Optimal experimental design

Builds on Design of experiments General linear model Determinant Eigenvalue Covariance matrix

Definition

Let $\mathcal{X}$ be a compact set, the design region, and let $f \colon \mathcal{X} \to \R^p$ be continuous. Consider the linear model $y = f(x)^T \beta + \varepsilon$ with unknown $\beta \in \R^p$ and uncorrelated errors of mean $0$ and common variance $\sigma^2$ . An approximate design is a probability measure $\xi$ on $\mathcal{X}$ with finite support, and its information matrix is $M(\xi) = \int_{\mathcal{X}} f(x) f(x)^T \, d\xi(x) = \sum_{x \in \operatorname{supp} \xi} \xi(\set{x}) f(x) f(x)^T.$ A design $\xi^*$ is:

D-optimal if it maximises $\det M(\xi)$ over all approximate designs;
A-optimal if $M(\xi^*)$ is invertible and it minimises $\tr M(\xi)^{-1}$ over the designs with $M(\xi)$ invertible;
E-optimal if it maximises the smallest eigenvalue of $M(\xi)$ ;
G-optimal if it minimises $\max_{x \in \mathcal{X}} f(x)^T M(\xi)^{-1} f(x)$ over the designs with $M(\xi)$ invertible.

Remarks

For an exact design with $n$ runs at $x_1, \dots, x_n$ and design matrix $X$ with rows $f(x_i)^T$ , one has $M = \frac{1}{n} X^T X$ . If $M$ is invertible, the least squares estimator has covariance matrix $\frac{\sigma^2}{n} M^{-1}$ . So a D-optimal design minimises the volume of the confidence ellipsoid for $\beta$ under normal errors.

Example: $\mathcal{X} = [-1, 1]$ and $f(x) = (1, x)^T$ , a straight line. With moments $m_1 = \int x \, d\xi$ and $m_2 = \int x^2 \, d\xi$ , one gets $\det M(\xi) = m_2 - m_1^2$ , the variance of $x$ under $\xi$ . On $[-1, 1]$ this variance is at most $1$ , with equality exactly for the design with mass $1/2$ at $-1$ and $1/2$ at $1$ , which is therefore D-optimal.

Non-example: the design with all mass at $x = 0$ has $M = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ , which is singular, so $\det M = 0$ and the slope cannot be estimated; it fails clause 1.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`MKAMVQJXBP`
URI	`m2://MKAMVQJXBP`
Kind	definition
Writing	W0 stub
MSC	62K05
Defines	Optimal experimental design
Uses	Design of experiments, General linear model, Determinant, Eigenvalue, Covariance matrix

Optimal experimental design

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here