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  "body": "\n## Definition\n\nLet $\\mathcal{X}$ be a compact set, the *design region*, and let $f \\colon \\mathcal{X} \\to \\R^p$ be continuous.\nConsider the linear model $y = f(x)^T \\beta + \\varepsilon$ with unknown $\\beta \\in \\R^p$ and uncorrelated errors of mean $0$ and common variance $\\sigma^2$.\nAn *approximate design* is a probability measure $\\xi$ on $\\mathcal{X}$ with finite support, and its *information matrix* is\n$$M(\\xi) = \\int_{\\mathcal{X}} f(x) f(x)^T \\, d\\xi(x) = \\sum_{x \\in \\operatorname{supp} \\xi} \\xi(\\set{x}) f(x) f(x)^T.$$\nA design $\\xi^*$ is:\n\n1. *D-optimal* if it maximises $\\det M(\\xi)$ over all approximate designs;\n2. *A-optimal* if $M(\\xi^*)$ is invertible and it minimises $\\tr M(\\xi)^{-1}$ over the designs with $M(\\xi)$ invertible;\n3. *E-optimal* if it maximises the smallest [[eigenvalue|0BQAA2BK3Q]] of $M(\\xi)$;\n4. *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.\n\n## Remarks\n\nFor an exact [[design|2TE42ZWJ3E]] with $n$ runs at $x_1, \\dots, x_n$ and design matrix $X$ with rows $f(x_i)^T$, one has $M = \\frac{1}{n} X^T X$.\nIf $M$ is invertible, the least squares estimator has [[covariance matrix|0V7G6WMFK0]] $\\frac{\\sigma^2}{n} M^{-1}$.\nSo a D-optimal design minimises the volume of the confidence ellipsoid for $\\beta$ under normal errors.\n\nExample: $\\mathcal{X} = [-1, 1]$ and $f(x) = (1, x)^T$, a straight line.\nWith 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 $\\xi$.\nOn $[-1, 1]$ 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.\n\nNon-example: the design with all mass at $x = 0$ has $M = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$, which is singular, so $\\det M = 0$ and the slope cannot be estimated; it fails clause 1.\n",
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