R/indep_MCAR_rgeneric.R

In INLAMSM: Multivariate Spatial Models with 'INLA'

#' @name inla.rgeneric.indep.MCAR.model
#' @rdname indepmcar
#'
#' @title \eqn{MCAR(\alpha, \Lambda)}: Proper multivariate CAR latent effect
#' without correlation parameters.
#'
#' @description Multivariate generalization of the proper conditional
#' autorregresive model with a common spatial autocorrelation parameter.
#' No correlation parameters are considered between the different diseases,
#' so the matrix which models the variability between diseases will be a
#' diagonal matrix.
#'
#' @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
#' diagonal 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 and the common spatial autocorrelation parameter. So the total
#' number of hyperpameters is equal to the number of diseases plus one.
#'
#' @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 independent MCAR model
#'model <- inla.rgeneric.define(inla.rgeneric.indep.MCAR.model,
#'                              debug = FALSE, k = k, W = W,
#'                              alpha.min = alpha.min, alpha.max = alpha.max)
#'
#'
#'#Fit model
#'r <- inla(OBS ~ 1 + f(idx, model = model), # + NWPROP,
#'          data = d, E = EXP, family = "poisson",
#'          control.predictor = list(compute = TRUE))
#'
#'summary(r)
#'
#' # Transformed parameters
#' r.hyperpar <- inla.MCAR.transform(r, k = 2, model = "INDPMCAR",
#'   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$summary.fitted.values[100 + 1:100, "mean"]
#'
#'#Display fitted relative risks
#'dev.new()
#'spplot(nc.sids, c("SMR74", "FITTED74", "SMR79", "FITTED79"))
#'
#'# Showing results
#'
#'#Show marginals of alpha, tau1, tau2
#'marg.alpha <- inla.tmarginal(
#'  function(x) alpha.min + (alpha.max - alpha.min) / (1 + exp(-x)),
#'  r$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$OBS)
#'
#'#Fit model
#'r.uni <- inla(OBS ~ 1 + f(idx, model = "besag", graph = W), # + NWPROP,
#'              data = d, E = EXP, family = "poisson",
#'              control.predictor = list(compute = TRUE))
#'
#'summary(r.uni)
#'
#'nc.sids$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$FITTED74, main="Relative Risk estimations",
#'       xlab="Univariate RR estimations"
#'     , ylab="Multivariate RR estimations")#, xlim=c(0.5, 2.5), ylim=c(0, 2))
#'abline(h=0, col="grey")
#'abline(v=0, col="grey")
#'abline(a=0, b=1, col="red")
#'
#'#Plot posterior mean of the spatial effects univ VS multi
#'
#'nc.sids$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")#, xlim=c(-1,1), ylim=c(-7,1))
#'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.indep.MCAR.model(cmd, theta)

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

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


  #theta: autocorrelation param. alpha, tau1, tau2, ..., tauk = k+1 hyperparams
  interpret.theta <- function()
  {
    # Function for changing from internal scale to external scale
    # also, build the precion matrix.

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

    # Next k parameters are the marginal precisions.
    mprec <- sapply(theta[as.integer(2:(k+1))], function(x) { exp(x) })

    # Diagonal precion matrix
    PREC <- diag(mprec, k)

    return (list(alpha = alpha, mprec = mprec, PREC = PREC))
  }


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

    PREC <- diag(1, 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

    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]))

    #Uniform priors on the standard deviations
    # log(constant_uniform) is ignored
     val <- val - sum(theta[as.integer(2:(k+1))]) / 2 - k * log(2)

    return (val)
  }

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

    #Initial values
    return ( c(0, rep(log(1), k)) )

  }

  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 indepmcar
##' @param ...  Arguments to be passed to 'inla.rgeneric.define'.
##' @export
##' @usage inla.INDMCAR.model(...)

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