We propose nonparametric estimators for the second-order central moments of spherical random fields within a functional data context. We consider a measurement framework where each field among an identically distributed collection of spherical random fields is sampled at a few random directions, possibly subject to measurement error. The collection of fields could be i.i.d. or serially dependent. Though similar setups have already been explored for random functions defined on the unit interval, the nonparametric estimators proposed in the literature often rely on local polynomials, which do not readily extend to the (product) spherical setting. We therefore formulate our estimation procedure as a variational problem involving a generalized Tikhonov regularization term. The latter favours smooth covariance/autocovariance functions, where the smoothness is specified by means of suitable Sobolev-like pseudo-differential operators. Using the machinery of reproducing kernel Hilbert spaces, we establish representer theorems that fully characterizing the form of our estimators. We determine their uniform rates of convergence as the number of fields diverges, both for the dense (increasing number of spatial samples) and sparse (bounded number of spatial samples) regimes. We moreover validate and demonstrate the practical feasibility of our estimation procedure in a simulation setting, assuming a fixed number of samples per field. Our numerical estimation procedure leverages the sparsity and second-order Kronecker structure of our setup to reduce the computational and memory requirements by approximately three orders of magnitude compared to a naive implementation would require.
翻译:我们为功能数据范围内球体随机字段的第二阶中央时点提出非参数估计值。 我们考虑一个测量框架, 将相同分布的球体随机字段的每个字段在几个随机方向进行抽样, 可能存在测量错误。 字段的收集可以是i. id. 或序列依赖。 虽然已经为单位间隔上定义的随机功能探索过类似的设置, 文献中提议的非参数估计值往往依赖于本地多数值级, 这不易延伸到( 产品) 球体设置。 因此, 我们制定我们的估算程序是一个变异性问题, 涉及通用的 Tikhonov 正规化术语。 后者有利于平滑的任意性/ 自动变异性功能, 以合适的 Sobolev 类似的假相操作者的方式来指定。 虽然对于单位间隔上定义的随机功能, 文献中提议的非参数往往依赖于本地多数值, 并不轻易延伸到( 产品) 球体环境设置。 因此, 我们确定它们统一的趋同率, 是一个差异性的问题,, 包括通用的 Tikon roal roupal roup roal roup roup roup roup roup roup rout rout roup rout 。