AlgDesign (version 1.2.1.1)

optMonteCarlo: Optimal design via Monte Carlo

Description

Finds a design using the specified criterion via Federov's algorithm applied to a random subset of all possible candidate points.

Usage

optMonteCarlo(frml,data,nTrials,approximate=FALSE,criterion="D",evaluateI=FALSE,
	space=NULL,mixtureSum=1,constraints=NULL,RandomStart=TRUE,nRepeats=5,nCand,
	nCandNull,DFrac=1,CFrac=1,args=FALSE)

Value

The output is the same list as from optFederov, but the criteria values are relative to the randomly chosen subsets of the putative candidate space. In general, they should not differ greatly from those obtained by an exhaustive search.

D

The kth root of the generalized variance: \(det(M)^{1/k}\), where \(det(M)\) is the determinant of the normalized dispersion matrix \(M\) -- i.e. \(M=Z'Z/n\), where \(Z=X[rows,]\)

A

The average coefficient variance: \(trace(Mi)/k\), where \(Mi\) is the inverse of \(M\).

I

The average prediction variance over X, which can be shown to be \(trace((X'X*Mi)/N)\). This is calculated only when I is the criterion or when evaluateI is TRUE.

Ge

The minimax normalized variance over X, expressed as an efficiency with respect to the optimal approximate theory design. It is defined as \(k/max(d)\), where \(max(d)\) is the maximum normalized variance over \(X\) -- i.e. the max of \(x'(Mi)x\), over all rows \(x'\) of \(X\).

Dea

A lower bound on D efficiency for approximate theory designs. It is equal to \(exp(1-1/Ge)\).

Design

The design.

args

A list of the actual arguments used in this call.

Arguments

frml

Required: A formula starting with ~ which will be used with model.matrix() to create a model matrix. If there are mixture variables, the constant term is suppressed.

data

Required: A data frame with 7 or 8 columns. See details below for specifics

nTrials

number trials in design -- must be greater than the number of terms in the model, if missing will be set to the number of model terms in the model plus five.

approximate

When FALSE, an exact design in nTrails will be calculated. When TRUE the proportions for an approximate theory design will be calculated. If nTrials is set, any proportion less than 1/(2*maxIteration) will be discarded before the proportions are efficiently rounded, otherwise all non-zero proportions will be shown: these are the support points.

criterion

"D", "A", or "I"

evaluateI

TRUE if I is to be evaluated in addition to the other criteria -- slower because of calculations for I

space

If the criterion is "I" or evaluate I is true, the space over which the I criterion is to be evaluated may be input. It should be a matrix with the same column types and names as in data. If space is not input the evaluation will be done over the space described by data.

constraints

A function taking a vector argument with length equal to the number of variables, and returning TRUE if the vector is inside the constrained region

mixtureSum

The mixture variables, if any, will sum to this value.

RandomStart

When TRUE, the starting design will be chosen at random, otherwise nullification will be used. Note: the nullifcation used here is different and much slower than that in optFederov().

nRepeats

number of times to retry the entire process

nCand

number of candidate points to generate, if missing, it will be 10 times the number of terms

nCandNull

Number of candidate points to use for nullification. If missing it will be set to nCand

DFrac

Fraction of design used in search: 1 uses all of them, 0 only the one with the smallest variance

CFrac

Fraction of candidate set searched : 1 uses all of them, 0 only the one with the largest variance

args

If TRUE, the actual arguments to the function including the starting random number seed will be output.

Author

Bob Wheeler

Please cite this program as follows:

Wheeler, R.E. (2004). optMonteCarlo. AlgDesign. The R project for statistical computing https://www.r-project.org/

Details

The columns of the input data frame are as follows. The columns need not be named. It is probably best to avoid naming the variables with single letters, especially "I" -- use paste(), as in the examples. For each variable nLevels are randomly generated between low and high, inclusive, and then rounded with round. For integer levels, round should be set to 0.

var:

The names of the variables.

low:

The lower limit of the range for each variable. Ignored for mixtures.

high:

The upper limit of the range for each variable. Ignored for mixtures.

center:

The centering value for each variable. Ignored for mixtures.

nLevels:

The number of levels for each variable. Ignored for mixture variables.

round:

The number of decimal digits for the levels. The levels are randomly and uniformly chosen between low and high, and this parameter controls the number of trailing digits. The max value for mixture variables in this vector is used to round all mixture variables.

factor:

TRUE, FALSE depending on whether or not the variable is a factor. Note: other columns will be reset to conform to a nLevels factor.

mix:

TRUE if the variable is a mixture variable. This column may be omitted if there are no mixture variables.

Candidate lists required by optFederov() increase with the number of variables, and can easily exceed storage capacity and can require excessive amounts of time to process. To overcome this problem, optMonteCarlo(), generates at random nCand points from a putative candidate list.

For non-mixture variables, optMonteCarlo() samples from the putative candidate list by choosing random levels inside the limits given by low and high in data. These are rounded to the number of levels given by nLevels in data and to the number of decimal digits given by round in data.

For mixture variables, optMonteCarlo() samples from the putative candidate list by choosing random levels between 0 and 1, rounded to the maximum in the round column of data, and such that the sum over all variables is equal to mixtureSum.

If a constraint function is supplied in Constraints, it is applied, and results which do not meet the constraint are discarded. The constraint function should be written to process uncentered variables.

The above procedures are repeated until nCand candidate points are found.

Nullification, successively adds points to a design until n points are found. This is the same procedure that is in optFederov except that each new point is selected from a new sampling of the putative candidate points. In general, this will produce better designs that those from a random start.

The entire process is repeated nRepeats times, and the best result is reported. The methodology compares favorably with an exhaustive search where the entire candidate list is searched by optFederov().

The random numbers used in these calculations are controlled by the usual R random number mechanism.

A vignette giving further details is availble. To access it, type

vignette("AlgDesign")

Examples

# EXAMPLE 1
# The data.frame in data might look like the following:
data<-data.frame(var=paste("X",1:6,sep=""),low=c(1,1,1,0,0,0),
high=c(3,3,3,1,1,1),center=c(2,2,2,0,0,0),nLevels=3,
round=1,factor=0,mix=c(FALSE,FALSE,FALSE,TRUE,TRUE,TRUE))
data

# and the design:

optMonteCarlo(~(X1+X2+X3)^2+X4+X5+X6,data)

# Example 2
# Standard designs will often be produced, just as 
# they will with optFederov(). For example,
# a half fraction of a 2^4:
data<-data.frame(paste("X",1:4,sep=""),-1,1,0,2,0,0)
data
optMonteCarlo(~.,data,nTrials=8)

# Example 3
# optMonteCarlo() can treat much larger problems than can 
# optFederov().  For example, optFederov()
# requires a candidate list of 3^20 points for
# a 20 variable, 3 level candidate list -- about
# 25 gigabytes. If the model is quadratic, this must
# be multiplied by about 12. There are other storage
# requirements internal to optFederov() which easily
# double this value. optMonteCarlo() since it only samples
# from the putative candidate list, has no difficulty 
# with a problem of this size. The criterion values
# appearing in the output of optMonteCarlo() are based on
# these samples, but their values seem to be reasonable
# correct, as the following shows: (These are commented
# out for those who have a slow machine.)

dat<-gen.factorial(levels=3,nVar=8)
#desF<-optFederov(~quad(.),dat,eval=TRUE)
#desF[1:5]

data<-data.frame(paste("X",1:8,sep=""),-1,1,0,3,0,0)
#desH<-optMonteCarlo(~quad(.),data,Rand=FALSE,eval=TRUE)
#desH[1:5]

# The following is a 20 variable quadratic. Uncomment
# and wait a while, even if you have a fast machine.
# Note: nRepeats has been changed from its default.
# Note: criterion values for exact designs are often
# far from approximate theory optima; hence, Ge and De
# will be small.

data<-data.frame(paste("X",1:20,sep=""),-1,1,0,3,0,0)
#desBig<-optMonteCarlo(~quad(.),data,nRepeats=1)

# The following will produce improved criterion values

#desNBig<-optMonteCarlo(~quad(.),data,Rand=FALSE,nRepeats=1)

# EXAMPLE 4
# Practically infeasible combinations of variable are 
# common. Designs may be produced which avoid such
# combinations by using a constraint function. Suppose,
# for example that one corner of a cubic box is not
# feasible, then the following will produce a design
# that makes no use of this corner.

Constraints<-function(x){!(x[1]>0.75 && x[2]>0.75)}
data<-data.frame(paste("X",1:4,sep=""),-1,1,0,3,0,0)
desC<-optMonteCarlo(~.,data,con=Constraints)

# The above just removes a corner. Increasing the
# number of levels will remove points along the
# boundary.

data<-data.frame(paste("X",1:4,sep=""),-1,1,0,11,3,0)
desC2<-optMonteCarlo(~.,data,con=Constraints)