Construction of anisotropic covariance functions using Riesz-representers
Construction of anisotropic covariance functions using Riesz-representers
复制标题
使用 Riesz 表示器构建各向异性协方差函数
DOI：
10.1007/s001900050250
复制
发表时间：
1999
影响因子：
4.4
通讯作者：
C. Tscherning
中科院分区：
文献类型：
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作者：
C. Tscherning
抽象的。考虑半径为R0的球外集合中函数调和的再生核Hilbert空间(RKHS)，具有再生核K0(P，Q)(P，Q，随后的Pn是调和性集合中的点)。该核的阶方差将被表示为σ0n。与涉及不同点Pn，n=1，…的评估泛函(或重力泛函)相关联的Riesz表示子集，N，在围绕边界球体的二维曲面上，将是线性独立的。这些函数用于定义具有核(AN>；0)的新的N维RKHS
如果所有点都位于半径为R1&R0的同心球体上，并形成覆盖该球体的ε网，并且An是合适的面元素(取决于N)，则该核将收敛到具有阶方差的各向同性核
因此，如果需要Kn(P，Q)来表示地球引力势的各向同性协方差函数Cov(P，Q)，则可以选择σ0n，使得σn变得等于经验度方差。如果在变化的径向距离Rn和R0处选择点，则可以构造各向异性核或等价协方差函数表示。如果这些点位于有界区域内，则可以使用该核来修改原始核
计算了基于这些思想构造的各向异性协方差函数的值，并对如何选择点Pn提出了一些初步的想法。
Abstract. A reproducing-kernel Hilbert space (RKHS) of functions harmonic in the set outside a sphere with radius R0, having a reproducing kernel K0(P,Q) is considered (P, Q, and later Pn being points in the set of harmonicity). The degree variances of this kernel will be denoted σ0n.The set of Riesz representers associated with the evaluation functionals (or gravity functionals) related to distinct points Pn,n = 1,…,N, on a two-dimensional surface surrounding the bounding sphere, will be linearly independent. These functions are used to define a new N-dimensional RKHS with kernel (an>0)
If the points all are located on a concentric sphere with radius R1>R0, and form an ε-net covering the sphere, and an are suitable area elements (depending on N), then this kernel will converge towards an isotropic kernel with degree variances
Consequently, if KN(P,Q) is required to represent an isotropic covariance function of the Earth's gravity potential, COV(P,Q), σ0n can be selected so that σn becomes equal to the empirical degree variances.If the points are chosen at varying radial distances Rn>R0, then an anisotropic kernel, or equivalent covariance function representation, can be constructed. If the points are located in a bounded region, the kernel may be used to modify the original kernel
Values of anisotropic covariance functions constructed based on these ideas are calculated, and some initial ideas are presented on how to select the points Pn.