R/MCAR_rgeneric.R

In INLAMSM: Multivariate Spatial Models with 'INLA'

#' @name inla.rgeneric.MCAR.model
#' @rdname mcar
#'
#' @title \eqn{MCAR(\alpha, \Lambda)}: Proper multivariate CAR latent effect
#' with a common autorcorrelation parameter.
#'
#' @description Multivariate generalization of the proper conditional
#' autorregresive model with one common correlation parameter. The matrix
#' which models the variability between diseases is a symmetric matrix with
#' the inverse of the marginal precisions on the diagonal elements and the
#' correlation parameters divided by the square root of the precisions on
#' the off-diagonal elements.
#'
#' @param cmd Arguments used by latent effects defined using the 'rgeneric'
#' latent effect.
#'
#' @param theta Vector of hyperparameters.
#'
#' @return This is used internally by the 'INLA::inla()'.
#'
#' @details This function is used to define a latent effect that is a
#' multivariate spatial effect with a proper conditional autorregresive
#' distribution (with a common spatial autocorrelation parameter) and
#' a symmetric matrix in order to model the whitin-disease and the
#' between-diseases variability, respectively. Due to this effect is a
#' multivariate spatial latent effect this function requires the following
#' arguments when defining the latent effect:
#' \itemize{
#'
#'   \item \emph{W} Adjacency SPARSE matrix for spatial effect in the basic
#'   binary code.
#'
#'   \item \emph{k} Number of diseases of the multivariate study.
#'
#'   \item \emph{alpha.min} Minimum value of the spatial autocorrelation
#'   parameter.
#'
#'   \item \emph{alpha.max} Maximum value of the spatial autocorrelation
#'   parameter.
#'
#'}
#'
#' This model is defined using the 'f()' function and an index in order to
#' identify the spatial areas. See the example.
#'
#' @references Palmí-Perales F, Gómez-Rubio V, Martinez-Beneito MA (2021). “Bayesian
#' Multivariate Spatial Models for Lattice Data with INLA.” _Journal of
#' Statistical Software_, *98*(2), 1-29. doi: 10.18637/jss.v098.i02 (URL:
#' https://doi.org/10.18637/jss.v098.i02).
#'
#' @section Prior distributions of the hyperparameters:
#' The hyperparamenters of this lattent effect are the marginal precisions of
#' each disease which are equal to the number of diseases, the correlation
#' parameters for the whole pair of diseases and the common spatial
#' autocorrelation parameter.
#'
#' @examples
#'
#' \donttest{
#' if (require("INLA", quietly = TRUE)) {
#' require(spdep)
#' require(spData)
#' require(rgdal)
#'
#' #Load SIDS data
#' nc.sids <- readOGR(system.file("shapes/sids.shp", package="spData")[1])
#' proj4string(nc.sids) <- CRS("+proj=longlat +ellps=clrk66")
#'
#' #Compute adjacency matrix, as nb object 'adj' and sparse matrix 'W'
#' adj <- poly2nb(nc.sids)
#' W <- as(nb2mat(adj, style = "B"), "Matrix")
#'
#' #Compute expected cases
#' r74 <- sum(nc.sids$SID74) / sum(nc.sids$BIR74)
#' nc.sids$EXP74 <- r74 * nc.sids$BIR74
#' nc.sids$SMR74 <- nc.sids$SID74 / nc.sids$EXP74
#' nc.sids$NWPROP74 <- nc.sids$NWBIR74 / nc.sids$BIR74
#'
#' r79 <- sum(nc.sids$SID79) / sum(nc.sids$BIR79)
#' nc.sids$EXP79 <- r79 * nc.sids$BIR79
#' nc.sids$SMR79 <- nc.sids$SID79 / nc.sids$EXP79
#' nc.sids$NWPROP79 <- nc.sids$NWBIR79 / nc.sids$BIR79
#'
#' # Data (replicated to assess scalability)
#'
#' #Real data
#' n.rep <- 1
#' d <- list(OBS = c(nc.sids$SID74, nc.sids$SID79),
#'           NWPROP = c(nc.sids$NWPROP74, nc.sids$NWPROP79),
#'           EXP = c(nc.sids$EXP74, nc.sids$EXP79))
#' d <- lapply(d, function(X) { rep(X, n.rep)})
#' d$idx <- 1:length(d$OBS)
#'
#' # Model parameters
#' k <- 2 * n.rep #Number of diseases
#' alpha.min <- 0
#' alpha.max <- 1
#'
#' #Define MCAR model
#' #model <- inla.rgeneric.define(inla.rgeneric.MCAR.model, debug = FALSE,
#' #  k = k, W = W, alpha.min = alpha.min, alpha.max = alpha.max)
#' model <- inla.MCAR.model(k = k, W = W, alpha.min = alpha.min, alpha.max = alpha.max)
#'
#'
#' #Fit model
#' r <- inla(OBS ~ 1 + f(idx, model = model),
#'   data = d, E = EXP, family = "poisson",
#'   control.compute = list(config = TRUE),
#'   control.predictor = list(compute = TRUE))
#'
#' summary(r)
#'
#' # Transformed parameters
#' r.hyperpar <- inla.MCAR.transform(r, k = 2, model = "PMCAR",
#'   alpha.min = alpha.min, alpha.max = alpha.max)
#' r.hyperpar$summary.hyperpar
#'
#' #Get fitted data, i.e., relative risk
#' nc.sids$FITTED74 <- r$summary.fitted.values[1:100, "mean"]
#' nc.sids$FITTED79 <- r$summary.fitted.values[100 + 1:100, "mean"]
#'
#' #Display fitted relative risks
#' dev.new()
#' spplot(nc.sids, c("SMR74", "FITTED74", "SMR79", "FITTED79"))
#'
#'
#' # Showing results of the MCAR: multivariate proper CAR.
#'
#' #Show marginals of alpha, tau1, tau2
#' marg.alpha <- inla.tmarginal(
#'   function(x) alpha.min + (alpha.max - alpha.min) / (1 + exp(-x)),
#'   r$marginals.hyperpar[[1]])
#'
#' marg.tau1 <- inla.tmarginal(
#'   function(x) exp(x),
#'   r$marginals.hyperpar[[2]])
#'
#' marg.tau2 <- inla.tmarginal(
#'   function(x) exp(x),
#'   r$marginals.hyperpar[[3]])
#'
#' dev.new()
#'
#' oldpar <- par(mfrow = c(2, 2))
#'
#' plot(marg.alpha, main="alpha", type="l")
#' plot(marg.tau1, main = "tau1", type = "l")
#' plot(marg.tau2, main = "tau2", type = "l")
#'
#' par(oldpar)
#'
#' ## Running UNIVARIATE MODEL
#'
#' #Real data
#' n.rep <- 1
#' d <- list(OBS = nc.sids$SID74,
#'           NWPROP = nc.sids$NWPROP74,
#'           EXP = nc.sids$EXP74)
#' d <- lapply(d, function(X) { rep(X, n.rep)})
#' d$idx <- 1:length(d$OBS)
#'
#' #Fit model
#' r.uni <- inla(OBS ~ 1 + f(idx, model = "besag", graph = W),
#'               data = d, E = EXP, family = "poisson",
#'               control.predictor = list(compute = TRUE))
#'
#' summary(r.uni)
#'
#' nc.sids$FITTED74.uni <- r.uni$summary.fitted.values[ , "mean"]
#'
#' #Display univariate VS multivariate  fitted relative risks.
#' dev.new()
#' spplot(nc.sids, c("SMR74", "FITTED74", "FITTED74.uni"))
#' spplot(nc.sids, c("FITTED74", "FITTED74.uni"),
#'        main=list(label="Relative risk estimation",cex=2))
#' dev.new()
#' plot(nc.sids$FITTED74.uni, nc.sids$FITTED74,
#'      main="Relative Risk estimations", xlab="Univariate RR estimations"
#'      , ylab="Multivariate RR estimations")
#' abline(h=0, col="grey")
#' abline(v=0, col="grey")
#' abline(a=0, b=1, col="red")
#'
#' #Plot posterior mean of the spatial effects univ VS multi
#'
#' nc.sids$m.uni <- r.uni$summary.random$idx[, "mean"]
#' nc.sids$m.mult <- r$summary.random$idx[1:100, "mean"]
#' dev.new()
#' plot(nc.sids$m.uni, nc.sids$m.mult,
#'      main="Posterior mean of the spatial effect",
#'      xlab="Uni. post. means", ylab="Mult. post. means")
#' abline(h=0, col="grey")
#' abline(v=0, col="grey")
#' abline(a=0, b=1, col="red")
#'
#' dev.new()
#' spplot(nc.sids, c("m.mult", "m.uni"),
#'        main=list(label="Post. mean spatial effect",cex=2))
#' }
#' }
#'
#' @export
#' @usage inla.rgeneric.MCAR.model(cmd, theta)

# Define previous variables as global to avoid warnings()
utils::globalVariables(c("k", "W", "alpha.min", "alpha.max"))


'inla.rgeneric.MCAR.model' <-
  function(cmd = c("graph", "Q", "mu", "initial", "log.norm.const",
                   "log.prior", "quit"), theta = NULL)
{
  ## MCAR implementation MCAR(alpha, Lambda) ->
  ## ->  PROPER CAR, common alpha, dense Lambda
  ## k: number of diseases/blocks
  ## W: adjacency matrix
  ## alpha.min: minimum value for alpha
  ## alpha.max: maximum value for alpha


  #theta: 1 common correlation parameter alpha,
  #        (k + 1) * k / 2  for lower-tri matrix by col.
  interpret.theta <- function()
  {
    #Function for changing from internal scale to external scale
    # also, build the inverse of the matrix used to model in the external scale
    # the between-disease variability and then this matrix is inverted.

    # First parameter is the common autocorrelation parameter
    alpha <- alpha.min + (alpha.max - alpha.min) / (1 + exp(-theta[1L]))

    # The next k parameters are the marginal precisions,
    # the other parameters are the correlation parameters ordered by columns.
    mprec <- sapply(theta[as.integer(2:(k+1))], function(x) { exp(x) })
    corre <- sapply(theta[as.integer(-(1:(k+1)))], function(x) {
      (2 * exp(x))/(1 + exp(x)) - 1 })

    param <- c(alpha, mprec, corre)

    #length non-diagonal elements
    n <- (k - 1) * k / 2

    # intial matrix with 1s at the diagonal
    M <- diag(1, k)

    #Adding correlation parameters (lower.tri) and (upper.tri)
    M[lower.tri(M)] <- param[k + 2:(n+1)]
    M[upper.tri(M)] <- t(M)[upper.tri(M)]

    #Preparing the st. dev matrix
    st.dev <- 1 / sqrt(param[2:(k+1)])

    # Matrix of st. dev.
    st.dev.mat <- matrix(st.dev, ncol = 1) %*% matrix(st.dev, nrow = 1)

    # Final inversed matrix
    M <- M * st.dev.mat

    # Inverting matrix
    #PREC <- MASS::ginv(M)# Generalized inverse
    PREC <- solve(M)

    return (list(alpha = alpha, param = param, VACOV = M, PREC = PREC))
  }


  #Graph of precision function; i.e., a 0/1 representation of precision matrix
  graph <- function()
  {

    PREC <- matrix(1, ncol = k, nrow = k)
    G <- kronecker(PREC, Matrix::Diagonal(nrow(W), 1) + W)
    return (G)
  }

  #Precision matrix
  Q <- function()
  {
    #Parameters in model scale
    param <- interpret.theta()

    #Precision matrix
    Q <- kronecker(param$PREC,
                   Matrix::Diagonal(nrow(W), apply(W, 1, sum)) - param$alpha * W
                   )

    return (Q)
  }

  #Mean of model
  mu <- function() {
    return(numeric(0))
  }

  log.norm.const <- function() {
    ## return the log(normalising constant) for the model
    #param = interpret.theta()
    #
    #val = n * (- 0.5 * log(2*pi) + 0.5 * log(prec.innovation)) +
    #0.5 * log(1.0 - param$alpha^2)

    val <- numeric(0)
    return (val)
  }

  log.prior <- function() {
    ## return the log-prior for the hyperparameters.
    ## Uniform prior in (alpha.min, alpha.max) on model scale
    param <- interpret.theta()

    # log-Prior for the autocorrelation parameter
    val <- - theta[1L] - 2 * log(1 + exp(-theta[1L]))

    # Whishart prior for joint matrix of hyperparameters
    val <- val + 
      log(MCMCpack::dwish(W = param$PREC, v =  k, S = diag(rep(1, k)))) +
      sum(theta[as.integer(2:(k + 1))]) +  # This is for precisions
      sum(log(2) + theta[-as.integer(1:(k + 1))] - 2 * log(1 + exp(theta[-as.integer(1:(k + 1))])))  # This is for correlation terms

    return (val)
  }

  initial <- function() {
    ## return initial values

    # The Initial values form a diagonal matrix
    return ( c(0, rep(log(1), k), rep(0, (k * (k - 1) / 2))))

  }

  quit <- function() {
    return (invisible())
  }

  # FIX for rgeneric to work on R >= 4
  # Provided by E. T. Krainski
  if (as.integer(R.version$major) > 3) {
    if (!length(theta))
      theta = initial()
  } else {
    if (is.null(theta)) {
      theta <- initial()
    }
  }


  val <- do.call(match.arg(cmd), args = list())
  return (val)
  }

##' @rdname mcar
##' @param ...  Arguments to be passed to 'inla.rgeneric.define'.
##' @export
##' @usage inla.MCAR.model(...)

inla.MCAR.model <- function(...) {
  INLA::inla.rgeneric.define(inla.rgeneric.MCAR.model, ...)
}