Nonparametric Rotations for Sphere-Sphere Regression

Fits sphere-sphere regression models by estimating locally weighted rotations. Simulation of sphere-sphere data according to non-rigid rotation models. Provides methods for bias reduction applying iterative procedures within a Newton-Raphson learning scheme. Cross-validation is exploited to select smoothing parameters. See Marco Di Marzio, Agnese Panzera & Charles C. Taylor (2018) .

output: github_document

This is nprotreg, an R package that exploits nonparametric rotations in the analysis of Sphere-Sphere regression models.

The package implements methods proposed by Di Marzio, Panzera & Taylor (2018).

Thanks to package nprotreg, regressing data represented as points on a hypersphere you can

• simulate a very flexible regression model where, for each location of the manifold, a specific rotation matrix is applied to obtain a spherical response;
• fit Sphere-Sphere regression models by allowing for approximations of rotation matrices based on a series expansion;
• reduce estimation bias applying iterative estimation procedures within a Newton-Raphson learning scheme;
• use cross-validation to select smoothing parameters.

Getting Started

The following script shows how to fit a Sphere-Sphere regression model using simulated data via package nprotreg.

library(nprotreg)
 
# Define a matrix of explanatory points.
 
number_of_explanatory_points <- 50
 
explanatory_points <- get_equally_spaced_points(
  number_of_explanatory_points)
 
### Define a matrix of response points by simulation.
 
# define the reponse _local_ rotation model (eg Model 2 in Table 1 of [Di Marzio, Panzera & Taylor (2018)])
 
local_rotation_composer <- function(point) {
  independent_components <- (1 / 2) *
    c(exp(2.0 * point[3]), - exp(2.0 * point[2]), exp(2.0 * point[1]))
}
 
# define rotation (error) perturbation model using random skew symmetric matrix:
 
local_error_sampler <- function(point) {
  rnorm(3,mean=0,sd=.25)
}
 
response_points <- simulate_regression(explanatory_points,
                                       local_rotation_composer,
                                       local_error_sampler)
 
# Define a matrix of evaluation points for prediction.
 
evaluation_points <- rbind(
  cbind(.5, 0, .8660254),
  cbind(-.5, 0, .8660254),
  cbind(1, 0, 0),
  cbind(0, 1, 0),
  cbind(-1, 0, 0),
  cbind(0, -1, 0),
  cbind(.5, 0, -.8660254),
  cbind(-.5, 0, -.8660254)
)
 
# Use a default weights generator.
 
weights_generator <- weight_explanatory_points
 
# Set the concentration parameter (kappa).
 
concentration <- 5
 
# Fit regression.
 
fit_info <- fit_regression(
  evaluation_points,
  explanatory_points,
  response_points,
  concentration,
  weights_generator,
  number_of_expansion_terms = 1,
  number_of_iterations = 2
)

See the documentation for addressing additional scenarios.

Installation

To download and install the package from the CRAN repository, execute the following command:

install.packages("nprotreg")