{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This routine constructs the smoothing spline function, " }, { "__type": "InlineMath", "__tag": 4057, "value": "g(x)" }, { "__type": "Text", "__tag": 4046, "value": ", to minimize the sum of jumps, " }, { "__type": "InlineMath", "__tag": 4057, "value": "D_j" }, { "__type": "Text", "__tag": 4046, "value": ", of the " }, { "__type": "InlineCode", "__tag": 4051, "value": "k" }, { "__type": "Text", "__tag": 4046, "value": "-th derivative at the internal knots (" }, { "__type": "InlineMath", "__tag": 4057, "value": "x_b < t_i < x_e" }, { "__type": "Text", "__tag": 4046, "value": "), where" } ] }, { "__type": "Math", "__tag": 4058, "value": "D_i = g^{(k)}(t_i + 0) - g^{(k)}(t_i - 0)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Specifically, the routine constructs the spline function " }, { "__type": "InlineMath", "__tag": 4057, "value": "g(x)" }, { "__type": "Text", "__tag": 4046, "value": " which minimizes" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\sum_i | D_i |^2 \\to \\mathrm{min}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "provided that" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\sum_{j=1}^m (w_j \\times (g(x_j) - y_j))^2 \\leqslant s ," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "s > 0" }, { "__type": "Text", "__tag": 4046, "value": " is the input parameter." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In other words, we balance maximizing the smoothness (measured as the jumps of the derivative, the first criterion), and the deviation of " }, { "__type": "InlineMath", "__tag": 4057, "value": "g(x_j)" }, { "__type": "Text", "__tag": 4046, "value": " from the data " }, { "__type": "InlineMath", "__tag": 4057, "value": "y_j" }, { "__type": "Text", "__tag": 4046, "value": " (the second criterion)." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that the summation in the second criterion is over all data points, and in the first criterion it is over the internal spline knots (i.e. those with " }, { "__type": "InlineCode", "__tag": 4051, "value": "xb < t[i] < xe" }, { "__type": "Text", "__tag": 4046, "value": "). 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