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  "body": "\n## Definition\n\nLet $x = (x_1, \\dots, x_k)$ be the settings of $k$ controllable factors in a region $R \\subseteq \\R^k$.\nAssume that the observed response is $y = \\eta(x) + \\varepsilon$, where $\\eta \\colon R \\to \\R$ is an unknown smooth function and $\\varepsilon$ is a random error with mean $0$.\n*Response surface methodology* is the sequential strategy for locating an optimum of $\\eta$ that repeats these steps:\n\n1. Run an experiment, planned by the methods of [[design of experiments|2TE42ZWJ3E]], in a small neighbourhood of the current point.\n2. While curvature is negligible, fit by [[least squares|0BYTQ4X1HC]] a first-order model $\\hat y = b_0 + \\sum_i b_i x_i$. Then move the current point along the path of steepest ascent, the direction $(b_1, \\dots, b_k)$.\n3. When the first-order model shows lack of fit, fit a second-order model $\\hat y = b_0 + \\sum_i b_i x_i + \\sum_{i \\le j} b_{ij} x_i x_j$, and locate and classify its stationary point.\n\n## Remarks\n\nExample: a fitted second-order model can be written $\\hat y = b_0 + x^T b + x^T B x$ with a symmetric $k \\times k$ matrix $B$.\nWhen $B$ is invertible, its stationary point is $x_s = -\\frac12 B^{-1} b$, since the gradient $b + 2 B x$ vanishes there.\nThe [[Hessian matrix|0693Y2METT]] of $\\hat y$ is $2B$, so if all eigenvalues of $B$ are negative, $x_s$ is the maximum of the fitted surface.\nThis eigenvalue check is called canonical analysis.\n\nNon-example: a single [[factorial experiment|EHJ3R589ZX]] analysed once, with no move to new settings, is not response surface methodology, because step 2 is missing.\n\nTypical designs are two-level factorial designs with centre points for the first-order stage, and central composite designs or Box–Behnken designs for the second-order stage.\n",
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