اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe Turning Arcs: a Computationally Efficient Algorithm to Simulate Isotropic Vector-Valued Gaussian Random Fields on the $d$-Sphere
50
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Random fields on the sphere play a fundamental role in the natural sciences. This paper presents a simulation algorithm parenthetical to the spectral turning bands method used in Euclidean spaces, for simulating scalar- or vector-valued Gaussian random fields on the $d$-dimensional unit sphere. The simulated random field is obtained by a sum of Gegenbauer waves, each of which is variable along a randomly oriented arc and constant along the parallels orthogonal to the arc. Convergence criteria based on the Berry-Esseen inequality are proposed to choose suitable parameters for the implementation of the algorithm, which is illustrated through numerical experiments. A by-product of this work is a closed-form expression of the Schoenberg coefficients associated with the Chentsov and exponential covariance models on spheres of dimensions greater than or equal to 2.
قيم البحث
اقرأ أيضاً
Series expansions of isotropic Gaussian random fields on $mathbb{S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations provide an alternative to the standard Karhunen-Lo`eve expansions of iso
tropic random fields in terms of spherical harmonics. Their multilevel localized structure of basis functions is especially useful in adaptive algorithms. The basis functions are obtained by applying the square root of the covariance operator to spherical needlets. Localization of the resulting covariance-dependent multilevel basis is shown under decay conditions on the angular power spectrum of the random field. In addition, numerical illustrations are given and an application to random elliptic PDEs on the sphere is analyzed.
We study the problem of estimating the mean of a multivariatedistribution based on independent samples. The main result is the proof of existence of an estimator with a non-asymptotic sub-Gaussian performance for all distributions satisfying some mild moment assumptions.
Gaussian process modeling is a standard tool for building emulators for computer experiments, which are usually used to study deterministic functions, for example, a solution to a given system of partial differential equations. This work investigates
applying Gaussian process modeling to a deterministic function from prediction and uncertainty quantification perspectives, where the Gaussian process model is misspecified. Specifically, we consider the case where the underlying function is fixed and from a reproducing kernel Hilbert space generated by some kernel function, and the same kernel function is used in the Gaussian process modeling as the correlation function for prediction and uncertainty quantification. While upper bounds and optimal convergence rate of prediction in the Gaussian process modeling have been extensively studied in the literature, a thorough exploration of convergence rates and theoretical study of uncertainty quantification is lacking. We prove that, if one uses maximum likelihood estimation to estimate the variance in Gaussian process modeling, under different choices of the nugget parameter value, the predictor is not optimal and/or the confidence interval is not reliable. In particular, lower bounds of the prediction error under different choices of the nugget parameter value are obtained. The results indicate that, if one directly applies Gaussian process modeling to a fixed function, the reliability of the confidence interval and the optimality of the predictor cannot be achieved at the same time.
A new acceptance-rejection method is proposed and investigated for the Bingham distribution on the sphere using the angular central Gaussian distribution as an envelope. It is shown to have high efficiency and to be straightfoward to use. The method
can also be extended to Fisher and Fisher-Bingham distributions on spheres and related manifolds.
In massive data analysis, training and testing data often come from very different sources, and their probability distributions are not necessarily identical. A feature example is nonparametric classification in posterior drift model where the condit
ional distributions of the label given the covariates are possibly different. In this paper, we derive minimax rate of the excess risk for nonparametric classification in posterior drift model in the setting that both training and testing data have smooth distributions, extending a recent work by Cai and Wei (2019) who only impose smoothness condition on the distribution of testing data. The minimax rate demonstrates a phase transition characterized by the mutual relationship between the smoothness orders of the training and testing data distributions. We also propose a computationally efficient and data-driven nearest neighbor classifier which achieves the minimax excess risk (up to a logarithm factor). Simulation studies and a real-world application are conducted to demonstrate our approach.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
