R/smoothing.R

In geoFKF: Kriging Method for Spatial Functional Data

#' Smoothed curve in Fourier Series.
#'
#' This function computes the smoothed curve using Fourier coefficients.
#'
#' @param coef Fourier coefficients.
#' @param x a time series to evaluate the smoothed curve.
#'
#' @return a time series with the smoothed curve.
#' @export
#' @examples
#' v_coef <- rnorm(23)
#' fourier_b(v_coef)
fourier_b <- function(coef, x) {
  if (missing(x)) x <- seq(from = -pi, to = pi, by = 0.01)

  #degree of Fourier Series
  ordem <- (length(coef) - 1) / 2

  apply(matrix(x, ncol = 1), 1, function(u) {
      valor <- coef[1] / 2
      for (k in 1:ordem) valor <- valor +
          coef[k + 1] * cos(k * u) + coef[k + ordem + 1] * sin(k * u)
      return(valor)
    })
}

#' Computing coefficients Fourier.
#'
#' This function computes minimum square estimates for Fourier coefficients.
#'
#' @param f A time series to be smoothed.
#' @param m Order of the Fourier polynomial. Default value is computed using
#' the Sturge's rule.
#'
#' @return A vector with the fourier coefficients.
#' @export
#' @examples
#' x <- seq(from = -pi, to = pi, by = 0.01)
#' y <- x^2 + rnorm(length(x), sd = 0.1)
#' v_coef <- coef_fourier(y)
coef_fourier <- function(f, m) {
  #m: ordem in the fourier approximation
  #f: time series to smooth

  if (missing(m)) m <- ceiling(1 + log2(length(f)))

  n <- length(f) #length of time series

  u <- pi + 2 * pi * (((1:n) - 1) / n - 1)

  m_a <- matrix(0, nrow = 2 * m + 1, ncol = 2 * m + 1)
  b <- matrix(0, nrow = 2 * m + 1, ncol = 1)

  #filling A
  #a_0
  m_a[1, 1] <- n / 2
  for (k in 1:n) m_a[1, 2:(m + 1)] <- m_a[1, 2:(m + 1)] + cos((1:m) * u[k])
  for (k in 1:n) m_a[1, (m + 2):(2 * m + 1)] <- m_a[1, (m + 2):(2 * m + 1)] +
    sin((1:m) * u[k])
  #a_1, ..., a_m and b_1, ..., b_m
  for (j in 1:m) {
    m_a[j + 1, 1] <- sum(cos(u) * j / 2)

    m_a[m + 1 + j, 1] <- sum(sin(u) * j / 2)
    for (k in 1:n) {
      m_a[j + 1, 2:(m + 1)] <- m_a[j + 1, 2:(m + 1)] + cos((1:m) * u[k]) *
        cos(j * u[k])
      m_a[j + 1, (m + 2):(2 * m + 1)] <- m_a[j + 1, (m + 2):(2 * m + 1)] +
        sin((1:m) * u[k]) * cos(j * u[k])

      m_a[m + j + 1, 2:(m + 1)] <- m_a[m + j + 1, 2:(m + 1)] +
        cos((1:m) * u[k]) *  sin(j * u[k])
      m_a[m + j + 1, (m + 2):(2 * m + 1)] <- m_a[m + j + 1,
                                                 (m + 2):(2 * m + 1)] +
        sin((1:m) * u[k]) * sin(j * u[k])
    }
  }

  #filling b
  b[1] <- sum(f)
  for (k in 1:n)  b[2:(m + 1)] <- b[2:(m + 1)] + f[k] * cos((1:m) * u[k])
  for (k in 1:n)  b[(m + 2):(2 * m + 1)] <- b[(m + 2):(2 * m + 1)] + f[k] *
    sin((1:m) * u[k])

  #solving the linear system
  solve(m_a, b)
}