EMC (version 1.0)

placeTempers: Place the intermediate temperatures between the temperature limits

Description

Multiple MCMC chains based algorithms (e.g., parallel tempering, evolutionary Monte Carlo) need a temperature ladder. This function places the intermediate temperatures between the minimum and the maximum temperature for the ladder.

Usage

placeTempers(nIters,                           
             acceptRatioLimits,                      
             ladderLenMax,                           
             startingVals,                           
             logTarDensFunc,                         
             MHPropNewFunc,                          
             logMHPropDensFunc = NULL,               
             temperLadder      = NULL,               
             temperLimits      = NULL,               
             ladderLen         = 15,                 
             scheme            = 'exponential',      
             schemeParam       = 1.5,                
             guideMe           = TRUE,               
             levelsSaveSampFor = NULL,               
             saveFitness       = FALSE,              
             verboseLevel      = 0,
             ...)

Arguments

nIters

integer $>$ 0.

acceptRatioLimits

double vector of two probabilities.

ladderLenMax

integer $>$ 0.

startingVals

double matrix of dimension temperLadderLen $\times$ sampDim or vector of length sampDim, in which case the same starting values are used for every temperature level.

logTarDensFunc

function of two arguments (draw, ...) that returns the target density evaluated in the log scale.

MHPropNewFunc

function of four arguments (temperature, block, currentDraw, ...) that returns new Metropolis-Hastings proposals. See details below on the argument block.

logMHPropDensFunc

function of five arguments (temperature, block, currentDraw, proposalDraw, ...) that returns the proposal density evaluated in the log scale. See details below on the argument block.

temperLadder

double vector with all positive entries, in decreasing order.

temperLimits

double vector with two positive entries.

ladderLen

integer $>$ 0.

scheme

character.

schemeParam

double $>$ 0.

guideMe

logical.

levelsSaveSampFor

integer vector with positive entries.

saveFitness

logical.

verboseLevel

integer, a value $\ge$ 2 produces a lot of output.

...

optional arguments to be passed to logTarDensFunc, MHPropNewFunc and logMHPropDensFunc.

Value

• This function returns a list with the following components:
• finalLadder
  the final temperature ladder found by placing the intermediate temperatures to be used in parallelTempering or evolMonteCarlo.
• temperLadder
  the temperature ladder used for the underlying MCMC run.
• acceptRatiosEst
  the estimated acceptance ratios for the random exchange move for the consecutive temperature levels of temperLadder.
• CVSqWeights
  this is the square of the coefficient of variation of the weights of the importance sampling estimators used to estimate the acceptance ratios, namely, estAcceptRatios.
• temperLimits
  the sorted temperLimits argument.
• acceptRatioLimits
  the sorted acceptRatioLimits argument.
• nIters
  the post burn-in nIters.
• levelsSaveSampFor
  the levelsSaveSampFor argument.
• draws
  array of dimension nIters $\times$ sampDim $\times$ levelsSaveSampForLen, if saveFitness = FALSE. If saveFitness = TRUE, then the returned array is of dimension nIters $\times$ (sampDim + 1) $\times$ levelsSaveSampForLen; i.e., each of the levelsSaveSampForLen matrices contain the fitness values in their last column.
• startingVals
  the startingVals argument.
• time
  the time taken by the run.

Details

This function is based on the temperature placement method introduced in section 4.2 of Goswami and Liu (2007).

[object Object],[object Object],[object Object],[object Object],[object Object],rl{ scheme schemeParam ======== ============= linear NA log NA geometric NA mult-power NA add-power $\ge$ 0 reciprocal NA exponential $\ge$ 0 tangent $\ge$ 0 }

We recommended using scheme = 'exponential' and schemeParam in [0.3, 0.5].,[object Object],[object Object]

References

Gopi Goswami and Jun S. Liu (2007). On learning strategies for evolutionary Monte Carlo. Statistics and Computing 17:1:23-38.

See Also

findMaxTemper, parallelTempering, evolMonteCarlo

Examples

## The V-shaped distribution
VShapedFuncGenerator <-
    function (seed = 13579)
{
    set.seed(seed)    
    dd     <- 2
    ARDisp <-
        function (rho)
        {
            tmp <- rep(1, dd)
            diag((1 - rho) * tmp) + rho * tmp %*% t(tmp)
        }

    nMixComps  <- 2
    logWeights <- log(rep(1 / nMixComps, nMixComps))
    meanMat    <- matrix(c(1, 1, 15, 1), nMixComps, byrow = TRUE)
    dispParam  <- c(-0.95, 0.95)
    dispArr    <- array(dim = c(2, 2, nMixComps))
    for (ii in seq_len(nMixComps)) {
        dispArr[ , , ii] <- ARDisp(dispParam[ii])
    }
    logTarDensFunc <-
        function (draw, ...)
        {
            ld <- sapply(seq_len(nMixComps), FUN =
                         function (ii)
                     {
                         dmvnorm(draw, meanMat[ii, ], dispArr[ , , ii], log = TRUE)
                     })
            ww <- logWeights + ld
            mm <- max(ww)
            mm + log(sum(exp(ww - mm)))
        }
    
    MHProposalSD  <- c(1.0, 1.0)
    MHPropNewFunc <-
        function (temperature, block, currentDraw, ...)
        {
            proposalDraw        <- currentDraw
            proposalDraw[block] <- rnorm(1, currentDraw[block],
                                         sqrt(temperature) * MHProposalSD[block])
            proposalDraw
        }

    list(logTarDensFunc = logTarDensFunc,
         MHPropNewFunc  = MHPropNewFunc)
}


placeTempersObj <-
    with(VShapedFuncGenerator( ),
         placeTempers(nIters            = 10000,
                      acceptRatioLimits = c(0.5, 0.6),
                      ladderLenMax      = 50,
                      startingVals      = c(0, 0),
                      logTarDensFunc    = logTarDensFunc,
                      MHPropNewFunc     = MHPropNewFunc,
                      temperLimits      = c(1, 5),
                      ladderLen         = 10,
                      levelsSaveSampFor = seq_len(10),
                      verboseLevel      = 1))
print(placeTempersObj)
print(names(placeTempersObj))
with(placeTempersObj,
 {
     par(mfcol = c(3, 3))
     for (ii in seq_along(levelsSaveSampFor)) {
         main <- paste('temper:', round(temperLadder[levelsSaveSampFor[ii]], 3))
         plot(draws[ , , ii],
              xlim = c(-4, 20),
              ylim = c(-8, 8),
              pch  = '.',
              ask  = FALSE,
              main = as.expression(main),
              xlab = as.expression(substitute(x[xii], list(xii = 1))),
              ylab = as.expression(substitute(x[xii], list(xii = 2))))   
     }
 })