LocSpheReg {frechet}

Local Fréchet Regression for Spherical Data

Description

Local Fréchet regression for spherical data with respect to the geodesic distance.

Usage

LocSpheReg(xin = NULL, yin = NULL, xout = NULL, optns = list())

Arguments

xin

A vector of length n with input measurement points.

yin

An n by m matrix holding the spherical data, of which the sum of squares of elements within each row is 1.

xout

A vector of length k with output measurement points; Default: xout = xin.

optns

A list of options control parameters specified by list(name=value). See ‘Details’.

Details

Available control options are

bw

A scalar used as the bandwidth or "CV" (default).

kernel

A character holding the type of kernel functions for local Fréchet regression for densities; "rect", "gauss", "epan", "gausvar", "quar" - default: "gauss".

Value

A list containing the following components:

xout

Input xout.

yout

A k by m matrix holding the fitted responses, of which each row is a spherical vector, corresponding to each element in xout.

xin

Input xin.

yin

Input yin.

optns

A list of control options used.

References

Petersen, A., & Müller, H.-G. (2019). "Fréchet regression for random objects with Euclidean predictors." The Annals of Statistics, 47(2), 691–719.

Examples

set.seed(1)
n <- 200
# simulate the data according to the simulation in Petersen & Müller (2019)
xin <- runif(n)
err_sd <- 0.2
xout <- seq(0,1,length.out = 51)

phi_true <- acos(xin)
theta_true <- pi * xin
ytrue <- cbind(
  sin(phi_true) * cos(theta_true),
  sin(phi_true) * sin(theta_true),
  cos(phi_true)
)
basis <- list(
  b1 = cbind(
    cos(phi_true) * cos(theta_true),
    cos(phi_true) * sin(theta_true),
    -sin(phi_true)
  ),
  b2 = cbind(
    sin(theta_true),
    -cos(theta_true),
    0
  )
)
yin_tg <- basis$b1 * rnorm(n, mean = 0, sd = err_sd) + 
  basis$b2 * rnorm(n, mean = 0, sd = err_sd)
yin <- t(sapply(seq_len(n), function(i) {
  tgNorm <- sqrt(sum(yin_tg[i,]^2))
  if (tgNorm < 1e-10) {
    return(ytrue[i,])
  } else {
    return(sin(tgNorm) * yin_tg[i,] / tgNorm + cos(tgNorm) * ytrue[i,])
  }
}))

res <- LocSpheReg(xin=xin, yin=yin, xout=xout, optns = list(bw = 0.15, kernel = "epan"))