This paper illustrates the use of diffusion kernels to estimate smooth density and regression functions defined on highly complex domains. We generalize the two-dimensional lattice-based estimators of Barry and McIntyre (2011) and McIntyre and Barry (2018) to estimate any function defined on a domain that may be embedded in \(\mathbb {R}^d\), \(d\ge 1\). Examples include function estimation on the surface of a sphere, a sphere with boundaries and holes, a sphere over multiple time periods, a linear network, the surface of cylinder, a three-dimensional volume with boundaries, and a union of one- and two-dimensional subregions.

中文翻译：

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基于格的复杂多维区域回归和密度估计方法

本文说明了使用扩散核估计高度复杂域上定义的平滑密度和回归函数。我们对Barry和McIntyre（2011）以及McIntyre和Barry（2018）的基于二维晶格的估计器进行一般化，以估计在可能嵌入\（\ mathbb {R} ^ d \），\ （d \ ge 1 \）。示例包括球体表面上的函数估计，带边界和孔的球体，多个时间段上的球体，线性网络，圆柱体表面，带边界的三维体积以及一和二的并集维子区域。