{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For the signal " }, { "__type": "InlineMath", "__tag": 4057, "value": "y = (y_1, y_2, \\ldots, y_n)" }, { "__type": "Text", "__tag": 4046, "value": " and weights " }, { "__type": "InlineMath", "__tag": 4057, "value": "w = (w_1, w_2, \\ldots, w_n)" }, { "__type": "Text", "__tag": 4046, "value": ", WH of order " }, { "__type": "InlineMath", "__tag": 4057, "value": "p" }, { "__type": "Text", "__tag": 4046, "value": " with smoothing or penalty parameter " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\lambda" }, { "__type": "Text", "__tag": 4046, "value": " solves the following optimization problem for " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_i" }, { "__type": "Text", "__tag": 4046, "value": ":" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\operatorname{argmin}_{x_i} \\sum_i^n w_i (y_i - x_i)^2\n+ \\lambda \\sum_i^{n-p} (\\Delta^p x_i)^2 \\,," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\Delta^p" }, { "__type": "Text", "__tag": 4046, "value": " is the forward difference operator of order " }, { "__type": "InlineMath", "__tag": 4057, "value": "p" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\Delta x_i = x_{i+1} - x_i" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\Delta^2 x_i = \\Delta(\\Delta x_i) = x_{i+2} - 2x_{i+1} + x_i" }, { "__type": "Text", "__tag": 4046, "value": ". For every input value " }, { "__type": "InlineMath", "__tag": 4057, "value": "y_i" }, { "__type": "Text", "__tag": 4046, "value": ", it generates a smoothed value " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_i" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "One of the nice properties of WH is that it automatically performs inter- and extrapolation of data. Interpolation means filling in values when signal data is only available on the left and on the right. Extrapolation means filling in values after the observed signal ends. Set " }, { "__type": "InlineCode", "__tag": 4051, "value": "weights = 0" }, { "__type": "Text", "__tag": 4046, "value": " for regions where you want to extra- or interpolate. 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WH automatically\n# inter- or extrapolates for all data points with w = 0.\nplt.plot(x[w==0], res.x[w==0], color=\"red\", label=\"inter-/extrapolation\")\nplt.xlabel(\"year\")\nplt.ylabel(\"temperature deviation [°C]\")\nplt.title(\"Global Temperature Anomalies (ref. 1951-1980)\")\nplt.legend()\n", "execution_status": "failure", "out": "\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success", "out": "" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "2.0.0.dev0+git20260907.41eeb59", "kind": "assets", "path": "fig-da1233f539b9a18a.png" } }, { "__type": "Text", "__tag": 4046, "value": "\nWe can see that extrapolation has occurred at the right end of the signal, meaning\nthat NaNs existed in the data for the most recent dates, in particular months of\n2026 that have not yet happened at the time of the data download in March 2026." } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "signal", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "lamb", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "reml" }, { "__type": "SigParam", "__tag": 4030, "name": "order", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "2" }, { "__type": "SigParam", "__tag": 4030, "name": "weights", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "whittaker_henderson" }, "references": [ ".. [1] Whittaker-Henderson smoothing,", " https://en.wikipedia.org/wiki/Whittaker%E2%80%93Henderson_smoothing", ".. [2] Eilers, P.H.C. (2003).", " \"A perfect smoother\". Analytical Chem. 75, 3631-3636.", " :doi:`10.1021/AC034173T`", ".. [3] Weinert, Howard L. (2007).", " \"Efficient computation for Whittaker-Henderson smoothing\".", " Computational Statistics and Data Analysis 52:959-74.", " :doi:`10.1016/j.csda.2006.11.038`", ".. [4] Hodrick, R. J., and Prescott, E. C. (1997).", " \"Postwar U.S. Business Cycles: An Empirical Investigation\".", " :doi:`10.2307/2953682`" ], "qa": "scipy.signal._whittaker:whittaker_henderson", "arbitrary": [], "local_refs": [ "lamb", "order", "res", "signal", "weights" ] }