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\title[]
{ポスター数式}

% \subtitle[short subtitle]{{\color{black}Developing Web Applications to Support Data Analysis \\
% by Visual Programming for Decision Making}
% }
\author[]
{
川口　晏璃
}

%\author[]{}
\institute[]
{
\scriptsize{
% Graduate School of Engineering\\
% Department of Information Systems Engineering\\
% Toyama Prefectural University\\


%\vspace{10mm}

\begin{large}

\end{large}
}
}
% \date{Teams, 9:50-10:15 Friday., December 4, 2020. }

%\date{L205, AM 9:25-9:50 Friday., December 8, 2018,\\
 %Toyama Prefectural Univ. }
%\date{\today}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle


%%%%%%%%%%%%%%%%%%%% 図2 %%%%%%%%%%%%%%%%%%%%%

\section{poster図2}
\begin{frame}[t]
    \frametitle{数式}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize
        % \setbeamercolor{structure}{fg=red}
        \begin{block}{Shapley値の一般的な求め方}
            \begin{itemize}
                \item[$\triangleright$] $N$人のプレイヤー$N=\{1,2,\cdots,N\}$がゲームに参加する
                \item[$\triangleright$] $S$は$N$からプレイヤー$j$を除いた参加者の組み合わせ（例えば，A君B君C君のケースで，A君のShapley値を考える場合，$\{\varnothing\}$，$\{B\}$，$\{C\}$，$\{B,C\}$が該当．）

                      \begin{equation*}
                          \phi_j=\sum_{S\subseteq N\backslash  \{j\}}\frac{|S|!(|N|-|S|-1)!}{|N|!}(v(S\cup \{j\})-v(S))
                      \end{equation*}
            \end{itemize}
        \end{block}
    \end{scriptsize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \section{図1}
% \begin{frame}[t]
%     \frametitle{}
%     \begin{scriptsize}
%         \vspace{-2mm}
%         \normalsize

%         % \setbeamercolor{structure}{fg=yellow}
%         % \begin{block}{安全在庫量の計算}
%         %     ${安全係数}\times{使用量の標準偏差}\times\sqrt{発注リードタイム+発注間隔} $
%         % \end{block}
%         \begin{block}<>{安全在庫量の計算}<>

%         \end{block}


%     \end{scriptsize}
% \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{}
\begin{frame}[t]
    \frametitle{}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize
        データセットから$x_i$と$y_i$を参照サンプルの値とするサンプル$\alpha$を考える．このとき，データ間の誤差を次の式で求めることができる．
        \begin{equation*}
            e_\alpha = f_\alpha(\mathbf{A},\mathbf{M},\mathbf{\textit{v}})= (\mathbf{A}^T\mathbf{M}\mathbf{\textit{v}} - \mathbf{A}^T\mathbf{M}\mathbf{A}\mathbf{\textit{z}})_\alpha
        \end{equation*}

        ここで $()_\alpha$ は $\alpha$ 行の値の選択を表す．したがって，すべての誤差関数の総和は，残差の絶対値の総和を最小化する重回帰モデルを用いて，$E = \displaystyle\sum_{\alpha} |e_\alpha|$ で与えられる．以下は，上の等式の最適化問題である．

    \end{scriptsize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{}
\begin{frame}[t]
    \frametitle{}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize
        \begin{exampleblock}{線形計画法によるshapley値}
            minimize $\epsilon$ \\
            subject to $A^TMv+ s^+ - s^- = A^TMAz$ \\
            \hspace{15mm}
            $\displaystyle\sum_{c\in S}{z_{\alpha}(S,v)}=v(S)$ \\
            \hspace{15mm}
            $0 \leq s^{+}\leq \epsilon,\; 0 \leq s^{-} \leq \epsilon$ \\

            \vspace{3mm}
            ここで $s^+ = [s_1^{+},s_2^{+},\cdots,s_n^{+}]^T$ と $s^- = [s_1^{-},s_2^{-},\cdots,s_n^{-}]^T$ はスラック変数とサープラス変数の集合を表し， $\mathbf{A}$ は超加法性の性質を持つ行列は超加法性を持つ行列，$\mathbf{v}$は$\mathbf{v}(S)$の実数値を要素とする列行列，$\mathbf{M}$は以下の式で得られる対角線行列である．

            \[
                M_{n,s} =\frac{1}{n - 2} \binom{n -2}{s - 1}^{-1}
            \]

            ここで，$n$はプレイヤーの数を表し，$s$は集合$S$における提携を表す．
        \end{exampleblock}

    \end{scriptsize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{}
\begin{frame}[t]
    \frametitle{}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize
        \begin{exampleblock}{二次形式のshapley値}
            minimize $\displaystyle\sum_{s \subset N}\big( v(S) - x(S) \big) ^2m(s)$ \\
            subject to $\displaystyle\sum_{i \in N}{x_i}{=v(N)}$ \\
            \hspace{15mm}
            $x_i \geq v(i),\; (\forall \;i \in N)$ \\

            \vspace{3mm}
            ここで，$m(s)=M$は重み関数であり，$x$は任意の集合$S$に対する$x(S) = \displaystyle\sum_{i \in S} x_i$のペイオフ・ベクトルである．二次プログラムの最適解は，個人とグループの合理性を制約条件として，shapley値を表す．

            % \[
            %     M_{n,s} =\frac{1}{n - 2} \binom{n -2}{s - 1}^{-1}
            % \]

            % ここで，$n$はプレイヤーの数を表し，$s$は集合$S$における提携を表す．
        \end{exampleblock}

    \end{scriptsize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%　図2ここまで　%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%　図4　%%%%%%%%%%%%%%%%%%%%%%%%%
\section{図4}
\begin{frame}[t]
    \frametitle{}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize
        計画期間を$n$期間とする．$i(\leq n)$期の不確実な需要量を$d_i$，需要量の期待値を$\overline{d}_i$，需要量の分散を$\omega_i^2$とし，互いに独立な正規分布に従うとする．
        また，$1$から$n$期までの累積需要量を$D$とし，互いに独立な正規分布に従うとする．

        \vspace{3mm}
        特性関数$v(s)$を求める．\\
        有意水準$\alpha$とし，$i$期の在庫量に影響を及ぼす需要量$D_i$とその特性を求めておく．
        \vspace{-6mm}
        \begin{equation*}
            D_i\equiv d_1 + d_2 + \cdots + d_i=\sum_{n = 1}^{i}d_n
        \end{equation*}
        \vspace{-15mm}
        である．
    \end{scriptsize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{}
\begin{frame}[t]
    \frametitle{}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize

        $D_i$の特性は，$i<j$として，\\
        \vspace{-4mm}
        \begin{center}
            $ E[D_i]=\displaystyle\sum_{n = 1}^{i}\overline{d}_n$ \\
            $Var[D_i]=\omega_1^2 + \omega_2^2 + \cdots + \omega_i^2 = \displaystyle\sum_{n = 1}^{i}\omega_n^2 \equiv\sigma_{ii} $ \\
            $Cov[D_i,D_j]=\omega_1^2 + \omega_2^2 + \cdots + \omega_i^2 = \displaystyle\sum_{n = 1}^{i}\omega_n^2 \equiv\sigma_{ij} $\\
        \end{center}
        となる．すなわち，$D_i$の分散共分散行列$\Sigma$は，\\

        % \begin{eqnarray}
        % \Sigma 
        % =
        % \begin{bmatrix}
        % \omega_1^2&\omega_1^2&\cdots&\omega_1^2\\
        % \omega_1^2&\omega_1^2 + \omega_2^2&\ddots&\vdots\\
        % \vdots&\ddots&\ddots&\omega_1^2 + \omega_2^2 + \cdots + \omega_{n-1}^2\\
        % \omega_1^2&\cdots&\omega_1^2 + \omega_2^2 + \cdots + \omega_{n-1}^2&\omega_1^2 + \omega_2^2+ \cdots + \omega_n^2\\
        % \end{bmatrix}
        % \hspace{}
        % \\
        % \equiv 
        % \begin{bmatrix}
        % \sigma_{11} & \sigma_{12} & \cdots & \sigma_{1n} \\
        % \sigma_{21} & \sigma_{22} & \ddots & \vdots \\
        % \vdots & \ddots & \ddots &  \\
        % \sigma_{n1} & \cdots &  & \sigma_{nn} \\
        % \end{bmatrix}
        % \end{eqnarray}
    \end{scriptsize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{}
\begin{frame}[t]
    \frametitle{}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize

        \begin{eqnarray*}
            \begin{split}
                \Sigma
                &=
                \begin{bmatrix}
                    \omega_1^2 & \omega_1^2              & \cdots                                            & \omega_1^2                                        \\
                    \omega_1^2 & \omega_1^2 + \omega_2^2 & \ddots                                            & \vdots                                            \\
                    \vdots     & \ddots                  & \ddots                                            & \omega_1^2 + \omega_2^2 + \cdots + \omega_{n-1}^2 \\
                    \omega_1^2 & \cdots                  & \omega_1^2 + \omega_2^2 + \cdots + \omega_{n-1}^2 & \omega_1^2 + \omega_2^2+ \cdots + \omega_n^2      \\
                \end{bmatrix}
                \\
                &\equiv
                \begin{bmatrix}
                    \sigma_{11} & \sigma_{12} & \cdots & \sigma_{1n} \\
                    \sigma_{21} & \sigma_{22} & \ddots & \vdots      \\
                    \vdots      & \ddots      & \ddots &             \\
                    \sigma_{n1} & \cdots      &        & \sigma_{nn} \\
                \end{bmatrix}
            \end{split}
        \end{eqnarray*}
        となる．
        %     これらを用いて，提携$S$の特性関数は，\\
        %     \begin{equation*}
        %     v(S)=-AVaR_{D(S)|D}(1-\alpha)=-\left({\sum_{i \in S}^{}\overline{d}_i}+{K\sqrt{\sum_{i \in S}^{} \sum_{j \in S}^{}\sigma_{ij}}}\right)
        % \end{equation*}


    \end{scriptsize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{}
\begin{frame}[t]
    \frametitle{}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize
        これらを用いて，提携$S$の特性関数は，\\
        \vspace{-6mm}
        \begin{eqnarray*}
            \begin{split}
                v(S)=-AVaR_{D(S)|D}(1-\alpha)
                &=
                E[D|D > VaR_{D}(1-\alpha)] \\
                &=
                -\left({\sum_{i \in S}^{}\overline{d}_i}+{K\sqrt{\sum_{i \in S}^{} \sum_{j \in S}^{}\sigma_{ij}}}\right)
            \end{split}
        \end{eqnarray*}
        \\
        \begin{equation*}
            K=\frac{\varphi (z_{1-\alpha})}{1-\Phi (z_{1-\alpha})}
        \end{equation*}
        \begin{equation*}
            z_{1-\alpha}=\frac{VaR_{D(S)}(1-\alpha)-\sum_{i \in S}^{}\overline{d}_i}{\sqrt{\sum_{i \in S}^{} \sum_{j \in S}^{}\sigma_{ij}   } }
        \end{equation*}

        である．
    \end{scriptsize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%% 図6 %%%%%%%%%%%%%%%%%%%%%%%
\section{poster図6}
\begin{frame}[t]
    \frametitle{数式}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize
        %%%%%%%%%% 表ここから %%%%%%%%%%
        \begin{table}[htb]
            \scriptsize
            \caption{$5$期間の特性関数$v(S)$}
            \scalebox{1.3}[1.3]{
                \begin{tabular}{|c|c||c|c|} \hline
                    提携               & $v(S)$    & 提携         & $v(S)$    \\ \hline \hline
                    $v\{\varnothing\}$ & 0         & $v\{123\}$   & -51.4360  \\ \hline
                    $v\{1\}$           & -22.1255  & $v\{124\}$   & -65.6824  \\ \hline
                    $v\{2\}$           & --15.8343 & $v\{125\}$   & -74.5018  \\ \hline
                    $v\{3\}$           & -15.1455  & $v\{134\}$   & -67.5026  \\ \hline
                    $v\{4\}$           & -32.2508  & $v\{135\}$   & -73.9824  \\ \hline
                    $v\{5\}$           & -39.2248  & $v\{145\}$   & -91.3499  \\ \hline
                    $v\{12\}$          & -37.2248  & $v\{234\}$   & -61.7849  \\ \hline
                    $v\{13\}$          & -36.1051  & $v\{235\}$   & -66.9050  \\ \hline

                    $v\{14\}$          & -52.9148  & $v\{245\}$   & -85.4364  \\ \hline
                    $v\{15\}$          & -59.6685  & $v\{345\}$   & -85.3370  \\ \hline
                    $v\{23\}$          & -30.3762  & $v\{1234\}$  & -82.5958  \\ \hline

                    $v\{24\}$          & -47.0458  & $v\{1235\}$  & -88.9693  \\ \hline
                    $v\{25\}$          & -53.6824  & $v\{1245\}$  & -106.0551 \\ \hline
                    $v\{34\}$          & -46.8745  & $v\{1345\}$  & -105.7632 \\ \hline
                    $v\{35\}$          & -53.4360  & $v\{2345\}$  & -95.3370  \\ \hline
                    $v\{45\}$          & -71.0095  & $v\{12345\}$ & -120.5949 \\ \hline
                \end{tabular}
            }
        \end{table}

        %%%%%%%%%% 表ここまで %%%%%%%%%%

    \end{scriptsize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{}
\begin{frame}[t]
    \frametitle{}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize
        $1$から$5$期間の重み関数$\textbf{M}$は，

        \begin{eqnarray*}
            \begin{split}
                \textbf{M}
                =
                \begin{bmatrix}
                    0.2500 & 0      & 0      & 0      \\
                    0      & 0.0833 & 0      & 0      \\
                    0      & 0      & 0.0833 & 0      \\
                    0      & 0      & 0      & 0.2500 \\
                \end{bmatrix}
                \\
            \end{split}
        \end{eqnarray*}
        である．また分散共分散行列$\Sigma$は，

        \begin{eqnarray*}
            \begin{split}
                \Sigma
                &=
                \begin{bmatrix}
                    4 & 4 & 4  & 4  & 4  \\
                    4 & 8 & 8  & 8  & 8  \\
                    4 & 8 & 12 & 12 & 12 \\
                    4 & 8 & 12 & 16 & 16 \\
                    4 & 8 & 12 & 16 & 20
                \end{bmatrix}
                \\
            \end{split}
        \end{eqnarray*}

        となる．
    \end{scriptsize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{}
\begin{frame}[t]
    \frametitle{}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize
        \begin{alertblock}{安全在庫量の計算}
            \vspace{-6mm}
            \begin{eqnarray*}
                \mbox{安全係数}
                \times
                \mbox{使用量の標準偏差}
                \times
                \sqrt{\mbox{発注リードタイム}+\mbox{発注間隔}}
            \end{eqnarray*}
        \end{alertblock}

    \end{scriptsize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{}
\begin{frame}[t]
    \frametitle{}
    \begin{scriptsize}
        \vspace{-2mm}
        \normalsize
        計画目標未達率を$\beta$とすると，期別の未達率$\beta_i$は，
        \vspace{-2mm}
        \begin{equation*}
            \beta_i=1-\sqrt[n]{1-\beta}(\forall _i\leq n) 
        \end{equation*}
        \vspace{-2mm}
        と与えられる．
    \end{scriptsize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

%%%%%%%%%%　スライド　%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{}
% \begin{frame}[t]
%   \frametitle{}
%   \begin{scriptsize}
%     \vspace{-2mm}
%     \normalsize

%   \end{scriptsize}
% \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%　ブロック　%%%%%%%%%%
% \begin{block}{}

% \end{block}


% \begin{exampleblock}{}

% \end{exampleblock}


% \begin{alertblock}{}

% \end{alertblock}

%%%%%%%%%%  図挿入  %%%%%%%%%%
% \begin{figure}[h]
%   \begin{center}
%     \includegraphics[width=100mm]{img/mpich.png}
%     \vspace{-3mm}
%     \caption{\small{実行結果}}
%   \end{center}
%   \vspace{-7mm}
% \end{figure}

%%%%%%%%%%　表　%%%%%%%%%%
% \begin{table}[htb]
%     \scriptsize
%     \scalebox{0.7}[1.2]{
%       \begin{tabular}{|c|c|c|} \hline
%         $\bm{M}(\theta)\in \Re^{n \times n}$: 慣性行列 & $\bm{h}(\theta,\dot{\theta})\in \Re^{n \times 1}$: コリオリ力・遠心力 & $\bm{g}(\theta)\in \Re^{n \times 1}$: 重力トルク \\ \hline
%         $\theta$: 関節角度 & $\bm{\tau}\in \Re^{n \times 1}$: 関節トルク & $\bm{J}(\theta)\in \Re^{m \times n}$: ヤコビ行列 \\ \hline
%         $\bm{F}_{int}\in \Re^{n \times 1}$: 手先に作用する力 & $\bm{M}_e\in \Re^{m \times m}$: 慣性& $\bm{B}_e\in \Re^{m \times m}$: 粘性  \\ \hline
%         $\bm{K}_e\in \Re^{m \times m}$: 剛性 & $\bm{X}$ $\in$ $ \Re^{m \times 1}$: 手先位置 & $\bm{X}_e\in \Re^{m \times 1}$: 対象物の平衡点の位置 \\ \hline
%         $\bm{F}_{act}\in \Re^{m \times 1}$: 作業空間で表現した制御入力 & $\bm{M}_d\in \Re^{m \times m}$: 目標慣性行列 & $\bm{B}_d \in \Re^{m \times m}$: 目標粘性行列 \\ \hline
%         $\bm{K}_d \in \Re^{m \times m}$: 目標剛性行列 & $\bm{X}_d\in \Re^{m \times 1}$: 目標軌道 & $\bm{F}_d\in \Re^{m \times 1}$: 目標手先力 \\ \hline
%         $d\bm{X} = \bm{X} - \bm{X}_d$ &  &  \\ \hline
%       \end{tabular}
%       }
%     \end{table}

%%%%%%%%%%　minipage　%%%%%%%%%%
% \begin{figure}[htbp]
%   \begin{minipage}{0.4\hsize}
%     \begin{center}
%       % \fbox{
%         \includegraphics[width=45mm]{img/table1.png}
%         % }
%     \end{center}
%     \vspace{-5mm}
%     % \caption{}
%   \end{minipage}
%   \hspace{7mm}
%   \begin{minipage}{0.4\hsize}
%     \begin{center}
%       % \fbox{
%         \includegraphics[width=45mm]{img/table2.PNG}
%         % }
%     \end{center}
%     \vspace{-5mm}
%     % \caption{}
%   \end{minipage}
%   \vspace{3mm}
% \end{figure}

%%%%%%%%%%　footnote　%%%%%%%%%%
% \footnote{\scriptsize{Muhammad Alkarouri, \lq\lq Parallel Genetic Algorithms: A Typical MPI Application\rq\rq,\\\hspace{5.3mm}Internatinal Centre for Theoretical Physics, Student Presentation}}


% \fontsize{7pt}{0pt}\selectfont
% [1] FRONTEO株式会社, ``Patent Explorer | 人工知能KIBIT ビジネスソリューション｜FRONTEO'', https://kibit.fronteo.com/products/patent-explorer/
% [2] 津村拓海, 斎藤史哲, 石津昌平, ランダムフォレストを用いた特許に関する文書デー
% タからの技術適用領域に関する知識抽出", 日本経営工学会論文誌, vol. 68, No. 3, pp.
% 161-170, 2017.

%%%%%%%%%%　箇条書き　%%%%%%%%%%

% \begin{block}{既存研究のアルゴリズム}
%   \setbeamertemplate{items}[square]
%   \begin{enumerate}\setcounter{enumi}{0}
%     \item 欲張り法
%     \item 帰着スケジュール生成
%     \item 局所探索法
%   \end{enumerate}
% \end{block}

%%%%%%%%%%　三角箇条　%%%%%%%%%%
% \begin{block}{まとめ}
%   \begin{itemize}
%     \item[$\triangleright$] 最新のpythonのバージョン(python3.8)を入れた．
%     \item[$\triangleright$] GAのプログラムを探して，実行できることを確認した．
%     \item[$\triangleright$] python用mpiライブラリを入れた．
%   \end{itemize}
% \end{block}

%%%%%%%%%%　blockの色指定　%%%%%%%%%%
% \setbeamercolor{structure}{fg=red}

%%%%%%%%%%　式　番号なし　%%%%%%%%%%
% \begin{equation*}
%     \phi_j=\sum_{S\subseteq N\backslash  \{j\}}\frac{|S|!(|N|-|S|-1)!}{|N|!}(v(S\cup \{j\})-v(S))  
% \end{equation*}

%%%%%%%%%%　式番号　%%%%%%%%%%
% \begin{equation}
%   {g_{i}(x)}\leq 0
%   \hspace{4mm}
%   (1,2,\cdots,m)
% \end{equation}