# R/de_mc.R In tpilz/HydroBayes: Bayesian statistics for use in hydrological applications

#### Documented in de_mc

```#' Differential Evolution Markov Chain (DE-MC) algorithm
#' @param prior A function(N,d) that draws N samples from a d-variate prior distribution.
#' Returns an N-by-d matrix.
#' @param pdf A function(prior) that calculates the density of the target distribution for given prior.
#' Returns an N-variate vector.
#' @param nc \code{numeric}. Number of chains evolved in parallel.
#' @param t \code{numeric}. Number of samples from the Markov chain.
#' @param d \code{numeric}. Number of parameters.
#'
#' @return \code{list} with named elements 'chain': a t-by-d-by-nc array of parameter realisations for each iteration
#' and Markov chain; and 'density': a t-by-nc matrix of densities computed by \code{pdf} at each iteration for each
#' Markov chain.
#'
#' @references Code from 'Algorithm 4' of:
#'
#' Vrugt, J. A.: "Markov chain Monte Carlo simulation using the DREAM software package:
#' Theory, concepts, and MATLAB implementation." Environmental Modelling & Software, 2016, 75, 273 -- 316,
#' \url{http://dx.doi.org/10.1016/j.envsoft.2015.08.013}.
#'
#' @author Tobias Pilz \email{[email protected]@uni-potsdam.de}
#'
#' @export
de_mc <- function(prior, pdf, nc, t, d) {

# Default jump rate
gamma_RWM <- 2.38/sqrt(2*d)

# allocate chains and density
x <- array(NaN, dim=c(t,d,nc))
p_x <- array(NaN, dim=c(t,nc))

# initialize chains by sampling from prior
x[1,,] <- t(prior(nc,d))
p_x[1,] <- pdf(t(x[1,,]))

# evolution of nc chains
for(i in 2:t) {

# initialise chains for current i based on i-1
x[i,,] <- x[i-1,,]

# proposal and accept/reject for each chain
for (j in 1:nc) {
# sample chains used to calculate prposal for current chain, a != b != j
sam <- sample((1:nc)[-j], 2, replace = FALSE)
a <- sam[1]
b <- sam[2]

# select jump rate gamma: weighted random sample of gamma_RWM or 1 with probabilities 0.9 and 0.1, respectively
g <- sample(x = c(gamma_RWM, 1), size = 1, replace = TRUE, prob = c(0.9, 0.1))

# create proposal via differential evolution
xp <- x[i,,j] + g * (x[i,,a] - x[i,,b]) + rnorm(d, sd=1e-6)

# calculate density at proposal
p_xp <- pdf(xp)

# probability of acceptance (Metropolis acceptance ratio)
p_acc <- min(1, p_xp/p_x[i-1,j])
if(p_acc > runif(1)) { # larger than sample point from U[0,1]?
x[i,,j] <- xp # accept candidate; NOTE: accepted candidate for chain j at i already used to calculate proposal for next chain j+1 at i, otherwise multi-modal target pdfs will not sampled correctly
p_x[i,j] <- p_xp # accept density accordingly
} else {
p_x[i,j] <- p_x[i-1,j] # retain previous density accordingly
}
}
}

return(list(chain=x, density=p_x))
} # EOF
```
tpilz/HydroBayes documentation built on Dec. 21, 2017, 11:43 p.m.