Package 'mixexp'

Title: Design and Analysis of Mixture Experiments
Description: Functions for creating designs for mixture experiments, making ternary contour plots, and making mixture effect plots.
Authors: John Lawson [aut, cre], Cameron Willden [aut], Greg Piepel [ctb]
Maintainer: John Lawson <[email protected]>
License: GPL (>= 2)
Version: 1.2.7.1
Built: 2024-10-24 04:26:11 UTC
Source: CRAN

Help Index

This package contains functions for creating designs for mixture experiments and making graphical display of results of mixture experiments.

Description

The mixexp package provides functions for creating mixture experiment designs in an unconstrained simplex or constrained mixture space. Functions are also provided for making ternary contour plots, pictures of constrained regions, design points, and mixture effect plots.

Details

Package: mixexp
Type: Package
Version: 1.2-1
Date: 2011-05-31
License: GPL2.0
Dependencies: gdata, lattice, grid
LazyLoad: yes
Packaged: 2011-05-31 19:54:07 UTC; Lawson
Built: R 2.12.1; i386-pc-mingw32; 2011-05-31 19:54:08 UTC; windows

Index: 

conmx                   example constraint matrix used as input to function 
                        crvtave
crvtave                 function for creating extreme vertices designs and 
                        centroids; this function calls Eflags, Nrows, and Vertcen
DesignPoints            function for plotting design points and or mixture 
                        constraint in the simplex
Eflags                  function for calling Piepel''s fortran code cnvrt 
                        to create extreme vertices designs and prints any 
						                        error messages
Effplot                 function for making mixture effect plots given a design
MixturePlot             function for making contour plots in simplex region given a design
MixModel                function for fitting mixture models to data
ModelPlot               function for making contour plots of an equation in an lm object 
                        created by the lm function or the MixModel function
Nrows                   function for calling Piepel's fortran code cnvrt to 
                        create extreme vertices designs and returns the 
						                        number of rows in the resulting design
SCD                     function for creating Simplex Centroid Designs
SLD                     function for creating Simplex Lattice Designs
Vertcen                 function for calling Piepel's fortran code cnvrt 
                        to create extreme vertices designs and returns the 
						                        resulting design
Xvert                   function for creating extreme vertices design 
                        and centroids, this function calls crvtave

Author(s)

John Lawson <[email protected]> and Cameron Willden <[email protected]>

Maintainer: John Lawson <[email protected]>

References

1. "John Lawson, Cameron Willden (2016).", "Mixture Experiments in R Using mixexp.", "Journal of Statistical Software, Code Snippets, 72(2), 1-20.", "doi:10.18637/jss.v072.c02"

Data from Table 4 in Gallant,Prickett, Cesarec, and Bruck(2008)

Description

This is an .rda file containing a mixture-process variable experiment with 3 mixture components and one process variable. z is the coded value of RPM, and y is average Burning rate for test presure 500(psig)

Usage

data(Burn)

Format

An 15 x 5 data frame

Source

source

References

Gallant, F.M., Prickett, S.E. Cesarec, M. and Bruck, H.A.(2008) Ingredients and processing effects on burning rates of composite rocket propellants utilizing a reduced-run mixture-process experiment design, Chemometrics and intellegent laboratory systems, Vol. 90, pp 49-63.

Example constraint matrix from Piepel 1988

Description

This is an .rda file containing the constraint matrix.

Usage

data(conmx)

Format

An 8 x 4 matrix

Source

source

References

Piepel, G. F. (1988) Programs for Generating Extreme Vertices and Centroids of Linearly Constrained Experimental Regions, Journal of Quality Technology, Vol. 20, No. 2.

This function creates an extreme vertices design

Description

This function calls the function Vertcen which uses Piepel's (1988) fortran code (cnvrt) for generating extreme vertices and centroids of linearly constrained mixture experimental regions.

Usage

crvtave(ndm,conmx)

Arguments

ndm

is an integer representing the highest order of centroids requested. An overall centroid is always included, 0 indicates no other centroids will be created, 1 indicates edge centroids are requested, 2 indicates face centroids, etc.
conmx

This is the matrix of constraints.

Value

vtcn

This is a data frame containing the extreme vertices design. The columns are labeled x1, x2 ...xn, where n is the number of mixture variables. The last column is labeled dimen and it indicates the order of centroid where 0 is an extreme vertex, 1 is an edge centroid, 2 is a face centroid, and n is the overall centroid.

Note

This function calls the function Eflags to get error messages from cnvrt, the function Vertcen to get the extreme vertices and centroids from cnvrt, and the function Nrows to get the number of vertices and centroids from cnvrt.

Author(s)

John S. Lawson [email protected]

References

1. Piepel, G. F. "Programs for Generating Extreme Vertices and Centroids of Linearly Constrained Experimental Regions" Journal of Quality Technology, Vol 20, No. 2, pp. 125-139, 1988.

Examples

data(conmx)
crvtave(1,conmx)

Creates cubic terms for Scheffe' full cubic model (3)

Description

Creates cubic terms that are used by the function MixModel when fitting model (3)

Usage

cubic(a, b)

Arguments

a

input - vector of mixture components in a column in the data frame

b

input - another vector of mixture components in a column in the data frame

Value

vector of elementwise a^2*b-a*b^2 function of terms in the a and b vectors

Author(s)

John Lawson

This function plots design points and or constraints in the simplex mixture space, given a data frame containing the design or vextors x, y, and z of the same length that contain the mixture components in the design.

Description

This function plots design points and or constraints in the simplex mixture space. It calls the function MixturePlot that does the actual plotting.

Usage

DesignPoints(des = NULL,nmxcmp=3, x = NULL, y = NULL, z = NULL,x1lower=0,x1upper=0,
                         x2lower=0, x2upper=0, x3lower=0,x3upper=0,
                         cornerlabs = c("x3","x2","x1"),
                         axislabs=c("x1","x2","x3"),pseudo=FALSE)

Arguments

des

data frame containing x1 x2 and x3 coordinates of data points to be plotted
nmxcmp

interger indicating the number of mixture components in the design
x

vector of x3 coordinates of design points to be plotted
y

vector of x2 coordinates of design points to be plotted
z

vector of x1 coordinates of design points to be plotted
x1lower

lower constraint on x1
x1upper

upper constraint on x1
x2lower

lower constraint on x2
x2upper

upper constraint on x2
x3lower

lower constraint on x3
x3upper

upper constraint on x3
axislabs

This is a vector of text labels for the x1, x2 and x3 axis.
cornerlabs

This is a vector of text labels for the x1, x2 and x3 vertices.
pseudo

logical variable, when TRUE plot is made in pseudo component space bounded by the lower constraints of each component.

Note

This function calls MixturePlot. If either des and x,y,z are missing no design points will be plotted, and if x1lower, x1upper, etc. are all zero no constraints will be plotted. If there are more than 3 columns of mixture components in des, only the first 3 will be plotted ignoring the others.

Author(s)

John S. Lawson [email protected]

References

1. Piepel, G. F. "Programs for Generating Extreme Vertices and Centroids of Linearly Consrtained Experimental Regions" Journal of Quality Technology, Vol 20, No. 2, pp. 125-139, 1988.

2. "John Lawson, Cameron Willden (2016).", "Mixture Experiments in R Using mixexp.", "Journal of Statistical Software, Code Snippets, 72(2), 1-20.", "doi:10.18637/jss.v072.c02"

Examples

dat<-SCD(3)
DesignPoints(des=dat)

x1<-c(1,0,0,.5,.5, 0,.33333)
x2<-c(0,1,0,.5,0,.5,.33333)
x3<-c(0,0,1,0,.5,.5,.33333)
DesignPoints(x=x3,y=x2,z=x1)

dat<-data.frame(x1,x2,x3)
DesignPoints(des=dat)

This function creates mixture effect plots

Description

This function makes effect plots using the Cox or Piepel directions in constrained mixture space.

Usage

EffPlot(des=NULL,nfac=3,mod=1,dir=1)

Arguments

des

data frame containing the design points and response data for a mixture experiment. The data frame must contain the variables x1, x2 ...xn for the mixture variables, and y for the response. n must be between 2 and 12. Only effect plots for linear models can be made when the number of factors is greater than 6.
nfac

The number of mixture components in the model.
mod

an interger representing the model to be traced: 1 for a linear model, 2 for a quadratic model, and 4 for a special cubic model. For models other than these, use the ModelEff function.
dir

an integer representing the direction for which the effect plot is made: 1 for Piepel direction, 2 for Cox direction.

Value

PX

This is a matrix containing the coordinates of the effect plot traces that are plotted.

Note

This function calls the function crvtave to get the design centroid from cnvrt.

Author(s)

John S. Lawson [email protected]

References

1. Piepel, G. F. "Measuring Component Effects in Constrained Mixture Experiments" Technometrics, Vol 25, pp. 97-105, 1982.

2. "John Lawson, Cameron Willden (2016).", "Mixture Experiments in R Using mixexp.", "Journal of Statistical Software, Code Snippets, 72(2), 1-20.", "doi:10.18637/jss.v072.c02".

Examples

#Example from Li, Tolley, Lee(2010) response is perm
x1<-c(.572,.358,.286,.286,.286,.143,.357)
x2<-c(.214,.428,.500,.357,.214,.500,.500)
x3<-c(.214,.214,.214,.357,.500,.357,.143)
y<-c(7.7,18.4,24.2,9.8,5.9,23.0,19.4)
des<-data.frame(x1,x2,x3,y)
EffPlot(des,2,2)


#Example from Snee, Marquart(1976)
x1<-c(.1,.1,.1,.15,.1,.1,.1,.4,.35,.30,.1,.45,.45,.45,.45,.45,.259,.259,.259,.259)
x2<-c(.5,.05,.5,.05,.05,.5,.05,.05,.05,.5,.5,.05,.2,.15,.25,.1,.222,.222,.222,.222)
x3<-c(0,0,0,0,.1,.1,.1,.1,.1,0,.1,0,0,0,.1,.1,.05,.05,.05,.05)
x4<-c(0,0,.1,.1,0,.1,.1,.1,.1,0,0,0,.1,.1,0,0,.05,.05,.05,.05)
x5<-c(.1,.55,.1,.6,.55,.1,.55,.1,.1,.1,.2,.45,.1,.1,.1,.1,.244,.244,.244,.244)
x6<-c(.2,.2,.2,.05,.2,.05,.05,.2,.2,.05,.05,.05,.05,.2,.05,.2,.125,.125,.125,.125)
x7<-c(.05,.05,0,.05,0,0,0,.05,.05,0,.05,0,.05,0,.05,0,.025,.025,.025,.025)
x8<-c(.05,.05,0,0,0,.05,.05,0,.05,.05,0,0,.05,0,0,.05,.025,.025,.025,.025)
y<-c(30,113,17,94,89,18,90,20,21,15,28,48,18,7,16,19,38,30,35,40)
des<-data.frame(x1,x2,x3,x4,x5,x6,x7,x8,y)
EffPlot(des,mod=1,dir=1)



# Weed control example from Lawson & Erjavec
x1<-c(1,0,0,.5,.5,0,.33333,.33333,.33333)
x2<-c(0,1,0,.5,0,.5,.33333,.33333,.33333)
x3<-c(0,0,1,0,.5,.5,.33333,.33333,.33333)
y<-c(73,68,80,77,86,75,92,93,88)
des<-data.frame(x1,x2,x3,y)
EffPlot(des,3)



# Polvoron Example from Lawson
des<-Xvert(3,uc=c(.8,.95,.50),lc=c(0,.10,.05),ndm=1,plot=FALSE)
dat<-as.matrix(des)
# remove the edge centroid at the top
dat<-dat[c(1:6,8:11), ]
# add two more centroids
dat<-rbind(dat,dat[10, ],dat[10,])
# response vector
y<-c(5.75,3.69,5.33,5.68,3.85,3.83,5.88,5.87,5.23,6.54,6.82,6.41)
# make the data frame for plotting
des<-data.frame(dat[,1:3],y)
EffPlot(des,3)

# Cornell's example of blending pesticides for control of mites (special cubic model)
mite<-SCD(4)
yavg<-c(1.8,25.4,28.6,38.5,4.9,3.1,28.7,3.4,37.4,10.7,22.0,2.6,2.4,
        11.1,0.8)
mite<-cbind(mite,yavg)
mite2<-mite
names(mite2)<-c("x1","x2","x3","x4","y")
EffPlot(des=mite2,mod=4,dir=2)

Loads compiled fortran in shared file cnvrt and returns the error messages

Description

This function loads and runs the compiled fortran code cnvrt and prints error messages. cnvrt is Piepel's 1988 JQT fortran code for extreme vertices designs.

Usage

Eflags(ndm,nvrr,ncon2,rtheta2)

Arguments

ndm

This is the order of centroids desired (0=none, 1=edge centroids, 2=face centroids etc.)
nvrr

This is the number of mixture variables ( maximum is 12)
ncon2

This is the number of constraints (maximum is 45)
rtheta2

This is the constraint matrix stored as a vector of columns.

Value

ifa

This is the vector of error flags. A negative value for flag 1 indicates that there are inconsistent constraints. A negative value for flag2 indicates there are two many vertices and centroids, this program only works when # vertices + # centroids <=1000. A negative value for flag 3 indicates an error encountered when calling subroutine allnr.

Note

This function is called by the function crtave.

Author(s)

John S. Lawson [email protected]

References

1. Piepel, G. F. "Programs for Generating Extreme Vertices and Centroids of Linearly Consrtained Experimental Regions" Journal of Quality Technology, Vol 20, No. 2, pp. 125-139, 1988.

Data from Etch rate experiment in Table 12.4 of Myers and Montgomery(2002)

Description

This is an .rda file containing a mixture experiment with 3 mixture components. The mixture components are x1, x2, and x3. The response is erate.

Usage

data(etch)

Format

An 14 x 4 data frame

Source

source

References

Myers, R. H. and Montgomery D. C. (2002) Response Surface Methodology - Product and Process Optimization Using Designed Experiments John Wiley and Sons, New York.

This function Creates interior points in an existing mixture design.

Description

This function creates interior points in a mixture design by averaging all possible pairs of design points. It duplicates SAS macro adxfill.

Usage

Fillv(nfac,des)

Arguments

nfac

an integer representing the number of mixture variables in the design
des

a data frame containing a mixture design created by one of the functions SLD, SCD or Xvert

Author(s)

John S. Lawson [email protected]

Examples

# Example 1 fills interior of Simplex Lattice Design
des<-SLD(3,3)
DesignPoints(des)
des2<-Fillv(3,des)
DesignPoints(des2)

# Example 2 fills interior of Simplex Centroid Design
des<-SCD(4)
Fillv(4,des)

# Example 3 fills interior of Extreme vertices design
ev<-Xvert(3,uc=c(.1,.1,1.0),lc=c(0,0,0),ndm=1)
ev2<-Fillv(3,ev)

Data from Cornell's famous fish patty mixture process variable experiment

Description

This is an .rda file design and response.

Usage

data(fishp)

Format

An 56 x 7 data frame

Source

source

References

Cornell, J. A., Experiments with Mixtures, Third Edition, John Wiley and Sons, 2002, pp 361-365.

Fit mixture and mixture process variable models.

Description

This function fits mixture models (1)-(4) and mixture process models (5)-(6) described in Lawson and Willden(2015) "Mixture Experiments in R, using mixexp", Journal Statistical Software http://www/jstatsoft.org/, and prints the correct R square and standard errors of model coefficients.

Usage

MixModel(frame, response, mixcomps=NULL,model,procvars=NULL)

Arguments

frame

a data frame containing columns with the mixture components, process variables, and responses
response

a character variable containing the column name of the response variable in frame to be fit
mixcomps

a character vector of column names of the mixture components in frame
model

an integer in the range of 1 to 6, indicating the model to be fit:

1.y=∑i=1qβixi+ϵ.1. y=\sum_{i=1}^{q} \beta_i x_i + \epsilon..

2.y=∑i=1qβixi+∑i=1q−1∑j=i+1qβijxixj+ϵ.2. y=\sum_{i=1}^{q} \beta_i x_i + \sum_{i=1}^{q-1} \sum_{j=i+1}^{q} \beta_{ij} x_i x_j + \epsilon..

3.y=∑i=1qβixi+∑i=1q−1∑j=i+1qβijxixj+∑i=1q−1∑j=i+1qδijxixj(xi−xj)+∑i=1q−2∑j=i+1q−1∑k=j+1qβijkxixjxk+ϵ.3. y=\sum_{i=1}^{q} \beta_i x_i + \sum_{i=1}^{q-1} \sum_{j=i+1}^{q} \beta_{ij} x_i x_j + \sum_{i=1}^{q-1} \sum_{j=i+1}^{q} \delta_{ij} x_i x_j (x_i - x_j) + \sum_{i=1}^{q-2} \sum_{j=i+1}^{q-1} \sum_{k=j+1}^{q} \beta_{ijk} x_i x_j x_k + \epsilon..

4.y=∑i=1qβixi+∑i=1q−1∑j=i+1qβijxixj+∑i=1q−2∑j=i+1q−1∑k=j+1qβijkxixjxk+ϵ.4. y=\sum_{i=1}^{q} \beta_i x_i + \sum_{i=1}^{q-1} \sum_{j=i+1}^{q} \beta_{ij} x_i x_j + \sum_{i=1}^{q-2} \sum_{j=i+1}^{q-1} \sum_{k=j+1}^{q} \beta_{ijk} x_i x_j x_k + \epsilon..

5.y=(∑i=1qβixi+∑i=1q−1∑j=i+1qβijxixj)(α0+∑l=1pαlzl+∑l=1p−1∑m=l+1pαlmzlzm)+ϵ5. y = ( \sum_{i=1}^{q} \beta_i x_i + \sum_{i=1}^{q-1} \sum_{j=i+1}^{q} \beta_{ij} x_i x_j )(\alpha_0 + \sum_{l=1}^{p} \alpha_l z_l + \sum_{l=1}^{p-1} \sum_{m=l+1}^{p} \alpha_{lm} z_l z_m ) + \epsilon.

6.y=∑i=1qβi(0)xi+∑i=1q−1∑j=i+1qβij(0)xixj+∑k=1m[∑i=1qβi(1)xi]zk+∑k=1m−1∑l=k+1mαklzkzl+∑k=1mαkkzk2+ϵ.6. y=\sum_{i=1}^{q} \beta_i^{(0)} x_i + \sum_{i=1}^{q-1} \sum_{j=i+1}^{q} \beta_{ij}^{(0)} x_i x_j + \sum_{k=1}^m \left[\sum_{i=1}^{q} \beta_i^{(1)} x_i \right]z_k + \sum_{k=1}^{m-1} \sum_{l=k+1}^m \alpha_{kl} z_k z_l + \sum_{k=1}^m \alpha_{kk} z_k^2 + \epsilon.

where xix_i are mixture components, and zjz_j are process variables.

procvars

a character vector of column names of the process variables in frame to be included in the model. Leave this out if there are no process variables in the frame

Author(s)

John S. Lawson [email protected]

References

1. "John Lawson, Cameron Willden (2016).", "Mixture Experiments in R Using mixexp.", "Journal of Statistical Software, Code Snippets, 72(2), 1-20.", "doi:10.18637/jss.v072.c02"

Examples

# example from Lawson(2014), quadratic model
library(daewr)
data(pest)
mixvars<-c("x1","x2","x3")
MixModel(pest,"y",mixvars,2)

# example from Myers and Montgomery(2002), special cubic model
library(mixexp)
etch<-SCD(3)
etch<-Fillv(3,etch)
etch<-rbind(etch[1:7, ],etch[1:3, ],etch[7, ], etch[etch$x1==2/3, ],
etch[etch$x2==2/3, ],etch[etch$x3==2/3, ])
erate<-c(540,330,295,610,425,330,800,560,350,260,850,710,640,460)
etch<-cbind(etch,erate)
mixvars<-c("x1","x2","x3")
response<-c("erate")
MixModel(etch,response,mixvars,4)

# example Mixture process variable model from Sahni, Pieple and Naes(2009)
library(daewr)
mixvars<-c("x1","x2","x3")
procvars<-c("z1","z2")
data(MPV)
MixModel(MPV,"y",mixvars,5,procvars)

#### Kowalski Cornell and Vining Simplified model on data from Gallant et. al. (2008)
data(Burn)
testBNM<-MixModel(Burn,"y",mixcomps=c("Course","Fine","Binder"),model=6,procvars=c("z"))

This function makes contour plots in the simplex mixture space.

Description

This function makes contour plots in the simplex mixture space, it also can draw constraint lines and show design points.

Usage

MixturePlot(x=NULL,y=NULL,z=NULL,w=NULL,des=NULL,
                       res=400,lims=c(rep(0,6)),color.palette = heat.colors,
                       constrts=FALSE,contrs=TRUE,n.breaks=10,levels=NULL,
                       cols=FALSE, despts=TRUE, mod=NA,x3lab="Fraction X3",
                       x2lab="Fraction X2", x1lab="Fraction X1", 
                       corner.labs = NULL,
                       colorkey=list(dx=0.04,x0=0.95,y0=0.45,y1=0.90,add=TRUE,mode="all"),
                       pseudo=FALSE)

Arguments

x

x3 locations for known points
y

x2 locations for known points
z

x1 locations for known points
w

y locations for known points
des

data frame with x1,x2,x3, and y locations for known points
res

number of color blocks between 0 and 1 of x
lims

vector of lower and upper constraints for x1,x2,x3
color.palette

is the color palette to use
constrts

if TRUE constraints found in lines will be added to the graph
contrs

if TRUE contour lines will be added to the graph
n.breaks

number of breaks between levels, this is used if levels is not specified
levels

vector of contour levels to be plotted
cols

if TRUE regions between contour lines will be colored
despts

if TRUE plots the design points in data frame des
mod

is an indicator for the model 1=linear, 2=quadratic, 4=special cubic. for other Models use the ModelEff function.
x3lab

label for the x3 axis
x2lab

label for the x2 axis
x1lab

label for the x1 axis
corner.labs

labels for x3, x2 and x1 vertices
colorkey

a list with the location of the color key
pseudo

if pseudo=TRUE uses pseudo components to zoom in on constrained region. By default pseudo=FALSE

Author(s)

John S. Lawson [email protected]

References

1. Cornell, J. A. Experiments with Mixtures: Models and Analysis of Mixture Data, John Wiley & Sons, New York, third edition, 2002.

2. See R Ternary Level Plot Function http://www.siftp.net/index.shtml

3. "John Lawson, Cameron Willden (2016).", "Mixture Experiments in R Using mixexp.", "Journal of Statistical Software, Code Snippets, 72(2), 1-20." "doi:10.18637/jss.v072.c02"

Examples

##Usage and Examples - Example from page 458 DAE with SAS
dat = data.frame(
          "x1"=c(1,.8,.6,.5,.5,.33333,.3,.3,.1,.1,0,0,0),
          "x2"=c(0,.1,.2,0,.5,.33333,.2,.5,.1,.8,0,.5,1),
          "x3"=c(0,.1,.2,.5,0,.33333,.5,.2,.8,.1,1.0,.5,0),
          "y"=c(48.7,49.5,50.2,52.8,49.3,51.1,52.7,50.3,60.7,49.9,64.9,53.5,50.6) 
                 )
MixturePlot(dat$x3,dat$x2,dat$x1,dat$y, x3lab="Fraction x3",
  x2lab="Fraction x2", x1lab="Fraction x1", corner.labs=c("x3","x2","x1"),
  constrts=FALSE,contrs=TRUE,cols=TRUE, mod=2,n.breaks=9)

# Weed control example from Lawson & Erjavec
x1<-c(1,0,0,.5,.5,0,.33333,.33333,.33333)
x2<-c(0,1,0,.5,0,.5,.33333,.33333,.33333)
x3<-c(0,0,1,0,.5,.5,.33333,.33333,.33333)
y<-c(73,68,80,77,86,75,92,93,88)
des<-data.frame(x1,x2,x3,y)
MixturePlot(des=des,x3lab="Fraction C",x2lab="Fraction B",
   x1lab="Fraction A",corner.labs=c("C","B","A"),mod=4,n.breaks=5,cols=TRUE)

This function creates mixture effect plots

Description

This function makes effect plots using the Cox or Piepel directions in constrained mixture space.

Usage

ModelEff(nfac=3,mod=1,nproc=0,dir=1,ufunc=mod,dimensions = list(NULL),
pvslice=c(1,1,1),lc=c(0,0,0,0,0,0,0,0,0,0,0,0), uc=c(1,1,1,1,1,1,1,1,1,1,1,1))

Arguments

nfac

The number of mixture components in the model (a number between 2 and 12)
mod

An integer representing the model to be traced: 1 for a linear model, 2 for a quadratic model, and 3 for a cubic model, 4 for a special cubic model, 5 for a mixture process variable model consisting of a full cross of quadratic model in up to 5 mixture components and a linear model in up to 3 process variables, 6 for Kowalski, Cornell and Vining's (2000) more parsimonious model for mixture process variable experiments. See the documentation for the MixModel function for a description of the models.
nproc

The number of process variables in the model (a number between 1 and 3 for models 5 and 6)
dir

an integer representing the direction for which the effect plot is made: 1 for Piepel direction, 2 for Cox direction.
ufunc

A user function, this should an lm object created by the MixModel function. Any lm object will work if the terms are in the same order as the model produced by the MixModel function.
dimensions

A vector of names of mixture components in the lm object.
pvslice

A vector giving fixed values of the process variables.
lc

A vector giving the lower bounds of the mixture components.
uc

A vector giving the upper bounds of the mixture components.

Value

PX

This is a matrix containing the coordinates of the effect plot traces that are plotted.

Note

This function calls the function crvtave to get the design centroid from cnvrt.

Author(s)

John S. Lawson [email protected]

References

1. Piepel, G. F. "Measuring Component Effects in Constrained Mixture Experiments" Technometrics, Vol 25, pp. 97-105, 1982.

2. "John Lawson, Cameron Willden (2016).", "Mixture Experiments in R Using mixexp.", "Journal of Statistical Software, Code Snippets, 72(2), 1-20.", "doi:10.18637/jss.v072.c02"

3. Kowalski, S. M., Cornell, J. A. and Vining, G. G. "A Model and Class of Designs for Mixture Experiments with Process Variables" Communication in Statistics: Theory and Methods, Vol 29, pp. 2255-2280.

Examples

#Example p. 63-65 Cornell (control of Mites)
# Four Component Mixture
mite<-SCD(4)
yavg<-c(1.8,25.4,28.6,38.5,4.9,3.1,28.7,3.4,37.4,10.7,22.0,2.6,2.4,
        11.1,0.8)
mite<-cbind(mite,yavg)
miteSC<-MixModel(mite,"yavg",mixcomps=c("x1","x2","x3","x4"),model=4)
ModelEff(nfac=4,mod=4,nproc=0,dir=2,ufunc=miteSC,lc=c(0,0,0,0),uc=c(1,1,1,1))


# Cornell's (2002) Yarn elongation
x1<-c(1,1,.5,.5,.5,0,0,0,0,0,0,0,.5,.5,.5)
x2<-c(0,0,.5,.5,.5,1,1,.5,.5,.5,0,0,0,0,0)
x3<-c(0,0,0,0,0,0,0,.5,.5,.5,1,1,.5,.5,.5)
y<-c(11,12.4,15,14.8,16.1,8.8,10,10,9.7,11.8,16.8,16,17.7,16.4,16.6)
elong<-data.frame(x1,x2,x3,y)
testQ2<-MixModel(elong,"y",mixcomps=c("x1","x2","x3"),model=2)
ModelEff(nfac=3,mod=2,nproc=0,dir=2,ufunc=testQ2,lc=c(0,0,0),uc=c(1,1,1))

#### Kowalski Cornell and Vining Simplified model on data from Gallant et. al. (2008)
data(Burn)
testBNM<-MixModel(Burn,"y",mixcomps=c("Course","Fine","Binder"),model=6,procvars=c("z"))
ModelEff(nfac=3,mod=6,nproc=1,dir=1,ufunc=testBNM,dimensions = list(NULL), pvslice=c(1),
lc=c(.403,.166,.130),uc=c(.704,.412,.210))
ModelEff(nfac=3,mod=6,nproc=1,dir=1,ufunc=testBNM,dimensions = list(NULL), pvslice=c(-1),
lc=c(.403,.166,.130),uc=c(.704,.412,.210))

This function makes contour plots of a user-supplied model in the simplex mixture space.

Description

This function makes contour plots in the simplex mixture space. It also can draw constraint lines and zoom in on pseudo component region.

Usage

ModelPlot(model=NULL,user.func = NULL, dimensions = list(x1=NULL,x2=NULL,x3=NULL),
          slice=NULL,lims=rep(0,6), constraints = FALSE,
          constraint.pars = list(lty=2,lwd=2),
          contour = FALSE, contour.pars = list(lwd=0.5,cex.lab=1.3),
          cuts = 10,at = NULL, res=300, pseudo=FALSE, 
          fill=FALSE, color.palette = heat.colors, 
          main=NULL, axislabs=c("Fraction X1","Fraction X2","Fraction X3"),
          axislab.pars = list(),
          axislab.offset=0,
          cornerlabs = c("X1", "X2", "X3"),
          cornerlab.pars = list(),
          grid=TRUE, grid.pars = list(col='darkgrey',lty=3,lwd=0.5),
          colorkey = FALSE,
          labels=TRUE, label.style="align", ...)

Arguments

model

an lm object, MixModel object, or any other model object that is compatible with the predict function, which is the mixture model to be plotted.
user.func

function supplied by the user that takes as arguments a dataframe called 'grid' and returns the predictions. This argument has been deprecated in favor of the model argument. Typically, this will be a wrapper function for predict() (e.g. predict(model,newdata=grid)). Additional arguments for user.func can be passed using the ellipsis argument for ModelPlot. Overrides model argument if both are specified.
dimensions

list argument that specifies the mixture variables to be plotted on the ternary plot. Values must correspond to variable names from the user-supplied model.
slice

list argument that specifies the value of fixed mixture components.
lims

vector of lower and upper constraints for ternary plot components (TopLower, TopUpper, LeftLower, LeftUpper, RightLower, RightUpper).
constraints

if TRUE constraints found in lims will be added to the graph.
constraint.pars

list of graphical parameters controlling the appearance of the constraint lines.
contour

if TRUE contour lines will be added to the graph.
contour.pars

list of graphical parameters controlling the appearance of the contour lines.
cuts

number of breaks between levels (used for contours if 'at' not specified).
at

list of contour levels (e.g. at=c(1,3,5,10) will draw contours at those heights). Overrides cuts argument.
res

resolution of the grid. Corresponds to number equally spaced values along the baseline of the simplex.
pseudo

if TRUE uses pseudo components to zoom in on constrained region. Will create the smallest equilateral triangle that still contains the whole constrained region.
fill

if TRUE regions between contour lines will be colored.
color.palette

is the color palette to use.
main

character value for main title or list containing character value and graphical parameters (e.g. main=list("main title",cex=2)).
axislabs

character vector of axis labels for ternary components.
axislab.pars

list of graphical parameters controlling the appearance of the axislabels.
axislab.offset

numeric value that creates or eliminates space between the angled axislabels and the tickmarks. Prevents axis labels from overlapping with tickmarks. Typically, absolute value would not exceed 0.05.

cornerlabs

character vector of corner labels for x1, x2 and x3 vertices.
cornerlab.pars

list of graphical parameters controlling the appearance of the axis labels.
grid

logical argument. If true, adds gridlines to the ternary plot.
grid.pars

list of graphical parameters controlling the appearance of the gridlines.
colorkey

logical or list of parameters. See levelplot documentation for more details.
labels

logical argument. If true, labels contour lines.
label.style

controls placement of contour labels. Choose from "mixed","flat", or "align." See panel.levelplot documentation for more details.
...

additional arguments passed to user.func

Author(s)

Cameron Willden [email protected]

References

1. Cornell, J. A. Experiments with Mixtures: Models and Analysis of Mixture Data, John Wiley & Sons, New York, third edition, 2002.

2. See R Ternary Level Plot Function http://www.siftp.net/index.shtml

3. "John Lawson, Cameron Willden (2016).", "Mixture Experiments in R Using mixexp.", "Journal of Statistical Software, Code Snippets, 72(2), 1-20.", "doi:10.18637/jss.v072.c02"

Examples

# Cornell's (2002) Yarn elongation
x1<-c(1,1,.5,.5,.5,0,0,0,0,0,0,0,.5,.5,.5)
x2<-c(0,0,.5,.5,.5,1,1,.5,.5,.5,0,0,0,0,0)
x3<-c(0,0,0,0,0,0,0,.5,.5,.5,1,1,.5,.5,.5)
y<-c(11,12.4,15,14.8,16.1,8.8,10,10,9.7,11.8,16.8,16,17.7,16.4,16.6)
elong<-data.frame(x1,x2,x3,y)
testQ<-lm(y~-1+x1+x2+x3+x1:x2+x1:x3+x2:x3,data=elong)
ModelPlot(model = testQ,dimensions = list(x1="x1",x2="x2",x3="x3"), 
           main="Thread Elongation",constraints=FALSE,contour=TRUE,
           at=c(12, 13, 14, 15, 16, 17),fill=FALSE,
           axislabs=c("X1", "X2", "X3"),
           cornerlabs = c("X1", "X2", "X3"),pseudo=FALSE)
           

# Cornells famous fish patty experiment 
data(fishp)
fishmod2<-MixModel(fishp, "y", mixcomps=c("x1","x2","x3"),model=5,procvars=c("z1","z2","z3"))
ModelPlot(fishmod2,dimensions = list(x1="x1",x2="x2",x3="x3"), 
          slice = list(process.vars=c(z1=-1, z2=-1, z3=-1)), main="z1=-1, z2=-1, z3=-1",
          constraints=FALSE,contour=TRUE,cuts=10,fill=FALSE,
          axislabs=c("Fraction X1","Fraction X2","Fraction X3"),
          cornerlabs = c("X1", "X2", "X3"),pseudo=FALSE)
          
          
#### Kowalski Cornell and Vining Simplified model on data from Gallant et. al. (2008)
data(Burn)
testBNM<-MixModel(Burn,"y",mixcomps=c("Course","Fine","Binder"),model=6,procvars=c("z"))
ModelPlot(testBNM,dimensions = list(x1="Course",x2="Fine",x3="Binder"), 
          slice = list(process.vars=c(z=1)), lims=c(.403,.704,.166,.467,.130,.431), main="z=1",
          constraints=TRUE,contour=TRUE,cuts=5,fill=FALSE,
          axislabs=c("Fraction Course","Fraction Fine","Fraction Binder"),
          cornerlabs = c("Course", "Fine", "Binder"),pseudo=TRUE)

ModelPlot(testBNM,dimensions = list(x1="Course",x2="Fine",x3="Binder"), 
          slice = list(process.vars=c(z=-1)),  lims=c(.403,.704,.166,.467,.130,.431),main="z=-1",
          constraints=TRUE,contour=TRUE,cuts=5,fill=FALSE,
          axislabs=c("Fraction Course","Fraction Fine","Fraction Binder"),
          cornerlabs = c("Course", "Fine", "Binder"),pseudo=TRUE)

Loads compiled fortran in shared file cnvrt and returns the number of rows in the resulting design

Description

This function loads and runs the compiled fortran code cnvrt. cnvrt is Piepel's 1988 JQT fortran code for extreme vertices designs.

Usage

Nrows(ndm,nvrr,ncon2,rtheta2)

Arguments

ndm

This is the order of centroids desired (0=none, 1=edge centroids, 2=face centroids etc.)
nvrr

This is the number of mixture variables ( maximum is 12)
ncon2

This is the number of constraints (maximum is 45)
rtheta2

This is the constraint matrix stored as a vector of columns.

Value

nvrtr

nvrtr

This is the number of rows in rxvt the matrix of extreme vertices and centroids

Note

This function is called by the function crtave.

Author(s)

John S. Lawson [email protected]

References

1. Piepel, G. F. "Programs for Generating Extreme Vertices and Centroids of Linearly Consrtained Experimental Regions" Journal of Quality Technology, Vol 20, No. 2, pp. 125-139, 1988.

This function creates simplex centroid mixture designs

Description

This function creates simplex centroid designs in unconstrained mixture experiment space.

Usage

SCD(fac)

Arguments

fac

This is the number of factors

Value

SC

This is a data frame containing the simplex centroid design. The columns are labeled x1, x2 ...xn, where n is the number of mixture variables.

Author(s)

John S. Lawson [email protected]

References

1. Cornell, J. A. Experiments with Mixtures: Models and Analysis of Mixture Data, John Wiley & Sons, New York, third edition, 2002.

2. "John Lawson, Cameron Willden (2016).", "Mixture Experiments in R Using mixexp.", "Journal of Statistical Software, Code Snippets, 72(2), 1-20.", "doi:10.18637/jss.v072.c02"

Examples

SCD(3)

des<-SCD(5)

des<-SCD(12)

This function creates simplex lattice mixture designs

Description

This function creates simplex lattice designs in unconstrained mixture experiment space.

Usage

SLD(fac, lev)

Arguments

fac

This is the number of factors, this must be between 2 and 12
lev

This is the number of levels, this must be between 2, and 5.

Value

SL

This is a data frame containing the simplex lattice design. The columns are labeled x1, x2 ...xn, where n is the number of mixture variables.

Author(s)

John S. Lawson [email protected]

References

1. Cornell, J. A. Experiments with Mixtures: Models and Analysis of Mixture Data, John Wiley \& Sons, New York, third edition, 2002.

2. "John Lawson, Cameron Willden (2016).", "Mixture Experiments in R Using mixexp.", "Journal of Statistical Software, Code Snippets, 72(2), 1-20.", "doi:10.18637/jss.v072.c02"

Examples

des<-SLD(3,2)

des<-SLD(4,3)

Data from Snee and Marquart's Screening Experiment with contsrained mixture components

Description

This is an .rda file design and response.

Usage

data(SneeMq)

Format

An 16 x 9 data frame

Source

source

References

Snee, D. D. and Marquardt D. W. (1976) Screening Concepts and designs for experiments with mixtures, Technometrics, Vol. 18, pp 19-29.

Loads compiled fortran in shared file cnvrt

Description

This function loads and runs the compiled fortran code cnvrt. cnvrt is Piepel's 1988 JQT fortran code for extreme vertices designs.

Usage

Vertcen(ndm,nvrr,ncon2,rtheta2)

Arguments

ndm

This is the order of centroids desired (0=none, 1=edge centroids, 2=face centroids etc.)
nvrr

This is the number of mixture variables ( maximum is 12)
ncon2

This is the number of constraints (maximum is 45)
rtheta2

This is the constraint matrix stored as a vector of columns.

Value

rxvt

This is the matrix of vertices and centroids stored as a vector of columns.

Note

This function is called by the function crtave.

Author(s)

John S. Lawson [email protected]

References

1. Piepel, G. F. "Programs for Generating Extreme Vertices and Centroids of Linearly Consrtained Experimental Regions" Journal of Quality Technology, Vol 20, No. 2, pp. 125-139, 1988.

This function creates an extreme vertices design in a constrained mixture space.

Description

This function calls the function crvtave to create an extreme vertices design in a constrained mixture space. If there are only three factors the function DesignPoints is called to plot the results.

Usage

Xvert(nfac=3,uc=c(0,0),lc=c(0,0),nlc=0,lb=c(0,0),ub=c(0,0),coef,ndm=0,plot=TRUE,
                 cornerlabs = c("x1","x2","x3"), axislabs = c("x1","x2","x3"),
                 pseudo=TRUE)

Arguments

nfac

an integer representing the number of mixture variables in the design. Maximum nfac=12
uc

a vector of length nfac containing upper constraints on each mixture component
lc

a vector of length nfac containing lower constraints on each mixture component
nlc

the number of linear constraints, the default is zero
lb

a vector of length nlc containing the lower bounds for the linear constraints
ub

a vector of length nlc containing the upper bounds for the linear constraints
coef

an nlc by nfac matrix containing the coefficients of the components of the linear constraints
ndm

an integer representing the highest order of centroids requested. An overall centroid is always included, 0 indicates no other centroids will be created, 1 indicates edge centroids are requested, etc.
plot

a logical variable indicating whether a plot of the design is desired when there are only 3 components. Default is TRUE
cornerlabs

This is a vector of text labels for the x1, x2 and x3 vertices. Use when there are only 3 components for plotting.
axislabs

This is a vector of text labels for the x1, x2 and x3 axis. Use when there are only 3 components for plotting.
pseudo

logical variable, when TRUE plot in pseudo component space when there are lower constraints.

Note

This function calls crvtave. If the number of factors is 3, the function DesignPoints is called to graph the results.

Author(s)

John S. Lawson [email protected]

References

1. Piepel, G. F. "Programs for Generating Extreme Vertices and Centroids of Linearly Consrtained Experimental Regions" Journal of Quality Technology, Vol 20, No. 2, pp. 125-139, 1988.

2. "John Lawson, Cameron Willden (2016).", "Mixture Experiments in R Using mixexp.", "Journal of Statistical Software, Code Snippets, 72(2), 1-20.", "doi:10.18637/jss.v072.c02"

Examples

# Polvoron Example from Lawson
des<-Xvert(3,uc=c(.8,.95,.50),lc=c(0,.10,.05),ndm=1,plot=FALSE)


#Snee Marquardt(1976) example
Xvert(8,uc=c(.45,.50,.10,.4,.6,.2,.05,.05),lc=c(.1,.05,0,0,.1,.05,0,0),ndm=0)
    
# Example page 465
exvert<-Xvert(4,uc=c(.188,.128,.438,.438),lc=c(.124,.064,.374,.374),ndm=2)

# Example from Piepel 1988
coef<-matrix(c(.85,.9,1,.7,0,1),ncol=3,byrow=TRUE)
des<-Xvert(3,lc=c(.1,.1,0),uc=c(.5,.7,.7),nlc=2,lb=c(.9,.4),ub=c(.95,0),coef,ndm=1,plot=FALSE)