6.7 KB · ⇊ 4 parts · ⤓ Micron 6.7 KB · ⤓ HTML 55.2 KB

top

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.cite-ref-1[1]

Cox processes are used to generate simulations of spike trains (the sequence of action potentials generated by a neuron),cite-ref-2[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."cite-ref-3[3]

• Definition

• Laplace transform

• See also

• References

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

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

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

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

L η η ( f ) = exp ⁡ ⁡ ( − − ∫ ∫ 1 − − exp ⁡ ⁡ ( − − f ( x ) ) ξ ξ ( d x ) ) {\displaystyle {\mathcal {L}}_{\eta }(f)=\exp \left(-\int 1-\exp(-f(x))\;\xi (\mathrm {d} x)\right)}

for any positive, measurable function 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

cite-note-11. ↑ citerefcox1955Cox, 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.

cite-note-22. ↑ citerefkruminshoham2009Krumin, 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.

cite-note-33. ↑ citereflando1998Lando, David (1998). "On cox processes and credit risky securities". Review of Derivatives Research. 2 (2–3): 99–120. doi:10.1007/BF01531332.

• 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)