • intensity_args – the arguments to be passed to the intensity function

  • intensity_kwargs – the keyword arguments to be passed to the intensity function

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

Methods

__init__(intensity[, intensity_args, ...])

parameter callable intensity:

a callable function which defines the structure distribution \(\Lambda\)

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

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

sample([jumps, T])

simulate(N[, jumps, T])

Simulate paths/trajectories from the instanced stochastic process.

Attributes

intensity

rng