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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $a \\le x_1 \u003c x_2 \u003c \\cdots \u003c x_n \\le b$ be real nodes, let $y_1, \\dots, y_n$ be real numbers, and let $\\lambda \u003e 0$.\nLet $W$ be the space of functions $f \\colon [a, b] \\to \\R$ such that $f'$ is absolutely continuous and $f''$ is square integrable.\nThe *smoothing spline* for these data is the function $f \\in W$ that minimizes\n$$J_\\lambda(f) = \\sum_{i=1}^{n} \\big(y_i - f(x_i)\\big)^2 + \\lambda \\int_a^b f''(x)^2 \\, dx.$$\n\n## Remarks\n\nThe minimizer exists and is unique, and it is a natural [[cubic spline|V42H0R3EKX]] with knots $x_1, \\dots, x_n$: a cubic on each $[x_i, x_{i+1}]$, of class $C^2$, and linear on $[a, x_1]$ and $[x_n, b]$.\nUniqueness uses $n \\ge 2$: then a function with $f'' = 0$ that vanishes at all nodes is zero.\nExample: if the data lie on a line, $y_i = \\alpha + \\beta x_i$, then $f(x) = \\alpha + \\beta x$ gives $J_\\lambda(f) = 0$, so it is the smoothing spline for every $\\lambda$.\nNon-example: take $n \\ge 3$ data not on a line, and let $g$ be the natural cubic spline through them.\nIts [[second derivative|0Y9D2FJRQ4]] penalty is positive, and moving $g$ slightly toward a line lowers $J_\\lambda$, so $g$ is not the minimizer.\nAs $\\lambda \\to 0$ the smoothing spline tends to the natural interpolating cubic spline, and as $\\lambda \\to \\infty$ it tends to the [[least squares|0BYTQ4X1HC]] regression line.\nIn practice $\\lambda$ is often chosen by generalized cross-validation.\n",
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