Optimal Response Surface Designs

A short description of the post.

Readings 11.4, Rmarkdown

library(dplyr)
library(ggplot2)
library(readr)
library(EBImage)
library(reshape2)
theme_set(theme_bw() + theme(legend.position = "bottom"))

display(readImage("https://uwmadison.box.com/shared/static/6xoe5onf01gdpaef6g0qtcstbhb9w72n.png"))

Figure 1: The optimality definitions from our earlier discussion on factorial design.

In Chapter 6, we saw that \(2^{K}\) factorial designs are optimal in the linear setting. These results don’t immediately apply to response surfaces, though, for two reasons,

• Second-order response surfaces are not necessarily linear.

• The experimental region might be irregularly shaped, due to known constraints on operating conditions

In this setting, there will be no single design that clearly optimal, like there was before. Instead, the typical strategy is to compute the same optimality criteria from before, but to designs constructed through various heuristics.

Reminder: Optimality Measures

We can use the same optimality measures that were studied for linear regression.

• \(D\)-optimality reflects the variance in the coefficients of the associated linear model. A \(D\)-optimal design has minimal value of \(\left|X^{T}X\right|^{-1}\).

• \(G\)-optimality reflects the pointwise variance of the fitted surface. A \(G\)-optimal design minimizes the maximal value of \(V\left(\hat{y}\left(x\right)\right)\).

• \(V\)-optimality also reflects the pointwise variance of the fitted surface, but with less focus on the worst case \(x\). A \(V\)-optimal design minimizes the average variance, \(\int_{R} V\left(\hat{y}\left(x\right)\right)dx\) over the experiment space \(R\).

Heuristics

display(readImage("https://uwmadison.box.com/shared/static/9pgowo8mgly6d8mm97t8sm324ggcio0y.png"))

Figure 2: One iteration of the point exchange algorithm, for a constrained response surface design.

display(readImage("https://uwmadison.box.com/shared/static/yr2dyd75we3ebp1gwe36r5xe3p8841h8.png"))

Figure 3: One iteration of the point coordinate exchange algorithm, in the same setup.

Once a candidate design is proposed, we can evaluate its quality using the measures above. There are various heuristics for proposing new candidate designs,

• Point exchange

  • Start with a grid of points to consider performing runs at.

  • Select a subset (possibly at random). Call this the design set and the complement the candidate set.

  • Compute an optimality criterion on the design set.

  • Try swapping a pair of points from the design and candidate sets

    • Is the optimality criterion is improved?

    • If it is, keep the swap in the next iteration.

  • Repeat until the optimality criterion has converged.

• Coordinate exchange

  • Start with a grid of points to consider performing runs at, call this the design set.

  • For each point in the design set,

    • For each factor \(k\),

      • Vary the value of factor \(k\) until it maximizes the chosen optimality criterion.

    • Repeat until convergence