Some notions of optimality in experimental design.
A design is \(D\)-optimal if \(\left|\text{Cov}\left(\hat{\beta}\right)\right|\) is minimized. In our picture, this happens when the width of the histogram of \(\hat{\beta}\) is minimized. The determinant generalizes the notion of width to higher-dimensions (specifically, it gives the volume of a high-dimensional parallelogram.)
A design is \(G\)-optimal if \(\max_{x} \text{Var}\left(\hat{y}\left(x\right)\right)\) is minimized. In our picture, this happens when the maximum vertical spread of the prediction band is minimized.
A design is \(V\)-optimal if \(\int_{\left[-1, 1\right]^{K}}. \text{Var}\left(\hat{y}\left(x\right)\right)dx\) is minimized. In our picture, this happens when the area of the prediction band is minimized