A method of modelling log Gaussian Cox processes. (Ukrainian, English) Zbl 1199.60114

Kozachenko, Yu. V.; Pogorilyak, O. O.

Teor. Jmovirn. Mat. Stat. 77, 82-95 (2007); translation in Theory Probab. Math. Stat. 77, 91-105 (2008).

Cox processes are a natural generalization of Poisson processes for which intensities are random measures. In particular, an integral of a nonnegative stochastic process \(Z(\omega, t)\) is an example of the mentioned random measure. Given a realization of \(Z(\omega, t)\), we have the corresponding Poisson process with intensity \(\mu(B)=\int_BZ(\omega, t) dt\). Log Gaussian Cox processes and their models are studied in the papers by A. Brix and J. Møller [Scand. J. Stat. 28, No. 3, 471–488 (2001; Zbl 0981.62079)], J. Møller, A. R. Syversveen and R. P. Waagepetersen [Scand. J. Stat. 25, No. 3, 451–482 (1998; Zbl 0931.60038)], J. Møller and R. P. Waagepetersen [Statistical inference and simulation for spatial point processes, Chapman and Hall (2004; Zbl 1044.62101)] and J. Møller [Spatial statistics and computational methods, Lecture Notes in Statistics 173, Springer (2003; Zbl 1010.62507)]. In contrast to the methods discussed in the papers mentioned above, the authors of this paper propose a method of construction of models with a given accuracy and reliability. A similar approach for modelling stochastic processes is developed in the papers [A. Pogorilyak, Visn., Mat. Mekh., Kyïv. Univ. Im. Tarasa Shevchenka 2006, No. 16, 94–100 (2006; Zbl 1150.62424); Yu. V. Kozachenko and O. O. Pogorilyak [Teor. Jmovirn. Mat. Stat. 76, 70–83 (2007; Zbl 1199.60115)].
For more results and references see the book by Yu. V. Kozachenko, A. O. Pashko and I. V. Rozora [Modelling of random processes and fields, Kyïv: Zadruga (2007; Zbl 1199.60003)].

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60G10	Stationary stochastic processes
65C40	Numerical analysis or methods applied to Markov chains

Keywords:

log Gaussian process; Cox process; accuracy; reliability

Citations:

Zbl 0981.62079; Zbl 0931.60038; Zbl 1044.62101; Zbl 1150.62424; Zbl 1010.62507; Zbl 1199.60115; Zbl 1199.60003