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


# 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 graphon distribution
#'
#' This is a thin wrapper around the generic infimum‑type quantile routine
#' \code{qinf()}.  It builds the CDF of the graphon,
#' \eqn{p_{\text{graphon}}(x) = \frac{1}{n}\sum_{i=1}^{n}F_v\bigl(x-a^\top X_i\bigr)},
#' and then finds the quantile(s) for the supplied probability(ies) \code{p}.
#'
#' @param p Numeric vector of probabilities in \eqn{[0,1]}.  Values outside this
#'   interval are rejected.
#' @param a Numeric coefficient vector.  Its length must equal the number of
#'   columns of \code{X_matrix}.
#' @param Fv Function.  The CDF of the latent variable $v$.  It must be
#'   vectorised (i.e. accept a numeric vector and return a numeric vector of
#'   the same length).  Typical examples are \code{pnorm}, \code{pexp}, etc.
#' @param X_matrix Numeric matrix of dimension \eqn{n\times p}.  Each row
#'   corresponds to an observation $X_i$.
#' @param lower,upper Numeric scalars giving the search interval for the root
#'   finder.  By default they are set to \code{-Inf} and \code{Inf}.
#' @param tol Numeric tolerance for the bisection algorithm used inside
#'   \code{qinf()}.  The default is the square‑root of machine epsilon.
#' @param max.iter Maximum number of bisection iterations (safety guard).
#' @param ... Additional arguments that are passed **directly** to the
#'   internal CDF \code{pgraphon}.  This makes the wrapper flexible – you do not
#'   have to list every argument (e.g. \code{a}, \code{X_matrix}, \code{Fv})
#'   explicitly.
#'
#' @return A numeric vector of the same length as \code{p} containing the
#'   quantiles.  The vector is named by the probabilities.
#'
#' @examples
#' ## ---- simple normal example ------------------------------------------------
#' set.seed(123)
#' X <- matrix(rnorm(200), ncol = 2)          # n = 100, p = 2
#' a <- c(0.7, -0.3)
#' qgraphon(p = c(0.25, 0.5, 0.75),
#'          a = a,
#'          Fv = pnorm,
#'          X_matrix = X)
#'
#' ## ---- mixture example ------------------------------------------------------
#' mix_cdf <- function(x, w = 0.4, mu = 0, sigma = 1) {
#'   w * (x >= 0) + (1 - w) * pnorm(x, mean = mu, sd = sigma)
#' }
#' qgraphon(p = seq(0, 1, 0.2),
#'          a = a,
#'          Fv = mix_cdf,
#'          X_matrix = X)
#'
#' @export
qgraphon <- function(
    p,                               # probabilities
    a,                               # coefficient vector
    Fv,                              # CDF of the v's
    X_matrix,                         # X_i samples, matrix
    lower = -Inf,                     # lower bound roots
    upper =  Inf,                     # upper bound roots
    tol   = .Machine$double.eps^0.5,  # parameter root finding
    max.iter = 100) {
  
  ## 3.1 Check inputs ----------------------------------------------------------
  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)")
  }
  
  ## 3.2 Call the generic quantile function ------------------------------------
  out <- qinf(F = pgraphon,
            p = p,
            lower = lower,
            upper = upper,
            tol = tol,
            max.iter = max.iter,
            a = a,
            X_matrix = X_matrix,
            Fv = Fv)
  
  # Name the result by the probabilities – this mirrors the behaviour of
  # base‑R quantile functions (e.g. qnorm, qbeta).
  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)
}