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]
Definition
Let [math]\displaystyle{ \xi }[/math] be a random measure.
A random measure [math]\displaystyle{ \eta }[/math] is called a Cox process directed by [math]\displaystyle{ \xi }[/math], if [math]\displaystyle{ \mathcal L(\eta \mid \xi=\mu) }[/math] is a Poisson process with intensity measure [math]\displaystyle{ \mu }[/math].
Here, [math]\displaystyle{ \mathcal L(\eta \mid \xi=\mu) }[/math] is the conditional distribution of [math]\displaystyle{ \eta }[/math], given [math]\displaystyle{ \{ \xi=\mu\} }[/math].
Laplace transform
If [math]\displaystyle{ \eta }[/math] is a Cox process directed by [math]\displaystyle{ \xi }[/math], then [math]\displaystyle{ \eta }[/math] has the Laplace transform
- [math]\displaystyle{ \mathcal L_\eta(f)=\exp \left(- \int 1-\exp(-f(x))\; \xi(\mathrm dx)\right) }[/math]
for any positive, measurable function [math]\displaystyle{ f }[/math].
See also
- 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
References
- Notes
- ↑ Cox, D. R. (1955). "Some Statistical Methods Connected with Series of Events". Journal of the Royal Statistical Society 17 (2): 129–164. doi:10.1111/j.2517-6161.1955.tb00188.x.
- ↑ Krumin, M.; Shoham, S. (2009). "Generation of Spike Trains with Controlled Auto- and Cross-Correlation Functions". Neural Computation 21 (6): 1642–1664. doi:10.1162/neco.2009.08-08-847. PMID 19191596.
- ↑ Lando, David (1998). "On cox processes and credit risky securities". Review of Derivatives Research 2 (2–3): 99–120. doi:10.1007/BF01531332.
- 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)
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