R/medists.R

In msm: Multi-State Markov and Hidden Markov Models in Continuous Time

#' Measurement error distributions
#' 
#' Truncated Normal and Uniform distributions, where the response is also
#' subject to a Normally distributed measurement error.
#' 
#' The normal distribution with measurement error has density
#' 
#' \deqn{
#' \frac{\Phi(u, \mu_2, \sigma_3) - \Phi(l, \mu_2, \sigma_3)}{\Phi(u, \mu_2, \sigma_3) -
#' \Phi(l, \mu_2, \sigma_3)} \phi(x, \mu_0 + \mu_\epsilon, \sigma_2)}{(Phi(upper, mu2, sigma3) - Phi(lower, mu2, sigma3)) / (Phi(upper,
#' mean, sd) - Phi(lower, mean, sd)) * phi(x, mean + meanerr, sigma2)}
#'
#' where
#'
#' \deqn{\sigma_2^2 = \sigma_0^2 + \sigma_\epsilon^2,}{sigma2*sigma2 = sd*sd +
#' sderr*sderr,}
#'
#' \deqn{\sigma_3 = \sigma_0 \sigma_\epsilon / \sigma_2,}{sigma3
#' = sd*sderr / sigma2,}
#'
#' \deqn{\mu_2 = (x - \mu_\epsilon) \sigma_0^2 + \mu_0 \sigma_\epsilon^2,
#' }{mu2 = (x - meanerr)*sd*sd + mean*sderr*sderr,}
#' 
#' \eqn{\mu_0}{mean} is the mean of the original Normal distribution before
#' truncation, \cr \eqn{\sigma_0}{sd} is the corresponding standard deviation,
#' \cr \eqn{u} is the upper truncation point, \cr \eqn{l} is the lower
#' truncation point, \cr \eqn{\sigma_\epsilon}{sderr} is the standard deviation
#' of the additional measurement error, \cr \eqn{\mu_\epsilon}{meanerr} is the
#' mean of the measurement error (usually 0). \cr \eqn{\phi(x)}{phi(x)} is the
#' density of the corresponding normal distribution, and \cr
#' \eqn{\Phi(x)}{Phi(x)} is the distribution function of the corresponding
#' normal distribution.
#' 
#' The uniform distribution with measurement error has density
#' 
#' \deqn{(\Phi(x, \mu_\epsilon+l, \sigma_\epsilon) - \Phi(x, \mu_\epsilon+u,
#' \sigma_\epsilon)) }{(Phi(x, meanerr+l, sderr) - Phi(x, meanerr+u, sderr)) /
#' (upper - lower)}\deqn{ / (u - l)}{(Phi(x, meanerr+l, sderr) - Phi(x,
#' meanerr+u, sderr)) / (upper - lower)}
#' 
#' These are calculated from the original truncated Normal or Uniform density
#' functions \eqn{f(. | \mu, \sigma, l, u)}{f(. | mu, sd)} as
#' 
#' \deqn{ \int f(y | \mu, \sigma, l, u) \phi(x, y + \mu_\epsilon, \sigma_\epsilon) dy }{
#' \int f(y | mu, sd, l, u) \phi(x, y + meanerr, sderr) dy }
#' 
#' If \code{sderr} and \code{meanerr} are not specified they assume the default
#' values of 0, representing no measurement error variance, and no constant
#' shift in the measurement error, respectively.
#' 
#' Therefore, for example with no other arguments, \code{dmenorm(x)}, is simply
#' equivalent to \code{dtnorm(x)}, which in turn is equivalent to
#' \code{dnorm(x)}.
#' 
#' These distributions were used by Satten and Longini (1996) for CD4 cell
#' counts conditionally on hidden Markov states of HIV infection, and later by
#' Jackson and Sharples (2002) for FEV1 measurements conditionally on states of
#' chronic lung transplant rejection.
#' 
#' These distribution functions are just provided for convenience, and are not
#' optimised for numerical accuracy or speed.  To fit a hidden Markov model
#' with these response distributions, use a \code{\link{hmmMETNorm}} or
#' \code{\link{hmmMEUnif}} constructor. See the \code{\link{hmm-dists}} help
#' page for further details.
#'
#' @name medists
#' @aliases medists dmenorm pmenorm qmenorm rmenorm dmeunif pmeunif qmeunif
#' rmeunif
#' @param x,q vector of quantiles.
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is
#' taken to be the number required.
#' @param mean vector of means.
#' @param sd vector of standard deviations.
#' @param lower lower truncation point.
#' @param upper upper truncation point.
#' @param sderr Standard deviation of measurement error distribution.
#' @param meanerr Optional shift for the measurement error distribution.
#' @param log,log.p logical; if TRUE, probabilities \eqn{p} are given as
#' \eqn{\log(p)}{log(p)}, or log density is returned.
#' @param lower.tail logical; if TRUE (default), probabilities are \eqn{P[X <=
#' x]}, otherwise, \eqn{P[X > x]}.
#' @return \code{dmenorm}, \code{dmeunif} give the density, \code{pmenorm},
#' \code{pmeunif} give the distribution function, \code{qmenorm},
#' \code{qmeunif} give the quantile function, and \code{rmenorm},
#' \code{rmeunif} generate random deviates, for the Normal and Uniform versions
#' respectively.
#' @author C. H. Jackson \email{chris.jackson@@mrc-bsu.cam.ac.uk}
#' @seealso \code{\link{dnorm}}, \code{\link{dunif}}, \code{\link{dtnorm}}
#' @references Satten, G.A. and Longini, I.M.  Markov chains with measurement
#' error: estimating the 'true' course of a marker of the progression of human
#' immunodeficiency virus disease (with discussion) \emph{Applied Statistics}
#' 45(3): 275-309 (1996)
#' 
#' Jackson, C.H. and Sharples, L.D. Hidden Markov models for the onset and
#' progression of bronchiolitis obliterans syndrome in lung transplant
#' recipients \emph{Statistics in Medicine}, 21(1): 113--128 (2002).
#' @keywords distribution
#' @examples
#' 
#' ## what does the distribution look like?
#' x <- seq(50, 90, by=1)
#' plot(x, dnorm(x, 70, 10), type="l", ylim=c(0,0.06)) ## standard Normal
#' lines(x, dtnorm(x, 70, 10, 60, 80), type="l")       ## truncated Normal
#' ## truncated Normal with small measurement error
#' lines(x, dmenorm(x, 70, 10, 60, 80, sderr=3), type="l")
#' 
NULL

#' @rdname medists
#' @export
dmenorm <- function(x, mean=0, sd=1, lower=-Inf, upper=Inf, sderr=0, meanerr=0, log = FALSE)
{
    sumsq <- sd*sd + sderr*sderr
    sigtmp <- sd*sderr / sqrt(sumsq)
    mutmp <- ((x - meanerr)*sd*sd + mean*sderr*sderr) / sumsq
    nc <- 1/(pnorm(upper, mean, sd) - pnorm(lower, mean, sd))
    nctmp <- pnorm(upper, mutmp, sigtmp) - pnorm(lower, mutmp, sigtmp)
    if (log)
      log(nc) + log(nctmp) + log(dnorm(x, meanerr + mean, sqrt(sumsq), 0))
    else
      nc * nctmp * dnorm(x, meanerr + mean, sqrt(sumsq), 0)
}

#' @rdname medists
#' @export
pmenorm <- function(q, mean=0, sd=1, lower=-Inf, upper=Inf, sderr=0, meanerr=0, lower.tail = TRUE, log.p = FALSE)
{
    ret <- numeric(length(q))
    dmenorm2 <- function(x)dmenorm(x, mean=mean, sd=sd, lower=lower, upper=upper, sderr=sderr, meanerr=meanerr)
    for (i in 1:length(q)) {
        ret[i] <- integrate(dmenorm2, -Inf, q[i])$value
    }
    if (!lower.tail) ret <- 1 - ret
    if (log.p) ret <- log(ret)
    ret[upper < lower] <- NaN
    ret
}


#' @rdname medists
#' @export
qmenorm <- function(p, mean=0, sd=1, lower=-Inf, upper=Inf, sderr=0, meanerr=0, lower.tail=TRUE, log.p=FALSE)
{
    qgeneric(pmenorm, p=p, mean=mean, sd=sd, lower=lower, upper=upper, sderr=sderr, meanerr=meanerr,
             lbound=lower, ubound=upper, lower.tail=lower.tail, log.p=log.p)
}

#' @rdname medists
#' @export
rmenorm <- function(n, mean=0, sd=1, lower=-Inf, upper=Inf, sderr=0, meanerr=0)
{
    rnorm(n, meanerr + rtnorm(n, mean, sd, lower, upper), sderr)
}

### Uniform distribution with measurement error

#' @rdname medists
#' @export
dmeunif <- function(x, lower=0, upper=1, sderr=0, meanerr=0, log = FALSE)
{
    if (log)
      log( pnorm(x, meanerr + lower, sderr) - pnorm(x, meanerr + upper, sderr) ) - log(upper - lower)
    else
      ( pnorm(x, meanerr + lower, sderr) - pnorm(x, meanerr + upper, sderr) ) / (upper - lower)
}

#' @rdname medists
#' @export
pmeunif <- function(q, lower=0, upper=1, sderr=0, meanerr=0, lower.tail = TRUE, log.p = FALSE)
{
    ret <- numeric(length(q))
    dmeunif2 <- function(x)dmeunif(x, lower=lower, upper=upper, sderr=sderr, meanerr=meanerr)
    for (i in 1:length(q)) {
        ret[i] <- integrate(dmeunif2, -Inf, q[i])$value
    }
    if (!lower.tail) ret <- 1 - ret
    if (log.p) ret <- log(ret)
    ret
}

#' @rdname medists
#' @export
qmeunif <- function(p, lower=0, upper=1, sderr=0, meanerr=0, lower.tail=TRUE, log.p=FALSE)
{
    qgeneric(pmeunif, p=p, lower=lower, upper=upper, sderr=sderr, meanerr=meanerr,
             lbound=lower, ubound=upper, lower.tail=lower.tail, log.p=log.p)
}

#' @rdname medists
#' @export
rmeunif <- function(n, lower=0, upper=1, sderr=0, meanerr=0)
{
    rnorm(n, meanerr + runif(n, lower, upper), sderr)
}