latticeDensity (version 1.2.7)

createNparReg: Performs nonparametric regression on irregular regions.

Description

This function takes the lattice from formLattice (which fills the region of interest) along with the list of responses and their locations, and creates a prediction surface. The approach is kernel non-parametric regression with the kernels created by a k-step diffusion on the lattice about each location where a response was collected.

Usage

createNparReg(formLatticeOutput, Z, PointPattern = NULL, M = 0.5, k)

Value

A list of class NparRegOut with elements:

• EW_locs Vector of EW locations.

• NS_locs Vector of NS locations.

• nodes Matrix of node locations in lattice.

• boundaryPoly Matrix showing bounding polynomial.

• hole_list List of polygons, holes in region.

• PointPattern Matrix of the locations of the data.

• which_nodes Matrix locations of nodes closest to data.

• NparRegNum Vector of numerators of the regression estimates

• NparRegDenom Vector of denominators of the regression estimates

• sigma2 Numeric, estimate of the noise variance.

Arguments

formLatticeOutput

An object returned by formLattice or editLattice.

Z

Vector of responses to be smoothed.

PointPattern

A 2 column matrix or data frame of locations.

M

The maximum probability that the random walk will move.

k

Number of steps.

Variance Estimation

We use the variance estimator \(\sum e_{i,-i}^2/n\), where \(e_{i,-i}\) is the ith deleted residual.

Details

We denote by \(K_{ik}(s)\) the kernel obtained by assigning the node nearest to the ith response and then running a k-step diffusion on the lattice and evaluating the resulting density at location s. Then the estimator \(\hat{f}(s) = (\sum_i K_{ik}(s)*Z_i)/\sum_i K_{ik}(s)\) which is the traditional kernal regression estimator with diffusion kernels. This approach leads to a non-parametric regression that respects the boundaries of the polygonal region. The construction of the kernels is detailed in Barry and McIntyre (2011). Using kernels to perform nonparametric regression is described in many publications, including Wasserman (2006).

References

Larry Wasserman. All of Nonparametric Statistics. Springer Science + Business Media, Inc. N.Y. 2006.

Julie McIntyre, Ronald P. Barry (2018) A Lattice-Based Smoother for Regions with Irregular Boundaries and Holes. Journal of Computational and Graphical Statistics. <doi:10.1080/10618600.2017.1375935>

Examples

data(nparExample)
attach(nparExample)
plot.new()
#  Simulate a response variable
index1 = (grid2[,2]<0.8)|(grid2[,1]>0.6)
Z = rep(NA,length(grid2[,1]))
n1 = sum(index1)
n2 = sum(!index1)
Z[index1] <- 3*grid2[index1,1] + 4 + rnorm(n1,0,sd=0.4)
Z[!index1] <- -2*grid2[!index1,1] + 4 + rnorm(n2,0,sd=0.4)
#
coords=rbind(polygon2,polygon2[1,])
plot(coords,type="l")
points(grid2,pch=19,cex=0.5,xlim=c(-0.1,1))
text(grid2,labels=round(Z,1),pos=4,cex=0.5)
#
nodeFillingOutput <- nodeFilling(poly=polygon2, node_spacing=0.025)
plot(nodeFillingOutput)
formLatticeOutput <- formLattice(nodeFillingOutput)
NparRegOut <- createNparReg(formLatticeOutput,Z,PointPattern=grid2,k=2)
plot(NparRegOut)