Statistical modeling of inhomogeneous random functions on the basis of Poisson point fields

“Poisson” models of random fields with applications in transport theory

“…Below, new representations (compared to [2,4]) of correlation functions of grid Poisson fields are con structed and their comparative analysis is presented. Since the scale can be suitably changed with the cor responding change in λ, we assume for presentational simplicity that d = 1; i.e., L l is the unit coordinate hypercube.…”

“…The space T is equipped with the measure τ defines as the product of the Lebesgue measure in ‫ޒ‬ and the Hausdorff measure Ᏼ l -1 on S (l) ("area" on the sphere); i.e., dτ = dpdᏴ (l -1) = dpds. Consider a Poisson point field (see, e.g., [6,4]) in T with a parameter λ. Identifying each point of this field with a hyperplane as described above yields the required Poisson field Γ of random hyperplanes. It can only be said a priori that the distribution of this field is invariant under rotations about the fixed origin.…”

“…The use of such models for the numerical solution of stochastic problems related to particle transfer in random scattering media also seems difficult. Therefore, it can be reasonable to use the "grid" models of random fields proposed in [2][3][4], which are based on a partition of the space by grids of coordinate planes passing through points of a Poisson field. An advantageous property of such models is that they are relatively easy to use in applications.…”

“…A special role is played by piece wise constant models based on a random partition of the phase space with the field values in each subdo main chosen independently according to a given distribution (see, e.g., [1][2][3][4][5]). The one dimensional dis tribution of such model fields can be arbitrary, while the correlation function is defined in some class of positive definite functions.…”

Calculation of Weight Coefficients in Continuous Particle Filter

“…При применении методики приближенного моделирования траекторий с обрывами и ветвлениями на основе метода «максимального сечения» [9,10], или метода прореживания пуассоновского потока, моменты времени τ k , в которые могут реализоваться события пуассо-новского потока (обрывы или ветвления), определяются с помощью моделирования случайных величин Δτ k , имеющих показательное распределение с параметром μ * . Величина μ * определяет-ся условием |μ(t)| ≤ μ * , t[t 0 , T], т. е. τ 0 = t 0 , τ k+1 = τ k + Δτ k , k = 1, 2, … Напомним, что необходимо получить реализацию случайной величины α, имеющей рав-номерное распределение на интервале (0, 1), и проверить условие α ≤ |μ(τ k )| / μ * , где μ(t) = μ(t, X(t), Z(t)) задает значения функции μ(t, x, z) на траектории случайного процесса X(t) с учетом измерений Z(t) оцениваемой траектории.…”