MaxPro package

Description

Maximum projection (MaxPro) designs for quantitative and/or qualitative factors

Details

Important functions in this package are: MaxProLHD generates the MaxPro Latin hypercube design for continuous factors, MaxPro searches for the locally optimal MaxPro design in the neighborhood of a given initial design for continuous factors, MaxProQQ generates the MaxPro design with both quantitative and qualitative factors, MaxProAugment sequentially augments an existing design with additional design points based on the MaxPro criterion for all types of factors and constraints, MaxProRunOrder finds the optimal order to run a given design in a sequential manner based on the MaxPro criterion, and MaxProMeasure reports the MaxPro criterion value for a given design.

Author(s)

Shan Ba and V. Roshan Joseph

Maintainer: Shan Ba <shanbatr@gmail.com>

References

Joseph, V. R., Gul, E., and Ba, S. (2015) "Maximum Projection Designs for Computer Experiments," Biometrika, 102, 371-380.

Joseph, V. R., Gul, E., and Ba, S. (2018) "Designing Computer Experiments with Multiple Types of Factors: The MaxPro Approach," Journal of Quality Technology, to appear.

Generate Candidate Design Points Randomly for Various Types of Factors

Description

Quickly generate random candidate design points for different types of factors which can be used as an input to the MaxProAugment function.

Usage

CandPoints(N,p_cont,l_disnum=NULL,l_nom=NULL)

Arguments

Details

This function generates uniform random numbers for each continuous factor and randomly sampled levels for each discrete numeric and nominal factor. The result can be used as the candidate design in the MaxProAugment function.

Value

The value returned from the function is a matrix containing the candidate design points.

Author(s)

Shan Ba <shanbatr@gmail.com> and V. Roshan Joseph <roshan@isye.gatech.edu>

See Also

MaxProAugment

Examples

CCC<-CandPoints(N=10000,p_cont=3,l_disnum=c(3,3,4),l_nom=c(2,2,3))

Locally Optimal Maximum Projection Designs for Continuous Factors

Description

Find the locally optimal maximum projection (MaxPro) design in the neighborhood of a given initial design for continuous factors.

Usage

MaxPro(InitialDesign,s=2,iteration=10)

Arguments

Details

This function applies a continuous optimization algorithm in nloptr (Ypma 2014) to find the locally optimal MaxPro design in the neighborhood of the initial design. A MaxPro Latin hypercube design generated by the MaxProLHD function is a good choice for the initial design. Please refer to Joseph, Gul and Ba (2015) for details.

Value

The value returned from the function is a list containing the following components:

Author(s)

Shan Ba <shanbatr@gmail.com> and V. Roshan Joseph <roshan@isye.gatech.edu>

References

Joseph, V. R., Gul, E., and Ba, S. (2015) "Maximum Projection Designs for Computer Experiments," Biometrika, 102, 371-380.

Ypma, J. (2014) "Introduction to nloptr: an R interface to NLopt", R Package Version 1.0.0.

See Also

MaxProLHD, MaxProRunOrder, MaxProAugment

Examples

InitialDesign<-MaxProLHD(n = 10, p = 4)$Design 
DOX<-MaxPro(InitialDesign)
DOX$Design

Augment an Existing Design based on the Maximum Projection Criterion

Description

Select the best set of design points to augment a given design matrix by optimizing the maximum projection (MaxPro) criterion sequentially. When multiple types of factors exist, all continuous factors should be placed as the leftmost columns in the design matrix, discrete numeric factors should be placed as middle columns, and all nominal factors should be placed at the rightmost columns in the design matrix. The ordinal factors, if any, should be pre-converted into discrete numeric factors through the scoring method (see, e.g., Wu and Hamada 2009, Section 14.10). This function can also be used for creating a nonadaptive sequential design, designs to run in batches, validation runs, designs in constrained regions, and nested designs.

Usage

MaxProAugment(ExistDesign, CandDesign, nNew, 
p_disnum=0, l_disnum=NULL, p_nom=0, l_nom=NULL)

Arguments

Details

This function uses a greedy search algorithm to select the best nNew points out of the CandDesign matrix to augment the ExistDesign matrix based on the MaxPro criterion. Factor constraints (irregular-shaped design space), if any, can be incorporated through the specification of valid candidate design points in CandDesign (please see the example below).

Value

The value returned from the function is a list containing the following components:

Author(s)

Shan Ba <shanbatr@gmail.com> and V. Roshan Joseph <roshan@isye.gatech.edu>

References

Joseph, V. R., Gul, E., and Ba, S. (2015) "Maximum Projection Designs for Computer Experiments," Biometrika, 102, 371-380.

Joseph, V. R. (2016) "Rejoinder," Quality Engineering, 28, 42-44.

Joseph, V. R., Gul, E., and Ba, S. (2018) "Designing Computer Experiments with Multiple Types of Factors: The MaxPro Approach," Journal of Quality Technology, to appear.

Wu, C. F. J., and Hamada, M. (2009), Experiments: Planning, Analysis, and Parameter Design Optimization, 2nd Edition, New York: Wiley.

See Also

CandPoints, MaxProLHD, MaxProQQ

Examples

ExistDesign=matrix(c(0.11,0.96,0.42,0.99),ncol=2,byrow=TRUE)

#Generate candidate design points that satisfy the factor constraints
S=CandPoints(N=10000,p_cont=2)
S=S[S[,1]+S[,2]<=1.5,] #constraint 1
S=S[(S[,1]-.5)^2+(S[,2]-.5)^2>=(2*.25^2),] #constraint 2
plot(S[,1],S[,2],bty="n",cex=.5,xlab=expression(x[1]),ylab=expression(x[2]),xlim=c(0,1),ylim=c(0,1))

result=MaxProAugment(ExistDesign, CandDesign=S, nNew=20)
D=result$Design
points(D[,1],D[,2],col='red',pch=16)
text(D[,1],D[,2],1:nrow(D),col=2,cex=2)

Maximum Projection Latin Hypercube Designs for Continuous Factors

Description

Generate the maximum projection (MaxPro) Latin hypercube design for continuous factors based on a simulated annealing algorithm. If nominal, discrete numeric, or ordinal factors exist, please see the function MaxProQQ.

Usage

MaxProLHD(n, p, s=2, temp0=0, nstarts = 1, itermax = 400, total_iter = 1e+06)

Arguments

Details

This function utilizes a version of the simulated annealing algorithm to efficiently generate the optimal Latin hypercube designs for continuous factors based on the MaxPro criterion. Parameters in the algorithm may need to be properly tuned to achieve global convergence. Please refer to Joseph, Gul and Ba (2015) for details.

Value

The value returned from the function is a list containing the following components:

Author(s)

Shan Ba <shanbatr@gmail.com> and V. Roshan Joseph <roshan@isye.gatech.edu>

References

Joseph, V. R., Gul, E., and Ba, S. (2015) "Maximum Projection Designs for Computer Experiments," Biometrika, 102, 371-380.

See Also

MaxProRunOrder, MaxProAugment, MaxProQQ

Examples

obj<-MaxProLHD(n = 10, p = 4) 
obj$Design

Maximum Projection Criterion Measure

Description

Calculate the maximum projection (MaxPro) criterion measure for a given design.

Usage

MaxProMeasure(Design, p_nom=0)

Arguments

Details

This function computes the MaxPro criterion measure for a given design. Please refer to Joseph, Gul and Ba (2015) and Joseph, Gul and Ba (2018) for details.

Value

The value returned from the function is the MaxPro criterion measure

Author(s)

Shan Ba <shanbatr@gmail.com> and V. Roshan Joseph <roshan@isye.gatech.edu>

References

Joseph, V. R., Gul, E., and Ba, S. (2015) "Maximum Projection Designs for Computer Experiments," Biometrika, 102, 371-380.

Joseph, V. R., Gul, E., and Ba, S. (2018) "Designing Computer Experiments with Multiple Types of Factors: The MaxPro Approach," Journal of Quality Technology, to appear.

Wu, C. F. J., and Hamada, M. (2009), Experiments: Planning, Analysis, and Parameter Design Optimization, 2nd Edition, New York: Wiley.

See Also

MaxProLHD, MaxProQQ, MaxProAugment

Examples

D<-MaxProLHD(10,2)$Design
MaxProMeasure(D,p_nom=0)

Maximum Projection Designs with Quantitative and Qualitative Factors

Description

Generate the maximum projection (MaxPro) design with quantitative and qualitative (QQ) factors. The quantitative factors can be continuous factors and/or discrete numeric factors, and the qualitative factors can be nominal factors and/or ordinal factors. If all factors are continuous, please use MaxProLHD directly.

To use this function, first convert all the ordinal factors into discrete numeric factors through the scoring method (see, e.g., Wu and Hamada 2009, Section 14.10). That is, if the ordinal factor has levels, "poor", "fair" and "good", then depending on the nature of the classification, the experimenter can choose some discrete numeric levels such as (1,2,3) or (1,4,5) to represent the three ordinal levels. Also, all columns of the continuous and discrete numeric factors should be standardized into the unit range of [0,1]

The users need to provide their desirable sub-design matrix for the nominal factors as part of the initial design input to this function. This function only optimizes the sub-design matrix of the continuous and discrete numeric factors as well as their row orders in joining with the pre-determined sub-design matrix of the nominal factors.

Usage

MaxProQQ(InitialDesign, p_nom=0, temp0=0, nstarts=1, itermax=400, total_iter=1e+06)

Arguments

Details

This function utilizes a version of the simulated annealing algorithm to efficiently generate the optimal design for continuous, nominal, discrete numeric and ordinal types of factors, where the design columns for nominal factors are pre-determined by the users and all the other columns are optimized based on the MaxProQQ criterion for the combined design. Parameters in the algorithm may need to be properly tuned to achieve global convergence. Please refer to Joseph, Gul and Ba (2018) for details.

Value

The value returned from the function is a list containing the following components:

Author(s)

Shan Ba <shanbatr@gmail.com> and V. Roshan Joseph <roshan@isye.gatech.edu>

References

Joseph, V. R., Gul, E., and Ba, S. (2015) "Maximum Projection Designs for Computer Experiments," Biometrika, 102, 371-380.

Joseph, V. R., Gul, E., and Ba, S. (2018) "Designing Computer Experiments with Multiple Types of Factors: The MaxPro Approach," Journal of Quality Technology.

Wu, C. F. J., and Hamada, M. (2009), Experiments: Planning, Analysis, and Parameter Design Optimization, 2nd Edition, New York: Wiley.

See Also

MaxProRunOrder, MaxProAugment, MaxProLHD

Examples

#18-run design
#3 continuous factors, 1 discrete numeric factor (6 levels), 2 nominal factors (3 levels)

#Generate a random Latin hypercube design (each factor standardized into [0,1]) 
#as the initial design for continuous factors
rand_design_part1=apply(matrix(rep(seq(from=0,to=1,length=18),3),ncol=3),2,sample)

#Generate a random initial design (standardized into [0,1]) for discrete numeric factors
rand_design_part2=sample(rep(seq(from=0,to=1,length=6),each=3))

#Construct an optimal design for the two nominal factors
OA_matrix=cbind(rep(1:3,each=6),rep(1:3,6))

#Initial design matrix
InitialDesign=cbind(rand_design_part1,rand_design_part2,OA_matrix)

#Optimize the design based on MaxProQQ criterion
obj=MaxProQQ(InitialDesign, p_nom=2) 
obj$Design

Find the Optimal Sequential Order to Run a Given Experimental Design

Description

Given a computer experimental design matrix, this function searches for an optimal run (row) order based on the maximum projection (MaxPro) criterion. This optimal order enables the given design to be run in a sequential manner: when terminated at any step, the previous design points form a nearly optimal subset based on the MaxPro criterion.

Usage

MaxProRunOrder(Design,p_nom=0,initial_row=1)

Arguments

Details

This function utilizes a greedy search algorithm to find the optimal row order to run the given experimental design based on the MaxPro criterion.

Value

The value returned from the function is a list containing the following components:

Author(s)

Shan Ba <shanbatr@gmail.com> and V. Roshan Joseph <roshan@isye.gatech.edu>

References

Joseph, V. R., Gul, E., and Ba, S. (2015) "Maximum Projection Designs for Computer Experiments," Biometrika, 102, 371-380.

Joseph, V. R. (2016) "Rejoinder," Quality Engineering, 28, 42-44.

Joseph, V. R., Gul, E., and Ba, S. (2018) "Designing Computer Experiments with Multiple Types of Factors: The MaxPro Approach," Journal of Quality Technology, to appear.

Wu, C. F. J., and Hamada, M. (2009), Experiments: Planning, Analysis, and Parameter Design Optimization, 2nd Edition, New York: Wiley.

See Also

MaxProLHD, MaxProQQ, MaxProAugment

Examples

D0=MaxProLHD(25,2)$Design 

#Assume the first two rows of the design have already been executed 
#Find the optimal run orders

D=MaxProRunOrder(D0,p_nom=0,initial_row=c(1,2))$Design
plot(D[,2],D[,3],xlim=c(0,1),ylim=c(0,1),type="n",
xlab=expression(x[1]),ylab=expression(x[2]),cex.lab=1.5)
text(D[,2],D[,3],labels=D[,1],col='red')