# Load the general quantile function, the loading can be improved later
# using the here package.
source(here::here("R", "qinf.R"))

# 0. Analytical test case for d = 1 -----------------------------
#' --------------------------------------------------------------
#'  Fa(y) = ∫ Fv(y - z) fZ(z) dz   with  Z = a * X
#' --------------------------------------------------------------
#' Compute the graphon CDF for a scalar X and scalar coefficient a.
#'
#' @param y   numeric vector of points where Fa(y) is required.
#' @param a   numeric scalar (coefficient).  a != 0.
#' @param fX  function(z) returning the pdf of X (vectorised).
#' @param Fv  function(x) returning the CDF of the noise v (vectorised).
#' @param lower,upper numeric bounds for the integration over z.
#'        By default they are set to the (practical) support of X
#'        transformed by a.  You can tighten them for speed.
#' @param rel.tol relative tolerance passed to `integrate`.
#' @return numeric vector of the same length as y.
#' @examples
#' ## Example 1: X ~ Gamma(2,1), a = 0.8, v ~ N(0,1)
#' fX  <- function(z) dgamma(z, shape = 2, rate = 1)   # pdf of X
#' Fv  <- function(x) pnorm(x, mean = 0, sd = 1)      # cdf of v
#' a   <- 0.8
#' y   <- seq(-3, 6, length.out = 200)
#' Fa_vals <- Fa_one_dim(y, a, fX, Fv)
#' plot(y, Fa_vals, type = "l", col = "steelblue",
#'      main = "Fa(y) for Gamma X, Normal noise",
#'      xlab = "y", ylab = expression(F[a](y)))
#'
#' ## Example 2: both X and v are normal – analytic check
#' fX2 <- function(z) dnorm(z, mean = 1, sd = 2)
#' Fv2 <- function(x) pnorm(x, mean = 0, sd = 1)
#' a2  <- -1.5
#' y2  <- seq(-5, 8, length.out = 100)
#' Fa2_num <- Fa_one_dim(y2, a2, fX2, Fv2)
#' ## analytic result: Y = a*X + v ~ N(a*mu_X, a^2*sd_X^2 + sd_v^2)
#' muY  <- a2 * 1
#' sdY  <- sqrt(a2^2 * 2^2 + 1^2)
#' Fa2_ana <- pnorm(y2, mean = muY, sd = sdY)
#' max(abs(Fa2_num - Fa2_ana))   # should be ~ 1e-6
#' --------------------------------------------------------------
pgraphon_analytical_1d <- function(y, a, fX, Fv,
                       lower = -Inf, upper = Inf,
                       rel.tol = .Machine$double.eps^0.5) {
  stopifnot(is.numeric(a), length(a) == 1, a != 0)
  stopifnot(is.function(fX), is.function(Fv))
  stopifnot(is.numeric(y))
  
  # pdf of Z = a*X
  fZ <- function(z) {
    # Jacobian |a|^{-1} and argument scaling
    (1 / abs(a)) * fX(z / a)
  }
  
  # Helper that integrates for a single y
  one_y <- function(yy) {
    integrand <- function(z) Fv(yy - z) * fZ(z)
    integrate(integrand, lower = lower, upper = upper,
              rel.tol = rel.tol)$value
  }
  
  # Vectorised over the whole y vector
  vapply(y, one_y, numeric(1))
}
# 1. Distribution Function -----------------------------------------------------

#' Empirical Distribution function for the graphon model.
#'
#' Computes the empirical expectation
#' \deqn{E[F_v(y - a^\top X_i)]}
#' using observed covariate samples \eqn{X_i}. This is done by evaluating the
#' supplied CDF \code{Fv} at the shifted values \eqn{y - a^\top X_i} and taking
#' their empirical average. See also Remark 2.1
#'
#' @param y Numeric scalar or vector of evaluation points.
#' @param a Numeric vector. Coefficient vector used to compute the inner products
#'   \eqn{a^\top X_i}. Its length must match the number of columns in
#'   \code{X_matrix}.
#' @param Fv Function. Cumulative distribution function of the noise variable
#'   \eqn{v} (e.g., \code{pnorm}, \code{pexp}, etc.). Must accept a numeric vector
#'   and return a numeric vector of the same length.
#' @param X_matrix Numeric matrix of dimension \eqn{n \times p}, where each row
#'   corresponds to an observed covariate vector \eqn{X_i}.
#'
#' @return A numeric scalar giving the empirical expectation
#'   \eqn{(1/n) \sum_{i=1}^n F_v(y - a^\top X_i)}.
#'
#' @examples
#' set.seed(1)
#' X <- matrix(rnorm(100), ncol = 2)
#' a <- c(1, -0.5)
#' y <- 0
#' pgraphon(y, a, pnorm, X)
#'
#' @export
pgraphon <- function(
    y,           # scalar
    a,           # vector (coefficient vector, can be one-dimensional)
    Fv,          # CDF function of the v's (e.g., pnorm, pexp, etc.)
    X_matrix     # n x p matrix: each row is X_i
) {
  ## 1.1 Check inputs ==========================================================
  if (ncol(X_matrix) != length(a)) {
    stop("Number of columns in X_matrix must match length of a")
  }
  if (!is.numeric(y)) {
    stop("'y' must be numeric (scalar or vector)")
  }
  if (!is.numeric(a) || !is.vector(a)) {
    stop("'a' must be a numeric vector")
  }
  if (!is.matrix(X_matrix) || !is.numeric(X_matrix)) {
    stop("'X_matrix' must be a numeric matrix")
  }
  if (!is.function(Fv)) {
    stop("'Fv' must be a function (a CDF)")
  }
  ## 1.2 Compute the expected value ============================================
  
  # Compute a^T X_i as in the paper. Here we switched to the R convention that each
  # observation is a row. The paper assumes that each observation is a vector
  inner_products <- as.vector(X_matrix %*% a)  # vector of length n
  
  # outer(y, inner_prod, "-") creates an |y| × n matrix where element (j,i)
  # equals y[j] - inner_prod[i].
  dev_mat <- outer(y, inner_products, "-")
  
  # Apply CDF Fv to each deviation
  cdf_vals <- Fv(dev_mat)
  
  # Row means give the empirical expectation for each y_j
  out <- rowMeans(cdf_vals)
  out
}

# 2. Density Function ----------------------------------------------------------

#' Empirical Graphon Density Estimate
#'
#' Computes the empirical expectation \eqn{E[f_v(y - a^\top X_i)]} using observed
#' covariate samples \eqn{X_i}. This serves as an empirical approximation of the
#' graphon density evaluated at \eqn{y}, where \eqn{f_v} is the density of the
#' latent variable \eqn{v}. See also Remark 2.1
#'
#' @param y Numeric scalar or vector of points at which the density should be 
#'   evaluated.
#' @param a Numeric vector. Coefficient vector used to compute the linear
#'   projection \eqn{a^\top X_i}. Its length must match the number of columns in
#'   \code{X_matrix}.
#' @param fv Function. Density function of the latent variable \eqn{v} (e.g.,
#'   \code{dnorm}, \code{dexp}). Must accept a numeric vector and return a
#'   numeric vector of the same length.
#' @param X_matrix Numeric matrix of size \eqn{n \times p}. Each row represents
#'   an observed covariate vector \eqn{X_i}. The number of columns must match the
#'   length of \code{a}.
#'
#' @return A numeric scalar giving the empirical average
#'   \eqn{\frac{1}{n} \sum_{i=1}^n f_v(y - a^\top X_i)}.
#'
#' @examples
#' set.seed(1)
#' X <- matrix(rnorm(50), ncol = 2)
#' a <- c(1, -0.5)
#' dgraphon(y = 0.2, a = a, fv = dnorm, X_matrix = X)
#'
#' @export
dgraphon <- function(
    y,           # scalar
    a,           # vector (coefficient vector, can be )
    fv,          # density function of the v's (e.g., dnorm, dexp, etc.)
    X_matrix     # n x p matrix: each row is X_i
) {
  ## 2.1 Check inputs ----------------------------------------------------------
  if (!is.numeric(y)) {
    stop("'y' must be numeric (scalar or vector)")
  }
  if (!is.numeric(a) || !is.vector(a)) {
    stop("'a' must be a numeric vector")
  }
  if (!is.matrix(X_matrix) || !is.numeric(X_matrix)) {
    stop("'X_matrix' must be a numeric matrix")
  }
  if (ncol(X_matrix) != length(a)) {
    stop("Number of columns in X_matrix must match length of a")
  }
  if (!is.function(fv)) {
    stop("'fv' must be a function (a density)")
  }
  
  ## 1.2 Compute the expected value ============================================
  
  # Compute a^T X_i as in the paper. Here we switched to the R convention that each
  # observation is a row. The paper assumes that each observation is a vector
  inner_products <- as.vector(X_matrix %*% a)  # vector of length n
  
  # outer(y, inner_prod, "-") creates an |y| × n matrix where element (j,i)
  # equals y[j] - inner_prod[i].
  dev_mat <- outer(y, inner_products, "-")
  
  # Apply the density function to each y[j]
  dens_vals <- fv(dev_mat)
  
  # Row means give the empirical expectation for each y_j
  out <- rowMeans(dens_vals)
  out
}

# 3. Quantile Function ---------------------------------------------------------

#' Quantile of the (empirical or analytic) graphon distribution
#'
#' If the covariate matrix has a single column and the coefficient vector
#' has length 1, the function can use an *analytic* CDF (provided the user
#' supplies the pdf of the covariate via `fX`).  Otherwise it falls back to the
#' original empirical estimator `pgraphon`.
#'
#' @param p Numeric vector of probabilities in \eqn{[0,1]}.
#' @param a Numeric coefficient vector (length = ncol(X_matrix)).
#' @param Fv Function. CDF of the latent variable $v$ (vectorised).
#' @param X_matrix Numeric matrix of size $n\times p$; each row is an observation.
#' @param fX Optional function(z) returning the *known* pdf of the *scalar*
#'            covariate when `ncol(X_matrix)==1 && length(a)==1`.  If supplied,
#'            the analytic CDF `pgraphon_analytical_1d` is used.
#' @param lower,upper Numeric bounds for the root‑finding algorithm.
#' @param tol Numeric tolerance for the bisection inside `qinf`.
#' @param max.iter Maximum number of bisection iterations.
#' @param ... Additional arguments passed to the internal CDF (`pgraphon` or the
#'            analytic version).  For the analytic case you may want to pass
#'            `lower`, `upper`, or `rel.tol`.
#' @return Numeric vector of quantiles, named by the probabilities.
#' @examples
#' ## 1. Empirical case
#' set.seed(123)
#' X <- matrix(rnorm(200), ncol = 2)
#' a <- c(0.7, -0.3)
#' qgraphon(p = c(0.25, 0.5, 0.75), a = a,
#'          Fv = pnorm, X_matrix = X)
#'
#' ## 2. Analytic 1‑D case
#' ## X ~ Gamma(2,1), a = 0.8, v ~ N(0,1)
#' fX  <- function(z) dgamma(z, shape = 2, rate = 1)
#' a1  <- 0.8
#' X1  <- matrix(rnorm(100), ncol = 1)   # dummy – not used by the analytic branch
#' qgraphon(p = c(0.1, 0.5, 0.9), a = a1,
#'          Fv = pnorm, X_matrix = X1,
#'          fX = fX, lower = -10, upper = 20)
#' @export
qgraphon <- function(p, a, Fv, X_matrix,
                     fX = NULL,               # <- new optional argument
                     lower = -Inf, upper = Inf,
                     tol = .Machine$double.eps^0.5,
                     max.iter = 100, ...) {
  
  ## ---- input checks -------------------------------------------------
  if (!is.numeric(p) || any(is.na(p))) {
    stop("'p' must be a numeric vector without NA values")
  }
  if (any(p < 0 | p > 1)) {
    stop("All probabilities in 'p' must lie in [0, 1]")
  }
  if (!is.numeric(a) || !is.vector(a)) {
    stop("'a' must be a numeric vector")
  }
  if (!is.matrix(X_matrix) || !is.numeric(X_matrix)) {
    stop("'X_matrix' must be a numeric matrix")
  }
  if (ncol(X_matrix) != length(a)) {
    stop("Number of columns of 'X_matrix' (", ncol(X_matrix),
         ") must equal length of 'a' (", length(a), ")")
  }
  if (!is.function(Fv)) {
    stop("'Fv' must be a function (the CDF of the latent variable)")
  }
  
  ## ---- decide which CDF to use --------------------------------------
  use_analytic <- (ncol(X_matrix) == 1L) && (length(a) == 1L) && (!is.null(fX))
  
  if (use_analytic) {
    # Build a *wrapper* that has the same signature as pgraphon()
    # (y, a, Fv, X_matrix, ...) so that qinf() can call it unchanged.
    analytic_cdf <- function(y, a, Fv, X_matrix, ...) {
      # X_matrix is ignored – the distribution of X is supplied via fX
      pgraphon_analytical_1d(y = y,
                             a = a,
                             fX = fX,
                             Fv = Fv,
                             lower = lower,
                             upper = upper,
                             rel.tol = tol,
                             ...)
    }
    CDF_to_use <- analytic_cdf
  } else {
    # Fall back to the empirical estimator 
    CDF_to_use <- pgraphon
  }
  
  ## ---- call the generic quantile routine ----------------------------
  out <- qinf(F = CDF_to_use,
              p = p,
              lower = lower,
              upper = upper,
              tol = tol,
              max.iter = max.iter,
              a = a,
              X_matrix = X_matrix,
              Fv = Fv,
              ...)   # forward any extra arguments (e.g., rel.tol)
  
  names(out) <- as.character(p)
  out
}

# 4. Create conditional density ------------------------------------------------

#' Create a Conditional Density Function for the Graphon Model.
#'
#' This function constructs and returns another function that computes the
#' conditional density of the graphon model at specified values.  
#' The returned function takes inputs \code{u} (probabilities) and \code{x}
#' (a scalar covariate value) and evaluates the graphon-based conditional
#' density using the supplied coefficient vector \code{a}, the density
#' \code{fv}, the distribution function \code{Fv}, and the empirical sample
#' \code{X_matrix}.
#'
#' @param a Numeric vector. Coefficient vector used in the graphon model; must
#'   have length equal to the number of columns in \code{X_matrix}.
#' @param fv Function. Density function of the latent variable \eqn{v}
#'   (e.g., \code{dnorm}, \code{dexp}).
#' @param Fv Function. Distribution function (CDF) corresponding to \code{fv}
#'   (e.g., \code{pnorm}, \code{pexp}).
#' @param X_matrix Numeric matrix. An \eqn{n \times p} matrix of covariates
#'   \eqn{X_i}, where each row corresponds to an observation.
#'
#' @return A function of two arguments \code{u} and \code{x}.  
#'   The returned function is vectorized in \code{u} and returns the conditional
#'   density:
#'   \deqn{ f_{U \mid X}(u \mid x) = \frac{f_v(q(u) - a^\top x)}{
#'           \mathbb{E}[f_v(q(u) - a^\top X_i)] },
#'   }
#'   where \eqn{q(u)} is computed via \code{qgraphon()}.
#'
#' @details
#' Internally, the function constructs a scalar evaluator
#' \code{scalar_conditional_density()} that computes the conditional density for a
#' single value of \code{u}.  
#' A wrapper function \code{conditional_density()} then applies this scalar
#' function to each element of the input vector \code{u}.
#'
#' Note: the function relies on the existence of \code{qgraphon()},
#' \code{pgraphon()}, and \code{dgraphon()} in the user's environment.
#'
#' @examples
#' \dontrun{
#' a <- c(1, -0.5)
#' fv <- dnorm
#' Fv <- pnorm
#' X_matrix <- matrix(rnorm(200), ncol = 2)
#'
#' cond_dens <- create_cond_density(a, fv, Fv, X_matrix)
#' cond_dens(c(0.2, 0.5, 0.8), x = 1.0)
#' }
#'
#' @export
create_cond_density <- function(
    a,
    fv,
    Fv,
    X_matrix){
  
  scalar_conditional_density <-function(u, x) {
    q_a  <- qgraphon(u, a, Fv, X_matrix)
    num  <- fv(q_a - x %*% a)
    den  <- dgraphon(q_a, a, fv, X_matrix)
    num / den
  }
  
  conditional_density <- function(u, x) {
    vapply(u, scalar_conditional_density, numeric(1), x = x)
  }
  
  return(conditional_density)
}