OptimaRegion (version 0.1.0)

OptRegionQuad: Computes Confidence Regions of Optima of Quadratic Polynomial Models

Description

Computes an displays the confidence region on the location of the optima of a quadratic response surface in 2 factors using bootstrapping.

Usage

OptRegionQuad(X, y, nosim=200, alpha=0.05, LB, UB,
triangularRegion=FALSE, vertex1=NULL, vertex2=NULL,
maximization=TRUE,
xlab="Protein eaten, mg",
ylab="Carbohydrates eaten, mg",
outputPDFFile="CRplot.pdf")

Arguments

X

nx2 matrix with the values of the 2 regressors (experimental factors) in the n observations

y

nx1 vector of response value observations

nosim

number of simulations (default=200)

alpha

confidence level (0

LB,UB

2x1 vectors of lower and upper bounds for search region where optima may lie

triangularRegion

logical: if TRUE it will constrain the optimum points to lie inside a triangle defined by the coordinates (0,0), and those in "vertex1", and "vertex2", see below (in addition to being constrained to lie inside the region defined by LB and UB). NOTE:

vertex1,vertex2

2x1 vectors with coordinates defining two of the 3 vertices of a triangular region. Must be provided if triangularRegion is TRUE (NOTE: vertices numbered clockwise, vertex0=c(0,0) always)

maximization

logical: if TRUE (default) it maximizes it FALSE it minimizes

xlab

text label for x axis in confidence region plot (default: "Protein eaten, mg")

ylab

text label for y axis in confidence region plot (default: "Carbohydrates eaten, mg")

outputPDFFile

name of the PDF file where the CR plot is saved (default: "CRplot.pdf")

Value

• meanPoint
  a 2x1 vector with the coordinates of the mean optimum point
• xin
  an mx2 matrix with the x,y coordinates of all simulated #points that belong to the confidence region (dim(m) is (1-alpha)*nosim)

Details

Computes and displays an approximated 100*(1-alpha) percent confidence region (CR) for the linear-constrained maximum of a quadratic polynomial regression model in 2 controllable factors. Grey region on output plot is the approximate CR. The CR is computed as the convex hull of the coordinates of the optima found from bootstrapping nosim quadratic polynomial regressions to the data (therefore, it is an approximate CR). The mean value of the optimum is shown as a red point, and a smoothed contour plot of the X,y data obtained via thin plate splines is shown as well.

Usage assuming all default options:

out<-OptRegionQuad(X=X,y=y,LB=LB,UB=UB)

References

Del Castillo, E., Hunt, J., and Rapkin, J. , "Data-depth bootstrapped confidence regions for the optima of parametric and nonparametric response surfaces".

Examples

## Example 1: randomly generated 2-variable response surface data

X<-cbind(runif(100,-2,2),runif(100,-2,2))

y<-as.matrix(72-11.78*X[,1]+0.74*X[,2]-7.25*X[,1]^2-7.55*X[,2]^2-4.85*X[,1]*X[,2]+
rnorm(100,0,8))

## Find a 95 percent confidence region for the maximum of a quadratic polynomial
fitted to these data

out<-OptRegionQuad(X=X,y=y,nosim=200,LB=c(-2,-2),UB=c(2,2), xlab="X1",ylab="X2",
outputPDFFile="CR_plot.pdf")

## Example 2: a mixture-amount experiment in two components (Drug dataset) with
non-normal data. Note triangular experimental region. Resulting 95p confidence
region is pushed against the constraint and results in a "thin line"

out<-OptRegionQuad(X=Drug[,1:2],y=Drug[,3],nosim=500,LB=c(0,0),UB=c(0.08,11),
 xlab="Component 1 (mg.)",ylab="Component 2 (mg.)",triangularRegion = TRUE,
 vertex1 = c(0.02,11),vertex2 = c(0.08,1.8),outputPDFFile="Mixture_plot.pdf")