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.

Appendix

Here we use a toy example to illustrate the basic steps for estimating a smooth function with the lattice-based smoother. R code for reproducing this example is provided in Supplementary Materials in the vignette ToyExample. Here we estimate a density function over a subregion of $\mathbb {R}^2$ from observations of a point process. This same basic squence of steps is followed to estimate a density or regression function on any arbitrary subregion of $\mathbb {R}^d$.

Figure 10 shows the region of interest, representing a lake. A single observation of a point process, for example the location where a certain fish species is observed, is recorded at $x=5.8$ and $y=1.2$. Here the subregion of interest is a polygonal region embedded in $\mathbb {R}^2$. The first step is to define a set of nodes over the lake. Typically nodes will be defined densely throughout the subregion, but for illustration we will use only ten nodes. Nodes are plotted in Fig. 10, and note that the observed point process location is not exactly on a node, so it is moved to the nearest node, node 10. The matrix of node coordinates ${{{\varvec{{L}}}}}$ is then the $10\times 2$ matrix shown below.

Fig. 10

Point process location (left) and nodes (right)

$$\begin{aligned} {{{\varvec{{L}}}}}^T = \left[ \begin{array}{cccccccccc} 1 &{} 2 &{} 1 &{} 2 &{} 3 &{} 4 &{} 5 &{} 6 &{} 5 &{} 6\\ 1 &{} 1 &{} 2 &{} 2 &{} 2 &{} 2 &{} 2 &{} 2 &{} 1 &{} 1\\ \end{array} \right] \end{aligned}$$

Next we define the neighbor matrix, ${{{\varvec{{B}}}}}$. The neighbor relationship is arbitrary, although most often, when the region of interest is spatial, it is determined by distance between nodes. For this example we declare points adjacent in the north-south and east-west directions to be neighbors. The neighbor matrix ${{{\varvec{{B}}}}}$ is shown below, along with a plot of the lattice in Fig. 11.

Fig. 11

Lattice structure on lake

Fig. 12

Colors represent density over the lake for the toy example

$$\begin{aligned} {{{\varvec{{B}}}}}= \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{}1 &{}1 &{}0 &{}0 &{}0 &{}0 &{}0 &{}0 &{}0\\ 1 &{}0 &{}0 &{}1 &{}0 &{}0 &{}0 &{}0 &{}0 &{}0\\ 1 &{}0 &{}0 &{}1 &{}0 &{}0 &{}0 &{}0 &{}0 &{}0\\ 0 &{}1 &{}1 &{}0 &{}1 &{}0 &{}0 &{}0 &{}0 &{}0\\ 0 &{}0 &{}0 &{}1 &{}0 &{}1 &{}0 &{}0 &{}0 &{}0\\ 0 &{}0 &{}0 &{}0 &{}1 &{}0 &{}1 &{}0 &{}0 &{}0\\ 0 &{}0 &{}0 &{}0 &{}0 &{}1 &{}0 &{}1 &{}1 &{}0\\ 0 &{}0 &{}0 &{}0 &{}0 &{}0 &{}1 &{}0 &{}0 &{}1\\ 0 &{}0 &{}0 &{}0 &{}0 &{}0 &{}1 &{}0 &{}0 &{}1\\ 0 &{}0 &{}0 &{}0 &{}0 &{}0 &{}0 &{}1 &{}1 &{}0\end{array} \right] \end{aligned}$$

Finally we specify the transition matrix ${{{\varvec{{T}}}}}$. Here we define ${{{\varvec{{T}}}}}$ to be isotropic, with the probability of movement between neighbors equal to 1/6 everywhere. Note then that the probabilities on the diagonal of staying put at any step in the random walk depend on the number of neighbhors of each node.

$$\begin{aligned} {{{\varvec{{T}}}}}=\left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {2\over 3} &{}{1\over 6}&{}{1\over 6}&{}0 &{}0 &{}0 &{}0 &{}0 &{}0 &{}0\\ {1\over 6} &{}{2\over 3} &{}0 &{}{1\over 6} &{}0 &{}0 &{}0 &{}0 &{}0 &{}0\\ {1\over 6} &{}0 &{}{2\over 3} &{}{1\over 6} &{}0 &{}0 &{}0 &{}0 &{}0 &{}0\\ 0 &{}{1\over 6} &{}{1\over 6} &{}{1\over 2} &{}{1\over 6} &{}0 &{}0 &{}0 &{}0 &{}0\\ 0 &{}0 &{}0 &{}{1\over 6} &{}{2\over 3} &{}{1\over 6} &{}0 &{}0 &{}0 &{}0\\ 0 &{}0 &{}0 &{}0 &{}{1\over 6} &{}{2\over 3} &{}{1\over 6} &{}0 &{}0 &{}0\\ 0 &{}0 &{}0 &{}0 &{}0 &{}{1\over 6} &{}{1\over 2} &{}{1\over 6} &{}{1\over 6} &{}0\\ 0 &{}0 &{}0 &{}0 &{}0 &{}0 &{}{1\over 6} &{}{2\over 3} &{}0 &{}{1\over 6}\\ 0 &{}0 &{}0 &{}0 &{}0 &{}0 &{}{1\over 6} &{}0 &{}{2\over 3} &{}{1\over 6}\\ 0 &{}0 &{}0 &{}0 &{}0 &{}0 &{}0 &{}{1\over 6} &{}{1\over 6} &{}{2\over 3}\end{array} \right] \end{aligned}$$

The estimate of density at each point on the lattice after k steps is computed using Equation 1. For the single point-process observation at (5.8,1.2) relocated to node 10, the initial density vector is ${{{\varvec{{p}}}}}_0 = [0,0,0,0,0,0,0,0,0,1]$. Taking $k=5$, the estimated density becomes

$$\begin{aligned} {{{\varvec{{T}}}}}^5{{{\varvec{{p}}}}}_0= \left[ \begin{array}{c} 0.0000000000\\ 0.0000000000\\ 0.0000000000\\ 0.0002572016\\ 0.0048868313\\ 0.0378086420\\ 0.1520061728\\ 0.2443415638\\ 0.2443415638\\ 0.3163580247\\ \end{array} \right] \end{aligned}$$

Note that three nodes have estimated density zero, because they are more than five steps from any observation. The estimated density at each node is plotted in Fig. 12. Commonly such an estimate would be displayed using a contour or image graphic, after estimating density on a more densely defined grid of nodes.