OptRegion: Confidence Regions for Response Surface Optima
In PengChenEngineer/OptimaRegion: Confidence Regions for Optima

Description Usage Arguments References Examples

This function bootstraps a Confidence Region (CR) on the resposne surface optima. One can choose from several popular Response Surface Models (RSM) to capture the true reponse pattern, such as polynomial model and Thin Plate Spline (TPS) model \insertCiteDelCastilloCROptimaRegion.

OptRegion(X, y, alpha = 0.05, num_boot = 200, RSM, lambda = 0.04,
  degree = 2, maximization = TRUE, constr_lb, constr_ub,
  constr_triangle = FALSE, constr_vertex_1 = NULL,
  constr_vertex_2 = NULL, verbose = FALSE)

`X`	A numeric matrix of shape (N, k), where N represents the number of data points and k represents the number of regressors. It contains the N design points from the experimental data. The value that k can take depends on the RSM argument.
`y`	A numeric vector of shape (N, 1), where N represents the number of data points. It contains the N response values from the experimental data.
`alpha`	A numeric scalor that specifies the confidence level, 1 - alpha, of the CR. Its value must be set between 0 and 1. Default is 0.05.
`num_boot`	An integer scalor that specifies how many times the bootstrap operation will be repeated to construct the CR. Default is 200.
`RSM`	A character string to choose the type of the RSM, which must be either "quad", "tps", or "poly". "quad" fits the experimental data to a quadratic polynomial model with 2 regressors; "tps" fits the experimental data to a TPS model with 2 regressors; "poly" fits the experimental data to a quadratic or cubic polynomial model (specified by the degree argument) with 2 - 5 regressors. For all model types, the optima of the response surface will be constrained by the bounds of the regressors (specified by the constr_lb and constr_ub arguments). For "quad" and "tps" models, the optima of the response surface can be further constrained to lie inside a triangle defined by the original (0,0) and two other vertices (specified by the constr_triangle, constr_vertex_1, and constr_vertex_2 arguments).
`lambda`	A numeric scalor that specifies the value of the penalty parameter if RSM is "tps". Default is 0.04.
`degree`	A integer scalor that specifies the degree of the polynomial model if RSM is "poly". It can be set to 2 or 3. Default is 2.
`maximization`	A boolean scalor. If TRUE, the function returns a CR on the maxima of the response surface; if FALSE, the function returns a CR on the minima of the response surface. Default is TRUE.
`constr_lb`	A numeric vector of shape (1, k) that specifies the lower bound constraint for each of the k regressors.
`constr_ub`	A numeric vector of shape (1, k) that specifies the upper bound constraint for each of the k regressors.
`constr_triangle`	A boolean scalor. If TRUE, the optima of the RSM will also be constrained to lie inside a triangle defined by the original (0,0) and two other vertices. Note only the "quad" model and the "tps" model have this option. Dafault is FALSE.
`constr_vertex_1`	A numeric vector of shape (1, 2) that specifies one of the other two vertices if constr_triangle is TRUE. (NOTE: vertices numbered clockwise)
`constr_vertex_2`	A numeric vector of shape (1, 2) that specifies one of the other two vertices if constr_triangle is TRUE. (NOTE: vertices numbered clockwise)
`verbose`	A boolean scalor. If TRUE, function status will be printed. Default is FALSE.

\insertAllCited

## Not run: 
# 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 <- OptRegion(
  X = X, y = y, B = 200, LB = c(-2, -2), UB = c(2, 2), RSM = "quad"
)
plot(out, 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 <- OptRegion(
  X = Drug[, 1:2], y = Drug[, 3], B = 500, LB = c(0, 0), UB = c(0.08, 11), RSM = "quad",
  triangularRegion = TRUE, vertex1 = c(0.02, 11), vertex2 = c(0.08, 1.8)
)
plot(out, xlab = "Component 1 (mg.)", ylab = "Component 2 (mg.)")

# Example 3: 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 Thin Plate Spline
# model fitted to these data
out <- OptRegion(X = X, y = y, B = 200, LB = c(-2, -2), UB = c(2, 2), RSM = "tps")
plot(out, xlab = "X1", ylab = "X2")

# Example 4: a mixture-amount experiment in two components (Drug dataset) with
# non-normal data. Note triangular experimental region. Resulting 95p confidence
# region of the maxima of a TPS model has area > 0. Contrast with region for
# quadratic polynomial model. Note: 500 bootstrap iterations may take a few minutes.
out <- OptRegion(
  X = Drug[, 1:2], y = Drug[, 3], B = 500, lambda = 0.05,
  LB = c(0, 0), UB = c(0.08, 11), RSM = "tps",
  triangularRegion = TRUE, vertex1 = c(0.02, 11), vertex2 = c(0.08, 1.8)
)
plot(out, xlab = "Component 1 (mg.)", ylab = "Component 2 (mg.)")

# Example 5: run GloptiPolyRegion on a quadratic, 3 vars example
out <- OptRegion(
  X = quad_3D[, 1:3], y = quad_3D[, 4], B = 500, alpha = 0.1,
  LB = c(-2, -2, -2), UB = c(2, 2, 2), RSM = "poly", degree = 2,
  maximization = TRUE, verbose = TRUE
)
plot(out, c("x1", "x2", "x3"))

# Example 6: run GloptiPolyRegion on a cubic, 5 vars example
out <- OptRegion(
  X = cubic_5D$design_matrix, y = cubic_5D$response, B = 200, alpha = 0.05,
  LB = rep(0, 5), UB = rep(5, 5), RSM = "poly", degree = 3,
  maximization = TRUE, verbose = TRUE
)
plot(out, c("x1", "x2", "x3", "x4", "x5"))

## End(Not run)