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  "body": "\n## Definition\n\nLet $y_1, \\dots, y_n \\in \\R$ be data observed at points $x_1 \u003c x_2 \u003c \\dots \u003c x_n$, and let $w_1, \\dots, w_n \u003e 0$ be weights.\nThe *isotonic regression* of $y$ is the vector $\\hat y \\in \\R^n$ that minimises\n$$\\sum_{i=1}^{n} w_i (y_i - z_i)^2$$\nover all $z \\in \\R^n$ with $z_1 \\le z_2 \\le \\dots \\le z_n$.\n\n## Remarks\n\nThe set of non-decreasing vectors is a closed [[convex|0K06X9QXDX]] cone, so $\\hat y$ is the projection of $y$ onto it for the weighted inner product, and it exists and is unique.\nIt has the min-max formula $\\hat y_i = \\max_{s \\le i} \\min_{t \\ge i} \\mathrm{Av}(s, t)$, where $\\mathrm{Av}(s, t) = \\sum_{k=s}^{t} w_k y_k / \\sum_{k=s}^{t} w_k$.\nThe pool adjacent violators algorithm computes $\\hat y$ in $O(n)$ steps: it merges adjacent blocks whose averages are out of order and replaces them by their weighted average.\nThe same definition works for any partial order on the index set, with $z_i \\le z_j$ required whenever $i$ precedes $j$.\n\nExample: $y = (1, 3, 2)$ with equal weights.\nThe pair $3, 2$ violates the order, and pooling gives $\\hat y = (1, 2.5, 2.5)$, which is non-decreasing.\nIts squared error is $0 + 0.25 + 0.25 = 0.5$.\n\nNon-example: $z = (1, 3, 2)$ itself has error $0$ but it is not non-decreasing, so it is not admissible; the [[monotone|0WFF6AZPBP]] constraint is what makes $\\hat y$ differ from $y$.\n",
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