riem.m2skreg {Riemann}

Manifold-to-Scalar Kernel Regression

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M} and scalars y_1, y_2, \ldots, y_N \in \mathbf{R}, perform the Nadaraya-Watson kernel regression by

\hat{m}_h (X) = \frac{\sum_{i=1}^n K \left( \frac{d(X,X_i)}{h} \right) y_i}{\sum_{i=1}^n K \left( \frac{d(X,X_i)}{h} \right)}

where the Gaussian kernel is defined as

K(x) := \frac{1}{\sqrt{2\pi}} \exp \left( - \frac{x^2}{2}\right)

with the bandwidth parameter h > 0 that controls the degree of smoothness.

Usage

riem.m2skreg(
  riemobj,
  y,
  bandwidth = 0.5,
  geometry = c("intrinsic", "extrinsic")
)

Arguments

riemobj

a S3 "riemdata" class for N manifold-valued data corresponding to X_1,\ldots,X_N.

y

a length-N vector of dependent variable values.

bandwidth

a nonnegative number that controls smoothness.

geometry

(case-insensitive) name of geometry; either geodesic ("intrinsic") or embedded ("extrinsic") geometry.

Value

a named list of S3 class m2skreg containing

ypred

a length-N vector of smoothed responses.

bandwidth

the bandwidth value that was originally provided, which is saved for future use.

inputs

a list containing both riemobj and y for future use.

Examples

#-------------------------------------------------------------------
#                    Example on Sphere S^2
#
#  X : equi-spaced points from (0,0,1) to (0,1,0)
#  y : sin(x) with perturbation
#-------------------------------------------------------------------
# GENERATE DATA
npts = 100
nlev = 0.25
thetas = seq(from=0, to=pi/2, length.out=npts)
Xstack = cbind(rep(0,npts), sin(thetas), cos(thetas))

Xriem  = wrap.sphere(Xstack)
ytrue  = sin(seq(from=0, to=2*pi, length.out=npts))
ynoise = ytrue + rnorm(npts, sd=nlev)

# FIT WITH DIFFERENT BANDWIDTHS
fit1 = riem.m2skreg(Xriem, ynoise, bandwidth=0.001)
fit2 = riem.m2skreg(Xriem, ynoise, bandwidth=0.01)
fit3 = riem.m2skreg(Xriem, ynoise, bandwidth=0.1)

# VISUALIZE
xgrd <- 1:npts
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(xgrd, fit1$ypred, pch=19, cex=0.5, "b", xlab="", ylim=c(-2,2), main="h=1e-3")
lines(xgrd, ytrue, col="red", lwd=1.5)
plot(xgrd, fit2$ypred, pch=19, cex=0.5, "b", xlab="", ylim=c(-2,2), main="h=1e-2")
lines(xgrd, ytrue, col="red", lwd=1.5)
plot(xgrd, fit3$ypred, pch=19, cex=0.5, "b", xlab="", ylim=c(-2,2), main="h=1e-1")
lines(xgrd, ytrue, col="red", lwd=1.5)
par(opar)