MD: Best Model Discrimination (MD) Follow-Up Experiments

In BsMD: Bayes Screening and Model Discrimination

Best Model Discrimination (MD) Follow-Up Experiments

Description

Best follow-up experiments based on the MD criterion are suggested to discriminate between competing models.

Usage

MD(X, y, nFac, nBlk = 0, mInt = 3, g = 2,  nMod, p, s2, nf, facs, nFDes = 4,
Xcand, mIter = 20, nStart = 5, startDes = NULL, top = 20, eps = 1e-05)

Arguments

X	matrix. Design matrix of the initial experiment.
y	vector. Response vector of the initial experiment.
nFac	integer. Number of factors in the initial experiment.
nBlk	integer >=1. The number of blocking factors in the initial experiment. They are accommodated in the first columns of matrix X.
mInt	integer. Maximum order of the interactions in the models.
g	vector. Variance inflation factor for main effects (g[1]) and interactions effects (g[2]). If vector length is 1 the same inflation factor is used for main and interactions effects.
nMod	integer. Number of competing models.
p	vector. Posterior probabilities of the competing models.
s2	vector. Competing model variances.
nf	vector. Factors considered in each of the models.
facs	matrix. Matrix [nMod x max(nf)] of factor numbers in the design matrix.
nFDes	integer. Number of runs to consider in the follow-up experiment.
Xcand	matrix. Candidate runs to be chosen for the follow-up design.
mIter	integer. If 0, then user-entered designs startDes are evaluated, otherwise the maximum number of iterations for each Wynn search.
nStart	integer. Number of starting designs.
startDes	matrix. Matrix [nStart x nFDes]. Each row has the row numbers of the user-supplied starting design.
top	integer. Highest MD follow-up designs recorded.
eps	numeric. A small number (1e-5 by default) used for computations.

Details

The MD criterion, proposed by Meyer, Steinberg and Box is used to discriminate among competing models. Random starting runs chosen from Xcand are used for the Wynn search of best MD follow-up designs. nStart starting points are tried in the search limited to mIter iterations. If mIter=0 then startDes user-provided designs are used. Posterior probabilities and variances of the competing models are obtained from BsProb. The function calls the FORTRAN subroutine ‘md’ and captures summary results. The complete output of the FORTRAN code is save in the ‘MDPrint.out’ file in the working directory.

Value

A list with all input and output parameters of the FORTRAN subroutine MD. Most of the variable names kept to match FORTRAN code.

NSTART	Number of starting designs.
NRUNS	Number of runs used in follow-up designs.
ITMAX	Maximum number of iterations for each Wynn search.
INITDES	Number of starting points.
NO	Numbers of runs already completed before follow-up.
IND	Indicator; 0 indicates the user supplied starting designs.
X	Matrix for initial data (nrow(X)=N0; ncol(X)=COLS+BL).
Y	Response values from initial experiment (length(Y)=N0).
GAMMA	Variance inflation factor.
GAM2	If IND=1, GAM2 was used for interaction factors.
BL	Number of blocks (>=1) accommodated in first columns of X and Xcand

.

COLS	Number of factors.
N	Number of candidate runs.
Xcand	Matrix of candidate runs. (nrow(Xcand)=N, ncol(Xcand)=ncol(X)).
NM	Number of models considered.
P	Models posterior probability.
SIGMA2	Models variances.
NF	Number of factors per model.
MNF	Maximum number of factor in models. (MNF=max(NF)).
JFAC	Matrix with the factor numbers for each of the models.
CUT	Maximum interaction order considered.
MBEST	If INITDES=0, the first row of the MBEST[1,] matrix has the first user-supplied starting design. The last row the NSTART-th user-supplied starting design.
NTOP	Number of the top best designs.
TOPD	The D value for the best NTOP designs.
TOPDES	Top NTOP design factors.
ESP	"Small number" provided to the ‘md’ FORTRAN subroutine. 1e-5 by default.
flag	Indicator = 1, if the ‘md’ subroutine finished properly, -1 otherwise.

Note

The function is a wrapper to call the FORTAN subroutine ‘md’, modification of Daniel Meyer's original program, ‘MD.f’, part of the mdopt bundle for Bayesian model discrimination of multifactor experiments.

Author(s)

R. Daniel Meyer. Adapted for R by Ernesto Barrios.

References

Meyer, R. D., Steinberg, D. M. and Box, G. E. P. (1996). "Follow-Up Designs to Resolve Confounding in Multifactor Experiments (with discussion)". Technometrics, Vol. 38, No. 4, pp. 303–332.

Box, G. E. P and R. D. Meyer (1993). "Finding the Active Factors in Fractionated Screening Experiments". Journal of Quality Technology. Vol. 25. No. 2. pp. 94–105.

See Also

print.MD, BsProb

Examples

### Injection Molding Experiment. Meyer et al. 1996, example 2.
library(BsMD)
data(BM93.e3.data,package="BsMD")
X <- as.matrix(BM93.e3.data[1:16,c(1,2,4,6,9)])
y <- BM93.e3.data[1:16,10]
p <- c(0.2356,0.2356,0.2356,0.2356,0.0566)
s2 <- c(0.5815,0.5815,0.5815,0.5815,0.4412)
nf <- c(3,3,3,3,4)
facs <- matrix(c(2,1,1,1,1,3,3,2,2,2,4,4,3,4,3,0,0,0,0,4),nrow=5,
    dimnames=list(1:5,c("f1","f2","f3","f4")))
nFDes <- 4
Xcand <- matrix(c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
                    -1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1,
                    -1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,
                    -1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,
                    -1,1,1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1),
                    nrow=16,dimnames=list(1:16,c("blk","f1","f2","f3","f4"))
                )
injectionMolding.MD <- MD(X = X, y = y, nFac = 4, nBlk = 1, mInt = 3,
            g = 2, nMod = 5, p = p, s2 = s2, nf = nf, facs = facs,
            nFDes = 4, Xcand = Xcand, mIter = 20, nStart = 25, top = 10)
summary(injectionMolding.MD)



### Reactor Experiment. Meyer et al. 1996, example 3.
par(mfrow=c(1,2),pty="s")
data(Reactor.data,package="BsMD")

# Posterior probabilities based on first 8 runs
X <- as.matrix(cbind(blk = rep(-1,8), Reactor.data[c(25,2,19,12,13,22,7,32), 1:5]))
y <- Reactor.data[c(25,2,19,12,13,22,7,32), 6]
reactor8.BsProb <- BsProb(X = X, y = y, blk = 1, mFac = 5, mInt = 3,
        p =0.25, g =0.40, ng = 1, nMod = 32)
plot(reactor8.BsProb,prt=TRUE,,main="(8 runs)")

# MD optimal 4-run design
p <- reactor8.BsProb$ptop
s2 <- reactor8.BsProb$sigtop
nf <- reactor8.BsProb$nftop
facs <- reactor8.BsProb$jtop
nFDes <- 4
Xcand <- as.matrix(cbind(blk = rep(+1,32), Reactor.data[,1:5]))
reactor.MD <- MD(X = X, y = y, nFac = 5, nBlk = 1, mInt = 3, g =0.40, nMod = 32,
        p = p,s2 = s2, nf = nf, facs = facs, nFDes = 4, Xcand = Xcand,
        mIter = 20, nStart = 25, top = 5)
summary(reactor.MD)

# Posterior probabilities based on all 12 runs
X <- rbind(X, Xcand[c(4,10,11,26), ])
y <- c(y, Reactor.data[c(4,10,11,26),6])
reactor12.BsProb <- BsProb(X = X, y = y, blk = 1, mFac = 5, mInt = 3,
        p = 0.25, g =1.20,ng = 1, nMod = 5)
plot(reactor12.BsProb,prt=TRUE,main="(12 runs)")

BsMD documentation built on Sept. 19, 2023, 5:07 p.m.