Cox process

In probability theory, a Cox process, also known as a doubly stochastic Poisson process is a point process which is a generalization of a Poisson process where the intensity that varies across the underlying mathematical space (often space or time) is itself a stochastic process. The process is named after the statistician David Cox, who first published the model in 1955.^[1]

Cox processes are used to generate simulations of spike trains (the sequence of action potentials generated by a neuron),^[2] and also in financial mathematics where they produce a "useful framework for modeling prices of financial instruments in which credit risk is a significant factor."^[3]

Let ${\displaystyle \xi }${\displaystyle \xi } be a random measure.

A random measure ${\displaystyle \eta }${\displaystyle \eta } is called a Cox process directed by ${\displaystyle \xi }${\displaystyle \xi }, if ${\displaystyle {\mathcal {L}}(\eta \mid \xi =\mu )}${\displaystyle {\mathcal {L}}(\eta \mid \xi =\mu )} is a Poisson process with intensity measure ${\displaystyle \mu }${\displaystyle \mu }.

Here, ${\displaystyle {\mathcal {L}}(\eta \mid \xi =\mu )}${\displaystyle {\mathcal {L}}(\eta \mid \xi =\mu )} is the conditional distribution of ${\displaystyle \eta }${\displaystyle \eta }, given ${\displaystyle \{\xi =\mu \}}${\displaystyle \{\xi =\mu \}}.

If ${\displaystyle \eta }${\displaystyle \eta } is a Cox process directed by ${\displaystyle \xi }${\displaystyle \xi }, then ${\displaystyle \eta }${\displaystyle \eta } has the Laplace transform

${\displaystyle {\mathcal {L}}_{\eta }(f)=\exp \left(-\int 1-\exp(-f(x))\;\xi (\mathrm {d} x)\right)}${\displaystyle {\mathcal {L}}_{\eta }(f)=\exp \left(-\int 1-\exp(-f(x))\;\xi (\mathrm {d} x)\right)}

for any positive, measurable function ${\displaystyle f}${\displaystyle f}.

• Poisson hidden Markov model
• Doubly stochastic model
• Inhomogeneous Poisson process, where λ(t) is restricted to a deterministic function
• Ross's conjecture
• Gaussian process
• Mixed Poisson process

Notes

Bibliography

• Cox, D. R. and Isham, V. Point Processes , London: Chapman & Hall, 1980 ISBN 0-412-21910-7
• Donald L. Snyder and Michael I. Miller Random Point Processes in Time and Space Springer-Verlag, 1991 ISBN 0-387-97577-2 (New York) ISBN 3-540-97577-2 (Berlin)

• Poisson point processes
• Statistics stubs

• Articles with short description
• Short description is different from Wikidata
• All stub articles