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最小二乗法 - Wikipedia
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method</span>）は、誤差を伴う測定値の処理において、その誤差の二乗の和を最小にするようにし、最も確からしい関係式を求める方法である。測定で得られた数値の組を、適当なモデルから想定される<a href="https://ja-m-wikipedia-org.translate.goog/wiki/1%E6%AC%A1%E9%96%A2%E6%95%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="1次関数">1次関数</a>、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%AF%BE%E6%95%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="対数">対数</a>曲線など特定の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E9%96%A2%E6%95%B0_(%E6%95%B0%E5%AD%A6)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="関数 (数学)">関数</a>を用いて<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E8%BF%91%E4%BC%BC?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="近似">近似</a>するときに、想定する関数が測定値に対してよい近似となるように、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%AE%8B%E5%B7%AE%E5%B9%B3%E6%96%B9%E5%92%8C?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="残差平方和">残差平方和</a>を最小とするような係数を決定する方法<sup id="cite_ref-ut_1-0" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-ut-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lh_2-0" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-lh-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bj_3-0" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-bj-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>、あるいはそのような方法によって近似を行うことである<sup id="cite_ref-ut_1-1" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-ut-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lh_2-1" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-lh-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bj_3-1" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-bj-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>。</p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:Linear_least_squares2.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Linear_least_squares2.png/220px-Linear_least_squares2.png" decoding="async" width="220" height="264" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/9/94/Linear_least_squares2.png/330px-Linear_least_squares2.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/9/94/Linear_least_squares2.png/440px-Linear_least_squares2.png 2x" data-file-width="889" data-file-height="1067"></a> <figcaption> データセットを4次関数で最小二乗近似した例 </figcaption> </figure> <style data-mw-deduplicate="TemplateStyles:r103029389">.mw-parser-output .sidebar{width:auto;max-width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output 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3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="ja" dir="ltr"> <h2 id="mw-toc-heading">目次</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E6%AD%B4%E5%8F%B2"><span class="tocnumber">1</span> <span class="toctext">歴史</span></a></li> <li class="toclevel-1 tocsection-2"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E8%A8%88%E7%AE%97%E3%81%AE%E6%A6%82%E8%A6%81"><span class="tocnumber">2</span> <span class="toctext">計算の概要</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E5%89%8D%E6%8F%90%E6%9D%A1%E4%BB%B6"><span class="tocnumber">2.1</span> <span class="toctext">前提条件</span></a></li> <li class="toclevel-2 tocsection-4"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E5%9F%BA%E7%A4%8E%E7%9A%84%E3%81%AA%E8%80%83%E3%81%88%E6%96%B9"><span class="tocnumber">2.2</span> <span class="toctext">基礎的な考え方</span></a></li> <li class="toclevel-2 tocsection-5"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E4%B8%80%E6%AC%A1%E6%96%B9%E7%A8%8B%E5%BC%8F%E3%81%AE%E5%A0%B4%E5%90%88"><span class="tocnumber">2.3</span> <span class="toctext">一次方程式の場合</span></a></li> </ul></li> <li class="toclevel-1 tocsection-6"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E8%A7%A3%E6%B3%95%E4%BE%8B"><span class="tocnumber">3</span> <span class="toctext">解法例</span></a></li> <li class="toclevel-1 tocsection-7"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E6%8B%A1%E5%BC%B5"><span class="tocnumber">4</span> <span class="toctext">拡張</span></a> <ul> <li class="toclevel-2 tocsection-8"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E5%A4%9A%E6%AC%A1%E5%85%83"><span class="tocnumber">4.1</span> <span class="toctext">多次元</span></a></li> <li class="toclevel-2 tocsection-9"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E6%B8%AC%E5%AE%9A%E3%81%AE%E8%AA%A4%E5%B7%AE%E3%81%8C%E6%97%A2%E7%9F%A5%E3%81%AE%E5%A0%B4%E5%90%88"><span class="tocnumber">4.2</span> <span class="toctext">測定の誤差が既知の場合</span></a></li> <li class="toclevel-2 tocsection-10"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E9%9D%9E%E7%B7%9A%E5%BD%A2%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span class="tocnumber">4.3</span> <span class="toctext">非線形最小二乗法</span></a></li> <li class="toclevel-2 tocsection-11"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E7%95%B0%E5%B8%B8%E5%80%A4%E3%81%AE%E9%99%A4%E5%8E%BB"><span class="tocnumber">4.4</span> <span class="toctext">異常値の除去</span></a></li> </ul></li> <li class="toclevel-1 tocsection-12"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E9%96%A2%E9%80%A3%E9%A0%85%E7%9B%AE"><span class="tocnumber">5</span> <span class="toctext">関連項目</span></a></li> <li class="toclevel-1 tocsection-13"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E8%84%9A%E6%B3%A8"><span class="tocnumber">6</span> <span class="toctext">脚注</span></a> <ul> <li class="toclevel-2 tocsection-14"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E6%B3%A8%E9%87%88"><span class="tocnumber">6.1</span> <span class="toctext">注釈</span></a></li> <li class="toclevel-2 tocsection-15"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E5%87%BA%E5%85%B8"><span class="tocnumber">6.2</span> <span class="toctext">出典</span></a></li> </ul></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="歴史"><span id=".E6.AD.B4.E5.8F.B2"></span>歴史</h2><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 歴史" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p><a href="https://ja-m-wikipedia-org.translate.goog/wiki/1805%E5%B9%B4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="1805年">1805年</a>に<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%82%A2%E3%83%89%E3%83%AA%E3%82%A2%E3%83%B3%EF%BC%9D%E3%83%9E%E3%83%AA%E3%83%BB%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="アドリアン＝マリ・ルジャンドル">アドリアン＝マリ・ルジャンドル</a>が出版したのが初出である。しかし、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/1809%E5%B9%B4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="1809年">1809年</a>に<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%82%AB%E3%83%BC%E3%83%AB%E3%83%BB%E3%83%95%E3%83%AA%E3%83%BC%E3%83%89%E3%83%AA%E3%83%92%E3%83%BB%E3%82%AC%E3%82%A6%E3%82%B9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="カール・フリードリヒ・ガウス">カール・フリードリヒ・ガウス</a>が出版した際に<a href="https://ja-m-wikipedia-org.translate.goog/wiki/1795%E5%B9%B4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="1795年">1795年</a>には最小二乗法を考案済みだったと主張したことで、最小二乗法の発明者が誰であるかについては不明になっている。</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="計算の概要"><span id=".E8.A8.88.E7.AE.97.E3.81.AE.E6.A6.82.E8.A6.81"></span>計算の概要</h2><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 計算の概要" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <div class="mw-heading mw-heading3"> <h3 id="前提条件"><span id=".E5.89.8D.E6.8F.90.E6.9D.A1.E4.BB.B6"></span>前提条件</h3><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 前提条件" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <p>最小二乗法では測定データ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> </noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> はモデル関数 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.418ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" data-alt="{\displaystyle f(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> と誤差 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ε<!-- ε --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \varepsilon } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> </noscript><span class="lazy-image-placeholder" style="width: 1.083ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" data-alt="{\displaystyle \varepsilon }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の和で</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)+\varepsilon }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> ε<!-- ε --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=f(x)+\varepsilon } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f385c9e8b0b5372dd00cfcaa57b1fe20bb812b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.595ex; height:2.843ex;" alt="{\displaystyle y=f(x)+\varepsilon }"> </noscript><span class="lazy-image-placeholder" style="width: 12.595ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f385c9e8b0b5372dd00cfcaa57b1fe20bb812b" data-alt="{\displaystyle y=f(x)+\varepsilon }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>と表せるとする。物理現象の測定データには、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E8%AA%A4%E5%B7%AE?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="誤差">誤差</a>が含まれ、それは<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%B3%BB%E7%B5%B1%E8%AA%A4%E5%B7%AE?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="系統誤差">系統誤差</a>と<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E8%AA%A4%E5%B7%AE?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E5%81%B6%E7%84%B6%E8%AA%A4%E5%B7%AE" title="誤差">偶然誤差</a>を含んでいる。この内、偶然誤差は、測定における信号経路の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%BE%AE%E8%A6%96%E7%9A%84?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="微視的">微視的</a>現象に由来するならば、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%AD%A3%E8%A6%8F%E5%88%86%E5%B8%83?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="正規分布">正規分布</a>であると期待されることが多い。また、社会調査などの誤差理由の特定が困難な場合でも誤差が正規分布になると期待する<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%AD%A3%E8%A6%8F%E5%88%86%E5%B8%83?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E6%AD%A3%E8%A6%8F%E5%88%86%E5%B8%83%E3%81%AE%E9%81%A9%E7%94%A8" title="正規分布">考え方</a>もある。</p> <p>誤差が正規分布に従わない場合、最小二乗法によって得られたモデル関数はもっともらしくないことに注意する必要がある。偶然誤差が正規分布していない場合、系統誤差が無視できない位大きくそれをモデル関数に含めていない場合、測定データに正規分布から大きく外れた<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%A4%96%E3%82%8C%E5%80%A4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="外れ値">外れ値</a>を含む場合などが該当する。</p> <p>上記を含め、最小二乗法の理論的基盤には次のような前提が設けられている<sup id="cite_ref-ut_1-2" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-ut-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lh_2-2" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-lh-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bj_3-2" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-bj-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>。</p> <ul> <li>測定値の誤差には偏りがない。すなわち誤差の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%B9%B3%E5%9D%87%E5%80%A4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="平均値">平均値</a>は 0 である。</li> <li>測定値の誤差の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%88%86%E6%95%A3_(%E7%A2%BA%E7%8E%87%E8%AB%96)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="分散 (確率論)">分散</a>は既知である。ただし測定データごとに異なる値でも良い。</li> <li>各測定は互いに<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%8B%AC%E7%AB%8B_(%E7%A2%BA%E7%8E%87%E8%AB%96)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="独立 (確率論)">独立</a>であり、誤差の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%85%B1%E5%88%86%E6%95%A3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="共分散">共分散</a>は 0 である。</li> <li>誤差は正規分布する。</li> <li><i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> 個のパラメータ<sup id="cite_ref-4" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-4"><span class="cite-bracket">[</span>注釈 1<span class="cite-bracket">]</span></a></sup>（フィッティングパラメータ）を含むモデル関数 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> が知られていて、測定量の真の値を近似誤差なく再現することのできるパラメータが存在する。</li> </ul> <div class="mw-heading mw-heading3"> <h3 id="基礎的な考え方"><span id=".E5.9F.BA.E7.A4.8E.E7.9A.84.E3.81.AA.E8.80.83.E3.81.88.E6.96.B9"></span>基礎的な考え方</h3><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 基礎的な考え方" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <p>話を簡単にするため、測定値は <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x,y} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"> </noscript><span class="lazy-image-placeholder" style="width: 3.519ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" data-alt="{\displaystyle x,y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の二次元の平面に分布するものとし、想定される分布（モデル関数）が <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=f(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 8.672ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" data-alt="{\displaystyle y=f(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の形である場合を述べる。想定している関数 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> は、既知の関数 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> g </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle g(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.255ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" data-alt="{\displaystyle g(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%B7%9A%E5%9E%8B%E7%B5%90%E5%90%88?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="線型結合">線型結合</a>で表されていると仮定する。すなわち、</p> <div style="margin-top:1ex; margin-left:2em; margin-bottom:1ex;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{k=1}^{m}a_{k}g_{k}(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mi> g </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f(x)=\sum _{k=1}^{m}a_{k}g_{k}(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651f4a98502ac197671868339b1dc46ff3b7e5e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.913ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{k=1}^{m}a_{k}g_{k}(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 18.913ex;height: 6.843ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651f4a98502ac197671868339b1dc46ff3b7e5e6" data-alt="{\displaystyle f(x)=\sum _{k=1}^{m}a_{k}g_{k}(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </div> <p>例えば、 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{k}(x)=x^{k-1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> g </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle g_{k}(x)=x^{k-1}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b882c659c29fb7b28576063a3c79f6d885d453" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.954ex; height:3.176ex;" alt="{\displaystyle g_{k}(x)=x^{k-1}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.954ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b882c659c29fb7b28576063a3c79f6d885d453" data-alt="{\displaystyle g_{k}(x)=x^{k-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> は、多項式近似であり、特に <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <mo> = </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m=2} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32de1b0dc05f6e525ad6a3e8ddeeb4321fd79e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=2}"> </noscript><span class="lazy-image-placeholder" style="width: 6.301ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32de1b0dc05f6e525ad6a3e8ddeeb4321fd79e5" data-alt="{\displaystyle m=2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> の時は <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=a_{1}+a_{2}x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f(x)=a_{1}+a_{2}x} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad8cc722c923f5766ab9030192dcd426da87c920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.254ex; height:2.843ex;" alt="{\displaystyle f(x)=a_{1}+a_{2}x}"> </noscript><span class="lazy-image-placeholder" style="width: 16.254ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad8cc722c923f5766ab9030192dcd426da87c920" data-alt="{\displaystyle f(x)=a_{1}+a_{2}x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> という直線による近似（<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%B7%9A%E5%BD%A2%E5%9B%9E%E5%B8%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="線形回帰">線形回帰</a>）になる。</p> <p>今、測定で得られた、次のような数値の組の集合があるとする。</p> <div style="margin-top:1ex; margin-left:2em; margin-bottom:1ex;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x_{1},y_{1}),\ (x_{2},y_{2}),\ \ldots ,\ (x_{n},y_{n})\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> <mtext> </mtext> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> <mtext> </mtext> <mo> …<!-- … --> </mo> <mo> , </mo> <mtext> </mtext> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{(x_{1},y_{1}),\ (x_{2},y_{2}),\ \ldots ,\ (x_{n},y_{n})\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b10e20858f6fa1c4328686c859188f6cd00b994f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.256ex; height:2.843ex;" alt="{\displaystyle \{(x_{1},y_{1}),\ (x_{2},y_{2}),\ \ldots ,\ (x_{n},y_{n})\}}"> </noscript><span class="lazy-image-placeholder" style="width: 33.256ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b10e20858f6fa1c4328686c859188f6cd00b994f" data-alt="{\displaystyle \{(x_{1},y_{1}),\ (x_{2},y_{2}),\ \ldots ,\ (x_{n},y_{n})\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </div> <p>これら <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x,y)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> </noscript><span class="lazy-image-placeholder" style="width: 5.328ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" data-alt="{\displaystyle (x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の分布が、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=f(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 8.672ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" data-alt="{\displaystyle y=f(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> というモデル関数に従うと仮定した時、想定される理論値は</p> <div style="margin-top:1ex; margin-left:2em; margin-bottom:1ex;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},f(x_{1})),(x_{2},f(x_{2})),...,(x_{n},f(x_{n}))}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> , </mo> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> , </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{1},f(x_{1})),(x_{2},f(x_{2})),...,(x_{n},f(x_{n}))} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70af058551abddc4ec3879ae04b1e726e911b1f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.629ex; height:2.843ex;" alt="{\displaystyle (x_{1},f(x_{1})),(x_{2},f(x_{2})),...,(x_{n},f(x_{n}))}"> </noscript><span class="lazy-image-placeholder" style="width: 38.629ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70af058551abddc4ec3879ae04b1e726e911b1f1" data-alt="{\displaystyle (x_{1},f(x_{1})),(x_{2},f(x_{2})),...,(x_{n},f(x_{n}))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </div> <p>ということになり、実際の測定値との残差は、各 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> i </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle i} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> </noscript><span class="lazy-image-placeholder" style="width: 0.802ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" data-alt="{\displaystyle i}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> につき <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |y_{i}-f(x_{i})|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> −<!-- − --> </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |y_{i}-f(x_{i})|} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afdeb7d330d60b7add24cdaf0e5da36ffc6a758b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.29ex; height:2.843ex;" alt="{\displaystyle |y_{i}-f(x_{i})|}"> </noscript><span class="lazy-image-placeholder" style="width: 11.29ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afdeb7d330d60b7add24cdaf0e5da36ffc6a758b" data-alt="{\displaystyle |y_{i}-f(x_{i})|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> ということになる。</p> <p>この残差の大きさは、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle xy} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"> </noscript><span class="lazy-image-placeholder" style="width: 2.485ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" data-alt="{\displaystyle xy}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> 平面上での <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i},y_{i})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{i},y_{i})} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6dbb919b91ccacf17ed47898048428a1baf9703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:2.843ex;" alt="{\displaystyle (x_{i},y_{i})}"> </noscript><span class="lazy-image-placeholder" style="width: 6.912ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6dbb919b91ccacf17ed47898048428a1baf9703" data-alt="{\displaystyle (x_{i},y_{i})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> と <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i},f(x_{i}))}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{i},f(x_{i}))} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535081ea2a4c9a79e3b1ab6e858e1d13fcfbfcaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.19ex; height:2.843ex;" alt="{\displaystyle (x_{i},f(x_{i}))}"> </noscript><span class="lazy-image-placeholder" style="width: 10.19ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535081ea2a4c9a79e3b1ab6e858e1d13fcfbfcaa" data-alt="{\displaystyle (x_{i},f(x_{i}))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> との距離でもある。</p> <p>ここで、理論値からの誤差の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%88%86%E6%95%A3_(%E7%A2%BA%E7%8E%87%E8%AB%96)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="分散 (確率論)">分散</a>の推定値は残差の平方和</p> <div style="margin-top:1ex; margin-left:2em; margin-bottom:1ex;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=\sum _{i=1}^{n}(y_{i}-f(x_{i}))^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> J </mi> <mo> = </mo> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> −<!-- − --> </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J=\sum _{i=1}^{n}(y_{i}-f(x_{i}))^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b1da814ede42a382f72a4aafcbf8949961f872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.785ex; height:6.843ex;" alt="{\displaystyle J=\sum _{i=1}^{n}(y_{i}-f(x_{i}))^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 20.785ex;height: 6.843ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b1da814ede42a382f72a4aafcbf8949961f872" data-alt="{\displaystyle J=\sum _{i=1}^{n}(y_{i}-f(x_{i}))^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </div> <p>で与えられるから、<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> J </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"> </noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" data-alt="{\displaystyle J}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> が最小になるように想定分布 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を（すなわち <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{k}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.009ex;" alt="{\displaystyle a_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.319ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" data-alt="{\displaystyle a_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を）、定めればよいということになる。</p> <p>それには、上式は <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{k}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.009ex;" alt="{\displaystyle a_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.319ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" data-alt="{\displaystyle a_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> を変数とする関数と見なすことができるので、<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> J </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"> </noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" data-alt="{\displaystyle J}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> を <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{k}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.009ex;" alt="{\displaystyle a_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.319ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" data-alt="{\displaystyle a_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> について偏微分したものを 0 と置く。こうして得られた <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> 個の連立方程式（正規方程式）を解き、<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{k}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.009ex;" alt="{\displaystyle a_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.319ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" data-alt="{\displaystyle a_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> を決定すればよい。</p> <div class="mw-heading mw-heading3"> <h3 id="一次方程式の場合"><span id=".E4.B8.80.E6.AC.A1.E6.96.B9.E7.A8.8B.E5.BC.8F.E3.81.AE.E5.A0.B4.E5.90.88"></span>一次方程式の場合</h3><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 一次方程式の場合" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <p>さらに簡単な例として、モデル関数を1次関数とし、</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ax+b\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> a </mi> <mi> x </mi> <mo> + </mo> <mi> b </mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f(x)=ax+b\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8e2f33372adf87767c8e2fd223670411164ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.301ex; height:2.843ex;" alt="{\displaystyle f(x)=ax+b\,}"> </noscript><span class="lazy-image-placeholder" style="width: 14.301ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8e2f33372adf87767c8e2fd223670411164ffe" data-alt="{\displaystyle f(x)=ax+b\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>とおくと、<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a,b} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"> </noscript><span class="lazy-image-placeholder" style="width: 3.261ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" data-alt="{\displaystyle a,b}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> は次式で求められる。</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {n\textstyle \sum \limits _{k=1}^{n}x_{k}y_{k}-\sum \limits _{k=1}^{n}x_{k}\sum \limits _{k=1}^{n}y_{k}}{n\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}-\left(\sum \limits _{k=1}^{n}x_{k}\right)^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> n </mi> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> −<!-- − --> </mo> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <mrow> <mi> n </mi> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> −<!-- − --> </mo> <msup> <mrow> <mo> ( </mo> <mrow> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a={\frac {n\textstyle \sum \limits _{k=1}^{n}x_{k}y_{k}-\sum \limits _{k=1}^{n}x_{k}\sum \limits _{k=1}^{n}y_{k}}{n\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}-\left(\sum \limits _{k=1}^{n}x_{k}\right)^{2}}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9984f801f13c187df3310e49f24092ce4cfc5ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:29.499ex; height:13.509ex;" alt="{\displaystyle a={\frac {n\textstyle \sum \limits _{k=1}^{n}x_{k}y_{k}-\sum \limits _{k=1}^{n}x_{k}\sum \limits _{k=1}^{n}y_{k}}{n\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}-\left(\sum \limits _{k=1}^{n}x_{k}\right)^{2}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 29.499ex;height: 13.509ex;vertical-align: -6.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9984f801f13c187df3310e49f24092ce4cfc5ce" data-alt="{\displaystyle a={\frac {n\textstyle \sum \limits _{k=1}^{n}x_{k}y_{k}-\sum \limits _{k=1}^{n}x_{k}\sum \limits _{k=1}^{n}y_{k}}{n\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}-\left(\sum \limits _{k=1}^{n}x_{k}\right)^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\frac {\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}\sum \limits _{k=1}^{n}y_{k}-\sum \limits _{k=1}^{n}x_{k}y_{k}\sum \limits _{k=1}^{n}x_{k}}{n\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}-\left(\sum \limits _{k=1}^{n}x_{k}\right)^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> b </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> −<!-- − --> </mo> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> <mrow> <mi> n </mi> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> −<!-- − --> </mo> <msup> <mrow> <mo> ( </mo> <mrow> <munderover> <mo movablelimits="false"> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle b={\frac {\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}\sum \limits _{k=1}^{n}y_{k}-\sum \limits _{k=1}^{n}x_{k}y_{k}\sum \limits _{k=1}^{n}x_{k}}{n\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}-\left(\sum \limits _{k=1}^{n}x_{k}\right)^{2}}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2925efa730cb842914cc7162ccf0dfec06548d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:34.688ex; height:13.509ex;" alt="{\displaystyle b={\frac {\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}\sum \limits _{k=1}^{n}y_{k}-\sum \limits _{k=1}^{n}x_{k}y_{k}\sum \limits _{k=1}^{n}x_{k}}{n\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}-\left(\sum \limits _{k=1}^{n}x_{k}\right)^{2}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 34.688ex;height: 13.509ex;vertical-align: -6.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2925efa730cb842914cc7162ccf0dfec06548d25" data-alt="{\displaystyle b={\frac {\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}\sum \limits _{k=1}^{n}y_{k}-\sum \limits _{k=1}^{n}x_{k}y_{k}\sum \limits _{k=1}^{n}x_{k}}{n\textstyle \sum \limits _{k=1}^{n}{x_{k}}^{2}-\left(\sum \limits _{k=1}^{n}x_{k}\right)^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd></dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="解法例"><span id=".E8.A7.A3.E6.B3.95.E4.BE.8B"></span>解法例</h2><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 解法例" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>当てはめたい関数 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> は、</p> <div style="margin-top:1ex; margin-left:2em; margin-bottom:1ex;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=(g_{1}(x),g_{2}(x),\ldots ,g_{m}(x))(a_{1},a_{2},\ldots ,a_{m})^{\textrm {T}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> g </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <msub> <mi> g </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mo> …<!-- … --> </mo> <mo> , </mo> <msub> <mi> g </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <mo> …<!-- … --> </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f(x)=(g_{1}(x),g_{2}(x),\ldots ,g_{m}(x))(a_{1},a_{2},\ldots ,a_{m})^{\textrm {T}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c2b84da6cd73874bf039176fd85c6bfa9c1bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.978ex; height:3.176ex;" alt="{\displaystyle f(x)=(g_{1}(x),g_{2}(x),\ldots ,g_{m}(x))(a_{1},a_{2},\ldots ,a_{m})^{\textrm {T}}}"> </noscript><span class="lazy-image-placeholder" style="width: 48.978ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c2b84da6cd73874bf039176fd85c6bfa9c1bae" data-alt="{\displaystyle f(x)=(g_{1}(x),g_{2}(x),\ldots ,g_{m}(x))(a_{1},a_{2},\ldots ,a_{m})^{\textrm {T}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </div> <p>と表すことができる。上付き添字 T は<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E8%BB%A2%E7%BD%AE%E8%A1%8C%E5%88%97?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="転置行列">転置行列</a>を表す。最小にすべき関数 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> J </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"> </noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" data-alt="{\displaystyle J}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> は</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}J({\boldsymbol {a}})&=(G{\boldsymbol {a}}-{\boldsymbol {y}})^{\textrm {T}}\,(G{\boldsymbol {a}}-{\boldsymbol {y}})\\&=\left({\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {a}}\\-1\end{bmatrix}}\right)^{\textrm {T}}\,\left({\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {a}}\\-1\end{bmatrix}}\right)\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi> J </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mi></mi> <mo> = </mo> <mo stretchy="false"> ( </mo> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo stretchy="false"> ( </mo> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> G </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> G </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}J({\boldsymbol {a}})&=(G{\boldsymbol {a}}-{\boldsymbol {y}})^{\textrm {T}}\,(G{\boldsymbol {a}}-{\boldsymbol {y}})\\&=\left({\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {a}}\\-1\end{bmatrix}}\right)^{\textrm {T}}\,\left({\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {a}}\\-1\end{bmatrix}}\right)\end{aligned}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f299472175af6b345c2bbb44626c9958c0fac7c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.123ex; margin-bottom: -0.215ex; width:45.123ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}J({\boldsymbol {a}})&=(G{\boldsymbol {a}}-{\boldsymbol {y}})^{\textrm {T}}\,(G{\boldsymbol {a}}-{\boldsymbol {y}})\\&=\left({\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {a}}\\-1\end{bmatrix}}\right)^{\textrm {T}}\,\left({\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {a}}\\-1\end{bmatrix}}\right)\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 45.123ex;height: 9.843ex;vertical-align: -4.123ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f299472175af6b345c2bbb44626c9958c0fac7c4" data-alt="{\displaystyle {\begin{aligned}J({\boldsymbol {a}})&=(G{\boldsymbol {a}}-{\boldsymbol {y}})^{\textrm {T}}\,(G{\boldsymbol {a}}-{\boldsymbol {y}})\\&=\left({\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {a}}\\-1\end{bmatrix}}\right)^{\textrm {T}}\,\left({\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {a}}\\-1\end{bmatrix}}\right)\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>と表される。ここに <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> G </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> </noscript><span class="lazy-image-placeholder" style="width: 1.827ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" data-alt="{\displaystyle G}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> は、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{ij}=g_{j}(x_{i})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> g </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G_{ij}=g_{j}(x_{i})} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d81b56448f5ef991f2be2054ae90e06b6b49278c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.36ex; height:3.009ex;" alt="{\displaystyle G_{ij}=g_{j}(x_{i})}"> </noscript><span class="lazy-image-placeholder" style="width: 12.36ex;height: 3.009ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d81b56448f5ef991f2be2054ae90e06b6b49278c" data-alt="{\displaystyle G_{ij}=g_{j}(x_{i})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> なる成分を持つ <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ×<!-- × --> </mo> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n\times m} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82325a2a02ad79bc7c347ba9702ad46eb0de824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle n\times m}"> </noscript><span class="lazy-image-placeholder" style="width: 6.276ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82325a2a02ad79bc7c347ba9702ad46eb0de824" data-alt="{\displaystyle n\times m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> 行列、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {y}}=(y_{1},y_{2},\ldots ,y_{n})^{\textrm {T}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <mo> …<!-- … --> </mo> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {y}}=(y_{1},y_{2},\ldots ,y_{n})^{\textrm {T}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318a5b5f10dc0064f9c44682d23a98ae347afc67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.655ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {y}}=(y_{1},y_{2},\ldots ,y_{n})^{\textrm {T}}}"> </noscript><span class="lazy-image-placeholder" style="width: 20.655ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318a5b5f10dc0064f9c44682d23a98ae347afc67" data-alt="{\displaystyle {\boldsymbol {y}}=(y_{1},y_{2},\ldots ,y_{n})^{\textrm {T}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>、係数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {a}}=(a_{1},a_{2},\ldots ,a_{m})^{\textrm {T}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <mo> …<!-- … --> </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {a}}=(a_{1},a_{2},\ldots ,a_{m})^{\textrm {T}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab406db8fec38378618fd46580dfa28ceae988a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.483ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {a}}=(a_{1},a_{2},\ldots ,a_{m})^{\textrm {T}}}"> </noscript><span class="lazy-image-placeholder" style="width: 21.483ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab406db8fec38378618fd46580dfa28ceae988a" data-alt="{\displaystyle {\boldsymbol {a}}=(a_{1},a_{2},\ldots ,a_{m})^{\textrm {T}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> である。</p> <p>これの最小解 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {a}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {a}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {a}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" data-alt="{\displaystyle {\boldsymbol {a}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> は、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}^{\textrm {T}}{\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}={\tilde {R}}^{\textrm {T}}{\tilde {R}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> G </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> G </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> R </mi> <mo stretchy="false"> ~<!-- ~ --> </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> R </mi> <mo stretchy="false"> ~<!-- ~ --> </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}^{\textrm {T}}{\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}={\tilde {R}}^{\textrm {T}}{\tilde {R}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02ce54ae83b69dffbe08383c9542decb0e90618c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.596ex; height:3.676ex;" alt="{\displaystyle {\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}^{\textrm {T}}{\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}={\tilde {R}}^{\textrm {T}}{\tilde {R}}}"> </noscript><span class="lazy-image-placeholder" style="width: 24.596ex;height: 3.676ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02ce54ae83b69dffbe08383c9542decb0e90618c" data-alt="{\displaystyle {\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}^{\textrm {T}}{\begin{bmatrix}G&{\boldsymbol {y}}\end{bmatrix}}={\tilde {R}}^{\textrm {T}}{\tilde {R}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を満たす上三角行列<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {R}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> R </mi> <mo stretchy="false"> ~<!-- ~ --> </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tilde {R}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9914b6b222f93964ba3227df7b7ef3d6db7cc342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.676ex;" alt="{\displaystyle {\tilde {R}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9914b6b222f93964ba3227df7b7ef3d6db7cc342" data-alt="{\displaystyle {\tilde {R}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>の計算を経て、解 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {a}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {a}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {a}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" data-alt="{\displaystyle {\boldsymbol {a}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を得ることができ、全体の計算量に無駄が少ない。下記の表式を用いると <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {R}}={\begin{bmatrix}R&Q^{\textrm {T}}{\boldsymbol {y}}\\{\boldsymbol {0}}^{\textrm {T}}&\alpha \end{bmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> R </mi> <mo stretchy="false"> ~<!-- ~ --> </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> R </mi> </mtd> <mtd> <msup> <mi> Q </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold"> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> </mtd> <mtd> <mi> α<!-- α --> </mi> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tilde {R}}={\begin{bmatrix}R&Q^{\textrm {T}}{\boldsymbol {y}}\\{\boldsymbol {0}}^{\textrm {T}}&\alpha \end{bmatrix}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b1f2120a393b8e8b65f852abc2427fd3d5c133b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.776ex; height:6.176ex;" alt="{\displaystyle {\tilde {R}}={\begin{bmatrix}R&Q^{\textrm {T}}{\boldsymbol {y}}\\{\boldsymbol {0}}^{\textrm {T}}&\alpha \end{bmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 17.776ex;height: 6.176ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b1f2120a393b8e8b65f852abc2427fd3d5c133b" data-alt="{\displaystyle {\tilde {R}}={\begin{bmatrix}R&Q^{\textrm {T}}{\boldsymbol {y}}\\{\boldsymbol {0}}^{\textrm {T}}&\alpha \end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> が得られ、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R{\boldsymbol {a}}=Q^{\textrm {T}}{\boldsymbol {y}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> <mo> = </mo> <msup> <mi> Q </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle R{\boldsymbol {a}}=Q^{\textrm {T}}{\boldsymbol {y}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648b816c25cf966d4a2aa1b6503d95272f8d36bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.962ex; height:3.009ex;" alt="{\displaystyle R{\boldsymbol {a}}=Q^{\textrm {T}}{\boldsymbol {y}}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.962ex;height: 3.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648b816c25cf966d4a2aa1b6503d95272f8d36bd" data-alt="{\displaystyle R{\boldsymbol {a}}=Q^{\textrm {T}}{\boldsymbol {y}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> から係数解 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {a}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {a}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {a}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" data-alt="{\displaystyle {\boldsymbol {a}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を求める<sup id="cite_ref-5" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-5"><span class="cite-bracket">[</span>注釈 2<span class="cite-bracket">]</span></a></sup>。</p> <p>また前節で述べたように <i>J</i> を<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {a}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {a}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {a}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" data-alt="{\displaystyle {\boldsymbol {a}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>のそれぞれの成分で偏微分して 0 と置いた <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> 個の式（正規方程式）は行列を用いて、</p> <div style="margin-top:1ex; margin-left:2em; margin-bottom:1ex;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{\textrm {T}}G{\boldsymbol {a}}=G^{\textrm {T}}{\boldsymbol {y}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> <mo> = </mo> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G^{\textrm {T}}G{\boldsymbol {a}}=G^{\textrm {T}}{\boldsymbol {y}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04bf16567d2dc17b9aa3645c2bc72d4ac12c89c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.259ex; height:3.009ex;" alt="{\displaystyle G^{\textrm {T}}G{\boldsymbol {a}}=G^{\textrm {T}}{\boldsymbol {y}}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.259ex;height: 3.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04bf16567d2dc17b9aa3645c2bc72d4ac12c89c" data-alt="{\displaystyle G^{\textrm {T}}G{\boldsymbol {a}}=G^{\textrm {T}}{\boldsymbol {y}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </div> <p>と表される。これを<b>正規方程式</b>（<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E8%8B%B1%E8%AA%9E?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="英語">英</a>: <span lang="en">normal equation</span>）と呼ぶ。この正規方程式を解けば係数解 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {a}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {a}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {a}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" data-alt="{\displaystyle {\boldsymbol {a}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> が求まる。</p> <p>係数解 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {a}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {a}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {a}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" data-alt="{\displaystyle {\boldsymbol {a}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の解法には以下のようないくつかの方法がある。</p> <ul> <li>逆行列で正規方程式を解く <dl> <dd> 行列 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{T}G}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> T </mi> </mrow> </msup> <mi> G </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G^{T}G} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e869543852e22dd03c7e1981fe2c945ae8d5a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.043ex; height:2.676ex;" alt="{\displaystyle G^{T}G}"> </noscript><span class="lazy-image-placeholder" style="width: 5.043ex;height: 2.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e869543852e22dd03c7e1981fe2c945ae8d5a89" data-alt="{\displaystyle G^{T}G}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> が<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%AD%A3%E5%89%87%E8%A1%8C%E5%88%97?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="正則行列">正則行列</a>（つまりフルランク）である場合は、解 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {a}}=(G^{\textrm {T}}G)^{-1}G^{\textrm {T}}{\boldsymbol {y}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> <mo> = </mo> <mo stretchy="false"> ( </mo> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mi> G </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {a}}=(G^{\textrm {T}}G)^{-1}G^{\textrm {T}}{\boldsymbol {y}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ea8bfeb7c2d28caafb05e30813a40282f4970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.401ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {a}}=(G^{\textrm {T}}G)^{-1}G^{\textrm {T}}{\boldsymbol {y}}}"> </noscript><span class="lazy-image-placeholder" style="width: 18.401ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81ea8bfeb7c2d28caafb05e30813a40282f4970" data-alt="{\displaystyle {\boldsymbol {a}}=(G^{\textrm {T}}G)^{-1}G^{\textrm {T}}{\boldsymbol {y}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> は一意に求まる。ただし <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{T}G}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> T </mi> </mrow> </msup> <mi> G </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G^{T}G} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e869543852e22dd03c7e1981fe2c945ae8d5a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.043ex; height:2.676ex;" alt="{\displaystyle G^{T}G}"> </noscript><span class="lazy-image-placeholder" style="width: 5.043ex;height: 2.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e869543852e22dd03c7e1981fe2c945ae8d5a89" data-alt="{\displaystyle G^{T}G}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の逆行列を明示的に求めることは通常は良い方法ではない。 </dd> </dl></li> </ul> <dl> <dd> <dl> <dd> 計算量が小さい方法として<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%82%B3%E3%83%AC%E3%82%B9%E3%82%AD%E3%83%BC%E5%88%86%E8%A7%A3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="コレスキー分解">コレスキー分解</a><sup id="cite_ref-Yamamoto1_6-0" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Yamamoto1-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{\textrm {T}}G=R^{\textrm {T}}R}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mi> G </mi> <mo> = </mo> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mi> R </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G^{\textrm {T}}G=R^{\textrm {T}}R} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0deea6dece50cb79039c44a10a771968ee6267" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.118ex; height:2.676ex;" alt="{\displaystyle G^{\textrm {T}}G=R^{\textrm {T}}R}"> </noscript><span class="lazy-image-placeholder" style="width: 13.118ex;height: 2.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0deea6dece50cb79039c44a10a771968ee6267" data-alt="{\displaystyle G^{\textrm {T}}G=R^{\textrm {T}}R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> R </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle R} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> </noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" data-alt="{\displaystyle R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> は <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <mo> ×<!-- × --> </mo> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m\times m} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367523981d714dcd9214703d654bfdedbe58d44a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.921ex; height:1.676ex;" alt="{\displaystyle m\times m}"> </noscript><span class="lazy-image-placeholder" style="width: 6.921ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367523981d714dcd9214703d654bfdedbe58d44a" data-alt="{\displaystyle m\times m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> 上<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E4%B8%89%E8%A7%92%E8%A1%8C%E5%88%97?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="三角行列">三角行列</a>）による三角行列分解を経て、最終的に <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R{\boldsymbol {a}}=R^{-{\textrm {T}}}G^{\textrm {T}}{\boldsymbol {y}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> <mo> = </mo> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </mrow> </msup> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle R{\boldsymbol {a}}=R^{-{\textrm {T}}}G^{\textrm {T}}{\boldsymbol {y}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04fa2c548f23be26562d5be84a5af0fde1805db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.412ex; height:3.009ex;" alt="{\displaystyle R{\boldsymbol {a}}=R^{-{\textrm {T}}}G^{\textrm {T}}{\boldsymbol {y}}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.412ex;height: 3.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04fa2c548f23be26562d5be84a5af0fde1805db0" data-alt="{\displaystyle R{\boldsymbol {a}}=R^{-{\textrm {T}}}G^{\textrm {T}}{\boldsymbol {y}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を解けばよい。 </dd> <dd> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%95%B0%E5%80%A4%E7%9A%84%E5%AE%89%E5%AE%9A%E6%80%A7?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="数値的安定性">数値的安定性</a>確保のためには、積 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{T}G}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> T </mi> </mrow> </msup> <mi> G </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G^{T}G} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e869543852e22dd03c7e1981fe2c945ae8d5a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.043ex; height:2.676ex;" alt="{\displaystyle G^{T}G}"> </noscript><span class="lazy-image-placeholder" style="width: 5.043ex;height: 2.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e869543852e22dd03c7e1981fe2c945ae8d5a89" data-alt="{\displaystyle G^{T}G}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を経ない三角行列分解が良い。すなわち以下と同じく<a href="https://ja-m-wikipedia-org.translate.goog/wiki/QR%E5%88%86%E8%A7%A3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="QR分解">QR分解</a>（直交分解）による <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=QR}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> G </mi> <mo> = </mo> <mi> Q </mi> <mi> R </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G=QR} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e57879f3461c4719f1d075d8bd4ae1b7ed549ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.528ex; height:2.509ex;" alt="{\displaystyle G=QR}"> </noscript><span class="lazy-image-placeholder" style="width: 8.528ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e57879f3461c4719f1d075d8bd4ae1b7ed549ab" data-alt="{\displaystyle G=QR}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> から、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R{\boldsymbol {a}}=Q^{\textrm {T}}{\boldsymbol {y}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> <mo> = </mo> <msup> <mi> Q </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> y </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle R{\boldsymbol {a}}=Q^{\textrm {T}}{\boldsymbol {y}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648b816c25cf966d4a2aa1b6503d95272f8d36bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.962ex; height:3.009ex;" alt="{\displaystyle R{\boldsymbol {a}}=Q^{\textrm {T}}{\boldsymbol {y}}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.962ex;height: 3.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648b816c25cf966d4a2aa1b6503d95272f8d36bd" data-alt="{\displaystyle R{\boldsymbol {a}}=Q^{\textrm {T}}{\boldsymbol {y}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を解く。 </dd> </dl> </dd> </dl> <ul> <li>直交分解で正規方程式を解く <dl> <dd> コレスキー分解の方法よりも計算量が大きいが、数値的に安定かつ汎用な方法として、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/QR%E5%88%86%E8%A7%A3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="QR分解">QR分解</a>や<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%89%B9%E7%95%B0%E5%80%A4%E5%88%86%E8%A7%A3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="特異値分解">特異値分解</a>（SVD）を用いる方法がある<sup id="cite_ref-Yamamoto1_6-1" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Yamamoto1-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>。これらの方法では計算の過程で積 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{T}G}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> T </mi> </mrow> </msup> <mi> G </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G^{T}G} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e869543852e22dd03c7e1981fe2c945ae8d5a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.043ex; height:2.676ex;" alt="{\displaystyle G^{T}G}"> </noscript><span class="lazy-image-placeholder" style="width: 5.043ex;height: 2.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e869543852e22dd03c7e1981fe2c945ae8d5a89" data-alt="{\displaystyle G^{T}G}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を必要としないため<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%95%B0%E5%80%A4%E7%9A%84%E5%AE%89%E5%AE%9A%E6%80%A7?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="数値的安定性">数値的安定性</a>が高い。また <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{T}G}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> T </mi> </mrow> </msup> <mi> G </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G^{T}G} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e869543852e22dd03c7e1981fe2c945ae8d5a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.043ex; height:2.676ex;" alt="{\displaystyle G^{T}G}"> </noscript><span class="lazy-image-placeholder" style="width: 5.043ex;height: 2.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e869543852e22dd03c7e1981fe2c945ae8d5a89" data-alt="{\displaystyle G^{T}G}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> が正則行列でない（ランク落ちしている）場合は正規方程式の解が不定となるが、その場合でも、これらの手法では解 <i>a</i> のうち<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%83%8E%E3%83%AB%E3%83%A0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ノルム">ノルム</a>が最も小さいものを求めることができる。特異値分解を用いる場合は、特異値のうち極めて小さい値を 0 とみなして計算することで数値計算上の大きな誤差の発生を防ぐことができる(<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E8%8B%B1%E8%AA%9E?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="英語">英</a>: <span lang="en">truncated SVD</span>)<sup id="cite_ref-7" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-7"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>。 </dd> </dl></li> <li><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%93%AC%E4%BC%BC%E9%80%86%E8%A1%8C%E5%88%97?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="擬似逆行列">擬似逆行列</a>を使う方法もあるが<sup id="cite_ref-8" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>、計算効率が悪いため、特殊な場合（解析的な数式が必要な場合など）を除いてあまり用いられない。</li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="拡張"><span id=".E6.8B.A1.E5.BC.B5"></span>拡張</h2><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 拡張" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <div class="mw-heading mw-heading3"> <h3 id="多次元"><span id=".E5.A4.9A.E6.AC.A1.E5.85.83"></span>多次元</h3><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 多次元" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <p>想定される分布が<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%AA%92%E4%BB%8B%E5%A4%89%E6%95%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="媒介変数">媒介変数</a> <i>t</i> を用いて <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)=(f(t),g(t))}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo stretchy="false"> ( </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x,y)=(f(t),g(t))} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e25730de0d089a509a435f58cba35750c3f6f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.963ex; height:2.843ex;" alt="{\displaystyle (x,y)=(f(t),g(t))}"> </noscript><span class="lazy-image-placeholder" style="width: 18.963ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e25730de0d089a509a435f58cba35750c3f6f2" data-alt="{\displaystyle (x,y)=(f(t),g(t))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の形（あるいは <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> , </mo> <mi> g </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f,g} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6ab1762925585cd7605809caa8b1b5284177b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.429ex; height:2.509ex;" alt="{\displaystyle f,g}"> </noscript><span class="lazy-image-placeholder" style="width: 3.429ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6ab1762925585cd7605809caa8b1b5284177b" data-alt="{\displaystyle f,g}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> は複数の媒介変数によって決まるとしても同様）であっても考察される。</p> <p>すなわち、測定値 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i},y_{i})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{i},y_{i})} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6dbb919b91ccacf17ed47898048428a1baf9703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:2.843ex;" alt="{\displaystyle (x_{i},y_{i})}"> </noscript><span class="lazy-image-placeholder" style="width: 6.912ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6dbb919b91ccacf17ed47898048428a1baf9703" data-alt="{\displaystyle (x_{i},y_{i})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> がパラメータ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle t_{i}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.639ex;height: 2.343ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" data-alt="{\displaystyle t_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> に対する <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f(t_{i}),g(t_{i}))}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (f(t_{i}),g(t_{i}))} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc3eac8a6b2613e485d85a8d43095094a171bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.135ex; height:2.843ex;" alt="{\displaystyle (f(t_{i}),g(t_{i}))}"> </noscript><span class="lazy-image-placeholder" style="width: 12.135ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc3eac8a6b2613e485d85a8d43095094a171bc2" data-alt="{\displaystyle (f(t_{i}),g(t_{i}))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を理論値として近似されているものと考えるのである。</p> <p>この場合、各点の理論値 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f(t_{i}),g(t_{i}))}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (f(t_{i}),g(t_{i}))} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc3eac8a6b2613e485d85a8d43095094a171bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.135ex; height:2.843ex;" alt="{\displaystyle (f(t_{i}),g(t_{i}))}"> </noscript><span class="lazy-image-placeholder" style="width: 12.135ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc3eac8a6b2613e485d85a8d43095094a171bc2" data-alt="{\displaystyle (f(t_{i}),g(t_{i}))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> と測定値 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i},y_{i})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{i},y_{i})} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6dbb919b91ccacf17ed47898048428a1baf9703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:2.843ex;" alt="{\displaystyle (x_{i},y_{i})}"> </noscript><span class="lazy-image-placeholder" style="width: 6.912ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6dbb919b91ccacf17ed47898048428a1baf9703" data-alt="{\displaystyle (x_{i},y_{i})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の間に生じる残差は</p> <div style="margin-top:1ex; margin-left:2em; margin-bottom:1ex;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {(x_{i}-f(t_{i}))^{2}+(y_{i}-g(t_{i}))^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> −<!-- − --> </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> −<!-- − --> </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\sqrt {(x_{i}-f(t_{i}))^{2}+(y_{i}-g(t_{i}))^{2}}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9849b4cae1d2cc083f94b6a08b78b9eece70cc45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:29.932ex; height:4.843ex;" alt="{\displaystyle {\sqrt {(x_{i}-f(t_{i}))^{2}+(y_{i}-g(t_{i}))^{2}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 29.932ex;height: 4.843ex;vertical-align: -1.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9849b4cae1d2cc083f94b6a08b78b9eece70cc45" data-alt="{\displaystyle {\sqrt {(x_{i}-f(t_{i}))^{2}+(y_{i}-g(t_{i}))^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </div> <p>である。故に、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%AE%8B%E5%B7%AE%E5%B9%B3%E6%96%B9%E5%92%8C?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="残差平方和">残差平方和</a>は</p> <div style="margin-top:1ex; margin-left:2em; margin-bottom:1ex;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}\left\{(x_{i}-f(t_{i}))^{2}+(y_{i}-g(t_{i}))^{2}\right\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mrow> <mo> { </mo> <mrow> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> −<!-- − --> </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> −<!-- − --> </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <msub> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> } </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sum _{i=1}^{n}\left\{(x_{i}-f(t_{i}))^{2}+(y_{i}-g(t_{i}))^{2}\right\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d60190fef69debe257cb9e1289f63645ceeadc05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.061ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}\left\{(x_{i}-f(t_{i}))^{2}+(y_{i}-g(t_{i}))^{2}\right\}}"> </noscript><span class="lazy-image-placeholder" style="width: 34.061ex;height: 6.843ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d60190fef69debe257cb9e1289f63645ceeadc05" data-alt="{\displaystyle \sum _{i=1}^{n}\left\{(x_{i}-f(t_{i}))^{2}+(y_{i}-g(t_{i}))^{2}\right\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </div> <p>となるから、この値が最小であるように、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> <mo> , </mo> <mi> g </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f,g} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6ab1762925585cd7605809caa8b1b5284177b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.429ex; height:2.509ex;" alt="{\displaystyle f,g}"> </noscript><span class="lazy-image-placeholder" style="width: 3.429ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6ab1762925585cd7605809caa8b1b5284177b" data-alt="{\displaystyle f,g}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を決定するのである。</p> <p>このように、<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> 組の <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x,y)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> </noscript><span class="lazy-image-placeholder" style="width: 5.328ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" data-alt="{\displaystyle (x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の測定値 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i},y_{i})(i=1,2,...,n)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <mi> n </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{i},y_{i})(i=1,2,...,n)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f205fd09db88ebcbed63ed0b7a34702bef88aa6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.545ex; height:2.843ex;" alt="{\displaystyle (x_{i},y_{i})(i=1,2,...,n)}"> </noscript><span class="lazy-image-placeholder" style="width: 22.545ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f205fd09db88ebcbed63ed0b7a34702bef88aa6f" data-alt="{\displaystyle (x_{i},y_{i})(i=1,2,...,n)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> を<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> 組の <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},...,x_{m})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{1},x_{2},...,x_{m})} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80ed3cec924e98510613b18f7f588227f3212bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.785ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},...,x_{m})}"> </noscript><span class="lazy-image-placeholder" style="width: 15.785ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80ed3cec924e98510613b18f7f588227f3212bf0" data-alt="{\displaystyle (x_{1},x_{2},...,x_{m})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> の測定値 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1i},x_{2i},...,x_{mi})(i=1,2,...,n)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mi> i </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> i </mi> </mrow> </msub> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <mi> n </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{1i},x_{2i},...,x_{mi})(i=1,2,...,n)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9950b83e4bd1facbea35e5ef48c970b8b424b51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.121ex; height:2.843ex;" alt="{\displaystyle (x_{1i},x_{2i},...,x_{mi})(i=1,2,...,n)}"> </noscript><span class="lazy-image-placeholder" style="width: 33.121ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9950b83e4bd1facbea35e5ef48c970b8b424b51" data-alt="{\displaystyle (x_{1i},x_{2i},...,x_{mi})(i=1,2,...,n)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> に拡張したものも考察することができる。</p> <div class="mw-heading mw-heading3"> <h3 id="測定の誤差が既知の場合"><span id=".E6.B8.AC.E5.AE.9A.E3.81.AE.E8.AA.A4.E5.B7.AE.E3.81.8C.E6.97.A2.E7.9F.A5.E3.81.AE.E5.A0.B4.E5.90.88"></span>測定の誤差が既知の場合</h3><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 測定の誤差が既知の場合" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <p><i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> 回の測定における<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E8%AA%A4%E5%B7%AE?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="誤差">誤差</a>があらかじめ分かっている場合を考える。異なる測定方法で測定した複数のデータ列を結合する場合などでは、測定ごとに誤差が異なることはしばしばある。誤差が正規分布していると考え、その<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%A8%99%E6%BA%96%E5%81%8F%E5%B7%AE?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="標準偏差">標準偏差</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}(i=1,2,\ldots ,n)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> σ<!-- σ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mo> …<!-- … --> </mo> <mo> , </mo> <mi> n </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma _{i}(i=1,2,\ldots ,n)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4a3b99baca008cc699590a977c55e810af119c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.769ex; height:2.843ex;" alt="{\displaystyle \sigma _{i}(i=1,2,\ldots ,n)}"> </noscript><span class="lazy-image-placeholder" style="width: 17.769ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4a3b99baca008cc699590a977c55e810af119c" data-alt="{\displaystyle \sigma _{i}(i=1,2,\ldots ,n)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> で、誤差の大きさを表す。すると、誤差が大きい測定より、誤差が小さい測定の結果により重みをつけて近似関数を与えるべきであるから、</p> <div style="margin-top:1ex; margin-left:2em; margin-bottom:1ex;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J'=\sum _{i=1}^{n}{\frac {(y_{i}-f(x_{i}))^{2}}{\sigma _{i}^{2}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> J </mi> <mo> ′ </mo> </msup> <mo> = </mo> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> −<!-- − --> </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <msubsup> <mi> σ<!-- σ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J'=\sum _{i=1}^{n}{\frac {(y_{i}-f(x_{i}))^{2}}{\sigma _{i}^{2}}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba31f887c5ded921b04777ec3d8f51ed5b9a3898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.747ex; height:7.009ex;" alt="{\displaystyle J'=\sum _{i=1}^{n}{\frac {(y_{i}-f(x_{i}))^{2}}{\sigma _{i}^{2}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 22.747ex;height: 7.009ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba31f887c5ded921b04777ec3d8f51ed5b9a3898" data-alt="{\displaystyle J'=\sum _{i=1}^{n}{\frac {(y_{i}-f(x_{i}))^{2}}{\sigma _{i}^{2}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </div> <p>を、最小にするように <i>f</i> を定める方がより正確な近似を与える。</p> <p>毎回の測定が<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%8B%AC%E7%AB%8B_(%E7%A2%BA%E7%8E%87%E8%AB%96)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="独立 (確率論)">独立</a>ならば、測定値の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%B0%A4%E5%BA%A6?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="尤度">尤度</a>は <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle exp(-J')}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> e </mi> <mi> x </mi> <mi> p </mi> <mo stretchy="false"> ( </mo> <mo> −<!-- − --> </mo> <msup> <mi> J </mi> <mo> ′ </mo> </msup> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle exp(-J')} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966651a31e8dedac72472343e34ed26eb05a5202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.411ex; height:3.009ex;" alt="{\displaystyle exp(-J')}"> </noscript><span class="lazy-image-placeholder" style="width: 9.411ex;height: 3.009ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966651a31e8dedac72472343e34ed26eb05a5202" data-alt="{\displaystyle exp(-J')}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> に比例する。そこで、上記の <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J'}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> J </mi> <mo> ′ </mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J'} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c76d5f08e40c83bfc959e27a75dd387278214e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.211ex; height:2.509ex;" alt="{\displaystyle J'}"> </noscript><span class="lazy-image-placeholder" style="width: 2.211ex;height: 2.509ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c76d5f08e40c83bfc959e27a75dd387278214e9c" data-alt="{\displaystyle J'}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> を最小にする <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> は、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%A4%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="最尤法">最尤推定値</a>であるとも解釈できる。また、<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J'}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> J </mi> <mo> ′ </mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J'} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c76d5f08e40c83bfc959e27a75dd387278214e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.211ex; height:2.509ex;" alt="{\displaystyle J'}"> </noscript><span class="lazy-image-placeholder" style="width: 2.211ex;height: 2.509ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c76d5f08e40c83bfc959e27a75dd387278214e9c" data-alt="{\displaystyle J'}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> は<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E8%87%AA%E7%94%B1%E5%BA%A6?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="自由度">自由度</a> <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> −<!-- − --> </mo> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n-m} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9677b812ea9ee4d4538767f9aef960c69aca59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.176ex;" alt="{\displaystyle n-m}"> </noscript><span class="lazy-image-placeholder" style="width: 6.276ex;height: 2.176ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9677b812ea9ee4d4538767f9aef960c69aca59" data-alt="{\displaystyle n-m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%82%AB%E3%82%A4%E4%BA%8C%E4%B9%97%E5%88%86%E5%B8%83?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="カイ二乗分布">カイ二乗分布</a><sup id="cite_ref-9" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-9"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>に従うので、それを用いてモデル <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> の妥当性を検定することもできる。</p> <p>毎回の測定誤差が同じ場合、<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J'}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> J </mi> <mo> ′ </mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J'} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c76d5f08e40c83bfc959e27a75dd387278214e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.211ex; height:2.509ex;" alt="{\displaystyle J'}"> </noscript><span class="lazy-image-placeholder" style="width: 2.211ex;height: 2.509ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c76d5f08e40c83bfc959e27a75dd387278214e9c" data-alt="{\displaystyle J'}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> を最小にするのは <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> J </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"> </noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" data-alt="{\displaystyle J}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> を最小にするのと同じ意味になる。</p> <div class="mw-heading mw-heading3"> <h3 id="非線形最小二乗法"><span id=".E9.9D.9E.E7.B7.9A.E5.BD.A2.E6.9C.80.E5.B0.8F.E4.BA.8C.E4.B9.97.E6.B3.95"></span>非線形最小二乗法</h3><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 非線形最小二乗法" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <div class="rellink" style="margin-bottom: 0.5em; padding-left: 2em; font-size: 90%;" role="note"> →詳細は「<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E9%9D%9E%E7%B7%9A%E5%BD%A2%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="非線形最小二乗法">非線形最小二乗法</a>」を参照 </div> <p>もし、<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> が、<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{k}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.009ex;" alt="{\displaystyle a_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.319ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" data-alt="{\displaystyle a_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> の線型結合で表されないときは、正規方程式を用いた解法は使えず、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%8F%8D%E5%BE%A9%E8%A7%A3%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="反復解法">反復解法</a>を用いて数値的に <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{k}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.009ex;" alt="{\displaystyle a_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.319ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" data-alt="{\displaystyle a_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> の近似値を求める必要がある。例えば、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%82%AC%E3%82%A6%E3%82%B9%E3%83%BB%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ガウス・ニュートン法">ガウス・ニュートン法</a><sup id="cite_ref-10" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-10"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>や<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%83%AC%E3%83%BC%E3%83%99%E3%83%B3%E3%83%90%E3%83%BC%E3%82%B0%E3%83%BB%E3%83%9E%E3%83%BC%E3%82%AB%E3%83%BC%E3%83%88%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="レーベンバーグ・マーカート法">レーベンバーグ・マーカート法</a><sup id="cite_ref-11" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-more_12-0" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-more-12"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-yu_13-0" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-yu-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-rang_14-0" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-rang-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>が用いられる。とくにレーベンバーグ・マーカート法（<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E8%8B%B1%E8%AA%9E?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="英語">英</a>: <span lang="en">Levenberg-Marquardt Method</span>）は多くの多次元非線形関数でパラメータを発散させずに効率よく収束させる（探索する）方法として知られている<sup id="cite_ref-more_12-1" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-more-12"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-yu_13-1" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-yu-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-rang_14-1" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-rang-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>。</p> <div class="mw-heading mw-heading3"> <h3 id="異常値の除去"><span id=".E7.95.B0.E5.B8.B8.E5.80.A4.E3.81.AE.E9.99.A4.E5.8E.BB"></span>異常値の除去</h3><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 異常値の除去" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <p>前提条件の節で述べたように、測定データを最小二乗法によって近似する場合、外れ値または異常値が含まれていると極端に近似の尤もらしさが低下することがある。また、様々な要因によって異常値を含む測定はしばしば得られるものである。</p> <p>誤差が正規分布から極端に外れた異常値を取り除くための方法として修正トンプソン-τ法が用いられる<sup id="cite_ref-16" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>。</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="関連項目"><span id=".E9.96.A2.E9.80.A3.E9.A0.85.E7.9B.AE"></span>関連項目</h2><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 関連項目" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <style data-mw-deduplicate="TemplateStyles:r94202605">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style> <div class="side-box side-box-right plainlinks sistersitebox noprint" style="width:22em;"> <div class="side-box-flex"> <div class="side-box-image"> <span class="noviewer" typeof="mw:File"><span> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" data-file-width="1024" data-file-height="1376"> </noscript><span class="lazy-image-placeholder" style="width: 30px;height: 40px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" data-alt="" data-width="30" data-height="40" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-class="mw-file-element"> </span></span></span> </div> <div class="side-box-text plainlist" style="font-size:100%;"> ウィキメディア・コモンズには、<b><span class="plainlinks"><a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://commons.wikimedia.org/wiki/Category:Least_squares?uselang%3Dja">最小二乗法</a></span></b>に関連するカテゴリがあります。 </div> </div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r94202605"> <div class="side-box side-box-right plainlinks sistersitebox noprint" style="width:22em;"> <div class="side-box-flex"> <div class="side-box-image"> <span class="noviewer" typeof="mw:File"><span> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" data-file-width="400" data-file-height="400"> </noscript><span class="lazy-image-placeholder" style="width: 40px;height: 40px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" data-alt="" data-width="40" data-height="40" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-class="mw-file-element"> </span></span></span> </div> <div class="side-box-text plainlist" style="font-size:100%;"> ウィキブックスに<b><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikibooks.org/wiki/%25E6%259C%2580%25E5%25B0%258F%25E4%25BA%258C%25E4%25B9%2597%25E6%25B3%2595" class="extiw" title="b:最小二乗法">最小二乗法</a></b>関連の解説書・教科書があります。 </div> </div> </div> <ul> <li><span title="リンク先の項目はまだありません。新規の執筆や他言語版からの翻訳が望まれます。"><a href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E7%B7%8F%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="総最小二乗法 (存在しないページ)">総最小二乗法</a><span style="font-size: 0.77em; font-weight: normal;" class="noprint">（<a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Total_least_squares" class="extiw" title="en:Total least squares">英語版</a>）</span></span> (<sup id="cite_ref-17" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>)</li> <li><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%82%AC%E3%82%A6%E3%82%B9%EF%BC%9D%E3%83%9E%E3%83%AB%E3%82%B3%E3%83%95%E3%81%AE%E5%AE%9A%E7%90%86?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ガウス＝マルコフの定理">ガウス＝マルコフの定理</a><sup id="cite_ref-19" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li> <li><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9B%B2%E7%B7%9A%E3%81%82%E3%81%A6%E3%81%AF%E3%82%81?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="曲線あてはめ">曲線あてはめ</a><sup id="cite_ref-20" class="reference"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="脚注"><span id=".E8.84.9A.E6.B3.A8"></span>脚注</h2><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 脚注" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <div class="noprint" style="float:right; font-size:90%;"> [<a href="https://ja-m-wikipedia-org.translate.goog/wiki/Help:%E8%84%9A%E6%B3%A8/%E8%AA%AD%E8%80%85%E5%90%91%E3%81%91?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Help:脚注/読者向け"><span title="この欄の操作法">脚注の使い方</span></a>] </div> <div class="mw-heading mw-heading3"> <h3 id="注釈"><span id=".E6.B3.A8.E9.87.88"></span>注釈</h3><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=14&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 注釈" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <div class="mw-references-wrap"> <ol class="references"> <li id="cite_note-4"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-4">^</a></b> <span class="reference-text"><i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i> は、測定データの数よりも小さいとする。</span></li> <li id="cite_note-5"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-5">^</a></b> <span class="reference-text"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{2}=\min J({\boldsymbol {a}})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> α<!-- α --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mo movablelimits="true" form="prefix"> min </mo> <mi> J </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> a </mi> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \alpha ^{2}=\min J({\boldsymbol {a}})} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a13780aeb1281de64bd8bc128239be1f679d66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.655ex; height:3.176ex;" alt="{\displaystyle \alpha ^{2}=\min J({\boldsymbol {a}})}"> </noscript><span class="lazy-image-placeholder" style="width: 14.655ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a13780aeb1281de64bd8bc128239be1f679d66" data-alt="{\displaystyle \alpha ^{2}=\min J({\boldsymbol {a}})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{\textrm {T}}G}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> T </mtext> </mrow> </mrow> </msup> <mi> G </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle G^{\textrm {T}}G} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1c6d00f0c5ef75d9a7a7398695eab7a0815a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.072ex; height:2.676ex;" alt="{\displaystyle G^{\textrm {T}}G}"> </noscript><span class="lazy-image-placeholder" style="width: 5.072ex;height: 2.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1c6d00f0c5ef75d9a7a7398695eab7a0815a1e" data-alt="{\displaystyle G^{\textrm {T}}G}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> は正則行列と仮定。</span></li> </ol> </div> <div class="mw-heading mw-heading3"> <h3 id="出典"><span id=".E5.87.BA.E5.85.B8"></span>出典</h3><span class="mw-editsection"> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95&action=edit&section=15&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 出典" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>編集</span> </a> </span> </div> <div class="mw-references-wrap mw-references-columns"> <ol class="references"> <li id="cite_note-ut-1">^ <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-ut_1-0"><sup><i><b>a</b></i></sup></a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-ut_1-1"><sup><i><b>b</b></i></sup></a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-ut_1-2"><sup><i><b>c</b></i></sup></a> <span class="reference-text"><cite style="font-style:normal" class="citation" id="CITEREF中川徹小柳義夫1982">中川徹; 小柳義夫『最小二乗法による実験データ解析』東京大学出版会、1982年、30頁。<style data-mw-deduplicate="TemplateStyles:r101121245">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation.cs-ja1 q,.mw-parser-output .citation.cs-ja2 q{quotes:"「""」""『""』"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:var(--color-success,#3a3);margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><a href="https://ja-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%89%B9%E5%88%A5:%E6%96%87%E7%8C%AE%E8%B3%87%E6%96%99/4-13-064067-4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="特別:文献資料/4-13-064067-4">4-13-064067-4</a>。</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95%E3%81%AB%E3%82%88%E3%82%8B%E5%AE%9F%E9%A8%93%E3%83%87%E3%83%BC%E3%82%BF%E8%A7%A3%E6%9E%90&rft.aulast=%E4%B8%AD%E5%B7%9D%E5%BE%B9&rft.au=%E4%B8%AD%E5%B7%9D%E5%BE%B9&rft.au=%E5%B0%8F%E6%9F%B3%E7%BE%A9%E5%A4%AB&rft.date=1982&rft.pages=30%E9%A0%81&rft.pub=%E6%9D%B1%E4%BA%AC%E5%A4%A7%E5%AD%A6%E5%87%BA%E7%89%88%E4%BC%9A&rft.isbn=4-13-064067-4&rfr_id=info:sid/ja.wikipedia.org:%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span style="display: none;"> </span></span></span></li> <li id="cite_note-lh-2">^ <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-lh_2-0"><sup><i><b>a</b></i></sup></a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-lh_2-1"><sup><i><b>b</b></i></sup></a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-lh_2-2"><sup><i><b>c</b></i></sup></a> <span class="reference-text"><cite style="font-style:normal" class="citation book">Lawson, Charles L.; Hanson, Richard J. (1995). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://epubs.siam.org/doi/10.1137/1.9781611971217"><i>Solving Least Squares Problems</i></a>. Society for Industrial and Applied Mathematics. <a href="https://ja-m-wikipedia-org.translate.goog/wiki/Doi_(%E8%AD%98%E5%88%A5%E5%AD%90)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (識別子)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1137%252F1.9781611971217">10.1137/1.9781611971217</a><span style="display:none;">. <a rel="nofollow" class="external free" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://epubs.siam.org/doi/10.1137/1.9781611971217">https://epubs.siam.org/doi/10.1137/1.9781611971217</a></span></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solving+Least+Squares+Problems&rft.aulast=Lawson%2C+Charles+L.%3B+Hanson%2C+Richard+J.&rft.au=Lawson%2C+Charles+L.%3B+Hanson%2C+Richard+J.&rft.date=1995&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft_id=info:doi/10.1137%2F1.9781611971217&rft_id=https%3A%2F%2Fepubs.siam.org%2Fdoi%2F10.1137%2F1.9781611971217&rfr_id=info:sid/ja.wikipedia.org:%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span style="display: none;"> </span></span><span style="font-size:0.95em; font-size: 90%; color: #555">(<span typeof="mw:File"><span title="要購読契約"> <noscript> <img alt="Paid subscription required" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/9px-Lock-red.svg.png" decoding="async" width="9" height="14" class="mw-file-element" data-file-width="512" data-file-height="813"> </noscript><span class="lazy-image-placeholder" style="width: 9px;height: 14px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/9px-Lock-red.svg.png" data-alt="Paid subscription required" data-width="9" data-height="14" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/14px-Lock-red.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/18px-Lock-red.svg.png 2x" data-class="mw-file-element"> </span></span></span>要購読契約)</span></span></li> <li id="cite_note-bj-3">^ <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-bj_3-0"><sup><i><b>a</b></i></sup></a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-bj_3-1"><sup><i><b>b</b></i></sup></a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-bj_3-2"><sup><i><b>c</b></i></sup></a> <span class="reference-text"><cite style="font-style:normal" class="citation book">Björck, Åke (1996). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://epubs.siam.org/doi/abs/10.1137/1.9781611971484"><i>Numerical Methods for Least Squares Problems</i></a>. Society for Industrial and Applied Mathematics. <a href="https://ja-m-wikipedia-org.translate.goog/wiki/Doi_(%E8%AD%98%E5%88%A5%E5%AD%90)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (識別子)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1137%252F1.9781611971484">10.1137/1.9781611971484</a><span style="display:none;">. <a rel="nofollow" class="external free" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://epubs.siam.org/doi/abs/10.1137/1.9781611971484">https://epubs.siam.org/doi/abs/10.1137/1.9781611971484</a></span></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Methods+for+Least+Squares+Problems&rft.aulast=Bj%C3%B6rck%2C+%C3%85ke&rft.au=Bj%C3%B6rck%2C+%C3%85ke&rft.date=1996&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft_id=info:doi/10.1137%2F1.9781611971484&rft_id=https%3A%2F%2Fepubs.siam.org%2Fdoi%2Fabs%2F10.1137%2F1.9781611971484&rfr_id=info:sid/ja.wikipedia.org:%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span style="display: none;"> </span></span></span></li> <li id="cite_note-Yamamoto1-6">^ <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Yamamoto1_6-0"><sup><i><b>a</b></i></sup></a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Yamamoto1_6-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"><cite style="font-style:normal" class="citation book">山本哲朗『数値解析入門』（増訂版）<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%82%B5%E3%82%A4%E3%82%A8%E3%83%B3%E3%82%B9%E7%A4%BE?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="サイエンス社">サイエンス社</a>〈サイエンスライブラリ 現代数学への入門 14〉、2003年6月。 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%89%B9%E5%88%A5:%E6%96%87%E7%8C%AE%E8%B3%87%E6%96%99/4-7819-1038-6?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="特別:文献資料/4-7819-1038-6">4-7819-1038-6</a>。</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%E6%95%B0%E5%80%A4%E8%A7%A3%E6%9E%90%E5%85%A5%E9%96%80&rft.aulast=%E5%B1%B1%E6%9C%AC%E5%93%B2%E6%9C%97&rft.au=%E5%B1%B1%E6%9C%AC%E5%93%B2%E6%9C%97&rft.date=2003-06&rft.series=%E3%82%B5%E3%82%A4%E3%82%A8%E3%83%B3%E3%82%B9%E3%83%A9%E3%82%A4%E3%83%96%E3%83%A9%E3%83%AA+%E7%8F%BE%E4%BB%A3%E6%95%B0%E5%AD%A6%E3%81%B8%E3%81%AE%E5%85%A5%E9%96%80+14&rft.edition=%E5%A2%97%E8%A8%82%E7%89%88&rft.pub=%5B%5B%E3%82%B5%E3%82%A4%E3%82%A8%E3%83%B3%E3%82%B9%E7%A4%BE%5D%5D&rft.isbn=4-7819-1038-6&rfr_id=info:sid/ja.wikipedia.org:%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span style="display: none;"> </span></span></span></li> <li id="cite_note-7"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-7">^</a></b> <span class="reference-text"><cite style="font-style:normal" class="citation journal">Hansen, Per Christian (1987). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007/BF01937276">“The truncated SVD as a method for regularization”</a>. <i>BIT Numerical Mathematics</i> (Springer) <b>27</b>: 534-553. <a href="https://ja-m-wikipedia-org.translate.goog/wiki/Doi_(%E8%AD%98%E5%88%A5%E5%AD%90)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (識別子)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252FBF01937276">10.1007/BF01937276</a><span style="display:none;">. <a rel="nofollow" class="external free" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007/BF01937276">https://doi.org/10.1007/BF01937276</a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+truncated+SVD+as+a+method+for+regularization&rft.jtitle=BIT+Numerical+Mathematics&rft.aulast=Hansen%2C+Per+Christian&rft.au=Hansen%2C+Per+Christian&rft.date=1987&rft.volume=27&rft.pages=534-553&rft.pub=Springer&rft_id=info:doi/10.1007%2FBF01937276&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%2FBF01937276&rfr_id=info:sid/ja.wikipedia.org:%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span style="display: none;"> </span></span> <span style="font-size:0.95em; font-size: 90%; color: #555">(<span typeof="mw:File"><span title="要購読契約"> <noscript> <img alt="Paid subscription required" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/9px-Lock-red.svg.png" decoding="async" width="9" height="14" class="mw-file-element" data-file-width="512" data-file-height="813"> </noscript><span class="lazy-image-placeholder" style="width: 9px;height: 14px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/9px-Lock-red.svg.png" data-alt="Paid subscription required" data-width="9" data-height="14" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/14px-Lock-red.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/18px-Lock-red.svg.png 2x" data-class="mw-file-element"> </span></span></span>要購読契約)</span></span></li> <li id="cite_note-8"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-8">^</a></b> <span class="reference-text">安川章. (2017). 科学実験/画像変換の近似計算に便利な 「疑似逆行列」 入門 できる人が使っている最小二乗法の一発フィット. インターフェース= Interface, 43(8), 142-146.</span></li> <li id="cite_note-9"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-9">^</a></b> <span class="reference-text">Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource. <span class="url"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathworld.wolfram.com/Chi-SquaredDistribution.html">mathworld<wbr>.wolfram<wbr>.com<wbr>/Chi-SquaredDistribution<wbr>.html</a></span></span></li> <li id="cite_note-10"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-10">^</a></b> <span class="reference-text"><cite style="font-style:normal" class="citation book">MAGREÑÁN, Alberto; Argyros, Ioannis (2018). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.co.jp/books?hl%3Dja%26lr%3Dlang_ja"><i>A contemporary study of iterative methods: convergence, dynamics and applications</i></a>. Academic Press<span style="display:none;">. <a rel="nofollow" class="external free" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.co.jp/books?hl%3Dja%26lr%3Dlang_ja">https://books.google.co.jp/books?hl=ja&lr=lang_ja</a></span></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+contemporary+study+of+iterative+methods%3A+convergence%2C+dynamics+and+applications&rft.aulast=MAGRE%C3%91%C3%81N%2C+Alberto%3B+Argyros%2C+Ioannis&rft.au=MAGRE%C3%91%C3%81N%2C+Alberto%3B+Argyros%2C+Ioannis&rft.date=2018&rft.pub=Academic+Press&rft_id=https%3A%2F%2Fbooks.google.co.jp%2Fbooks%3Fhl%3Dja%26lr%3Dlang_ja&rfr_id=info:sid/ja.wikipedia.org:%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span style="display: none;"> </span></span></span></li> <li id="cite_note-11"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-11">^</a></b> <span class="reference-text"> Weisstein, Eric W. "Levenberg-Marquardt Method." From MathWorld--A Wolfram Web Resource. <span class="url"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathworld.wolfram.com/Levenberg-MarquardtMethod.html">mathworld<wbr>.wolfram<wbr>.com<wbr>/Levenberg-MarquardtMethod<wbr>.html</a></span></span></li> <li id="cite_note-more-12">^ <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-more_12-0"><sup><i><b>a</b></i></sup></a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-more_12-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"><cite style="font-style:normal" class="citation journal">Moré Jorge J. (1978). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://link.springer.com/chapter/10.1007/BFb0067700">“The Levenberg-Marquardt algorithm : Implementation and theory”</a>. <i>Lecture Notes in Mathematics</i> <b>630</b>: 105-116. <a href="https://ja-m-wikipedia-org.translate.goog/wiki/Doi_(%E8%AD%98%E5%88%A5%E5%AD%90)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (識別子)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252FBFb0067700">10.1007/BFb0067700</a><span style="display:none;">. <a rel="nofollow" class="external free" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://link.springer.com/chapter/10.1007/BFb0067700">https://link.springer.com/chapter/10.1007/BFb0067700</a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Levenberg-Marquardt+algorithm+%3A+Implementation+and+theory&rft.jtitle=Lecture+Notes+in+Mathematics&rft.aulast=Mor%C3%A9+Jorge+J.&rft.au=Mor%C3%A9+Jorge+J.&rft.date=1978&rft.volume=630&rft.pages=105-116&rft_id=info:doi/10.1007%2FBFb0067700&rft_id=https%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2FBFb0067700&rfr_id=info:sid/ja.wikipedia.org:%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span style="display: none;"> </span></span> <span style="font-size:0.95em; font-size: 90%; color: #555">(<span typeof="mw:File"><span title="要購読契約"> <noscript> <img alt="Paid subscription required" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/9px-Lock-red.svg.png" decoding="async" width="9" height="14" class="mw-file-element" data-file-width="512" data-file-height="813"> </noscript><span class="lazy-image-placeholder" style="width: 9px;height: 14px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/9px-Lock-red.svg.png" data-alt="Paid subscription required" data-width="9" data-height="14" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/14px-Lock-red.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Lock-red.svg/18px-Lock-red.svg.png 2x" data-class="mw-file-element"> </span></span></span>要購読契約)</span></span></li> <li id="cite_note-yu-13">^ <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-yu_13-0"><sup><i><b>a</b></i></sup></a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-yu_13-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text">Yu, H., & Wilamowski, B. M. (2011). Levenberg-marquardt training. Industrial electronics handbook, 5(12), 1.</span></li> <li id="cite_note-rang-14">^ <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-rang_14-0"><sup><i><b>a</b></i></sup></a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-rang_14-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"><cite style="font-style:normal" class="citation journal">Ranganathan, Ananth (2004). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sites.cs.ucsb.edu/~yfwang/courses/cs290i_mvg/pdf/LMA.pdf">“The levenberg-marquardt algorithm”</a> (PDF). <i>Tutoral on LM algorithm</i> <b>11</b> (1): 101-110<span style="display:none;">. <a rel="nofollow" class="external free" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sites.cs.ucsb.edu/~yfwang/courses/cs290i_mvg/pdf/LMA.pdf">https://sites.cs.ucsb.edu/~yfwang/courses/cs290i_mvg/pdf/LMA.pdf</a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+levenberg-marquardt+algorithm&rft.jtitle=Tutoral+on+LM+algorithm&rft.aulast=Ranganathan%2C+Ananth&rft.au=Ranganathan%2C+Ananth&rft.date=2004&rft.volume=11&rft.issue=1&rft.pages=101-110&rft_id=https%3A%2F%2Fsites.cs.ucsb.edu%2F%7Eyfwang%2Fcourses%2Fcs290i_mvg%2Fpdf%2FLMA.pdf&rfr_id=info:sid/ja.wikipedia.org:%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span style="display: none;"> </span></span></span></li> <li id="cite_note-15"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-15">^</a></b> <span class="reference-text"><cite style="font-style:normal" class="citation journal">山下信雄, 福島雅夫「<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hdl.handle.net/2433/64462">Levenberg-Marquardt法の局所収束性について (最適化の数理科学)</a>」『数理解析研究所講究録』第1174巻、京都大学数理解析研究所、2000年10月、161-168頁、 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/CRID_(%E8%AD%98%E5%88%A5%E5%AD%90)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="CRID (識別子)">CRID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cir.nii.ac.jp/crid/1050001201691367552">1050001201691367552</a>、 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/Hdl_(%E8%AD%98%E5%88%A5%E5%AD%90)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Hdl (識別子)">hdl</a>:<span><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hdl.handle.net/2433%252F64462">2433/64462</a></span>、 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a href="https://ja-m-wikipedia-org.translate.goog/wiki/ISSN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISSN">ISSN</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://search.worldcat.org/ja/search?fq%3Dx0:jrnl%26q%3Dn2:1880-2818">1880-2818</a>。</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Levenberg-Marquardt%E6%B3%95%E3%81%AE%E5%B1%80%E6%89%80%E5%8F%8E%E6%9D%9F%E6%80%A7%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6+%28%E6%9C%80%E9%81%A9%E5%8C%96%E3%81%AE%E6%95%B0%E7%90%86%E7%A7%91%E5%AD%A6%29&rft.jtitle=%E6%95%B0%E7%90%86%E8%A7%A3%E6%9E%90%E7%A0%94%E7%A9%B6%E6%89%80%E8%AC%9B%E7%A9%B6%E9%8C%B2&rft.aulast=%E5%B1%B1%E4%B8%8B%E4%BF%A1%E9%9B%84%2C+%E7%A6%8F%E5%B3%B6%E9%9B%85%E5%A4%AB&rft.au=%E5%B1%B1%E4%B8%8B%E4%BF%A1%E9%9B%84%2C+%E7%A6%8F%E5%B3%B6%E9%9B%85%E5%A4%AB&rft.date=2000-10&rft.volume=1174&rft.pages=161-168%E9%A0%81&rft.pub=%E4%BA%AC%E9%83%BD%E5%A4%A7%E5%AD%A6%E6%95%B0%E7%90%86%E8%A7%A3%E6%9E%90%E7%A0%94%E7%A9%B6%E6%89%80&rft_dat=crid/1050001201691367552&rft_id=info:hdl/2433%2F64462&rft.issn=1880-2818&rft_id=https%3A%2F%2Fhdl.handle.net%2F2433%2F64462&rfr_id=info:sid/ja.wikipedia.org:%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span style="display: none;"> </span></span></span></li> <li id="cite_note-16"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-16">^</a></b> <span class="reference-text">Michele Rienzner (2020). Find Outliers with Thompson Tau (<span class="url"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.mathworks.com/matlabcentral/fileexchange/27553-find-outliers-with-thompson-tau">www<wbr>.mathworks<wbr>.com<wbr>/matlabcentral<wbr>/fileexchange<wbr>/27553-find-outliers-with-thompson-tau</a></span>), <a href="https://ja-m-wikipedia-org.translate.goog/wiki/MATLAB?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="MATLAB">MATLAB</a> Central File Exchange. Retrieved May 17, 2020.</span></li> <li id="cite_note-17"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-17">^</a></b> <span class="reference-text">Van Huffel, S., & Vandewalle, J. (1991). The total least squares problem: computational aspects and analysis (Vol. 9). SIAM.</span></li> <li id="cite_note-18"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-18">^</a></b> <span class="reference-text"><cite style="font-style:normal" class="citation journal">Golub, Gene H; Van Loan, Charles F (1980). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1137/0717073">“An analysis of the total least squares problem”</a>. <i>SIAM journal on numerical analysis</i> (SIAM) <b>17</b> (6): 883-893. <a href="https://ja-m-wikipedia-org.translate.goog/wiki/Doi_(%E8%AD%98%E5%88%A5%E5%AD%90)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (識別子)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1137%252F0717073">10.1137/0717073</a><span style="display:none;">. <a rel="nofollow" class="external free" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1137/0717073">https://doi.org/10.1137/0717073</a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=An+analysis+of+the+total+least+squares+problem&rft.jtitle=SIAM+journal+on+numerical+analysis&rft.aulast=Golub%2C+Gene+H%3B+Van+Loan%2C+Charles+F&rft.au=Golub%2C+Gene+H%3B+Van+Loan%2C+Charles+F&rft.date=1980&rft.volume=17&rft.issue=6&rft.pages=883-893&rft.pub=SIAM&rft_id=info:doi/10.1137%2F0717073&rft_id=https%3A%2F%2Fdoi.org%2F10.1137%2F0717073&rfr_id=info:sid/ja.wikipedia.org:%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95"><span style="display: none;"> </span></span></span></li> <li id="cite_note-19"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-19">^</a></b> <span class="reference-text">Drygas, H. (2012). The coordinate-free approach to Gauss-Markov estimation (Vol. 40). Springer Science & Business Media.</span></li> <li id="cite_note-20"><b><a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%97%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-20">^</a></b> <span class="reference-text">Motulsky, H., & Christopoulos, A. (2004). Fitting models to biological data using linear and nonlinear regression: a practical guide to curve fitting. Oxford University Press.</span></li> </ol> </div><!-- NewPP limit report Parsed by mw‐api‐int.codfw.main‐5b65fffc7d‐qcnlj Cached time: 20250214105442 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.589 seconds Real time usage: 0.845 seconds Preprocessor visited node count: 10613/1000000 Post‐expand include size: 131974/2097152 bytes Template argument size: 16845/2097152 bytes Highest expansion depth: 24/100 Expensive parser function count: 36/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 46939/5000000 bytes Lua time usage: 0.185/10.000 seconds Lua memory usage: 2667332/52428800 bytes Number of Wikibase entities loaded: 1/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 569.472 1 -total 34.76% 197.927 1 Template:回帰分析 34.01% 193.690 1 Template:Sidebar 16.55% 94.242 2 Template:Citation/core-ja-jp 14.60% 83.120 1 Template:Cite 14.12% 80.420 10 Template:Citation/identifier 13.74% 78.261 35 Template:仮リンク 13.51% 76.934 5 Template:Cite_journal 9.83% 55.964 2 Template:ISBN2 9.46% 53.883 7 Template:Citation/core --> <!-- Saved in parser cache with key jawiki:pcache:105617:|#|:idhash:canonical and timestamp 20250214105442 and revision id 101574986. 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class="interlanguage-link-target"><span>Български</span></a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/M%25C3%25A8tode_dels_m%25C3%25ADnims_quadrats" title="カタロニア語: Mètode dels mínims quadrats" lang="ca" hreflang="ca" data-title="Mètode dels mínims quadrats" data-language-autonym="Català" data-language-local-name="カタロニア語" class="interlanguage-link-target"><span>Català</span></a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Metoda_nejmen%25C5%25A1%25C3%25ADch_%25C4%258Dtverc%25C5%25AF" title="チェコ語: Metoda nejmenších čtverců" lang="cs" hreflang="cs" data-title="Metoda nejmenších čtverců" data-language-autonym="Čeština" data-language-local-name="チェコ語" class="interlanguage-link-target"><span>Čeština</span></a></li> <li class="interlanguage-link interwiki-da mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://da.wikipedia.org/wiki/Mindste_kvadraters_metode" title="デンマーク語: Mindste kvadraters metode" lang="da" hreflang="da" data-title="Mindste kvadraters metode" data-language-autonym="Dansk" data-language-local-name="デンマーク語" class="interlanguage-link-target"><span>Dansk</span></a></li> <li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="良質な記事"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Methode_der_kleinsten_Quadrate" title="ドイツ語: Methode der kleinsten Quadrate" lang="de" hreflang="de" data-title="Methode der kleinsten Quadrate" data-language-autonym="Deutsch" data-language-local-name="ドイツ語" class="interlanguage-link-target"><span>Deutsch</span></a></li> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Least_squares" title="英語: Least squares" lang="en" hreflang="en" data-title="Least squares" data-language-autonym="English" data-language-local-name="英語" class="interlanguage-link-target"><span>English</span></a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/M%25C3%25ADnimos_cuadrados" title="スペイン語: Mínimos cuadrados" lang="es" hreflang="es" data-title="Mínimos cuadrados" data-language-autonym="Español" data-language-local-name="スペイン語" class="interlanguage-link-target"><span>Español</span></a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/V%25C3%25A4himruutude_meetod" title="エストニア語: Vähimruutude meetod" lang="et" hreflang="et" data-title="Vähimruutude meetod" data-language-autonym="Eesti" data-language-local-name="エストニア語" class="interlanguage-link-target"><span>Eesti</span></a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Karratu_txikienen_erregresio" title="バスク語: Karratu txikienen erregresio" lang="eu" hreflang="eu" data-title="Karratu txikienen erregresio" data-language-autonym="Euskara" data-language-local-name="バスク語" class="interlanguage-link-target"><span>Euskara</span></a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25DA%25A9%25D9%2585%25D8%25AA%25D8%25B1%25DB%258C%25D9%2586_%25D9%2585%25D8%25B1%25D8%25A8%25D8%25B9%25D8%25A7%25D8%25AA" title="ペルシア語: کمترین مربعات" lang="fa" hreflang="fa" data-title="کمترین مربعات" data-language-autonym="فارسی" data-language-local-name="ペルシア語" class="interlanguage-link-target"><span>فارسی</span></a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Pienimm%25C3%25A4n_neli%25C3%25B6summan_menetelm%25C3%25A4" title="フィンランド語: Pienimmän neliösumman menetelmä" lang="fi" hreflang="fi" data-title="Pienimmän neliösumman menetelmä" data-language-autonym="Suomi" data-language-local-name="フィンランド語" class="interlanguage-link-target"><span>Suomi</span></a></li> <li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/M%25C3%25A9thode_des_moindres_carr%25C3%25A9s" title="フランス語: Méthode des moindres carrés" lang="fr" hreflang="fr" data-title="Méthode des moindres carrés" data-language-autonym="Français" data-language-local-name="フランス語" class="interlanguage-link-target"><span>Français</span></a></li> <li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gl.wikipedia.org/wiki/M%25C3%25ADnimos_cadrados" title="ガリシア語: Mínimos cadrados" lang="gl" hreflang="gl" data-title="Mínimos cadrados" data-language-autonym="Galego" data-language-local-name="ガリシア語" class="interlanguage-link-target"><span>Galego</span></a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%25A9%25D7%2599%25D7%2598%25D7%25AA_%25D7%2594%25D7%25A8%25D7%2599%25D7%2591%25D7%2595%25D7%25A2%25D7%2599%25D7%259D_%25D7%2594%25D7%25A4%25D7%2597%25D7%2595%25D7%25AA%25D7%2599%25D7%259D" title="ヘブライ語: שיטת הריבועים הפחותים" lang="he" hreflang="he" data-title="שיטת הריבועים הפחותים" data-language-autonym="עברית" data-language-local-name="ヘブライ語" class="interlanguage-link-target"><span>עברית</span></a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%25A8%25E0%25A5%258D%25E0%25A4%25AF%25E0%25A5%2582%25E0%25A4%25A8%25E0%25A4%25A4%25E0%25A4%25AE_%25E0%25A4%25B5%25E0%25A4%25B0%25E0%25A5%258D%25E0%25A4%2597_%25E0%25A4%25B5%25E0%25A4%25BF%25E0%25A4%25A7%25E0%25A4%25BF" title="ヒンディー語: न्यूनतम वर्ग विधि" lang="hi" hreflang="hi" data-title="न्यूनतम वर्ग विधि" data-language-autonym="हिन्दी" data-language-local-name="ヒンディー語" class="interlanguage-link-target"><span>हिन्दी</span></a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/Legkisebb_n%25C3%25A9gyzetek_m%25C3%25B3dszere" title="ハンガリー語: Legkisebb négyzetek módszere" lang="hu" hreflang="hu" data-title="Legkisebb négyzetek módszere" data-language-autonym="Magyar" data-language-local-name="ハンガリー語" class="interlanguage-link-target"><span>Magyar</span></a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Metodo_dei_minimi_quadrati" title="イタリア語: Metodo dei minimi quadrati" lang="it" hreflang="it" data-title="Metodo dei minimi quadrati" data-language-autonym="Italiano" data-language-local-name="イタリア語" class="interlanguage-link-target"><span>Italiano</span></a></li> <li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kk.wikipedia.org/wiki/%25D0%2595%25D2%25A3_%25D0%25BA%25D1%2596%25D1%2588%25D1%2596_%25D0%25BA%25D0%25B2%25D0%25B0%25D0%25B4%25D1%2580%25D0%25B0%25D1%2582%25D1%2582%25D0%25B0%25D1%2580_%25D3%2599%25D0%25B4%25D1%2596%25D1%2581%25D1%2596" title="カザフ語: Ең кіші квадраттар әдісі" lang="kk" hreflang="kk" data-title="Ең кіші квадраттар әдісі" data-language-autonym="Қазақша" data-language-local-name="カザフ語" class="interlanguage-link-target"><span>Қазақша</span></a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25EC%25B5%259C%25EC%2586%258C%25EC%25A0%259C%25EA%25B3%25B1%25EB%25B2%2595" title="韓国語: 최소제곱법" lang="ko" hreflang="ko" data-title="최소제곱법" data-language-autonym="한국어" data-language-local-name="韓国語" class="interlanguage-link-target"><span>한국어</span></a></li> <li class="interlanguage-link interwiki-la mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://la.wikipedia.org/wiki/Methodus_quadratorum_minimorum" title="ラテン語: Methodus quadratorum minimorum" lang="la" hreflang="la" data-title="Methodus quadratorum minimorum" data-language-autonym="Latina" data-language-local-name="ラテン語" class="interlanguage-link-target"><span>Latina</span></a></li> <li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mk.wikipedia.org/wiki/%25D0%259C%25D0%25B5%25D1%2582%25D0%25BE%25D0%25B4_%25D0%25BD%25D0%25B0_%25D0%25BD%25D0%25B0%25D1%2598%25D0%25BC%25D0%25B0%25D0%25BB%25D0%25B8_%25D0%25BA%25D0%25B2%25D0%25B0%25D0%25B4%25D1%2580%25D0%25B0%25D1%2582%25D0%25B8" title="マケドニア語: Метод на најмали квадрати" lang="mk" hreflang="mk" data-title="Метод на најмали квадрати" data-language-autonym="Македонски" data-language-local-name="マケドニア語" class="interlanguage-link-target"><span>Македонски</span></a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Kleinste-kwadratenmethode" title="オランダ語: Kleinste-kwadratenmethode" lang="nl" hreflang="nl" data-title="Kleinste-kwadratenmethode" data-language-autonym="Nederlands" data-language-local-name="オランダ語" class="interlanguage-link-target"><span>Nederlands</span></a></li> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Minste_kvadrats_metode" title="ノルウェー語(ニーノシュク): Minste kvadrats metode" lang="nn" hreflang="nn" data-title="Minste kvadrats metode" data-language-autonym="Norsk nynorsk" data-language-local-name="ノルウェー語(ニーノシュク)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li> <li class="interlanguage-link interwiki-no mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://no.wikipedia.org/wiki/Minste_kvadraters_metode" title="ノルウェー語(ブークモール): Minste kvadraters metode" lang="nb" hreflang="nb" data-title="Minste kvadraters metode" data-language-autonym="Norsk bokmål" data-language-local-name="ノルウェー語(ブークモール)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Metoda_najmniejszych_kwadrat%25C3%25B3w" title="ポーランド語: Metoda najmniejszych kwadratów" lang="pl" hreflang="pl" data-title="Metoda najmniejszych kwadratów" data-language-autonym="Polski" data-language-local-name="ポーランド語" class="interlanguage-link-target"><span>Polski</span></a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/M%25C3%25A9todo_dos_m%25C3%25ADnimos_quadrados" title="ポルトガル語: Método dos mínimos quadrados" lang="pt" hreflang="pt" data-title="Método dos mínimos quadrados" data-language-autonym="Português" data-language-local-name="ポルトガル語" class="interlanguage-link-target"><span>Português</span></a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Metoda_celor_mai_mici_p%25C4%2583trate" title="ルーマニア語: Metoda celor mai mici pătrate" lang="ro" hreflang="ro" data-title="Metoda celor mai mici pătrate" data-language-autonym="Română" data-language-local-name="ルーマニア語" class="interlanguage-link-target"><span>Română</span></a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%259C%25D0%25B5%25D1%2582%25D0%25BE%25D0%25B4_%25D0%25BD%25D0%25B0%25D0%25B8%25D0%25BC%25D0%25B5%25D0%25BD%25D1%258C%25D1%2588%25D0%25B8%25D1%2585_%25D0%25BA%25D0%25B2%25D0%25B0%25D0%25B4%25D1%2580%25D0%25B0%25D1%2582%25D0%25BE%25D0%25B2" title="ロシア語: Метод наименьших квадратов" lang="ru" hreflang="ru" data-title="Метод наименьших квадратов" data-language-autonym="Русский" data-language-local-name="ロシア語" class="interlanguage-link-target"><span>Русский</span></a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Least_squares" title="シンプル英語: Least squares" lang="en-simple" hreflang="en-simple" data-title="Least squares" data-language-autonym="Simple English" data-language-local-name="シンプル英語" class="interlanguage-link-target"><span>Simple English</span></a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Najmanj%25C5%25A1i_kvadrati" title="スロベニア語: Najmanjši kvadrati" lang="sl" hreflang="sl" data-title="Najmanjši kvadrati" data-language-autonym="Slovenščina" data-language-local-name="スロベニア語" class="interlanguage-link-target"><span>Slovenščina</span></a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/Najmanji_kvadrati" title="セルビア語: Najmanji kvadrati" lang="sr" hreflang="sr" data-title="Najmanji kvadrati" data-language-autonym="Српски / srpski" data-language-local-name="セルビア語" class="interlanguage-link-target"><span>Српски / srpski</span></a></li> <li class="interlanguage-link interwiki-su mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://su.wikipedia.org/wiki/Kuadrat_leutik" title="スンダ語: Kuadrat leutik" lang="su" hreflang="su" data-title="Kuadrat leutik" data-language-autonym="Sunda" data-language-local-name="スンダ語" class="interlanguage-link-target"><span>Sunda</span></a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Minstakvadratmetoden" title="スウェーデン語: Minstakvadratmetoden" lang="sv" hreflang="sv" data-title="Minstakvadratmetoden" data-language-autonym="Svenska" data-language-local-name="スウェーデン語" class="interlanguage-link-target"><span>Svenska</span></a></li> <li class="interlanguage-link interwiki-th mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://th.wikipedia.org/wiki/%25E0%25B8%25A7%25E0%25B8%25B4%25E0%25B8%2598%25E0%25B8%25B5%25E0%25B8%2581%25E0%25B8%25B3%25E0%25B8%25A5%25E0%25B8%25B1%25E0%25B8%2587%25E0%25B8%25AA%25E0%25B8%25AD%25E0%25B8%2587%25E0%25B8%2599%25E0%25B9%2589%25E0%25B8%25AD%25E0%25B8%25A2%25E0%25B8%25AA%25E0%25B8%25B8%25E0%25B8%2594" title="タイ語: วิธีกำลังสองน้อยสุด" lang="th" hreflang="th" data-title="วิธีกำลังสองน้อยสุด" data-language-autonym="ไทย" data-language-local-name="タイ語" class="interlanguage-link-target"><span>ไทย</span></a></li> <li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/En_k%25C3%25BC%25C3%25A7%25C3%25BCk_kareler_y%25C3%25B6ntemi" title="トルコ語: En küçük kareler yöntemi" lang="tr" hreflang="tr" data-title="En küçük kareler yöntemi" data-language-autonym="Türkçe" data-language-local-name="トルコ語" class="interlanguage-link-target"><span>Türkçe</span></a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259C%25D0%25B5%25D1%2582%25D0%25BE%25D0%25B4_%25D0%25BD%25D0%25B0%25D0%25B9%25D0%25BC%25D0%25B5%25D0%25BD%25D1%2588%25D0%25B8%25D1%2585_%25D0%25BA%25D0%25B2%25D0%25B0%25D0%25B4%25D1%2580%25D0%25B0%25D1%2582%25D1%2596%25D0%25B2" title="ウクライナ語: Метод найменших квадратів" lang="uk" hreflang="uk" data-title="Метод найменших квадратів" data-language-autonym="Українська" data-language-local-name="ウクライナ語" class="interlanguage-link-target"><span>Українська</span></a></li> <li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ur.wikipedia.org/wiki/%25D9%2584%25DA%25A9%25DB%258C%25D8%25B1%25DB%258C_%25D8%25A7%25D9%2582%25D9%2584_%25D9%2585%25D8%25B1%25D8%25A8%25D8%25B9%25D8%25A7%25D8%25AA" title="ウルドゥー語: لکیری اقل مربعات" lang="ur" hreflang="ur" data-title="لکیری اقل مربعات" data-language-autonym="اردو" data-language-local-name="ウルドゥー語" class="interlanguage-link-target"><span>اردو</span></a></li> <li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uz.wikipedia.org/wiki/Eng_kichik_kvadratlar_usuli" title="ウズベク語: Eng kichik kvadratlar usuli" lang="uz" hreflang="uz" data-title="Eng kichik kvadratlar usuli" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ウズベク語" 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