tag:crieit.net,2005:https://crieit.net/tags/%E3%80%90%E3%83%A9%E3%83%93%E3%83%83%E3%83%88%E3%83%BB%E3%83%81%E3%83%A3%E3%83%AC%E3%83%B3%E3%82%B8%E3%80%91%E5%8F%97%E8%AC%9B%E3%83%AC%E3%83%9D%E3%83%BC%E3%83%88/feed Crieitでタグ「【ラビット・チャレンジ】受講レポート」に投稿された最近の記事 2021-09-16T19:02:58+09:00 https://crieit.net/tags/%E3%80%90%E3%83%A9%E3%83%93%E3%83%83%E3%83%88%E3%83%BB%E3%83%81%E3%83%A3%E3%83%AC%E3%83%B3%E3%82%B8%E3%80%91%E5%8F%97%E8%AC%9B%E3%83%AC%E3%83%9D%E3%83%BC%E3%83%88/feed tag:crieit.net,2005:PublicArticle/17678 2021-09-16T19:02:58+09:00 2021-09-16T19:02:58+09:00 https://crieit.net/posts/9d475ee4332ab019aef4f43996497c33 <p><a target="_blank" rel="nofollow noopener" href="https://ai999.careers/rabbit/">ラビット・チャレンジ</a>の受講レポート。</p> <hr /> <h1 id="【線形代数学 (行列)】"><a href="#%E3%80%90%E7%B7%9A%E5%BD%A2%E4%BB%A3%E6%95%B0%E5%AD%A6+%28%E8%A1%8C%E5%88%97%29%E3%80%91">【線形代数学 (行列)】</a></h1> <h2 id="スカラーとベクトル"><a href="#%E3%82%B9%E3%82%AB%E3%83%A9%E3%83%BC%E3%81%A8%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB">スカラーとベクトル</a></h2> <ul> <li><strong>スカラー</strong>：普通の数</li> <li><strong>ベクトル</strong>：「大きさ」と「向き」をもつ</li> </ul> <h2 id="行列"><a href="#%E8%A1%8C%E5%88%97">行列</a></h2> <ul> <li>スカラーを表にしたもの</li> <li>ベクトルを並べたもの</li> </ul> <h2 id="連立方程式"><a href="#%E9%80%A3%E7%AB%8B%E6%96%B9%E7%A8%8B%E5%BC%8F">連立方程式</a></h2> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0Ax_1+%2B+2x_2+%3D+3%5C%5C+%0A++++2x_1+%2B+5x_2+%3D+5%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} x_1 + 2x_2 = 3\ 2x_1 + 5x_2 = 5 \end{align*} " /></p> <p>の式を <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0AA%5Cvec%7Bx%7D+%3D+%5Cvec%7Bb%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} A\vec{x} = \vec{b} \end{align*} " />の形にすると、以下のようになる</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cleft%28%0A++++++++%5Cbegin%7Barray%7D%7Bcc%7D%0A++++++++++++1+%26+2+%5C%5C%5C%5C++%0A++++++++++++2+%26+5%0A++++++++%5Cend%7Barray%7D%0A++++%5Cright%29%0A++++%5Cleft%28%0A++++++++%5Cbegin%7Barray%7D%7Bc%7D%0A++++++++++++x_1+%5C%5C%5C%5C++%0A++++++++++++x_2%0A++++++++%5Cend%7Barray%7D%0A++++%5Cright%29+%3D+%0A++++%5Cleft%28%0A++++++++%5Cbegin%7Barray%7D%7Bc%7D%0A++++++++++++3+%5C%5C%5C%5C++%0A++++++++++++5%0A++++++++%5Cend%7Barray%7D%0A++++%5Cright%29%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \left( \begin{array}{cc} 1 & 2 \\ 2 & 5 \end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) = \left( \begin{array}{c} 3 \\ 5 \end{array} \right) \end{align*} " /></p> <p>係数をまとめて表のようにした部分を<strong>行列</strong>という</p> <h3 id="行基本変形"><a href="#%E8%A1%8C%E5%9F%BA%E6%9C%AC%E5%A4%89%E5%BD%A2">行基本変形</a></h3> <p>= 行列の変形<br /> 　→行列を左からかけることで表現できる</p> <p>手順：</p> <p>(1) i行目をc倍する<br /> (2) s行目にt行目のc倍を加える<br /> (3) p行目とq行目を入れ替える<br /> 　　(→連立方程式での例：2行目に<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0Ax_1%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} x_1 \end{align*} " />, 1行目に<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0Ax_2%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} x_2 \end{align*} " />が残ってしまっているので入れ替える)</p> <p>参考：<a target="_blank" rel="nofollow noopener" href="https://math.005net.com/yoten/renrituKagen.php">連立方程式の解き方(加減法,代入法)</a></p> <p>各工程で使用する行列</p> <p>(1) i行目をc倍する<br /> + (i, i)番目の要素をc倍する<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0AQ_%7Bi%2C+c%7D+%3D%0A++++%5Cbegin%7Bpmatrix%7D%0A++++++++1+++%26+++++++++++%26+++%26+++%26+++%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+%5Cddots++++%26+++%26+++%26+++%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+1+%26+++%26+++%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+c+%26+++%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+++%26+1+%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+++%26+++%26+%5Cddots++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+++%26+++%26+++++++++++%26+1+%5C%5C%5C%5C++%0A++++%5Cend%7Bpmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} Q_{i, c} = \begin{pmatrix} 1 & & & & & & \\ & \ddots & & & & & \\ & & 1 & & & & \\ & & & c & & & \\ & & & & 1 & & \\ & & & & & \ddots & \\ & & & & & & 1 \\ \end{pmatrix} \end{align*} " /><br /> (2) s行目にt行目のc倍を加える<br /> + (s, t)の成分をcに変える<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0AR_%7Bs%2C+t%2C+c%7D+%3D%0A++++%5Cbegin%7Bpmatrix%7D%0A++++++++1+++%26+++++++++++%26+++%26+++++++++++%26+++%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+%5Cddots++++%26+++%26+++++++++++%26+++%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+1+%26+++++++++++%26+c+%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+%5Cddots++++%26+++%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+++++++++++%26+1+%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+++++++++++%26+++%26+%5Cddots++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+++++++++++%26+++%26+++++++++++%26+1+%5C%5C%5C%5C++%0A++++%5Cend%7Bpmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} R_{s, t, c} = \begin{pmatrix} 1 & & & & & & \\ & \ddots & & & & & \\ & & 1 & & c & & \\ & & & \ddots & & & \\ & & & & 1 & & \\ & & & & & \ddots & \\ & & & & & & 1 \\ \end{pmatrix} \end{align*} " /></p> <p>(3) p行目とq行目を入れ替える<br /> + (p, p), (q, q)の成分を0に変える<br /> + (p, q), (q, p)の成分を1に変える<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0AP_%7Bp%2C+q%7D+%3D%0A++++%5Cbegin%7Bpmatrix%7D%0A++++++++1+++%26+++++++++++%26+++%26+++++++++++%26+++%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+%5Cddots++++%26+++%26+++++++++++%26+++%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+0+%26+++++++++++%26+1+%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+%5Cddots++++%26+++%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+1+%26+++++++++++%26+0+%26+++++++++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+++++++++++%26+++%26+%5Cddots++++%26+++%5C%5C%5C%5C++%0A++++++++++++%26+++++++++++%26+++%26+++++++++++%26+++%26+++++++++++%26+1+%5C%5C%5C%5C++%0A++++%5Cend%7Bpmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} P_{p, q} = \begin{pmatrix} 1 & & & & & & \\ & \ddots & & & & & \\ & & 0 & & 1 & & \\ & & & \ddots & & & \\ & & 1 & & 0 & & \\ & & & & & \ddots & \\ & & & & & & 1 \\ \end{pmatrix} \end{align*} " /></p> <h3 id="単位行列"><a href="#%E5%8D%98%E4%BD%8D%E8%A1%8C%E5%88%97">単位行列</a></h3> <p>かけてもかけられても相手が変化しない行列<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0AI+%3D+%0A++++++++%5Cbegin%7Bpmatrix%7D%0A++++++++++++1+%26+++%26+++++++++%5C%5C%5C%5C++%0A++++++++++++++%26+1+%26+++++++++%5C%5C%5C%5C++%0A++++++++++++++%26+++%26+%5Cddots++%5C%5C%5C%5C++%0A++++++++%5Cend%7Bpmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} I = \begin{pmatrix} 1 & & \\ & 1 & \\ & & \ddots \\ \end{pmatrix} \end{align*} " /></p> <h3 id="逆行列"><a href="#%E9%80%86%E8%A1%8C%E5%88%97">逆行列</a></h3> <p>まるで逆数のような働きをする行列</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0AAA%5E%7B-1%7D+%3D+A%5E%7B-1%7DA+%3D+I%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} AA^{-1} = A^{-1}A = I \end{align*} " /><br /> (「-1乗」ではなく 「<strong>inverse</strong>」)</p> <p>掃き出し法などで求める</p> <h4 id="逆行列が存在しない行列"><a href="#%E9%80%86%E8%A1%8C%E5%88%97%E3%81%8C%E5%AD%98%E5%9C%A8%E3%81%97%E3%81%AA%E3%81%84%E8%A1%8C%E5%88%97">逆行列が存在しない行列</a></h4> <p>解がない/一組に定まらない連立方程式の係数を抜き出したような行列<br /> 形式的には</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cbegin%7Bpmatrix%7D%0A++++a+%26+b+++++++%5C%5C%5C%5C++%0A++++c+%26+d+++++++%5C%5C%5C%5C++%0A%5Cend%7Bpmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \end{align*} " /> という行列があったとき、<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0Aad+-+bc+%3D+0%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} ad - bc = 0 \end{align*} " /></p> <p>また<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cbegin%7Bpmatrix%7D%0A++++++++a+%26+b+++++++%5C%5C%5C%5C++%0A++++++++c+%26+d+++++++%5C%5C%5C%5C++%0A++++%5Cend%7Bpmatrix%7D+%3D%0A++++%5Cbegin%7Bpmatrix%7D%0A++++++++%5Cvec%7Bv_1%7D+++++++%5C%5C%5C%5C++%0A++++++++%5Cvec%7Bv_2%7D+++++++%5C%5C%5C%5C++%0A++++%5Cend%7Bpmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} = \begin{pmatrix} \vec{v_1} \\ \vec{v_2} \\ \end{pmatrix} \end{align*} " /><br /> と考えたとき、二つのベクトルに囲まれた<br /> <code>平行四辺形の面積 = 0</code><br /> の場合は逆行列が存在しない</p> <h3 id="行列式(determinant)"><a href="#%E8%A1%8C%E5%88%97%E5%BC%8F%28determinant%29">行列式(determinant)</a></h3> <p>上記の平行四辺形の面積が逆行列の有無を示す<br /> これを<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cbegin%7Bvmatrix%7D%0A++++++++a+%26+b+++++++%5C%5C%5C%5C++%0A++++++++c+%26+d+++++++%5C%5C%5C%5C++%0A++++%5Cend%7Bvmatrix%7D+%3D%0A++++%5Cbegin%7Bvmatrix%7D%0A++++++++%5Cvec%7Bv_1%7D+++++++%5C%5C%5C%5C++%0A++++++++%5Cvec%7Bv_2%7D+++++++%5C%5C%5C%5C++%0A++++%5Cend%7Bvmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} = \begin{vmatrix} \vec{v_1} \\ \vec{v_2} \\ \end{vmatrix} \end{align*} " /><br /> と表し、<strong>逆行列</strong>と呼ぶ</p> <h4 id="特徴"><a href="#%E7%89%B9%E5%BE%B4">特徴</a></h4> <ul> <li>同じ行ベクトルが含まれていると行列式は0</li> <li>1つのベクトルが<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Clambda%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \lambda \end{align*} " />倍されると行列式は<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Clambda%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \lambda \end{align*} " />倍される</li> <li>他の成分が全部同じで<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0Ai%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} i \end{align*} " />番目のベクトルだけが違う場合、行列式の足し合わせになる</li> </ul> <p>3つ以上のベクトルからできている行列式は展開できる</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cbegin%7Bvmatrix%7D%0A++++++++%5Cvec%7Bv_1%7D+%5C%5C%5C%5C++%0A++++++++%5Cvec%7Bv_2%7D+%5C%5C%5C%5C++%0A++++++++%5Cvec%7Bv_3%7D+%0A++++%5Cend%7Bvmatrix%7D+%3D+%0A++++%5Cbegin%7Bvmatrix%7D%0A++++++++a+%26+b+%26+c+%5C%5C%5C%5C++%0A++++++++d+%26+e+%26+f+%5C%5C%5C%5C++%0A++++++++g+%26+h+%26+i+%0A++++%5Cend%7Bvmatrix%7D+%3D+%0A++++%5Cbegin%7Bvmatrix%7D%0A++++++++a+%26+b+%26+c+%5C%5C%5C%5C++%0A++++++++0+%26+e+%26+f+%5C%5C%5C%5C++%0A++++++++0+%26+h+%26+i+%0A++++%5Cend%7Bvmatrix%7D+%2B+%0A++++%5Cbegin%7Bvmatrix%7D%0A++++++++0+%26+b+%26+c+%5C%5C%5C%5C++%0A++++++++d+%26+e+%26+f+%5C%5C%5C%5C++%0A++++++++0+%26+h+%26+i+%0A++++%5Cend%7Bvmatrix%7D+%2B+%0A++++%5Cbegin%7Bvmatrix%7D%0A++++++++0+%26+b+%26+c+%5C%5C%5C%5C++%0A++++++++0+%26+e+%26+f+%5C%5C%5C%5C++%0A++++++++g+%26+h+%26+i+%0A++++%5Cend%7Bvmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \begin{vmatrix} \vec{v_1} \\ \vec{v_2} \\ \vec{v_3} \end{vmatrix} = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = \begin{vmatrix} a & b & c \\ 0 & e & f \\ 0 & h & i \end{vmatrix} + \begin{vmatrix} 0 & b & c \\ d & e & f \\ 0 & h & i \end{vmatrix} + \begin{vmatrix} 0 & b & c \\ 0 & e & f \\ g & h & i \end{vmatrix} \end{align*} " /><br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%3D+a%0A++++%5Cbegin%7Bvmatrix%7D%0A++++++++e+%26+f+%5C%5C%5C%5C++%0A++++++++h+%26+i+%0A++++%5Cend%7Bvmatrix%7D+-+%0A++++d%0A++++%5Cbegin%7Bvmatrix%7D%0A++++++++b+%26+c+%5C%5C%5C%5C++%0A++++++++h+%26+i+%0A++++%5Cend%7Bvmatrix%7D+%2B%0A++++g%0A++++%5Cbegin%7Bvmatrix%7D%0A++++++++b+%26+c+%5C%5C%5C%5C++%0A++++++++e+%26+f+%0A++++%5Cend%7Bvmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - d \begin{vmatrix} b & c \\ h & i \end{vmatrix} + g \begin{vmatrix} b & c \\ e & f \end{vmatrix} \end{align*} " /></p> <h4 id="行列式の求め方"><a href="#%E8%A1%8C%E5%88%97%E5%BC%8F%E3%81%AE%E6%B1%82%E3%82%81%E6%96%B9">行列式の求め方</a></h4> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cbegin%7Bvmatrix%7D%0A++++++++a+%26+b+++++++%5C%5C%5C%5C++%0A++++++++c+%26+d+++++++%5C%5C%5C%5C++%0A++++%5Cend%7Bvmatrix%7D+%3D+ad+-+bc%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} = ad - bc \end{align*} " /><br /> 3つ以上のベクトルでできている場合は展開して求める</p> <p>参考：<a target="_blank" rel="nofollow noopener" href="https://risalc.info/src/determinant-formulas.html">行列式の基本的な性質と公式</a></p> <hr /> <h1 id="【線形代数学 (固有値)】"><a href="#%E3%80%90%E7%B7%9A%E5%BD%A2%E4%BB%A3%E6%95%B0%E5%AD%A6+%28%E5%9B%BA%E6%9C%89%E5%80%A4%29%E3%80%91">【線形代数学 (固有値)】</a></h1> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0AA%5Cvec%7Bx%7D+%3D+%5Clambda%5Cvec%7Bx%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} A\vec{x} = \lambda\vec{x} \end{align*} " /><br /> が成り立つような行列A, 特殊なベクトル<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cvec%7Bx%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \vec{x} \end{align*} " />, 右辺の係数<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Clambda%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \lambda \end{align*} " />があるとき、</p> <ul> <li><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cvec%7Bx%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \vec{x} \end{align*} " />: 行列Aに対する固有ベクトル <ul> <li>一つに定まらない</li> <li>「<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cvec%7Bx%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \vec{x} \end{align*} " />の定数倍」のように表す</li> </ul></li> <li><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Clambda%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \lambda \end{align*} " />: 行列Aに対する固有値 <ul> <li>一つに定まる</li> </ul></li> </ul> <h4 id="固有値と固有ベクトルの求め方："><a href="#%E5%9B%BA%E6%9C%89%E5%80%A4%E3%81%A8%E5%9B%BA%E6%9C%89%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E3%81%AE%E6%B1%82%E3%82%81%E6%96%B9%EF%BC%9A">固有値と固有ベクトルの求め方：</a></h4> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Cbegin%7Bvmatrix%7D%0A%09++++A+-+%5Clambda+I+%3D+0%09%5C%5C%5C%5C++%0A%09%5Cend%7Bvmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \begin{vmatrix} A - \lambda I = 0 \\ \end{vmatrix} \end{align*} " /><br /> となるような<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0A%5Clambda%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} \lambda \end{align*} " />を求め、<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0AA%0A%09%5Cbegin%7Bpmatrix%7D%0A%09%09x_1%5C%5C%5C%5C++%0A++++++++x_2%0A%09%5Cend%7Bpmatrix%7D%0A++++%3D+%5Clambda%0A++++%5Cbegin%7Bpmatrix%7D%0A%09%09x_1%5C%5C%5C%5C++%0A++++++++x_2%0A%09%5Cend%7Bpmatrix%7D%0A%5Cend%7Balign%2A%7D%0A" alt="\begin{align*} A \begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \lambda \begin{pmatrix} x_1\\ x_2 \end{pmatrix} \end{align*} " /><br /> を解いて<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+x_1%0A" alt="x_1 " />と<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+x_2%0A" alt="x_2 " />の比を求める</p> <h3 id="固有値分解"><a href="#%E5%9B%BA%E6%9C%89%E5%80%A4%E5%88%86%E8%A7%A3">固有値分解</a></h3> <ul> <li>ある実数を正方形に並べた行列<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+A%0A" alt="A " /></li> <li><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+A%0A" alt="A " />の固有値<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Clambda_1" alt="\lambda_1" />, <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Clambda_2" alt="\lambda_2" />, ...</li> <li><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+A%0A" alt="A " />の固有ベクトル<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cvec%7Bv_1%7D" alt="\vec{v_1}" />, <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cvec%7Bv_2%7D" alt="\vec{v_2}" />, ...</li> </ul> <p>があるとき、固有値を対角線上に並べた行列<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5CLambda+%3D+%0A%09%5Cbegin%7Bpmatrix%7D%0A%09%09%5Clambda_1+%26++%26++%5C%5C%5C%5C++%0A+++++++++%26+%5Clambda_2+%26+%5C%5C%5C%5C++%0A+++++++++%26+%26+%5Cddots%0A%09%5Cend%7Bpmatrix%7D" alt="\Lambda = \begin{pmatrix} \lambda_1 & & \\ & \lambda_2 & \\ & & \ddots \end{pmatrix}" /><br /> と、それに対応する固有ベクトルを並べた行列<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+V+%3D+%0A%09%5Cbegin%7Bpmatrix%7D%0A+++++++++%26+%26+%5C%5C%5C%5C++%0A%09%09%5Cvec%7Bv_1%7D+%26+%5Cvec%7Bv_2%7D+%26+%5Ccdots+%5C%5C%5C%5C++%0A+++++++++%26+%26+%5C%5C%5C%5C++%0A%09%5Cend%7Bpmatrix%7D" alt="V = \begin{pmatrix} & & \\ \vec{v_1} & \vec{v_2} & \cdots \\ & & \\ \end{pmatrix}" /><br /> を用意したとき、<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+AV+%3D+V%5CLambda" alt="AV = V\Lambda" />となる<br /> 変形すると<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+A+%3D+V+%5CLambda+V%5E%7B-1%7D" alt="A = V \Lambda V^{-1}" /><br /> + <strong>固有値分解</strong>：正方形の行列を上記のような3つの行列の積に分解すること<br /> + 利点：行列の累乗が容易になる　など<br /> + <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5CLambda" alt="\Lambda" />の中身は、<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Clambda" alt="\lambda" />を小さい順or大きい順に並べることが多い</p> <h3 id="特異値分解"><a href="#%E7%89%B9%E7%95%B0%E5%80%A4%E5%88%86%E8%A7%A3">特異値分解</a></h3> <p>正方行列以外の行列において<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cbegin%7Balign%2A%7D%0AM%5Cvec%7Bv%7D+%3D+%5Csigma%5Cvec%7Bu%7D+%5C%5C++++%0AM%5ET%5Cvec%7Bu%7D+%3D+%5Csigma%5Cvec%7Bv%7D%0A%5Cend%7Balign%2A%7D" alt="\begin{align*} M\vec{v} = \sigma\vec{u} \ M^T\vec{u} = \sigma\vec{v} \end{align*}" /><br /> となる特殊な単位ベクトルがある場合、<strong>特異値分解</strong>が可能<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+M+%3D+USV%5ET" alt="M = USV^T" /></p> <ul> <li><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+U" alt="U" />や<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+V" alt="V" />は直行行列 <ul> <li>複素数を要素に持つ場合はユニタリ行列</li> </ul></li> <li><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+S" alt="S" /> = Sigma</li> </ul> <h4 id="特異値の求め方"><a href="#%E7%89%B9%E7%95%B0%E5%80%A4%E3%81%AE%E6%B1%82%E3%82%81%E6%96%B9">特異値の求め方</a></h4> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+MV+%3D+US" alt="MV = US" />　→　<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+M+%3D+USV%5ET" alt="M = USV^T" /><br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+M%5ETU+%3D+VS%5ET" alt="M^TU = VS^T" />　→　<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+M%5ET+%3D+VS%5ETU%5ET" alt="M^T = VS^TU^T" /><br /> これらの積は<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+MM%5ET+%3D+USV%5ETVS%5ETU%5ET+%3D+USS%5ETU%5ET" alt="MM^T = USV^TVS^TU^T = USS^TU^T" /><br /> (<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+MM%5ET" alt="MM^T" />で正方行列を作って固有値分解する)</p> <h4 id="特異値分解の利用例"><a href="#%E7%89%B9%E7%95%B0%E5%80%A4%E5%88%86%E8%A7%A3%E3%81%AE%E5%88%A9%E7%94%A8%E4%BE%8B">特異値分解の利用例</a></h4> <ul> <li>画像データの圧縮</li> <li>機械学習の前処理 <ul> <li>特異値の大きい部分が似ている画像どうしは、画像の特徴も似ている</li> <li>画像の分類などができる</li> </ul></li> </ul> <hr /> <h1 id="【統計学1】"><a href="#%E3%80%90%E7%B5%B1%E8%A8%88%E5%AD%A61%E3%80%91">【統計学1】</a></h1> <h2 id="集合とは？"><a href="#%E9%9B%86%E5%90%88%E3%81%A8%E3%81%AF%EF%BC%9F">集合とは？</a></h2> <p>→ものの集まり</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+S+%3D+%5C%7B+a%2C+b%2C+c%2C+d%2C+e%2C+f%2C+g+%5C%7D" alt="S = { a, b, c, d, e, f, g }" /><br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+a+%5Cin+S" alt="a \in S" /> ← <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+a" alt="a" />は集合<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+S" alt="S" />の要素<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+h+%5Cnotin+S" alt="h \notin S" /> ← <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+h" alt="h" />は集合<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+S" alt="S" />の要素ではない<br /> (「要素」は「元(げん)」と呼ばれることもある)</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+M+%3D+%5C%7B+c%2C+d%2C+e+%5C%7D" alt="M = { c, d, e }" /><br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+M+%5Csubset+S" alt="M \subset S" /> ← <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+M" alt="M" />は<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+S" alt="S" />の一部</p> <ul> <li>和集合 <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+A+%5Ccap+B" alt="A \cap B" /></li> <li>共通部分 <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+A+%5Ccup+B" alt="A \cup B" /></li> <li>絶対補 <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+U+%5Csetminus+A+%3D+%5Cbar%7BA%7D" alt="U \setminus A = \bar{A}" /></li> <li>相対補 <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+B+%5Csetminus+A" alt="B \setminus A" /></li> </ul> <h2 id="確率"><a href="#%E7%A2%BA%E7%8E%87">確率</a></h2> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P%28A%29+%3D+%5Cdfrac%7Bn%28A%29%7D%7Bn%28U%29%7D" alt="P(A) = \dfrac{n(A)}{n(U)}" /></p> <ul> <li>頻度確率(客観確率)：発生する頻度</li> <li>ベイズ確率(主観確率)：信念の度合い</li> </ul> <h4 id="条件付き確率"><a href="#%E6%9D%A1%E4%BB%B6%E4%BB%98%E3%81%8D%E7%A2%BA%E7%8E%87">条件付き確率</a></h4> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P%28B+%7C+A%29" alt="P(B | A)" /> ←<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+A" alt="A" />という条件のもと、<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+B" alt="B" />である<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P%28B+%7C+A%29+%3D+%5Cdfrac%7BP%28A+%5Ccap+B%29%7D%7BP%28B%29%7D+%3D+%5Cdfrac%7Bn%28A+%5Ccap+B%29%7D%7Bn%28B%29%7D" alt="P(B | A) = \dfrac{P(A \cap B)}{P(B)} = \dfrac{n(A \cap B)}{n(B)}" /></p> <h4 id="独立な事象の同時確率"><a href="#%E7%8B%AC%E7%AB%8B%E3%81%AA%E4%BA%8B%E8%B1%A1%E3%81%AE%E5%90%8C%E6%99%82%E7%A2%BA%E7%8E%87">独立な事象の同時確率</a></h4> <p>事象Aと事象Bに因果関係がない場合、<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P%28A+%5Ccap+B%29+%3D+P%28A%29P%28B%7CA%29+%3D+P%28A%29P%28B%29" alt="P(A \cap B) = P(A)P(B|A) = P(A)P(B)" /></p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P%28A+%5Ccup+B%29+%3D+P%28A%29+%2B+P%28B%29+-+P%28A+%5Ccap+B%29" alt="P(A \cup B) = P(A) + P(B) - P(A \cap B)" /></p> <h4 id="ベイズ則"><a href="#%E3%83%99%E3%82%A4%E3%82%BA%E5%89%87">ベイズ則</a></h4> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P%28A%29P%28B%7CA%29+%3D+P%28B%29P%28A%7CB%29" alt="P(A)P(B|A) = P(B)P(A|B)" /></p> <hr /> <h1 id="【統計学2】"><a href="#%E3%80%90%E7%B5%B1%E8%A8%88%E5%AD%A62%E3%80%91">【統計学2】</a></h1> <ul> <li><strong>記述統計</strong>：集団の性質を要約し記述する</li> <li><p><strong>推測統計</strong>：集団から一部を取り出し(標本)、元の集団(母集団)の性質を推測する</p></li> <li><p><strong>確率変数</strong>：事象と結び付けられた数値</p></li> <li><strong>確率分布</strong>：事象の発生する確率の分布</li> </ul> <h4 id="期待値"><a href="#%E6%9C%9F%E5%BE%85%E5%80%A4">期待値</a></h4> <p>その分布における、確率変数の<br /> <em>平均の値</em> または <em>「ありえそう」な値</em></p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+E%28f%29+%3D+%5Csum_%7Bk%3D1%7D%5E%7Bn%7DP%28X+%3D+x_%7Bk%7D%29f%28X+%3D+x_%7Bk%7D%29%0A" alt="E(f) = \sum_{k=1}^{n}P(X = x_{k})f(X = x_{k}) " /></p> <p>連続する値なら、</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+E%28f%29+%3D+%5Cint+P%28X+%3D+x%29f%28X+%3D+x%29dx" alt="E(f) = \int P(X = x)f(X = x)dx" /></p> <h4 id="分散"><a href="#%E5%88%86%E6%95%A3">分散</a></h4> <p>データの散らばり具合</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+Var%28f%29+%3D+E%0A++++++++%5Cbiggl%28++%0A++++++++++++%5CBigl%28%0A++++++++++++++++f_%7B+%28X+%3D+x%29+%7D+-+E_%7B+%28f%29+%7D%0A++++++++++++%5CBigr%29+%5E+2%0A++++++++%5Cbiggr%29+%3D+E%0A++++++++%5CBigl%28%0A++++++++++++f+%5E+2+_%7B+%28X+%3D+x%29+%7D%0A++++++++%5CBigr%29+-+%0A++++++++%5CBigl%28%0A++++++++++++E+_%7B+%28f%29+%7D%0A++++++++%5CBigr%29+%5E+2" alt="Var(f) = E \biggl( \Bigl( f_{ (X = x) } - E_{ (f) } \Bigr) ^ 2 \biggr) = E \Bigl( f ^ 2 _{ (X = x) } \Bigr) - \Bigl( E _{ (f) } \Bigr) ^ 2" /><br /> →(二乗の平均) - (平均の二乗)</p> <h4 id="共分散"><a href="#%E5%85%B1%E5%88%86%E6%95%A3">共分散</a></h4> <p>2つのデータ系列の傾向の違い</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+Cov%28f%2C+g%29+%3D+E%0A++++++++%5Cbiggl%28++%0A++++++++++++%5CBigl%28%0A++++++++++++++++f+_%7B+%28X+%3D+x%29+%7D+-+E%28f%29%0A++++++++++++%5CBigr%29%0A++++++++++++%5CBigl%28%0A++++++++++++++++g+_%7B+%28Y+%3D+y%29+%7D+-+E%28g%29%0A++++++++++++%5CBigr%29%0A++++++++%5Cbiggr%29+%3D+E%28fg%29+-+E%28f%29E%28g%29" alt="Cov(f, g) = E \biggl( \Bigl( f _{ (X = x) } - E(f) \Bigr) \Bigl( g _{ (Y = y) } - E(g) \Bigr) \biggr) = E(fg) - E(f)E(g)" /></p> <h4 id="標準偏差"><a href="#%E6%A8%99%E6%BA%96%E5%81%8F%E5%B7%AE">標準偏差</a></h4> <p>分散の平方根</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Csigma+%3D+%5Csqrt%7B+Var%28f%29+%7D%0A++++%3D+%5Csqrt%7B+%0A++++++++E%0A++++++++%5Cbiggl%28++%0A++++++++++++%5CBigl%28%0A++++++++++++++++f_%7B+%28X+%3D+x%29+%7D+-+E_%7B+%28f%29+%7D%0A++++++++++++%5CBigr%29+%5E+2%0A++++++++%5Cbiggr%29%0A+++++%7D" alt="\sigma = \sqrt{ Var(f) } = \sqrt{ E \biggl( \Bigl( f_{ (X = x) } - E_{ (f) } \Bigr) ^ 2 \biggr) }" /></p> <h3 id="様々な確率分布"><a href="#%E6%A7%98%E3%80%85%E3%81%AA%E7%A2%BA%E7%8E%87%E5%88%86%E5%B8%83">様々な確率分布</a></h3> <h4 id="ベルヌーイ分布"><a href="#%E3%83%99%E3%83%AB%E3%83%8C%E3%83%BC%E3%82%A4%E5%88%86%E5%B8%83">ベルヌーイ分布</a></h4> <p>コイントスのイメージ(表か裏か？)<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P%28X%7C%5Cmu%29+%3D+%5Cmu+%5E+x+%281+-+%5Cmu%29+%5E+%7B1-x%7D" alt="P(X|\mu) = \mu ^ x (1 - \mu) ^ {1-x}" /></p> <h4 id="マルチヌーイ(カテゴリカル)分布"><a href="#%E3%83%9E%E3%83%AB%E3%83%81%E3%83%8C%E3%83%BC%E3%82%A4%28%E3%82%AB%E3%83%86%E3%82%B4%E3%83%AA%E3%82%AB%E3%83%AB%29%E5%88%86%E5%B8%83">マルチヌーイ(カテゴリカル)分布</a></h4> <p>サイコロを転がすイメージ(出る目が3種類以上)</p> <h4 id="二項分布"><a href="#%E4%BA%8C%E9%A0%85%E5%88%86%E5%B8%83">二項分布</a></h4> <p>ベルヌーイ分布の多試行版<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P%28x%7C%5Clambda%2C+n%29+%3D%5Cdfrac%7Bn%21%7D%7Bx%21%28n+-+x%29%21%7D+%5Clambda+%5E+x%281+-+%5Clambda%29+%5E+%7Bn-x%7D" alt="P(x|\lambda, n) =\dfrac{n!}{x!(n - x)!} \lambda ^ x(1 - \lambda) ^ {n-x}" /></p> <h4 id="ガウス分布"><a href="#%E3%82%AC%E3%82%A6%E3%82%B9%E5%88%86%E5%B8%83">ガウス分布</a></h4> <p>釣鐘型の連続分布<br /> →指数関数を2つ並べたような形<br /> →真の分布がわからなくても、大体の予想がつく</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Cmathcal%7BN%7D%28x%3B+%5Cmu%2C+%5Csigma%5E2%29+%3D+%5Csqrt%0A++++%7B+%0A++++++++%5Cdfrac%7B1%7D%7B2+%5Cpi+%5Csigma%5E2%7D%0A++++%7D+%5Cmathrm%7Bexp%7D%0A++++%5CBigl%28%0A++++++++-+%5Cdfrac%7B1%7D%7B2+%5Csigma+%5E+2%7D%28x+-+%5Cmu%29+%5E+2%0A++++%5CBigr%29" alt="\mathcal{N}(x; \mu, \sigma^2) = \sqrt { \dfrac{1}{2 \pi \sigma^2} } \mathrm{exp} \Bigl( - \dfrac{1}{2 \sigma ^ 2}(x - \mu) ^ 2 \Bigr)" /></p> <h2 id="推定"><a href="#%E6%8E%A8%E5%AE%9A">推定</a></h2> <p>母集団を特徴づける <strong>母数</strong> を統計学的に推測すること<br /> →母数：パラメータ(平均など)</p> <ul> <li><strong>点推定</strong>：平均値などを一つの値に推定</li> <li><strong>区間推定</strong>：平均値などが存在する範囲を推定</li> <li><strong>推定量</strong>(推定関数 / estimator)：パラメータ推定のための計算方法、計算式 <ul> <li>表記例：<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Chat%7B%5Ctheta%7D%28x%29" alt="\hat{\theta}(x)" /></li> </ul></li> <li><strong>推定値</strong>(estimate)：実際に試行した結果から計算した値 <ul> <li>表記例：<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Chat%7B%5Ctheta%7D" alt="\hat{\theta}" /></li> </ul></li> </ul> <h3 id="点推定の例"><a href="#%E7%82%B9%E6%8E%A8%E5%AE%9A%E3%81%AE%E4%BE%8B">点推定の例</a></h3> <h4 id="標本平均"><a href="#%E6%A8%99%E6%9C%AC%E5%B9%B3%E5%9D%87">標本平均</a></h4> <p>母集団から取り出した標本の平均値</p> <ul> <li><strong>一致性</strong>：サンプル数が大きいほど母集団の値に近くなる</li> <li><strong>不偏性</strong>：サンプル数がいくつでも、その期待値は母集団の値と同様 <ul> <li><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+E%28%5Chat%7B%5Ctheta%7D%29+%3D+%5Ctheta" alt="E(\hat{\theta}) = \theta" /></li> </ul></li> </ul> <h4 id="標本分散"><a href="#%E6%A8%99%E6%9C%AC%E5%88%86%E6%95%A3">標本分散</a></h4> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Chat%7B%5Csigma%7D+%5E+2+%3D+%5Cdfrac%7B1%7D%7Bn%7D+%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%28x_i+-++%5Cbar%7Bx%7D%29+%5E+2" alt="\hat{\sigma} ^ 2 = \dfrac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x}) ^ 2" /></p> <h4 id="普遍分散"><a href="#%E6%99%AE%E9%81%8D%E5%88%86%E6%95%A3">普遍分散</a></h4> <p>→標本分散のばらつきを修正(サンプル数に応じて変わるのを防ぐ)</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+s+%5E+2+%3D+%5Cdfrac%7Bn%7D%7Bn+-+1%7D+%5Ctimes+%5Cdfrac%7B1%7D%7Bn%7D+%5Csum_%7Bi%3D1%7D%5E%7Bn%7D+%28x_i+-+%5Cbar%7Bx%7D%29+%5E+2+%3D+%5Cdfrac%7B1%7D%7Bn-1%7D%28x_i+-+%5Cbar%7Bx%7D%29+%5E+2" alt="s ^ 2 = \dfrac{n}{n - 1} \times \dfrac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x}) ^ 2 = \dfrac{1}{n-1}(x_i - \bar{x}) ^ 2" /></p> <h2 id="情報科学"><a href="#%E6%83%85%E5%A0%B1%E7%A7%91%E5%AD%A6">情報科学</a></h2> <h4 id="自己情報量"><a href="#%E8%87%AA%E5%B7%B1%E6%83%85%E5%A0%B1%E9%87%8F">自己情報量</a></h4> <div class="table-responsive"><table> <thead> <tr> <th>対数の底</th> <th>単位</th> </tr> </thead> <tbody> <tr> <td>2</td> <td>bit</td> </tr> <tr> <td>e (ネイピア)</td> <td>nat (natural)</td> </tr> </tbody> </table></div> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+I%28x%29+%3D+-+%5Cmathrm%7Blog%7D%28P%28x%29%29+%3D+%5Cmathrm%7Blog%7D%28W%28x%29%29" alt="I(x) = - \mathrm{log}(P(x)) = \mathrm{log}(W(x))" /></p> <p>ON/OFFのスイッチで情報を伝えるとき、情報の種類数に対して必要なスイッチの数は？<br /> →事象の数<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+W" alt="W" />のlogを取ることで求められる</p> <h4 id="シャノンエントロピ"><a href="#%E3%82%B7%E3%83%A3%E3%83%8E%E3%83%B3%E3%82%A8%E3%83%B3%E3%83%88%E3%83%AD%E3%83%94">シャノンエントロピ</a></h4> <p>= <strong>微分エントロピ</strong> (微分しているわけではない)<br /> 自己情報量の期待値 (情報の珍しさの平均値みたいなもの)</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+H%28x%29+%3D+E%28+I%28x%29+%29+%5C%5C%5C%5C++%0A++++%3D+-E+%5CBigl%28+%5Cmathrm%7Blog%7D+%5Cbigl%28P%28x%29+%5Cbigr%29+%5CBigr%29+%5C%5C%5C%5C++%0A++++%3D+-+%5Csum+%5CBigl%28+P%28x%29+%5Cmathrm%7Blog%7D+%5Cbigl%28P%28x%29+%5Cbigr%29+%5CBigr%29" alt="H(x) = E( I(x) ) \\ = -E \Bigl( \mathrm{log} \bigl(P(x) \bigr) \Bigr) \\ = - \sum \Bigl( P(x) \mathrm{log} \bigl(P(x) \bigr) \Bigr)" /></p> <h4 id="カルバック・ライブラー ダイバージェンス"><a href="#%E3%82%AB%E3%83%AB%E3%83%90%E3%83%83%E3%82%AF%E3%83%BB%E3%83%A9%E3%82%A4%E3%83%96%E3%83%A9%E3%83%BC+%E3%83%80%E3%82%A4%E3%83%90%E3%83%BC%E3%82%B8%E3%82%A7%E3%83%B3%E3%82%B9">カルバック・ライブラー ダイバージェンス</a></h4> <p>同じ事象・確率変数における異なる確率分布<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P" alt="P" />, <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+Q" alt="Q" />の違いを表す<br /> →想定していた確率分布：<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+Q" alt="Q" />、実際の確率分布：<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P" alt="P" /><br /> 距離のようなもの(厳密には違う)<br /> 例：普通のコインと不正なコインの、表と裏が出る確率の違い</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+D_%7BKL%7D%28P%7C%7CQ%29+%3D+%5Coverbrace%7B%0A++++++++%5Cmathbb%7BE%7D_%7Bx+%5Csim+P%7D%0A++++%7D%5E%7B%281%29%7D+%5Cbiggl%5B+%0A++++++++%5Coverbrace%7B+%0A++++++++++++%5Cmathrm%7Blog%7D+%5Cdfrac%7BP%28x%29%7D%7BQ%28x%29%7D+%0A++++++++%7D%5E%7B%282%29%7D%0A++++%5Cbiggr%5D%0A%3D+%5Coverbrace%7B+%0A+++++++++%5Cmathbb%7BE%7D_%7Bx+%5Csim+P%7D+%0A++++%7D%5E%7B%281%29%7D+%5Cbigl%5B+%0A++++++++%5Coverbrace%7B+%0A++++++++++++%5Cmathrm%7Blog%7DP%28x%29+-+%5Cmathrm%7Blog%7DQ%28x%29+%0A++++++++%7D%5E%7B%282%29%7D%0A++++%5Cbigr%5D" alt="D_{KL}(P||Q) = \overbrace{ \mathbb{E}_{x \sim P} }^{(1)} \biggl[ \overbrace{ \mathrm{log} \dfrac{P(x)}{Q(x)} }^{(2)} \biggr] = \overbrace{ \mathbb{E}_{x \sim P} }^{(1)} \bigl[ \overbrace{ \mathrm{log}P(x) - \mathrm{log}Q(x) }^{(2)} \bigr]" /></p> <p><code>(1)</code> について、</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+E+%5Cbigl%28+f%28x%29+%5Cbigr%29+%3D+%5Csum_%7Bx%7D+P%28x%29f%28x%29" alt="E \bigl( f(x) \bigr) = \sum_{x} P(x)f(x)" /></p> <p><code>(2)</code>について、</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+I%28Q%28x%29%29+-+I%28P%28x%29%29+%3D+%0A++++%5CBigl%28%0A++++++++-+%5Cmathrm%7Blog%7D+%5Cbigl%28Q%28x%29+%5Cbigr%29+%0A++++%5CBigr%29+-%0A++++%5CBigl%28+%0A++++++++-+%5Cmathrm%7Blog%7D+%5Cbigl%28P%28x%29+%5Cbigr%29+%0A++++%5CBigr%29+%3D+%5Cmathrm%7Blog%7D+%5Cdfrac%7BP%28x%29%7D%7BQ%28x%29%7D" alt="I(Q(x)) - I(P(x)) = \Bigl( - \mathrm{log} \bigl(Q(x) \bigr) \Bigr) - \Bigl( - \mathrm{log} \bigl(P(x) \bigr) \Bigr) = \mathrm{log} \dfrac{P(x)}{Q(x)}" /></p> <p>よって</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+D_%7BKL%7D%28P%7C%7CQ%29+%3D+%5Csum_%7Bx%7D+P%28x%29+%0A++++%5Cbiggl%28+%0A++++++++%5CBigl%28%0A++++++++++++-+%5Cmathrm%7Blog%7D+%5Cbigl%28+Q%28x%29+%5Cbigr%29+%0A++++++++%5CBigr%29+-+%0A++++++++%5CBigl%28+%0A++++++++++++-+%5Cmathrm%7Blog%7D+%5Cbigl%28+P%28x%29+%5Cbigr%29+%0A++++++++%5CBigr%29+%0A++++%5Cbiggr%29+%0A++++%3D+%5Csum_%7Bx%7D+P%28x%29+%5Cmathrm%7Blog%7D+%5Cdfrac%7BP%28x%29%7D%7BQ%28x%29%7D++" alt="D_{KL}(P||Q) = \sum_{x} P(x) \biggl( \Bigl( - \mathrm{log} \bigl( Q(x) \bigr) \Bigr) - \Bigl( - \mathrm{log} \bigl( P(x) \bigr) \Bigr) \biggr) = \sum_{x} P(x) \mathrm{log} \dfrac{P(x)}{Q(x)} " /></p> <h4 id="交差エントロピー"><a href="#%E4%BA%A4%E5%B7%AE%E3%82%A8%E3%83%B3%E3%83%88%E3%83%AD%E3%83%94%E3%83%BC">交差エントロピー</a></h4> <p>KLダイバージェンスの一部を取り出したもの<br /> <img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+Q" alt="Q" />(想定していた信号)についての自己情報量を<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+P" alt="P" />(現実の信号)の分布で平均<br /> エントロピーは<img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+H" alt="H" />で表す</p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+D_%7BKL%7D%28P%7C%7CQ%29+%3D+%5Csum_%7Bx%7D+%0A++++%5Coverbrace%7B+P%28x%29+%7D%5E%7B%281%29%2C+%282%29%7D+%0A++++%5Cbiggl%28+%0A++++++++%5CBigl%28%0A++++++++++++%5Coverbrace%7B%0A++++++++++++++++-+%5Cmathrm%7Blog%7D+%5Cbigl%28+Q%28x%29+%5Cbigr%29+%0A++++++++++++%7D%5E%7B%281%29%7D%0A++++++++%5CBigr%29+-+%0A++++++++%5CBigl%28+%0A++++++++++++%5Coverbrace%7B%0A++++++++++++++++-+%5Cmathrm%7Blog%7D+%5Cbigl%28+P%28x%29+%5Cbigr%29+%0A++++++++++++%7D%5E%7B%282%29%7D%0A++++++++%5CBigr%29+%0A++++%5Cbiggr%29+" alt="D_{KL}(P||Q) = \sum_{x} \overbrace{ P(x) }^{(1), (2)} \biggl( \Bigl( \overbrace{ - \mathrm{log} \bigl( Q(x) \bigr) }^{(1)} \Bigr) - \Bigl( \overbrace{ - \mathrm{log} \bigl( P(x) \bigr) }^{(2)} \Bigr) \biggr) " /></p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+%5Coverbrace%7B%0A++++++++H%28P%2C+Q%29%0A+++++%7D%5E%7Bfrom+%281%29%7D+%3D+%0A+++++%5Coverbrace%7B%0A+++++++++H%28P%29%0A++++++%7D%5E%7Bfrom+%282%29%7D+%0A++++++%2B+D_%7BKL%7D%28P%7C%7CQ%29" alt="\overbrace{ H(P, Q) }^{from (1)} = \overbrace{ H(P) }^{from (2)} + D_{KL}(P||Q)" /></p> <p><img src="https://render.githubusercontent.com/render/math?math=%5Cdisplaystyle+H%28P%2C+Q%29+%0A++++%3D+-+%5Cmathbb%7BE%7D_%7Bx+%5Csim+P%7D+%5Cmathrm%7Blog%7D+Q%28x%29+%0A%3D+%5Csum_%7Bx%7D+P%28x%29+%5Cmathrm%7Blog%7D+Q%28x%29" alt="H(P, Q) = - \mathbb{E}_{x \sim P} \mathrm{log} Q(x) = \sum_{x} P(x) \mathrm{log} Q(x)" /></p> marshmallow444