OptimaRegion (version 1.2)

OptRegionQuad: Confidence region for optima of quadratic polynomial models (2 regressors)

Description

Computes and displays an approximated (1 - alpha)*100 the linear-constrained maximum of a quadratic polynomial regression model in 2 controllable factors DelCastilloCROptimaRegion. 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 simulating and optimizing nosim quadratic polynomial regressions from 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

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"
)

Value

Upon completion, a PDF file containing the CR plot with name as set in ouputPDFFile is created and the function also returns a list containing the following 2 components:

meanPoint

a 2x1 vector with the coordinates of the mean optimum point (displayed as a red dot in the CR plot in output PDF file)

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)

Arguments

X

n*2 matrix with the values of the 2 regressors (experimental factors) in the n observations. Note: this can have replicates. They will be eliminated by the program and the corresponding y-values averaged

y

n*1 vector of response value observations, in the same order corresponding to the rows of X

nosim

number of simulations (default = 200)

alpha

confidence level (0 < alpha < 1; default = 0.05)

LB

vector of lower bounds for x (2*1 vector) above which the maximum is sought

UB

vector of upper bounds for x (2*1 vector) below which the maximum is sought

triangularRegion

logical: if TRUE it will constrain the maximum 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: use TRUE when the treatments form a triangular experimental region in shape. If FALSE, maxima will only be constrained to lie inside the rectangular region defined by LB and UB. Default is FALSE.

vertex1

2 times 1 vector with coordinates defining one of the 3 vertices of a triangular region. Must be provided if triangularRegion is TRUE (NOTE: vertices numbered clockwise)

vertex2

2 times 1 vector with coordinates defining a second vertex of a triangular region (third vertex is (0,0) by default). Must be provided if triangularRegion is TRUE (NOTE: vertices numbered clockwise)

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: "CR_plot.pdf")

Author(s)

Enrique del Castillo , Peng Chen , Adam Meyers , John Hunt and James Rapkin .

Details

This program approximates the confidence region (CR) of the location of the optimum of a regression function in 2 regressors constrained inside a rectangular region defined by LB and UB. If triangularRegion = TRUE it will also contrain the optimum to lie inside the experimental region (assumed to be well approximated by a triangle). The CR is generated pointwise by simulating from the posterior of the regression parameters and solving the corresponding constrained maximization problem. The confidence region is approximated by the convex hull of all the solutions found. The simulation approach is based on the "CS" bootstrapping approach for building a confidence set described in WoutersenHam2013;textualOptimaRegion. This version of the program uses nonparamteric bootstrapping confidence regions to get the posteazrior of the parameters of the regression equation using the notion of data depth according to yeh1997balanced;textualOptimaRegion. Hence, this version does not rely on any normality assumption on the data.

References

Examples

if (FALSE) {
# 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"
)

# Example 2: a mixture-amount experiment in two components (Drug dataset) with
# non-normal data. Note triangular experimental region. Resulting 95%
# 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"
)
}