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arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = Object.assign({ h3: "h3", p: "p", a: "a", span: "span", math: "math", semantics: "semantics", mrow: "mrow", msub: "msub", mi: "mi", annotation: "annotation", div: "div", mo: "mo", mn: "mn", msup: "msup", mtext: "mtext", munderover: "munderover", mfrac: "mfrac", msqrt: "msqrt", svg: "svg", path: "path", mtable: "mtable", mtr: "mtr", mtd: "mtd", mstyle: "mstyle", mpadded: "mpadded", mspace: "mspace", pre: "pre", code: "code", img: "img" }, _provideComponents(), props.components); return _jsxs(_Fragment, { children: [_jsx(_components.h3, { children: "はじめに" }), "\n", _jsxs(_components.p, { children: [_jsx(_components.a, { href: "blog23.html", children: "先日のブログ" }), "で説明しましたが、最尤推定法は機械学習で多く使用されるパラメータ推定法です。"] }), "\n", _jsx(_components.p, { children: "月並みではありますが、今回は、線形回帰分析のパラメータを最尤推定法で解いてみたいと思います。" }), "\n", _jsx(_components.h3, { children: "線形回帰分析" }), "\n", _jsxs(_components.p, { children: ["まずは、線形回帰分析の定義を行います。 ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "y_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: 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children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "x" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "j" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "x_{ij}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "ij" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "を説明変数とすると、線形回帰分析の式は以下のように表されます。"] }), 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children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }) }) }), "\n", _jsxs(_components.p, { children: ["ただし、", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: 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height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "ε" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "∼" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.10903em" }, children: "N" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "σ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "で、独立同分布を仮定します。(", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "ε" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "∼" }), _jsx(_components.mi, { children: "N" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "σ" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\varepsilon_i\\sim N(0,\\sigma^2)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "ε" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "∼" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.10903em" }, children: "N" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "σ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "は、平均0、分散", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "σ" }), _jsx(_components.mn, { children: "2" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sigma^2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "σ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] }) })] }) }), "の正規分布にしたがっていることを表します。)"] }), "\n", _jsxs(_components.p, { children: ["これにより、説明変数を定数だとみなすと、", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "y_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "は、", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "∼" }), _jsx(_components.mi, { children: "N" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mn, { children: "1" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "x" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mo, { children: "⋯" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "k" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "x" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "k" })] })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "σ" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "y_i\\sim N(\\beta_0+\\beta_1x_{i1}+\\cdots+\\beta_kx_{ik},\\sigma^2)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "∼" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.10903em" }, children: "N" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "minner", children: "⋯" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "ik" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "σ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), " で、独立な確率変数と見なすことができます。(正確にいうと、", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "x" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "j" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "x_{ij}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "ij" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "で条件づけた", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "y_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "の条件付き確率を考えています。)"] }), "\n", _jsx(_components.h3, { children: "線形回帰分析と最尤推定法" }), "\n", _jsxs(_components.p, { children: ["さて、", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "y" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "y_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "の確率分布が決まったので、尤度関数を考えてみましょう。"] }), "\n", _jsx(_components.p, { children: "その前に、最尤推定法のおさらいです。" }), "\n", _jsx(_components.p, { children: "最尤推定法は、「パラメータをいろいろ変えて今あるデータである確率密度関数の値を出してみて、最も高いものをもとのパラメータとしよう」という考えのもと推定していきます。当然パラメータは無限の可能性がありいちいち代入していられないので、パラメータを変数とみた関数を考え、これが最大となるパラメータを最尤推定値とします。このパラメータを変数とみた関数を尤度関数といいます。" }), "\n", _jsx(_components.p, { children: "ということなので確率密度関数を出しましょう。" }), "\n", _jsxs(_components.p, { children: ["パラメータは、偏回帰係数の", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mo, { children: "⋯" }), _jsx(_components.mtext, { children: " " }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "k" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\beta_0,\\cdots,\\beta_k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "minner", children: "⋯" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "と、正規分布の分散", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "σ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sigma" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "σ" })] }) })] }) }), 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_jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "ik" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", 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{ children: "0" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mo, { children: "⋯" }), _jsx(_components.mtext, { children: " " }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "k" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\beta_0,\\cdots,\\beta_k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "minner", children: "⋯" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "を求めることを考えると、対数尤度関数を最大化するには、以下の式を最小化すれば良いことがわかります。"] }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "E" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.munderover, { 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_jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mo, { children: "⋯" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "k" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "x" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "k" })] })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: " E=\\sum_{i=1}^n(y_i-(\\beta_0+\\beta_1x_{i1}+\\cdots+\\beta_kx_{ik}))^2" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", 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className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "ik" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "これ、実は最小二乗法と式が同じです。つまり、最小二乗法は、最尤推定法の観点からも妥当なパラメータ推定方法であることがわかります。" }), "\n", _jsx(_components.h3, { children: "最尤推定値を求めてみる" }), "\n", _jsxs(_components.p, { children: ["一応", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "E" }) }), _jsx(_components.annotation, { encoding: 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_jsx(_components.mi, { children: "k" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\beta_0\\cdots\\beta_k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "minner", children: "⋯" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "も求めておきます。"] }), "\n", _jsxs(_components.p, { children: ["基本は、全ての", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }) }), "について", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "E" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "E" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "E" })] }) })] }) }), "を", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\beta_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "で偏微分して0となる", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\beta_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "を求めます。ちゃんとやるなら、「偏微分=0」だけでは最小値とは言えず、Eの凸性を議論する必要があります。(下記の「凸関数の最小化」が参考になります。)"] }), "\n", _jsx(_components.p, { children: _jsx(_components.a, { href: "http://www2.econ.tohoku.ac.jp/~ksuzuki/teaching/2005/ch5.pdf", children: "http://www2.econ.tohoku.ac.jp/~ksuzuki/teaching/2005/ch5.pdf" }) }), "\n", _jsxs(_components.p, { children: ["ここでは、この証明は省きます。(感覚的には", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "E" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "E" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "E" })] }) })] }) }), "は", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\beta_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "に対して2次の多項式で、2次の多項式はは凸関数となるため、凸性を持つことが想像できます。ヘッセ行列を計算してみてもいいでしょう。)"] }), "\n", _jsxs(_components.p, { children: ["さて、一つずつ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\beta_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "で偏微分するのは面倒臭いので、", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "β" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "(" }), _jsxs(_components.mtable, { rowspacing: "0.16em", columnalign: "center", columnspacing: "1em", children: [_jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mn, { children: "0" })] }) }) }) }), _jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "⋮" }), _jsx(_components.mpadded, { height: "0em", voffset: "0em", children: _jsx(_components.mspace, { mathbackground: "black", width: "0em", height: "1.5em" }) })] }) }) }) }), _jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "k" })] }) }) }) })] }), _jsx(_components.mo, { fence: "true", children: ")" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mathbf \\beta=\\left(\\begin{array}{c}\\beta_0\\\\\\vdots\\\\\\beta_k\\end{array}\\right)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "4.26em", verticalAlign: "-1.88em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen", children: _jsx(_components.span, { className: "delimsizing mult", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "2.35em" }, children: _jsxs(_components.span, { style: { top: "-4.35em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "6.2em" } }), _jsx(_components.span, { style: { width: "0.875em", height: "4.200em" }, children: _jsx(_components.svg, { xmlns: "http://www.w3.org/2000/svg", width: "0.875em", height: "4.200em", viewBox: "0 0 875 4200", children: _jsx(_components.path, { d: "M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 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className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0858em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0858em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0858em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0858em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "であり、以下の1.2の公式を使用すると、以下となります。" }), "\n", _jsx(_components.p, { children: _jsx(_components.a, { href: "https://qiita.com/AnchorBlues/items/8fe2483a3a72676eb96d", children: "https://qiita.com/AnchorBlues/items/8fe2483a3a72676eb96d" }) }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mfrac, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsx(_components.mi, { mathvariant: "bold", children: "β" })] })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "β" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsx(_components.mi, { mathvariant: "bold", children: "β" })] })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "β" }), _jsx(_components.mi, { children: "T" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "y" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: " \\frac{\\partial}{\\partial \\mathbf\\beta}(\\mathbf y^T\\mathbf X\\mathbf \\beta)=\\frac{\\partial}{\\partial \\mathbf\\beta}(\\mathbf \\beta^T \\mathbf X^T \\mathbf y)=\\mathbf X^T \\mathbf y" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.2519em", verticalAlign: "-0.8804em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3714em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8804em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.2519em", verticalAlign: "-0.8804em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3714em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8804em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0858em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] })] })] }) }) }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mfrac, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsx(_components.mi, { mathvariant: "bold", children: "β" })] })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "β" }), _jsx(_components.mi, { children: "T" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mi, { mathvariant: "bold", children: "β" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "β" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\frac{\\partial}{\\partial \\mathbf\\beta}(\\mathbf \\beta^T \\mathbf X^T \\mathbf X\\mathbf \\beta)=2(\\mathbf X^T \\mathbf X)\\beta " })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.2519em", verticalAlign: "-0.8804em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3714em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8804em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1413em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "したがって、以下が導けます。" }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mfrac, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∂" }), _jsx(_components.mi, { mathvariant: "bold", children: "β" })] })] }), _jsx(_components.mi, { children: "E" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "2" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "y" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "2" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "β" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: " \\frac{\\partial}{\\partial \\mathbf\\beta}E=-2\\mathbf X^T \\mathbf y+2(\\mathbf X^T \\mathbf X)\\beta=0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.2519em", verticalAlign: "-0.8804em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3714em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", style: { marginRight: "0.05556em" }, children: "∂" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8804em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "E" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0858em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "2" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1413em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }) }) }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "bold", children: "X" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { mathvariant: "bold", children: "y" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\beta=(\\mathbf X^T \\mathbf X)^{-1}\\mathbf X^T\\mathbf y " })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1413em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbf", children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathbf", style: { marginRight: "0.01597em" }, children: "y" })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "これで線形回帰の解析解が求められました。" }), "\n", _jsx(_components.h3, { children: "Pythonによる線形回帰" }), "\n", _jsx(_components.p, { children: "本当に一致するの？と疑問に思う人もいるかもしれないので、実際に計算してみましょう。" }), "\n", _jsx(_components.p, { children: "pythonのモジュール(statsmodels使用)" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " pandas ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " pd\n", _jsx(_components.span, { className: "hljs-keyword", children: "from" }), " sklearn.datasets ", _jsx(_components.span, { className: "hljs-keyword", children: "import" }), " load_boston\n", _jsx(_components.span, { className: "hljs-keyword", children: "import" }), " statsmodels.api ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " sm \n\n", _jsx(_components.span, { className: "hljs-comment", children: "#「boston house price」というデータをロード " }), "\nboston = load_boston() \ndf = pd.DataFrame(boston.data, columns=boston.feature_names) \ndf[", _jsx(_components.span, { className: "hljs-string", children: "'PRICE'" }), "] = boston.target \n\n", _jsx(_components.span, { className: "hljs-comment", children: "# 変数「CRIM」から「LSTAT」を説明変数として、「PRICE」を予測する " }), "\nX = df.drop(", _jsx(_components.span, { className: "hljs-string", children: "'PRICE'" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "1" }), ")\ny = boston.target \n\n", _jsx(_components.span, { className: "hljs-comment", children: "# add_constantで定数項を挿入 " }), "\nmodel = sm.OLS(y, sm.add_constant(X)) \nresult = model.fit() \nresult.summary()\n"] }) }), "\n", _jsx(_components.p, { children: "結果：" }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog25/img1.png", alt: "" }) }), "\n", _jsx(_components.p, { children: "numpyで先ほど求めた解析解を計算" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " numpy ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " np\n\nX_add = np.array(sm.add_constant(X))\nbeta = np.linalg.inv((X_add.T @ X_add)) @ X_add.T @ y\npd.Series(beta, index=sm.add_constant(X).columns)\n"] }) }), "\n", _jsx(_components.p, { children: "結果：" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: ["const ", _jsx(_components.span, { className: "hljs-number", children: "36.459488" }), " \nCRIM -", _jsx(_components.span, { className: "hljs-number", children: "0.108011" }), " \nZN ", _jsx(_components.span, { className: "hljs-number", children: "0.046420" }), " \nINDUS ", _jsx(_components.span, { className: "hljs-number", children: "0.020559" }), " \nCHAS ", _jsx(_components.span, { className: "hljs-number", children: "2.686734" }), " \nNOX -", _jsx(_components.span, { className: "hljs-number", children: "17.766611" }), " \nRM ", _jsx(_components.span, { className: "hljs-number", children: "3.809865" }), " \nAGE ", _jsx(_components.span, { className: "hljs-number", children: "0.000692" }), " \nDIS -", _jsx(_components.span, { className: "hljs-number", children: "1.475567" }), " \nRAD ", _jsx(_components.span, { className: "hljs-number", children: "0.306049" }), " \nTAX -", _jsx(_components.span, { className: "hljs-number", children: "0.012335" }), " \nPTRATIO -", _jsx(_components.span, { className: "hljs-number", children: "0.952747" }), " \nB ", _jsx(_components.span, { className: "hljs-number", children: "0.009312" }), " \nLSTAT -", _jsx(_components.span, { className: "hljs-number", children: "0.524758" }), " \ndtype: float64\n"] }) }), "\n", _jsx(_components.p, { children: "coefの部分の結果は一致していますね。" }), "\n", _jsx(_components.h3, { children: "線形回帰のパラメータ推定値の性質" }), "\n", _jsxs(_components.p, { children: ["最尤推定値の統計的性質より、一致性と漸近正規性を持ちます。", _jsx(_components.a, { href: "./blog2.html", children: "昔のブログ" }), "で書いた線形回帰分析のパラメータが一致性を持つというのは、パラメータが最尤推定値であることからも証明することができます。"] }), "\n", _jsx(_components.h3, { children: "まとめ" }), "\n", _jsx(_components.p, { children: "線形回帰分析において、最尤推定法によるパラメータ推定と最小二乗法によるパラメータ推定は同じ結果になることがわかります。最小二乗法は最尤推定法の観点からも妥当な推定法であると言えます。" }), "\n", _jsx(_components.p, { children: "他のポアソン回帰やロジスティック回帰等の一般線形回帰分析も最尤推定法が元になっています。これについても記事をかけたらなと思います。" })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = Object.assign({}, _provideComponents(), props.components); return MDXLayout ? _jsx(MDXLayout, Object.assign({}, props, { children: _jsx(_createMdxContent, props) })) : 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