cross_validate_concentration {nprotreg}

Cross-validates The Concentration Parameter In A 3D Spherical Regression.

Description

Returns a cross-validated value for the concentration parameter in a 3D regression, relating specific explanatory points to response ones, given a weighting scheme for the observed data set. This function supports the method for sphere-sphere regression proposed by Di Marzio et al. (2018).

Usage

cross_validate_concentration(
  concentration_upper_bound = 10,
  explanatory_points,
  response_points,
  weights_generator = weight_explanatory_points,
  number_of_expansion_terms = 1,
  number_of_iterations = 1,
  allow_reflections = FALSE
)

Arguments

concentration_upper_bound

A scalar numeric value representing the upper end-point of the interval to be searched for the required minimizer. Defaults to 10.

explanatory_points

An m-by-3 matrix whose rows contain the Cartesian coordinates of the explanatory points used to calculate the regression estimators.

response_points

An m-by-3 matrix whose rows contain the Cartesian coordinates of the response points corresponding to the explanatory points.

weights_generator

A function that, given a matrix of n evaluation points, returns an m-by-n matrix whose j-th column contains the weights assigned to the explanatory points while analyzing the j-th evaluation point. Defaults to weight_explanatory_points.

number_of_expansion_terms

The number of terms to be included in the expansion of the matrix exponential applied while approximating a local rotation matrix. Must be 1 or 2. Defaults to 1.

number_of_iterations

The number of rotation fitting steps to be executed. At each step, the points estimated during the previous step are exploited as the current explanatory points. Defaults to 1.

allow_reflections

A logical scalar value. If set to TRUE signals that reflections are allowed. Defaults to FALSE. It is ignored if number_of_expansion_terms is 2.

Details

Function weights_generator must be prototyped as having the following three arguments:

evaluation_points

a matrix whose n rows are the Cartesian coordinates of given evaluation points.

explanatory_points

a matrix whose m rows are the Cartesian coordinates of given explanatory points.

concentration

A non negative scalar whose reciprocal value is proportional to the bandwidth applied while estimating a spherical regression model.

It is also expected that weights_generator will return a non NULL numerical m-by-n matrix whose j-th column contains the weights assigned to the explanatory points while analyzing the j-th evaluation point.

Value

A list having two components, concentration, a scalar, numeric value representing the cross-validated concentration for the specified 3D regression, and objective, the value of the cross-validating objective function at argument concentration.

References

Marco Di Marzio, Agnese Panzera & Charles C. Taylor (2018) Nonparametric rotations for sphere-sphere regression, Journal of the American Statistical Association, <doi:10.1080/01621459.2017.1421542>.

See Also

Other Regression functions: fit_regression(), get_equally_spaced_points(), get_skew_symmetric_matrix(), simulate_regression(), simulate_rigid_regression(), weight_explanatory_points()

Examples

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.

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]))
}

local_error_sampler <- function(point) {
  rnorm(3)
}

response_points <- simulate_regression(explanatory_points,
                                       local_rotation_composer,
                                       local_error_sampler)

# Define an upper bound for concentration.

concentration_upper_bound <- 1

# Use default weights generator.

weights_generator <- weight_explanatory_points

# Cross-validate concentration parameter.

cv_info <- cross_validate_concentration(
  concentration_upper_bound,
  explanatory_points,
  response_points,
  weights_generator,
  number_of_expansion_terms = 1,
  number_of_iterations = 2,
  allow_reflections = FALSE
)

# Get the cross-validated concentration value.

cat("cross-validated concentration value: \n")
print(cv_info$concentration)