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#' xlin_fits_R
#'
#' Calculate out of sample linear fit predictions.
#'
#' @param x NumericVector, x-coords of values to group (length>=2).
#' @param y NumericVector, values to group in order.
#' @param w NumericVector, weights (positive).
#' @return vector of predictions.
#'
#' @keywords internal
#'
#' @examples
#'
#' xlin_fits_V(c(1, 2, 3, 4), c(1, 2, 2, 1), c(1, 1, 1, 1))
#'
#' @export
#'
xlin_fits_V <- function(x, y, w) {
n = length(y)
# build fitting data
regularization = 1.0e-5
xx_0_0 = numeric(n) + sum(w*1)
xx_1_0 = numeric(n) + sum(w*x)
xx_0_1 = numeric(n) + sum(w*x)
xx_1_1 = numeric(n) + sum(w*x*x)
xy_0 = numeric(n) + sum(w*y)
xy_1 = numeric(n) + sum(w*x*y)
xx_1_0 = xx_1_0 + regularization
xx_0_1 = xx_0_1 + regularization
# pull out k'th observation
xxk_0_0 = xx_0_0 - w*1
xxk_1_0 = xx_1_0 - w*x
xxk_0_1 = xx_0_1 - w*x
xxk_1_1 = xx_1_1 - w*x*x
xyk_0 = xy_0 - w*y
xyk_1 = xy_1 - w*x*y
# solve linear system
det = xxk_0_0*xxk_1_1 - xxk_0_1*xxk_1_0
c0 = (xxk_1_1*xyk_0 - xxk_0_1*xyk_1)/det
c1 = (-xxk_1_0*xyk_0 + xxk_0_0*xyk_1)/det
# form estimate
y_est = c0 + c1*x
return(y_est)
}