Lattice-based methods for regression and density estimation on complicated multidimensional regions

Abstract

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.