placeTempers: Place the intermediate temperatures between the temperature...

In EMC: Evolutionary Monte Carlo (EMC) algorithm

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.

Below sampDim refers to the dimension of the sample space, temperLadderLen refers to the length of the temperature ladder, and levelsSaveSampForLen refers to the length of levelsSaveSampFor. Note, this function calls evolMonteCarlo, so some of the arguments below have the same name and meaning as the corresponding ones for evolMonteCarlo. See details below for explanation on the arguments.

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 x 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 >= 2 produces a lot of output.
...	optional arguments to be passed to logTarDensFunc, MHPropNewFunc and logMHPropDensFunc.

Details

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

acceptRatioLimits

This is a range for the estimated acceptance ratios for the random exchange move for the consecutive temperature levels of the final ladder. It is recommended that specified range is between 0.3 and 0.6.

ladderLenMax

It is preferred that one specifies acceptRatioLimits for constructing the final temperature ladder. However, If one has some computational limitations then one could also specify ladderLenMax which will limit the length of the final temperature ladder produced. This also serves as an upper bound on the number of temperature levels while placing the intermediate temperatures using the acceptRatioLimits.

temperLadder

This is the temperature ladder needed for the second stage preliminary run. One can either specify a temperature ladder via temperLadder or specify temperLimits, ladderLen, scheme and schemeParam. For details on the later set of parameters, see below. Note, temperLadder overrides temperLimits, ladderLen, scheme and schemeParam.

temperLimits

temperLimits = c(lowerLimit, upperLimit) is a two-tuple of positive numbers, where the lowerLimit is usually 1 and upperLimit is a number in [100, 1000]. If stochastic optimization (via sampling) is the goal, then lowerLimit is taken to be in [0, 1]. Often the upperLimit is the maximum temperature as suggested by findMaxTemper.

ladderLen, scheme and schemeParam

These three parameters are required (along with temperLimits) if temperLadder is not provided. We recommend taking ladderLen in [15, 30]. The allowed choices for scheme and schemeParam are:

scheme	schemeParam
========	=============
linear	NA
log	NA
geometric	NA
mult-power	NA
add-power	>= 0
reciprocal	NA
exponential	>= 0
tangent	>= 0

We recommended using scheme = 'exponential' and schemeParam in [1.5, 2].

guideMe

If guideMe = TRUE, then the function suggests different modifications to alter the setting towards a re-run, in case there are problems with the underlying MCMC run.

levelsSaveSampFor

This is passed to evolMonteCarlo for the underlying MCMC run.

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 x sampDim x levelsSaveSampForLen, if saveFitness = FALSE. If saveFitness = TRUE, then the returned array is of dimension nIters x (sampDim + 1) x 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.

Note

The effect of leaving the default value NULL for some of the arguments above are as follows:

logMHPropDensFunc	the proposal density MHPropNewFunc is deemed symmetric.
temperLadder	valid temperLimits, ladderLen, scheme and schemeParam
	are provided, which are used to construct the temperLadder.
temperLimits	a valid temperLadder is provided.
levelsSaveSampFor	temperLadderLen.

Author(s)

Gopi Goswami [e-mail]

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

## Not run: 
placeTempersObj <-
    with(VShapedFuncGenerator(-13579),
         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))))   
     }
 })

## End(Not run)

EMC documentation built on May 2, 2019, 5:11 a.m.