weight_explanatory_points {nprotreg}

Weights the Specified Explanatory Points in a 3D Spherical Regression.

Description

Returns the weights assigned to the specified explanatory points for each evaluation point under study, given a concentration parameter.

Usage

weight_explanatory_points(evaluation_points, explanatory_points, concentration)

Arguments

evaluation_points

An n-by-3 matrix whose rows contain the Cartesian coordinates of the points on which the regression will be estimated.

explanatory_points

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

concentration

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

Details

Let X be the m-by-3 matrix of explanatory points, and E the n-by-3 matrix of evaluation points, and \kappa the concentration parameter. This function will return an m-by-n matrix whose (i,j) entry is defined as follows:

exp(\kappa (s(i,j) - 1))

where s(i,j) is the scalar product of the i-th row of X and the j-th row of E.

Value

An m-by-n matrix whose j-th column contains the weights assigned to the explanatory points while analyzing the j-th evaluation point.

See Also

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

Examples

library(nprotreg)

# Define a matrix of evaluation points.

north_pole <- cbind(0, 0, 1)
south_pole <- cbind(0, 0, -1)
evaluation_points <- rbind(north_pole, south_pole)

# Define a matrix of explanatory points

explanatory_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)
)

# Define a value for the concentration parameter.

concentration <- 1.0

# Get the corresponding 8-by-2 matrix of weights.
# Columns corresponds to evaluation points,
# rows to explanatory ones.

weights <- weight_explanatory_points(evaluation_points,
                                     explanatory_points,
                                     concentration)

# Get the weights assigned to the explanatory points
# while analyzing the second evaluation point.

cat("Weights assigned while analyzing the second evaluation point: \n")
cat(weights[, 2], "\n")