New Statistical Kernel-Projection Estimator in the Monte Carlo Method
- PDF / 346,651 Bytes
- 5 Pages / 612 x 792 pts (letter) Page_size
- 96 Downloads / 217 Views
UTER SCIENCE
New Statistical Kernel-Projection Estimator in the Monte Carlo Method Corresponding Member of the RAS G. A. Mikhailova,b,*, N. V. Trachevaa,b,**, and S. A. Ukhinova,b Received March 12, 2020; revised May 23, 2020; accepted May 23, 2020
Abstract—The statistical kernel estimator in the Monte Carlo method is usually optimized based on the preliminary construction of a “microgrouped” sample of values of the variable under study. Even for the twodimensional case, such optimization is very difficult. Accordingly, we propose a combined (kernel-projection) statistical estimator of the two-dimensional distribution density: a kernel estimator is constructed for the first (main) variable, and a projection estimator, for the second variable. In this case, for each kernel interval determined by the microgrouped sample, the coefficients of a particular orthogonal decomposition of the conditional probability density are statistically estimated based on preliminary results for the “micro intervals.” An important result of this work is the mean-square optimization of such an estimator under assumptions made about the convergence rate of the orthogonal decomposition in use. The constructed estimator is verified by evaluating the bidirectional distribution of a radiation flux passing through a layer of scattering and absorbing substance. Keywords: kernel density estimator, projection estimator, kernel-projection estimator, Monte Carlo method DOI: 10.1134/S1064562420040122
By way of introduction, suppose that we want to estimate the distribution density f(x) of particles (radiation quanta) with respect to the parameter x in a finite interval ( x1, x2 ). The universal Parzen–Rosenblatt kernel density estimator with a rectangular (“uniform”) kernel can be practically efficient for this purpose [1] (see also [2]). It is constructed by statistically estimating functionals of the form
JΔ =
f ( x ') I
Δ
( x ') dx ',
where I Δ ( x ' ) is the indicator of the interval Δ = Δ(x) = x − δ , x + δ . It is assumed that the formulation of 2 2 the problem admits the construction of a Bernoulli estimator of this functional by calculating the number nΔ of particle trajectories Ω ending in the interval Δ. In particle transport problems, f(x) is the stochastic distribution density of the number of particles at points of their “destruction,” for example, due to irretrievable escape from the medium. We have the statis-
(
a Institute
)
of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia b Novosibirsk State University, Novosibirsk, 630090 Russia *e-mail: [email protected] **e-mail: [email protected]
nΔ , where N is the size of the samN ple {Ωk } ( k = 1,..., N ). The mean square of the error of the estimator n f ( x) ≈ Δ is given by N ⋅δ
tical estimator J Δ ≈
2
n n 2 ε ( x; N , δ) = E f ( x ) − Δ(x ) = D Δ N δ Nδ (1) 2 4 f x ( ) 2 J + f ( x ) − Δ ≈ + ( f '' ( x )) δ Nδ 576 δ (see, e.g., [2]) with a relative error decreasing to ze
Data Loading...
