第6講 各種データ解析法

$$\begin{equation} \left\{ \begin{array}{l} z_{1 i} = a_{1 1} f_{1 i} + a_{1 2} f_{2i} + \cdots + a_{1 m} f_{m i} + e_{1 i} \\ ~~~~~~~~\vdots \\ z_{j i} = a_{j 1} f_{1 i} + a_{j 2} f_{2 i} + \cdots + a_{j m} f_{m i} + e_{j i} \\ ~~~~~~~~\vdots \\ z_{p i} = a_{p 1} f_{1 i} + a_{j 2} f_{2 i} + \cdots + a_{p m} f_{m i} + e_{p i} \\ \end{array} \right.~~~~\cdots\cdots~(1) \end{equation}$$

$$\begin{equation} i = 1, 2, \ldots, n \end{equation}$$

$$\begin{equation} f_{k} \sim (0, 1^2) \end{equation}$$

$$\begin{equation} e_{j} \sim (0, d_j^2) \end{equation}$$

$$\begin{equation} z_{j} = a_{j 1} f_{1} + a_{j 2} f_{2} + \cdots + a_{j m} f_{m} + e_{j}~~~~\cdots\cdots~(2) \end{equation}$$

$$\begin{align} E[z_{j}] &= E[a_{j 1} f_{1} + a_{j 2} f_{2} + \cdots + a_{j m} f_{m} + e_{j}] \\ &= E[a_{j 1} f_{1}] + E[a_{j 2} f_{2} + \cdots + E[a_{j m} f_{m}] + E[e_{j}] \\ &= a_{j 1} E[f_{1}] + a_{j 2} E[f_{2}] + \cdots + a_{j m} E[f_{m}] + E[e_{j}] \\ &= 0 \end{align}$$

$$\begin{align} Cov[f_k, f_{k'}] &= E[f_k f_{k'}] - E[f_k] E[f_{k'}] \\ &= E[f_k f_{k'}] \\ &= 0 \end{align}$$

$$\begin{align} E[{f_k}^2] &= V[f_k] + (E[f_k])^2 \\ &= V[f_k] \\ &= 1 \end{align}$$

$$\begin{align} E[f_k e_j] &= Cov[f_k, e_j] + E[f_k] E[e_j] \\ &= 0 \end{align}$$

$$\begin{align} \sigma_{j j} &= V[z_{j}] \\ &= V[a_{j 1} f_{1} + a_{j 2} f_{2} + \cdots + a_{j m} f_{m} + e_{j}] \\ &= V\left[\sum_{k=1}^{m} a_{j k} f_{k} + e_{j}\right] \\ &= \sum_{k=1}^{m} V[a_{j k} f_{k}] + V[e_{j}] \\ &= \sum_{k=1}^{m} {a_{j k}}^2 V[f_{k}] + V[e_{j}] \\ &= \sum_{k=1}^{m} {a_{j k}}^2 + {d_{j}}^2~~~~\cdots\cdots~(3) \end{align}$$

$$\begin{align} \sigma_{j j'} &= Cov[z_{j},~z_{j'}] \\ &= E[z_{j} z_{j'}] - E[z_{j}] E[z_{j'}] \\ &= E[z_{j} z_{j'}] \\ &= E[(a_{j 1} f_{1} + a_{j 2} f_{2} + \cdots + a_{j m} f_{m} + e_{j})(a_{j' 1} f_{1} + a_{j' 2} f_{2} + \cdots + a_{j' m} f_{m} + e_{j'})] \\ &= E\left[\sum_{k=1}^{m}a_{j' k} f_{k} \cdot a_{j k} f_{k} \right] + E\left[\sum_{k \ne k'} a_{j k} f_{k} \cdot a_{j' k'} f_{k'} \right] + E\left[\sum_{k=1}^{m}a_{j' k} f_{k} \cdot e_{j'} \right] + E\left[\sum_{k'=1}^{m}e_j \cdot a_{j' k'} f_{k'} \right] + E[e_j e_{j'}] \\ &= \sum_{k=1}^{m}E[a_{j' k} f_{k} \cdot a_{j k} f_{k}] + \sum_{k \ne k'} E[a_{j k} f_{k} \cdot a_{j' k'} f_{k'}] + \sum_{k=1}^{m}E[a_{j' k} f_{k} \cdot e_{j'}] + \sum_{k'=1}^{m}E[e_j \cdot a_{j' k'} f_{k'}] + E[e_j e_{j'}] \\ &= \sum_{k=1}^{m}a_{j' k}a_{j k} E[{f_{k}}^2] + \sum_{k \ne k'}a_{j k}a_{j' k'} E[ f_{k} f_{k'}] + \sum_{k=1}^{m} a_{j' k} E[f_{k} e_{j'}] + \sum_{k'=1}^{m} a_{j' k'} E[e_j f_{k'}] + E[e_j e_{j'}] \\ &= \sum_{k=1}^{m}a_{j' k}a_{j k}~~~~\cdots\cdots~(4) \end{align}$$

$$\begin{equation} \bm{z}_{i} = A \bm{f}_i + \bm{e}_i,~~~~i = 1, \ldots, n \end{equation}$$

$$\begin{equation} \bm{z}_{i} = \begin{pmatrix} z_{i 1} \\ \vdots \\ z_{i p} \end{pmatrix} \end{equation}$$

$$\begin{equation} A = \begin{pmatrix} a_{1 1} & \cdots & a_{1 m} \\ \vdots & \ddots & \vdots \\ a_{p 1} & \cdots & a_{p m} \\ \end{pmatrix} \end{equation}$$

$$\begin{equation} \bm{f}_{i} = \begin{pmatrix} f_{1 i} \\ \vdots \\ f_{m i} \end{pmatrix} \end{equation}$$

$$\begin{equation} \bm{e}_{i} = \begin{pmatrix} e_{i 1} \\ \vdots \\ e_{i p} \end{pmatrix} \end{equation}$$

$$\begin{equation} \varSigma = A A' + D~~~~\cdots\cdots~(5) \end{equation}$$

$$\begin{equation} \varSigma = \begin{pmatrix} \sigma_{11} & \cdots & \sigma_{1p} \\ \vdots & \ddots & \vdots \\ \sigma_{p1} & \cdots & \sigma_{pp} \end{pmatrix} \end{equation}$$

$$\begin{equation} D = \begin{pmatrix} {d_1}^2 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {d_p}^2 \\ \end{pmatrix} \end{equation}$$

$$\begin{align} \bm{z}_{i} &= A \bm{f}_i + \bm{e}_i \\ &= A (T T^{-1}) \bm{f}_i + \bm{e}_i \\ &= A^{*} \bm{f}_i^{*} + \bm{e}_i \\ \end{align}$$

$$\begin{equation} A^{*} = AT \end{equation}$$

$$\begin{equation} \bm{f}_i^{*} = T^{-1}\bm{f}_i \end{equation}$$