{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Any masked values in x is propagated in y, and vice-versa." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The solution minimizes the squared error" } ] }, { "__type": "Math", "__tag": 4058, "value": "E = \\sum_{j=0}^k |p(x_j) - y_j|^2" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "in the equations " } ] }, { "__type": "Code", "__tag": 4050, "value": "x[0]**n * p[0] + ... + x[0] * p[n-1] + p[n] = y[0]\nx[1]**n * p[0] + ... + x[1] * p[n-1] + p[n] = y[1]\n...\nx[k]**n * p[0] + ... + x[k] * p[n-1] + p[n] = y[k]", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The coefficient matrix of the coefficients " }, { "__type": "CrossRef", "__tag": 4002, "value": "p", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "p" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " is a Vandermonde matrix." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "polyfit", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy:polyfit" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " issues a " }, { "__type": "CrossRef", "__tag": 4002, "value": "RankWarning", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy.exceptions:RankWarning" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " when the least-squares fit is badly conditioned. 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`poly1d` objects for dealing with polynomials:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "p = np.poly1d(z)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "p(0.5)\np(3.5)\np(10)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nHigh-order polynomials may oscillate wildly:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "with warnings.catch_warnings():\n warnings.simplefilter('ignore', np.exceptions.RankWarning)\n p30 = np.poly1d(np.polyfit(x, y, 30))\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "p30(4)\np30(5)\np30(4.5)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nIllustration:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import matplotlib.pyplot as plt\nxp = np.linspace(-2, 6, 100)\n_ = plt.plot(x, y, '.', xp, p(xp), '-', xp, p30(xp), '--')\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": 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