source(here::here("R", "create_Q.R"))

calc_deriv_a <- function(k, i, matrix_X, a, K, qFa, fv){
  # Quantile levels
  q1 <- k / K
  q0 <- (k - 1) / K
  
  # F^{-1}_a at the two quantiles
  z1 <- qFa(q1)
  z0 <- qFa(q0)
  
  # a' X_j for all j
  aTX <- as.vector(matrix_X %*% a)
  
  # f_v(z - a'X_j)
  f1 <- fv(z1 - aTX)
  f0 <- fv(z0 - aTX)
  
  # X_j - X_i
  Xi <- matrix_X[i, ]
  XminusXi <- sweep(matrix_X, 2, Xi, "-")
  
  # compute fraction
  term1 <- (fv(z1 - aTX[i]) / sum(f1)) * colSums(XminusXi * f1)
  term0 <- (fv(z0 - aTX[i]) / sum(f0)) * colSums(XminusXi * f0)
  
  # Difference
  return(term1 - term0)
  
}
X <- matrix(seq(1,15, 1), ncol=3)
a <- c(1, -2, 1)
Fv <- pnorm
K <- 3

Fa <- make_distribution_func(a=a, X_matrix = X, Fv=Fv)
Q <- create_matrix_Q(Fa, a, K, Fv, matrix_X = X)

qFa <- make_quantile_function(Fa)

deriv <- calc_deriv_a(1, 1, X, a, K, qFa, dnorm)