ガウス・ニュートン法

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mw-parser-output" lang="ja" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> ガウス・ニュートン法（ガウス・ニュートンほう、<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>: Gauss–Newton method）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>を解く手法の一つである。これは関数の最大・最小値を見出す<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%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="ニュートン法">ニュートン法</a>の修正とみなすことができる。ニュートン法とは違い、ガウス・ニュートン法は二乗和の最小化にしか用いることができないが、計算するのが困難な2階微分が不要という長所がある。 非線形最小二乗法は<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E9%9D%9E%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="非線形回帰">非線形回帰</a>などで、観測データを良く表すようにモデルのパラメータを調整するために必要となる。 この手法の名称は<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>と<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%82%A2%E3%82%A4%E3%82%B6%E3%83%83%E3%82%AF%E3%83%BB%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="アイザック・ニュートン">アイザック・ニュートン</a>にちなむ。 <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><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><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#%E6%A6%82%E8%A6%81">1 概要</a></li> <li class="toclevel-1 tocsection-2"><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#%E6%B3%A8%E9%87%88">2 注釈</a></li> <li class="toclevel-1 tocsection-3"><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#%E4%BE%8B">3 例</a></li> <li class="toclevel-1 tocsection-4"><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#%E5%8F%8E%E6%9D%9F%E6%80%A7">4 収束性</a></li> <li class="toclevel-1 tocsection-5"><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#%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E6%B3%95%E3%81%8B%E3%82%89%E3%81%AE%E5%B0%8E%E5%87%BA">5 ニュートン法からの導出</a></li> <li class="toclevel-1 tocsection-6"><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#%E6%94%B9%E5%96%84%E3%83%90%E3%83%BC%E3%82%B8%E3%83%A7%E3%83%B3">6 改善バージョン</a></li> <li class="toclevel-1 tocsection-7"><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#%E9%96%A2%E9%80%A3%E3%81%99%E3%82%8B%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0">7 関連するアルゴリズム</a></li> <li class="toclevel-1 tocsection-8"><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#%E8%84%9A%E6%B3%A8">8 脚注</a></li> <li class="toclevel-1 tocsection-9"><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#%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE">9 参考文献</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="概要">概要</h2> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%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&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 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 "> 編集 </a> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <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">データフィッティング</a>において、与えられたモデル関数 y = f (x , β) がm 個のデータ点 {(xi , yi ); i = 1, ... , m } に最もよくフィットするようなn （≤ m ）個<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#cite_note-1">[1]</a>のパラメータβ = (β1 , ... , βn )を見つけることが目的である。 このとき、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%AE%8B%E5%B7%AE?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-disambig" title="残差">残差</a>を <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}({\boldsymbol {\beta }})=y_{i}-f(x_{i},{\boldsymbol {\beta }})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> = </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> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{i}({\boldsymbol {\beta }})=y_{i}-f(x_{i},{\boldsymbol {\beta }})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7435bbefdb472cb4ab07ebacc9a3f8b17f6470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.855ex; height:2.843ex;" alt="{\displaystyle r_{i}({\boldsymbol {\beta }})=y_{i}-f(x_{i},{\boldsymbol {\beta }})}"> </noscript><span class="lazy-image-placeholder" style="width: 20.855ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7435bbefdb472cb4ab07ebacc9a3f8b17f6470" data-alt="{\displaystyle r_{i}({\boldsymbol {\beta }})=y_{i}-f(x_{i},{\boldsymbol {\beta }})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> とする。 このとき、ガウス・ニュートン法は残差の平方和 <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S({\boldsymbol {\beta }})=\sum _{i=1}^{m}(r_{i}({\boldsymbol {\beta }}))^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mo stretchy="false"> ) </mo> <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> m </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <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 S({\boldsymbol {\beta }})=\sum _{i=1}^{m}(r_{i}({\boldsymbol {\beta }}))^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c5d60a0c525eb34e7dda6ced7f2f38d6040c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.351ex; height:6.843ex;" alt="{\displaystyle S({\boldsymbol {\beta }})=\sum _{i=1}^{m}(r_{i}({\boldsymbol {\beta }}))^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 19.351ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c5d60a0c525eb34e7dda6ced7f2f38d6040c1b" data-alt="{\displaystyle S({\boldsymbol {\beta }})=\sum _{i=1}^{m}(r_{i}({\boldsymbol {\beta }}))^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> の最小値を<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%8F%8D%E5%BE%A9%E8%A8%88%E7%AE%97?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="反復計算">反復計算</a>で求める<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#cite_note-ab-2">[2]</a>。初期推測値β(0) から初めて、この方法は以下の計算を繰り返す。 <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-({J_{r}}^{\mathrm {T} }{J_{r}})^{-1}{J_{r}}^{\mathrm {T} }{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)}),\quad (s=0,1,2,\dots ).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> − </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> r </mi> </mrow> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> , </mo> <mspace width="1em"></mspace> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mo> … </mo> <mo stretchy="false"> ) </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-({J_{r}}^{\mathrm {T} }{J_{r}})^{-1}{J_{r}}^{\mathrm {T} }{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)}),\quad (s=0,1,2,\dots ).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87ffa63b96404d3770cb34741ee4da2c5463120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.614ex; height:3.343ex;" alt="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-({J_{r}}^{\mathrm {T} }{J_{r}})^{-1}{J_{r}}^{\mathrm {T} }{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)}),\quad (s=0,1,2,\dots ).}"> </noscript><span class="lazy-image-placeholder" style="width: 55.614ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87ffa63b96404d3770cb34741ee4da2c5463120" data-alt="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-({J_{r}}^{\mathrm {T} }{J_{r}})^{-1}{J_{r}}^{\mathrm {T} }{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)}),\quad (s=0,1,2,\dots ).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> ここで <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{r}={\frac {\partial r_{i}({\boldsymbol {\beta }}^{(s)})}{\partial \beta _{j}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J_{r}={\frac {\partial r_{i}({\boldsymbol {\beta }}^{(s)})}{\partial \beta _{j}}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb388f20d5783df8469877b78f5d786ce9ce6c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.991ex; height:7.009ex;" alt="{\displaystyle J_{r}={\frac {\partial r_{i}({\boldsymbol {\beta }}^{(s)})}{\partial \beta _{j}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.991ex;height: 7.009ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb388f20d5783df8469877b78f5d786ce9ce6c9" data-alt="{\displaystyle J_{r}={\frac {\partial r_{i}({\boldsymbol {\beta }}^{(s)})}{\partial \beta _{j}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> はβ(s ) におけるr の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%83%A4%E3%82%B3%E3%83%93%E3%82%A2%E3%83%B3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ヤコビアン">ヤコビアン</a>、JrT は行列Jr の<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</a>を表す。 m = n ならば、この反復計算は <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-J_{r}^{-1}{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> − </mo> <msubsup> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> r </mi> </mrow> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-J_{r}^{-1}{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f22eab5305565c708b611413ff08f87a0c22237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.387ex; height:3.343ex;" alt="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-J_{r}^{-1}{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)})}"> </noscript><span class="lazy-image-placeholder" style="width: 26.387ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f22eab5305565c708b611413ff08f87a0c22237" data-alt="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-J_{r}^{-1}{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> のように簡略化される。これは1次元ニュートン法の直接的な一般化である。 ガウス・ニュートン法は関数f の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%83%A4%E3%82%B3%E3%83%93%E3%82%A2%E3%83%B3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ヤコビアン">ヤコビアン</a>Jf を用いて次のように表すこともできる： <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}+({J_{f}}^{\mathrm {T} }{J_{f}})^{-1}{J_{f}}^{\mathrm {T} }{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)}).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> r </mi> </mrow> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}+({J_{f}}^{\mathrm {T} }{J_{f}})^{-1}{J_{f}}^{\mathrm {T} }{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)}).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ee6b276d1df34718a558ed02bb9fda347ee6f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.435ex; height:3.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}+({J_{f}}^{\mathrm {T} }{J_{f}})^{-1}{J_{f}}^{\mathrm {T} }{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)}).}"> </noscript><span class="lazy-image-placeholder" style="width: 37.435ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ee6b276d1df34718a558ed02bb9fda347ee6f0" data-alt="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}+({J_{f}}^{\mathrm {T} }{J_{f}})^{-1}{J_{f}}^{\mathrm {T} }{\boldsymbol {r}}({\boldsymbol {\beta }}^{(s)}).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="注釈">注釈</h2> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%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&action=edit&section=2&_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 "> 編集 </a> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> ガウス・ニュートン法は関数ri を並べたベクトルの<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%B7%9A%E5%BD%A2%E8%BF%91%E4%BC%BC?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="線形近似">線形近似</a>で与えられる。<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%83%86%E3%82%A4%E3%83%A9%E3%83%BC%E3%81%AE%E5%AE%9A%E7%90%86?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="テイラーの定理">テイラーの定理</a>を用いれば、各反復において次式が成り立つ： <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {r}}({\boldsymbol {\beta }})\approx {\boldsymbol {r}}({\boldsymbol {\beta }}^{s})+J_{r}({\boldsymbol {\beta }}^{s}){\boldsymbol {\Delta }}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> r </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> r </mi> </mrow> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {r}}({\boldsymbol {\beta }})\approx {\boldsymbol {r}}({\boldsymbol {\beta }}^{s})+J_{r}({\boldsymbol {\beta }}^{s}){\boldsymbol {\Delta }}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4518f9b0dbb5cf9399fbdef140bece418c818035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.572ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {r}}({\boldsymbol {\beta }})\approx {\boldsymbol {r}}({\boldsymbol {\beta }}^{s})+J_{r}({\boldsymbol {\beta }}^{s}){\boldsymbol {\Delta }}.}"> </noscript><span class="lazy-image-placeholder" style="width: 25.572ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4518f9b0dbb5cf9399fbdef140bece418c818035" data-alt="{\displaystyle {\boldsymbol {r}}({\boldsymbol {\beta }})\approx {\boldsymbol {r}}({\boldsymbol {\beta }}^{s})+J_{r}({\boldsymbol {\beta }}^{s}){\boldsymbol {\Delta }}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> ここで Δ = β - βs である。右辺の残差平方和を最小化するΔを見つけること、すなわち <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {min} \left[{\|{\boldsymbol {r}}({\boldsymbol {\beta }}^{s})+J_{r}({\boldsymbol {\beta }}^{s}){\boldsymbol {\Delta }}\|_{2}}^{2}\right]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> min </mi> <mo> ⁡ </mo> <mrow> <mo> [ </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> r </mi> </mrow> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> ] </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {min} \left[{\|{\boldsymbol {r}}({\boldsymbol {\beta }}^{s})+J_{r}({\boldsymbol {\beta }}^{s}){\boldsymbol {\Delta }}\|_{2}}^{2}\right]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0cb09d02c2fbe4efc774bfcadfb1b1799e7352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.757ex; height:4.843ex;" alt="{\displaystyle \operatorname {min} \left[{\|{\boldsymbol {r}}({\boldsymbol {\beta }}^{s})+J_{r}({\boldsymbol {\beta }}^{s}){\boldsymbol {\Delta }}\|_{2}}^{2}\right]}"> </noscript><span class="lazy-image-placeholder" style="width: 27.757ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0cb09d02c2fbe4efc774bfcadfb1b1799e7352" data-alt="{\displaystyle \operatorname {min} \left[{\|{\boldsymbol {r}}({\boldsymbol {\beta }}^{s})+J_{r}({\boldsymbol {\beta }}^{s}){\boldsymbol {\Delta }}\|_{2}}^{2}\right]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> は線形<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" title="最小二乗法">最小二乗法</a>の問題であるため、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E9%99%BD%E9%96%A2%E6%95%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="陽関数">陽的</a>に解くことができ、正規方程式を与える。ここで ||*||2 は 2-ノルム（<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%83%A6%E3%83%BC%E3%82%AF%E3%83%AA%E3%83%83%E3%83%89%E3%83%8E%E3%83%AB%E3%83%A0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ユークリッドノルム">ユークリッドノルム</a>）である。 正規方程式は未知の増分Δについてのm 本の線形同次方程式である。これは<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>を用いることで、またはより良い方法としてはJr の<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>を用いることで、1ステップで解ける。大きな系に対しては、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%85%B1%E5%BD%B9%E5%8B%BE%E9%85%8D%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="共役勾配法">共役勾配法</a>のような<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>が有効である。Jr の列ベクトルが<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%B7%9A%E5%BD%A2%E5%BE%93%E5%B1%9E?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="線形従属">線形従属</a>である場合、JrTJr が非正則になるため反復解法は失敗する。 </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="例">例</h2> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%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&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 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 "> 編集 </a> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <figure class="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:Gauss_Newton_illustration.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Gauss_Newton_illustration.png/280px-Gauss_Newton_illustration.png" decoding="async" width="280" height="226" class="mw-file-element" data-file-width="1532" data-file-height="1236"> </noscript><span class="lazy-image-placeholder" style="width: 280px;height: 226px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Gauss_Newton_illustration.png/280px-Gauss_Newton_illustration.png" data-width="280" data-height="226" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Gauss_Newton_illustration.png/420px-Gauss_Newton_illustration.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Gauss_Newton_illustration.png/560px-Gauss_Newton_illustration.png 2x" data-class="mw-file-element"> </a> <figcaption> この例で得られているデータ（赤点）と、 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\beta }}_{1}=0.362,{\hat {\beta }}_{2}=0.556}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> β </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 0.362 </mn> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> β </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mn> 0.556 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\hat {\beta }}_{1}=0.362,{\hat {\beta }}_{2}=0.556} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88216c483a5eebc23ff67f61c6a98a2c62aea147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.835ex; height:3.343ex;" alt="{\displaystyle {\hat {\beta }}_{1}=0.362,{\hat {\beta }}_{2}=0.556}"> </noscript><span class="lazy-image-placeholder" style="width: 22.835ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88216c483a5eebc23ff67f61c6a98a2c62aea147" data-alt="{\displaystyle {\hat {\beta }}_{1}=0.362,{\hat {\beta }}_{2}=0.556}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> から計算されたモデル曲線（青線） </figcaption> </figure> ここでは例として、ガウス・ニュートン法を使ってデータとモデルによる予測値の間の残差平方和を最小化し、データにモデルをフィットさせる。 生物実験において、<a href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E9%85%B5%E7%B4%A0%E5%AA%92%E4%BB%8B%E5%8F%8D%E5%BF%9C&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="「酵素媒介反応」 (存在しないページ)">酵素媒介反応</a>における<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%9F%BA%E8%B3%AA?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-disambig" title="基質">基質</a>濃度[S] と<a href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E5%8F%8D%E5%BF%9C%E7%8E%87&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="「反応率」 (存在しないページ)">反応率</a>v の関係として次表のようなデータが得られたとする（右図の赤点）。 <dl> <dd> <table class="wikitable" style="text-align: center;"> <tbody> <tr> <td>i</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> </tr> <tr> <td>[S]</td> <td>0.038</td> <td>0.194</td> <td>0.425</td> <td>0.626</td> <td>1.253</td> <td>2.500</td> <td>3.740</td> </tr> <tr> <td>v</td> <td>0.050</td> <td>0.127</td> <td>0.094</td> <td>0.2122</td> <td>0.2729</td> <td>0.2665</td> <td>0.3317</td> </tr> </tbody> </table> </dd> </dl> これらのデータに対し、次の形のモデル曲線（<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%83%9F%E3%82%AB%E3%82%A8%E3%83%AA%E3%82%B9%E3%83%BB%E3%83%A1%E3%83%B3%E3%83%86%E3%83%B3%E5%BC%8F?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ミカエリス・メンテン式">ミカエリス・メンテン式</a>）のパラメータVmax とKM を、最小二乗の意味で最もよくフィットするように決定したい<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#cite_note-3">[3]</a>： <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\frac {V_{\mathrm {max} }[\mathrm {S} ]}{K_{\mathrm {M} }+[\mathrm {S} ]}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> m </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> x </mi> </mrow> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> S </mi> </mrow> <mo stretchy="false"> ] </mo> </mrow> <mrow> <msub> <mi> K </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> M </mi> </mrow> </mrow> </msub> <mo> + </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> S </mi> </mrow> <mo stretchy="false"> ] </mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v={\frac {V_{\mathrm {max} }[\mathrm {S} ]}{K_{\mathrm {M} }+[\mathrm {S} ]}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5661add5762d0cb155367651e130d3bf5f3a9bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.201ex; height:6.509ex;" alt="{\displaystyle v={\frac {V_{\mathrm {max} }[\mathrm {S} ]}{K_{\mathrm {M} }+[\mathrm {S} ]}}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.201ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5661add5762d0cb155367651e130d3bf5f3a9bf0" data-alt="{\displaystyle v={\frac {V_{\mathrm {max} }[\mathrm {S} ]}{K_{\mathrm {M} }+[\mathrm {S} ]}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> xi とyi (i = 1, ... , 7) で [S] とv のデータを表す。また、β1 = Vmax 、β2 = KM とする。残差 <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}=y_{i}-{\frac {\beta _{1}x_{i}}{\beta _{2}+x_{i}}}\quad (i=1,\dots ,7)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </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> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="1em"></mspace> <mo stretchy="false"> ( </mo> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mn> 7 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{i}=y_{i}-{\frac {\beta _{1}x_{i}}{\beta _{2}+x_{i}}}\quad (i=1,\dots ,7)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89cf232fa310eca8835e7c35953d900f6b81282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.438ex; height:5.843ex;" alt="{\displaystyle r_{i}=y_{i}-{\frac {\beta _{1}x_{i}}{\beta _{2}+x_{i}}}\quad (i=1,\dots ,7)}"> </noscript><span class="lazy-image-placeholder" style="width: 33.438ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89cf232fa310eca8835e7c35953d900f6b81282a" data-alt="{\displaystyle r_{i}=y_{i}-{\frac {\beta _{1}x_{i}}{\beta _{2}+x_{i}}}\quad (i=1,\dots ,7)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> の平方和を最小化するβ1 とβ2 を見つけることが目的となる。 未知パラメータβに関する残差ベクトルr のヤコビアンJr は7×2の行列で、i 番目の行は次の要素を持つ： <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}{\dfrac {\partial r_{i}}{\partial \beta _{1}}},&{\dfrac {\partial r_{i}}{\partial \beta _{2}}}\end{pmatrix}}={\begin{pmatrix}-{\dfrac {x_{i}}{\beta _{2}+x_{i}}},&{\dfrac {\beta _{1}x_{i}}{\left(\beta _{2}+x_{i}\right)^{2}}}\end{pmatrix}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> <mo> , </mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> <mo> , </mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{pmatrix}{\dfrac {\partial r_{i}}{\partial \beta _{1}}},&{\dfrac {\partial r_{i}}{\partial \beta _{2}}}\end{pmatrix}}={\begin{pmatrix}-{\dfrac {x_{i}}{\beta _{2}+x_{i}}},&{\dfrac {\beta _{1}x_{i}}{\left(\beta _{2}+x_{i}\right)^{2}}}\end{pmatrix}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66fe1fc7ec3955555c0b508411294ad289a48c53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.101ex; height:6.509ex;" alt="{\displaystyle {\begin{pmatrix}{\dfrac {\partial r_{i}}{\partial \beta _{1}}},&{\dfrac {\partial r_{i}}{\partial \beta _{2}}}\end{pmatrix}}={\begin{pmatrix}-{\dfrac {x_{i}}{\beta _{2}+x_{i}}},&{\dfrac {\beta _{1}x_{i}}{\left(\beta _{2}+x_{i}\right)^{2}}}\end{pmatrix}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 48.101ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66fe1fc7ec3955555c0b508411294ad289a48c53" data-alt="{\displaystyle {\begin{pmatrix}{\dfrac {\partial r_{i}}{\partial \beta _{1}}},&{\dfrac {\partial r_{i}}{\partial \beta _{2}}}\end{pmatrix}}={\begin{pmatrix}-{\dfrac {x_{i}}{\beta _{2}+x_{i}}},&{\dfrac {\beta _{1}x_{i}}{\left(\beta _{2}+x_{i}\right)^{2}}}\end{pmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> 初期推定値としてβ1 = 0.9、β2 = 0.2から始め、ガウス・ニュートン法による5回の反復計算を行うと、最適値 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\beta }}_{1}=0.362,{\hat {\beta }}_{2}=0.556}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> β </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 0.362 </mn> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> β </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mn> 0.556 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\hat {\beta }}_{1}=0.362,{\hat {\beta }}_{2}=0.556} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88216c483a5eebc23ff67f61c6a98a2c62aea147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.835ex; height:3.343ex;" alt="{\displaystyle {\hat {\beta }}_{1}=0.362,{\hat {\beta }}_{2}=0.556}"> </noscript><span class="lazy-image-placeholder" style="width: 22.835ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88216c483a5eebc23ff67f61c6a98a2c62aea147" data-alt="{\displaystyle {\hat {\beta }}_{1}=0.362,{\hat {\beta }}_{2}=0.556}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  が得られる。残差平方和は5回の反復計算で初期の1.445から0.00784まで減少する。右図はこれらの最適パラメータを用いたモデルで決まる曲線と、実験データとの比較を示す。 </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="収束性">収束性</h2> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%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&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 収束性" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 編集 </a> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> 増分ΔがS の減少方向を向いていることは証明されている<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#cite_note-4">[4]</a>。もしこのアルゴリズムが収束すれば、その極限はS の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%A5%B5%E5%80%A4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="極値">停留点</a>である。しかし収束については、ニュートン法では保証されている局所収束さえも保証されていない。 ガウス・ニュートン法の<a href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%E5%8F%8E%E6%9D%9F%E3%81%AE%E9%80%9F%E3%81%95&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="「収束の速さ」 (存在しないページ)">収束の速さ</a>（<a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/rate_of_convergence" class="extiw" title="en:rate of convergence">英語版</a>）は2次である<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#cite_note-5">[5]</a>。もし初期推測値が最小値から遠いか、または行列JrT Jr が<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9D%A1%E4%BB%B6%E6%95%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="条件数">悪条件</a>であれば収束は遅いか、全くしなくなる。例えば、m = 2本の方程式とn = 1個の変数のある次の問題を考える： <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}r_{1}(\beta )&=\beta +1\\r_{2}(\beta )&=\lambda \beta ^{2}+\beta -1.\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> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> β </mi> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mi></mi> <mo> = </mo> <mi> β </mi> <mo> + </mo> <mn> 1 </mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> β </mi> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mi></mi> <mo> = </mo> <mi> λ </mi> <msup> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> β </mi> <mo> − </mo> <mn> 1. </mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}r_{1}(\beta )&=\beta +1\\r_{2}(\beta )&=\lambda \beta ^{2}+\beta -1.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fbe5781945b7a51e65f651380ad7cdee75e5a86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.663ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}r_{1}(\beta )&=\beta +1\\r_{2}(\beta )&=\lambda \beta ^{2}+\beta -1.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 21.663ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fbe5781945b7a51e65f651380ad7cdee75e5a86" data-alt="{\displaystyle {\begin{aligned}r_{1}(\beta )&=\beta +1\\r_{2}(\beta )&=\lambda \beta ^{2}+\beta -1.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> この問題の最適値はβ = 0 である。もしλ = 0 なら実質的に線形問題であり、最適値は一回の計算で見つかる。もし|λ| < 1 なら、この手法は線形に収束し残差は係数|λ|で反復ごとに漸近的に減少する。しかし|λ| > 1 なら、この方法はもはや局所的にも収束しない<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#cite_note-6">[6]</a>。 </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="ニュートン法からの導出">ニュートン法からの導出</h2> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%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&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: ニュートン法からの導出" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 編集 </a> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> 後に示すように、ガウス・ニュートン法は近似関数の最適化に用いられるニュートン法から与えられる。その結果、ガウス・ニュートン法の収束の速さはほとんど2次である。 パラメータβを持つ関数S の最小化をするとき、ニュートン法による<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%BC%B8%E5%8C%96%E5%BC%8F?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="漸化式">漸化式</a>は <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-H^{-1}{\boldsymbol {g}}\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> − </mo> <msup> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> g </mi> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-H^{-1}{\boldsymbol {g}}\,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abb4456b76482ec02a79c94ec349b067132ac63e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.763ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-H^{-1}{\boldsymbol {g}}\,}"> </noscript><span class="lazy-image-placeholder" style="width: 21.763ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abb4456b76482ec02a79c94ec349b067132ac63e" data-alt="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}-H^{-1}{\boldsymbol {g}}\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> である。ここでg はS の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%8B%BE%E9%85%8D%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="勾配ベクトル">勾配ベクトル</a>、H はS の<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%83%98%E3%83%83%E3%82%B7%E3%82%A2%E3%83%B3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ヘッシアン">ヘッシアン</a>である。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\sum _{i=1}^{m}r_{i}^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </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> m </mi> </mrow> </munderover> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S=\sum _{i=1}^{m}r_{i}^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1664fb2547dc103fd93a01c4a8b3da9b61a52e5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.443ex; height:6.843ex;" alt="{\displaystyle S=\sum _{i=1}^{m}r_{i}^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.443ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1664fb2547dc103fd93a01c4a8b3da9b61a52e5d" data-alt="{\displaystyle S=\sum _{i=1}^{m}r_{i}^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> であるから、勾配g は次で与えられる： <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {g}}=2{J_{r}}^{\mathrm {T} }{\boldsymbol {r}},\quad {\text{or,}}\quad g_{j}=2\sum _{i=1}^{m}r_{i}{\frac {\partial r_{i}}{\partial \beta _{j}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> g </mi> </mrow> <mo> = </mo> <mn> 2 </mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> r </mi> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> or, </mtext> </mrow> <mspace width="1em"></mspace> <msub> <mi> g </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {g}}=2{J_{r}}^{\mathrm {T} }{\boldsymbol {r}},\quad {\text{or,}}\quad g_{j}=2\sum _{i=1}^{m}r_{i}{\frac {\partial r_{i}}{\partial \beta _{j}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302c55fb56757d6dbfbf9813f14826f6f710ac98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.124ex; height:6.843ex;" alt="{\displaystyle {\boldsymbol {g}}=2{J_{r}}^{\mathrm {T} }{\boldsymbol {r}},\quad {\text{or,}}\quad g_{j}=2\sum _{i=1}^{m}r_{i}{\frac {\partial r_{i}}{\partial \beta _{j}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 36.124ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302c55fb56757d6dbfbf9813f14826f6f710ac98" data-alt="{\displaystyle {\boldsymbol {g}}=2{J_{r}}^{\mathrm {T} }{\boldsymbol {r}},\quad {\text{or,}}\quad g_{j}=2\sum _{i=1}^{m}r_{i}{\frac {\partial r_{i}}{\partial \beta _{j}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> ヘッシアンH は勾配g をβ で微分することで計算される： <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{jk}=2\sum _{i=1}^{m}\left({\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}+r_{i}{\frac {\partial ^{2}r_{i}}{\partial \beta _{j}\partial \beta _{k}}}\right).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mi> k </mi> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo> + </mo> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle H_{jk}=2\sum _{i=1}^{m}\left({\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}+r_{i}{\frac {\partial ^{2}r_{i}}{\partial \beta _{j}\partial \beta _{k}}}\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/087b42a7146541cefd8c502b7474efe12a1c74f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.272ex; height:6.843ex;" alt="{\displaystyle H_{jk}=2\sum _{i=1}^{m}\left({\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}+r_{i}{\frac {\partial ^{2}r_{i}}{\partial \beta _{j}\partial \beta _{k}}}\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 38.272ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/087b42a7146541cefd8c502b7474efe12a1c74f6" data-alt="{\displaystyle H_{jk}=2\sum _{i=1}^{m}\left({\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}+r_{i}{\frac {\partial ^{2}r_{i}}{\partial \beta _{j}\partial \beta _{k}}}\right).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> 2階微分項（右辺第2項）を無視することでガウス・ニュートン法を得る。つまり、ヘッシアンは <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\approx 2{J_{r}}^{\mathrm {T} }J_{r},\quad {\text{or,}}\quad H_{jk}\approx 2\sum _{i=1}^{m}{\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}=2\sum _{i=1}^{m}J_{ij}J_{ik}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> H </mi> <mo> ≈ </mo> <mn> 2 </mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> T </mi> </mrow> </mrow> </msup> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> <mo> , </mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> or, </mtext> </mrow> <mspace width="1em"></mspace> <msub> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mi> k </mi> </mrow> </msub> <mo> ≈ </mo> <mn> 2 </mn> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo> = </mo> <mn> 2 </mn> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle H\approx 2{J_{r}}^{\mathrm {T} }J_{r},\quad {\text{or,}}\quad H_{jk}\approx 2\sum _{i=1}^{m}{\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}=2\sum _{i=1}^{m}J_{ij}J_{ik}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca720f4e6552647a5da5f6f37cd9f5239346367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.801ex; height:6.843ex;" alt="{\displaystyle H\approx 2{J_{r}}^{\mathrm {T} }J_{r},\quad {\text{or,}}\quad H_{jk}\approx 2\sum _{i=1}^{m}{\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}=2\sum _{i=1}^{m}J_{ij}J_{ik}}"> </noscript><span class="lazy-image-placeholder" style="width: 55.801ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca720f4e6552647a5da5f6f37cd9f5239346367" data-alt="{\displaystyle H\approx 2{J_{r}}^{\mathrm {T} }J_{r},\quad {\text{or,}}\quad H_{jk}\approx 2\sum _{i=1}^{m}{\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}=2\sum _{i=1}^{m}J_{ij}J_{ik}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> と近似される。ここで <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{ij}={\frac {\partial r_{i}}{\partial \beta _{j}}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J_{ij}={\frac {\partial r_{i}}{\partial \beta _{j}}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42f1bc5dda788a276fc23986b9f6d2f2caf2f3ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.245ex; height:6.176ex;" alt="{\displaystyle J_{ij}={\frac {\partial r_{i}}{\partial \beta _{j}}}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.245ex;height: 6.176ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42f1bc5dda788a276fc23986b9f6d2f2caf2f3ba" data-alt="{\displaystyle J_{ij}={\frac {\partial r_{i}}{\partial \beta _{j}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> はヤコビアンJr の成分である。 これらの表現を上述の漸化式に代入して、次式を得る： <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}+{\boldsymbol {\Delta }};\quad {\boldsymbol {\Delta }}=-({J_{r}}^{\mathrm {T} }J_{r})^{-1}{J_{r}}^{\mathrm {T} }{\boldsymbol {r}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> s </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <mo> ; </mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <mo> = </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> T </mi> </mrow> </mrow> </msup> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> r </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> r </mi> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}+{\boldsymbol {\Delta }};\quad {\boldsymbol {\Delta }}=-({J_{r}}^{\mathrm {T} }J_{r})^{-1}{J_{r}}^{\mathrm {T} }{\boldsymbol {r}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ed7cd693d2daab25d4c8ffe5cc2c569d071138" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.037ex; height:3.343ex;" alt="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}+{\boldsymbol {\Delta }};\quad {\boldsymbol {\Delta }}=-({J_{r}}^{\mathrm {T} }J_{r})^{-1}{J_{r}}^{\mathrm {T} }{\boldsymbol {r}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 44.037ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ed7cd693d2daab25d4c8ffe5cc2c569d071138" data-alt="{\displaystyle {\boldsymbol {\beta }}^{(s+1)}={\boldsymbol {\beta }}^{(s)}+{\boldsymbol {\Delta }};\quad {\boldsymbol {\Delta }}=-({J_{r}}^{\mathrm {T} }J_{r})^{-1}{J_{r}}^{\mathrm {T} }{\boldsymbol {r}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> ガウス・ニュートン法の収束は常に保証されているわけではない。2階微分項を無視するという近似、すなわち <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|r_{i}{\frac {\partial ^{2}r_{i}}{\partial \beta _{j}\partial \beta _{k}}}\right|\ll \left|{\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}\right|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> | </mo> <mrow> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo> | </mo> </mrow> <mo> ≪ </mo> <mrow> <mo> | </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo> | </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left|r_{i}{\frac {\partial ^{2}r_{i}}{\partial \beta _{j}\partial \beta _{k}}}\right|\ll \left|{\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}\right|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d087e3a5587fe8a4d5d5f80a5201e2325cc34859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.09ex; height:6.509ex;" alt="{\displaystyle \left|r_{i}{\frac {\partial ^{2}r_{i}}{\partial \beta _{j}\partial \beta _{k}}}\right|\ll \left|{\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}\right|}"> </noscript><span class="lazy-image-placeholder" style="width: 25.09ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d087e3a5587fe8a4d5d5f80a5201e2325cc34859" data-alt="{\displaystyle \left|r_{i}{\frac {\partial ^{2}r_{i}}{\partial \beta _{j}\partial \beta _{k}}}\right|\ll \left|{\frac {\partial r_{i}}{\partial \beta _{j}}}{\frac {\partial r_{i}}{\partial \beta _{k}}}\right|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> に正当性があるのは次の2つの条件の下であり、これらが成り立つ場合には収束が期待される<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#cite_note-7">[7]</a>： <ol> <li>ri は十分小さい。少なくとも最小値付近。</li> <li>関数の非線形性は穏やかであり、 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial ^{2}r_{i}}/{\partial \beta _{j}\partial \beta _{k}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\partial ^{2}r_{i}}/{\partial \beta _{j}\partial \beta _{k}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0451dd4517e80536a0a0d2f13b716e4205226f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.674ex; height:3.343ex;" alt="{\displaystyle {\partial ^{2}r_{i}}/{\partial \beta _{j}\partial \beta _{k}}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.674ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0451dd4517e80536a0a0d2f13b716e4205226f28" data-alt="{\displaystyle {\partial ^{2}r_{i}}/{\partial \beta _{j}\partial \beta _{k}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  が比較的小さくなる。</li> </ol> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="改善バージョン">改善バージョン</h2> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%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&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 改善バージョン" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 編集 </a> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> ガウス・ニュートン法は、初期推定値が真の解から大きく離れていたり、モデル関数の非線形性が大きい場合には安定性が悪い。また、残差平方和S は反復ごとに必ずしも減少するわけではない。そのため、実用上は安定化が必要である<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#cite_note-#1-8">[8]</a>。 S は反復ごとに必ずしも減少するわけではないが、増分ベクトルΔはS が減少する方向を向いているから、S (βs) が停留点にない限り、任意の十分に小さなα> 0 に対して S (βs + αΔ) < S (βs) が成り立つ。したがって、発散したときに、更新方程式に縮小因子<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#cite_note-#1-8">[8]</a>αを導入して <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}^{s+1}={\boldsymbol {\beta }}^{s}+\alpha {\boldsymbol {\Delta }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> β </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msup> <mo> + </mo> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\beta }}^{s+1}={\boldsymbol {\beta }}^{s}+\alpha {\boldsymbol {\Delta }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6068d625a9a3b63bef34bbd00953a940fc9268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.828ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\beta }}^{s+1}={\boldsymbol {\beta }}^{s}+\alpha {\boldsymbol {\Delta }}}"> </noscript><span class="lazy-image-placeholder" style="width: 16.828ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6068d625a9a3b63bef34bbd00953a940fc9268" data-alt="{\displaystyle {\boldsymbol {\beta }}^{s+1}={\boldsymbol {\beta }}^{s}+\alpha {\boldsymbol {\Delta }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> とすることが解決法の一つとなる。 言い換えれば、増分ベクトルΔは目的関数S の下り方向を指してはいるが長すぎるので、その道のほんの一部を行くことで、S を減少させようというアイディアである。縮小因子αの最適値は<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E7%9B%B4%E7%B7%9A%E6%8E%A2%E7%B4%A2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="直線探索">直線探索</a>で見つけることができる。つまり、直接探索法を（通常 0 < α ≤ 1 の区間で）用いて、S を最小化する値を探すことでαの大きさは決められる。 最適な縮小因子αが 0 に近いような場合、発散を回避する別の方法は<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%AB%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" title="レーベンバーグ・マルカート法">レーベンバーグ・マルカート法</a>（<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E4%BF%A1%E9%A0%BC%E9%A0%98%E5%9F%9F?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="信頼領域">信頼領域</a>法）を使うことである<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#cite_note-ab-2">[2]</a>。増分ベクトルが<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E6%80%A5%E9%99%8D%E4%B8%8B%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="最急降下法">最急降下</a>方向に向くように正規方程式は修正される。 <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (J^{\mathrm {T} }J+\lambda D)\Delta =J^{\mathrm {T} }{\boldsymbol {r}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msup> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> T </mi> </mrow> </mrow> </msup> <mi> J </mi> <mo> + </mo> <mi> λ </mi> <mi> D </mi> <mo stretchy="false"> ) </mo> <mi mathvariant="normal"> Δ </mi> <mo> = </mo> <msup> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> r </mi> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (J^{\mathrm {T} }J+\lambda D)\Delta =J^{\mathrm {T} }{\boldsymbol {r}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644add873cbfb977a47a8f5b8b925f46aae4b656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.201ex; height:3.176ex;" alt="{\displaystyle (J^{\mathrm {T} }J+\lambda D)\Delta =J^{\mathrm {T} }{\boldsymbol {r}},}"> </noscript><span class="lazy-image-placeholder" style="width: 22.201ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644add873cbfb977a47a8f5b8b925f46aae4b656" data-alt="{\displaystyle (J^{\mathrm {T} }J+\lambda D)\Delta =J^{\mathrm {T} }{\boldsymbol {r}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> ここでD は<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%AD%A3%E5%AE%9A%E5%80%A4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-disambig" title="正定値">正定値</a>対角行列である。λが 0 から増大するにつれて増分ベクトルΔは長さが単調に減少し、かつ方向は最急降下方向に近づくため、λを十分大きくすれば必ずより小さいS の値を見出せることが保証されている<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#cite_note-9">[9]</a>。 いわゆるマルカートパラメータλは直線探索により最適化されるが、λが変わるたびに毎回シフトベクトルの再計算をしなければならないため非効率的である。より効率的な方法は、発散が起きた時にλをS が減少するまで増加させる。そしてその値を1回の反復から次まで維持する、しかしλを 0 に設定することができるときにもしカットオフ値に届くまで可能なら減少させる。このときS の最小値は標準のガウス・ニュートン法の最小化になる。 </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <h2 id="関連するアルゴリズム">関連するアルゴリズム</h2> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%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&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 関連するアルゴリズム" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 編集 </a> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/DFP%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="DFP法">DFP法</a>や<a href="https://ja-m-wikipedia-org.translate.goog/wiki/BFGS%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="BFGS法">BFGS法</a>のような<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%BA%96%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="準ニュートン法">準ニュートン法</a>では、ヘッシアン <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial ^{2}S}/{\partial \beta _{j}\partial \beta _{k}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> S </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\partial ^{2}S}/{\partial \beta _{j}\partial \beta _{k}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8ab50aa5328619709363359b19fac567ac38ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.325ex; height:3.343ex;" alt="{\displaystyle {\partial ^{2}S}/{\partial \beta _{j}\partial \beta _{k}}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.325ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8ab50aa5328619709363359b19fac567ac38ed" data-alt="{\displaystyle {\partial ^{2}S}/{\partial \beta _{j}\partial \beta _{k}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> の推定は1階微分 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial r_{i}}/{\partial \beta _{j}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> β </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\partial r_{i}}/{\partial \beta _{j}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88fce34caedbe1fdeb45e5afaadb7e65deaf10e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.872ex; height:3.009ex;" alt="{\displaystyle {\partial r_{i}}/{\partial \beta _{j}}}"> </noscript><span class="lazy-image-placeholder" style="width: 7.872ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88fce34caedbe1fdeb45e5afaadb7e65deaf10e4" data-alt="{\displaystyle {\partial r_{i}}/{\partial \beta _{j}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> のみを用いて数値的になされる。したがってn 回の反復計算による修正の後、この方法はパフォーマンスにおいてニュートン法を近似する。ガウス・ニュートン法やレーベンバーグ・マルカート法などは非線形最小二乗問題にのみ適用できるのに対して、準ニュートン法は一般的な実数値関数を最小化できることに注意する。 1階微分のみを使って最小化問題を解く他の方法は、<a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E6%9C%80%E6%80%A5%E9%99%8D%E4%B8%8B%E6%B3%95?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="最急降下法">最急降下法</a>である。しかし、その方法は近似に2階微分を考慮していないので、多くの関数に対して、特にパラメータが強い相互作用を持っている場合には、計算効率が非常に悪い。 </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <h2 id="脚注">脚注</h2> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%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&action=edit&section=8&_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 "> 編集 </a> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <ol class="references"> <li id="cite_note-1"><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#cite_ref-1">^</a> アルゴリズム内のm ≥ n という仮定は必要である。そうでなければ、行列JrTJr の逆行列を計算できず、正規方程式の解（少なくとも唯一解）を求めることができない。</li> <li id="cite_note-ab-2">^ <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#cite_ref-ab_2-0">a</a> <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#cite_ref-ab_2-1">b</a> Björck (1996)</li> <li id="cite_note-3"><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#cite_ref-3">^</a> <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E3%83%9F%E3%82%AB%E3%82%A8%E3%83%AA%E3%82%B9%E3%83%BB%E3%83%A1%E3%83%B3%E3%83%86%E3%83%B3%E5%BC%8F?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E5%AE%9A%E6%95%B0%E3%81%AE%E6%B1%BA%E5%AE%9A" title="ミカエリス・メンテン式">ミカエリス・メンテン式#定数の決定</a>で説明するように、実際は変数に[S]-1とv-1 を選ぶことで、この問題は線形最小二乗法として解ける。</li> <li id="cite_note-4"><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#cite_ref-4">^</a> Björck (1996) p260</li> <li id="cite_note-5"><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#cite_ref-5">^</a> Björck (1996) p341, 342</li> <li id="cite_note-6"><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#cite_ref-6">^</a> Fletcher (1987) p.113</li> <li id="cite_note-7"><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#cite_ref-7">^</a> Nocedal (1997) [<a href="https://ja-m-wikipedia-org.translate.goog/wiki/Wikipedia:%E5%87%BA%E5%85%B8%E3%82%92%E6%98%8E%E8%A8%98%E3%81%99%E3%82%8B?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%E5%87%BA%E5%85%B8%E3%81%AE%E7%A4%BA%E3%81%97%E6%96%B9" title="Wikipedia:出典を明記する">要ページ番号</a>]</li> <li id="cite_note-#1-8">^ <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#cite_ref-#1_8-0">a</a> <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#cite_ref-#1_8-1">b</a> 中川、小柳 (1982) p.98</li> <li id="cite_note-9"><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#cite_ref-9">^</a> 中川、小柳 (1982) p.102</li> </ol> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"> <h2 id="参考文献">参考文献</h2> <a role="button" href="https://ja-m-wikipedia-org.translate.goog/w/index.php?title=%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&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="節を編集: 参考文献" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 編集 </a> </div> <section class="mf-section-9 collapsible-block" id="mf-section-9"> <ul> <li><cite style="font-style:normal" class="citation book">Björck, A. (1996). Numerical methods for least squares problems. SIAM, Philadelphia. <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%99/0-89871-360-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="特別:文献資料/0-89871-360-9">0-89871-360-9</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=Numerical+methods+for+least+squares+problems&rft.aulast=Bj%C3%B6rck&rft.aufirst=A.&rft.au=Bj%C3%B6rck%2C%26%2332%3BA.&rft.date=1996&rft.pub=SIAM%2C+Philadelphia&rft.isbn=0-89871-360-9&rfr_id=info:sid/ja.wikipedia.org:%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"> </li> <li><cite style="font-style:normal" class="citation book">Fletcher, Roger (1987). Practical methods of optimization (2nd ed.). New York: John Wiley & Sons. <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%99/978-0-471-91547-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="特別:文献資料/978-0-471-91547-8">978-0-471-91547-8</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=Practical+methods+of+optimization&rft.aulast=Fletcher&rft.aufirst=Roger&rft.au=Fletcher%2C%26%2332%3BRoger&rft.date=1987&rft.edition=2nd&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.isbn=978-0-471-91547-8&rfr_id=info:sid/ja.wikipedia.org:%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"> .</li> <li><cite style="font-style:normal" class="citation book">Nocedal, Jorge; Wright, Stephen (1999). Numerical optimization. New York: Springer. <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%99/0-387-98793-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="特別:文献資料/0-387-98793-2">0-387-98793-2</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=Numerical+optimization&rft.aulast=Nocedal&rft.aufirst=Jorge&rft.au=Nocedal%2C%26%2332%3BJorge&rft.date=1999&rft.pub=New+York%3A+Springer&rft.isbn=0-387-98793-2&rfr_id=info:sid/ja.wikipedia.org:%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"> </li> <li><cite style="font-style:normal" class="citation" id="CITEREF中川徹小柳義夫1982">中川徹; <a href="https://ja-m-wikipedia-org.translate.goog/wiki/%E5%B0%8F%E6%9F%B3%E7%BE%A9%E5%A4%AB?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="小柳義夫">小柳義夫</a>『最小二乗法による実験データ解析』東京大学出版会、1982年。 <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%99/4-13-064067-4?_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=%5B%5B%E5%B0%8F%E6%9F%B3%E7%BE%A9%E5%A4%AB%5D%5D&rft.date=1982&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:%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"> </li> </ul>   </section> </div> <noscript> <img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> 「<a dir="ltr" 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