本稿は、「 第1章 線形時系列解析と非線形時系列解析の違い 」を参照しています。
$\Large \displaystyle y(t) = \sum_{n=-\infty}^{\infty}L(n)w(t-n)$
$\Large \displaystyle l(t) = \sum_{n=-\infty}^{\infty}L(n)z^n, z\in G$
$\Large \displaystyle \sum_{n=0}^{p}A(n)y(t-n) = \sum_{n=1}^{r}D(n)z(t-n)+\sum_{n=0}^{q}B(n)e(t-n)$
$\Large \displaystyle T\{f(t)\} = g(t)$
$\Large \displaystyle L\{\alpha_1 f_1(t)+\alpha_2 f_2(t) \} = \alpha_1 L\{f_1(t)\}+\alpha_2 L\{f_2(t)\}$
$\Large \displaystyle L\{f(t)\} = \frac{df}{dt}$
$\Large \displaystyle L\{f(t)\} = f(t-t_1)$
$\Large \displaystyle L\{\delta(t-t_1)\} = h(t;t_1)$
$\Large \displaystyle f(x)=\int_{-\infty}^{\infty}f(t_1)\delta(t-t_1)dt_1$
$\Large \displaystyle L\{f(x)\}=\int_{-\infty}^{\infty}f(t_1)L\{\delta(t-t_1)\}dt_1 =\int_{-\infty}^{\infty}f(t_1)h(t;t_1)dt_1$
$\Large \displaystyle g(t) = \int_{-\infty}^{\infty}f(t_1)h(t;t_1)dt_1$
$\Large \displaystyle L\{f(t-t_1)\} = g(t-t_1)$
$\Large \displaystyle L\{\delta(t)\} = h(t)$
$\Large \displaystyle L\{\delta(t-t_1)\} =h(t:t_1)=h(t-t_1)$
$\Large \displaystyle g(t) = \int_{-\infty}^{\infty}f(\tau)h(t;\tau)d\tau =\int_{-\infty}^{\infty}f(\tau)h(t-\tau)d\tau =\int_{-\infty}^{\infty}f(t-\tau)h(\tau)d\tau =f(t)*g(t)$
$\Large \displaystyle F(\omega)=\int_{-\infty}^{\infty}p_T(t)e^{-j\omega t}dt=\int_{-T}^{T}e^{-j\omega t}dt$
$\Large \displaystyle =\frac{1}{-j\omega}[e^{-j\omega T}-e^{j\omega T}]=\frac{2\sin(\omega T)}{\omega}$
$\Large \displaystyle \int_{-\infty}^{\infty}f(t-t_0)e^{j\omega t}dt=\int_{-\infty}^{\infty}f(x)e^{-j\omega(t_0 + x)}dx=F(\omega)e^{-j t_0 \omega} $
$\Large \displaystyle p_T(t-t_0) \leftrightarrow \frac{2\sin(\omega T)}{\omega} e^{-j\omega t_0}$
$\Large \displaystyle F^*(\omega) = \sum_{n=-\infty}^{\infty}\frac{\pi}{\omega_c}f_n e^{-jn\pi\frac{\omega}{\omega_c}}$
$\Large \displaystyle f^*(t) = \sum_{n=-\infty}^{\infty}\frac{\pi}{\omega_c}f_n \delta(t-\frac{n\pi}{\omega_c})$
$\Large \displaystyle F(t) \leftrightarrow 2\pi f(-\omega)$
$\Large \displaystyle f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)e^{j\omega t}dw$
$\Large \displaystyle 2\pi f(-t)=\int_{-\infty}^{\infty} F(\omega)e^{-j\omega t}dw$
$\Large \displaystyle 2\pi f(-\omega)=\int_{-\infty}^{\infty} F(t)e^{-j\omega t}dt$
$\Large \displaystyle X(e^{j\omega})=\sum_{n=-\infty}^{\infty}x(n)e^{-j\omega}$
$\Large \displaystyle X(z)=\sum_{n=-\infty}^{\infty}x(n)z^{-n}$
$\Large \displaystyle x(n)=\frac{1}{2\pi i}\oint_C X(z)z^{n-1}dz$
$\Large \displaystyle X^T=(1,x_1)$
$\Large \displaystyle \hat{y}(x.w)=w_0+w_1 x_1$
$\Large \displaystyle \hat{y}(x,w)=w_0+w_1 x_1$
$\Large \displaystyle =(w_0,w_1) \begin{pmatrix} 1 \\ x_1 \end{pmatrix} $
$\Large \displaystyle =w^T \phi(x)$
$\Large \displaystyle X^T=(1,x_1,x_2,\cdots,x_D)$
$\Large \displaystyle \hat{y}(x.w)=w_0+w_1 x_1+w_2 x_2+\cdots+w_D x_D$
$\Large \displaystyle \hat{y}(x,w)=w_0+w_1 x_1+w_2 x_2+\vdots++w_D x_D$
$\Large \displaystyle =(w_0,w_1,w_2,\cdots,w_D) \begin{pmatrix} 1 \\ x_1 \\ x_2 \\ \vdots \\ x_D \end{pmatrix} $
$\Large \displaystyle =w^T \phi(x)$
$\Large \displaystyle \hat{y}(x,w)=w_0\phi(x_0)+w_1\phi(x_1)+\cdots+w_d\phi(x_D)$
$\Large \displaystyle \hat{y}(x,w)=w_0+w_1 x_1+w_2 x_1^2+w_3 \sin(x_1)$
$\Large \displaystyle \hat{y}(x,w)=w_0+w_1 x_1+w_2 x_1^2+w_3 \sin(x_1)$
$\Large \displaystyle =(w_0,w_1,w_2,w_3) \begin{pmatrix} 1 \\ x_1 \\ x_1^2 \\ \sin(x_1) \end{pmatrix} $
$\Large \displaystyle =w^T \phi(x)$
$\Large \displaystyle E=\sum_{n=1}^{N}(y_n-\hat{y}_n)^2$
$\Large \displaystyle =\sum_{n=1}^{N}(y_n-w^T \phi(x_n))^2$
$\Large \displaystyle \frac{\partial E}{\partial w}=\frac{\partial}{\partial w}\sum_{n=1}^{N}(y_n-\hat{y}_n)^2$
$\Large \displaystyle =\frac{\partial}{\partial w}\sum_{n=1}^{N}(y_n-w^T \phi(x_n))^2=0$
$\Large \displaystyle \begin{pmatrix} \hat{y}_1 \\ \hat{y}_2 \\ \vdots \\ \hat{y}_N \end{pmatrix} = \begin{pmatrix} w^T\phi(x_1) \\ w^T\phi(x_2) \\ \vdots \\ w^T\phi(x_N) \\ \end{pmatrix} $
= $\Large \displaystyle \begin{pmatrix} \phi(x_1)^T \\ \phi(x_2)^T \\ \vdots \\ \phi(x_N)^T \\ \end{pmatrix} w $
= $\Large \displaystyle \begin{pmatrix} \phi_0(x_1) & \phi_1(x_1) & \cdots & \phi_H(x_1) \\ \phi_0(x_2) & \phi_1(x_2) & \cdots & \phi_H(x_2) \\ \vdots \\ \phi_0(x_N) & \phi_1(x_N) & \cdots & \phi_H(x_N) \\ \end{pmatrix} w $
$\Large \displaystyle \hat{y}= \begin{pmatrix} \hat{y}_1 \\ \hat{y}_2 \\ \vdots \\ \hat{y}_N \end{pmatrix} $
$\Large \displaystyle \Phi= \begin{pmatrix} \phi_0(x_1) & \phi_1(x_1) & \cdots & \phi_H(x_1) \\ \phi_0(x_2) & \phi_1(x_2) & \cdots & \phi_H(x_2) \\ \vdots \\ \phi_0(x_N) & \phi_1(x_N) & \cdots & \phi_H(x_N) \\ \end{pmatrix} $
$\Large \displaystyle x_1^2+x_2^2+\cdots x_n^2= (x_1,x_2,\cdots,x_n) \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \\ \end{pmatrix} =x^T x $
$\Large \displaystyle w^T x=w_1 x_1+w_2 x_2+\cdots+w_n x_n$
$\Large \displaystyle \frac{\partial}{\partial w}w^T x= \begin{pmatrix} \frac{\partial}{\partial w_1}w^T x \\ \frac{\partial}{\partial w_2}w^T x \\ \vdots \\ \frac{\partial}{\partial w_n}w^T x \\ \end{pmatrix} $
$\Large \displaystyle = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \\ \end{pmatrix} $
$\Large \displaystyle =x$
$\Large \displaystyle y_t=\phi_0 + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \cdots + \phi_p y_{t-p} + \epsilon_t$
$\Large \displaystyle y_t=\phi_0 + \phi_1 y_{t-1} + \epsilon_t$
$\Large \displaystyle E[y_t]=\phi_0 + \phi_1E[y_{t-1}] + E[\epsilon_t]$
$\Large \displaystyle \mu=\phi_0 + \phi_1\mu$
$\Large \displaystyle \mu=\frac{\phi_0}{1-\phi_2}$
$\Large \displaystyle E[{y_t}^2]=\phi_0E[y_t] + \phi_1E[y_t y_{t-1}] + E[y_t\epsilon_t]$
$\Large \displaystyle x(n)=-a_{p1}x(n-1)-a_{p2}x(n-2)-\cdots-a_{pp}x(n-p)+ \epsilon(n)$
$\Large \displaystyle p(x_t|y_{1:t})\simeq \sum_{i=1}^{M}\frac{w_t^i}{\sum_{i=1}^{M}w_t^i}\delta(x_t-x_{t|t-1}^i)$
$\Large \displaystyle p(x)=\frac{1}{\sqrt{2\pi\sigma^2}}exp\{-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}\}$
$\Large \displaystyle \mu=E[X]$
$\Large \displaystyle V[X]=E[(X-\mu)^2]$
$\Large \displaystyle \sigma=\sqrt{V[X]}$
$\Large \displaystyle Cov[X,Y]=E[(X-\mu)(Y-\nu)]$
$\Large \displaystyle Cov[X,Y]=Cov[Y,X]$
$\Large \displaystyle Cov[X,X]=V[X]$
$\Large \displaystyle Cov[X+a,Y+b]=Cov[X,Y]$
$\Large \displaystyle Cov[aX,bY]=abCov[X,Y]$
$\Large \displaystyle \rho_{XY}=\frac{Cov[X,Y]}{\sqrt{\mathstrut V[X]}\sqrt{\mathstrut V[Y]}}$
$\Large \displaystyle \hat{X}=\frac{X}{\sqrt{V[X]}}$
$\Large \displaystyle \hat{Y}=\frac{Y}{\sqrt{V[Y]}}$
$\Large \displaystyle Cov[\hat{X},\hat{Y}]=Cov[\frac{X}{\sqrt{V[X]}},\frac{Y}{\sqrt{V[Y]}}]=\frac{Cov[X,Y]}{\sqrt{\mathstrut V[X]}\sqrt{\mathstrut V[Y]}}=\rho_{XY}$
$X_1$ | $X_2$ | $X_3$ | |
---|---|---|---|
$X_1$ | $Cov(X_1,X_1)$ | $Cov(X_1,X_2)$ | $Cov(X_1,X_3)$ |
$X_2$ | $Cov(X_2,X_1)$ | $Cov(X_2,X_2)$ | $Cov(X_2,X_3)$ |
$X_3$ | $Cov(X_3,X_1)$ | $Cov(X_3,X_2)$ | $Cov(X_3,X_3)$ |
$\Large \displaystyle V[X]= \begin{pmatrix} V[X_1] & Cov[X_1,X_2] & Cov[X_1,X_3] \\ Cov[X_2,X_1] & V[X_2] & Cov[X_2,X_3] \\ Cov[X_3,X_1] & Cov[X_3,X_2] & V[X_3] \\ \end{pmatrix} $
$\Large \displaystyle X= \begin{pmatrix} X_1 \\ X_2 \\ \vdots \\ X_n \end{pmatrix} $
$\Large \displaystyle V[X]=E[(X-\mu)(X-\mu)^T]$
$\Large \displaystyle \mu=E[X]$
$\Large \displaystyle V[X]= \begin{pmatrix} V[X_1] & Cov[X_1,X_2] & Cov[X_1,X_3] \\ Cov[X_2,X_1] & V[X_2] & Cov[X_2,X_3] \\ Cov[X_3,X_1] & Cov[X_3,X_2] & V[X_3] \\ \end{pmatrix} $
$\Large \displaystyle \gamma_k =\frac{1}{T}\sum_{t=k+1}^{T}(y_t-\bar{y})(y_{t-k}-\bar{y})$
$\Large \displaystyle \rho_f(t) =\int_{-\infty}^{\infty}f(t+\tau)f(\tau)d\tau$
$\Large \displaystyle f(x)=\int_{-\infty}^{\infty}f_1(y)f_2(x-y)dy=f_1(x)*f_2(x)$
$\Large \displaystyle \int_{-\infty}^{\infty}f_1(\tau)f_2(t-\tau)d\tau \leftrightarrow F_1(\omega)*F_2(\omega) $
$\Large \displaystyle f_1(t)f_2(t) \leftrightarrow \frac{1}{2\pi}\int_{-\infty}^{\infty}F_1(y)F_2(\omega-y)dy$
$\Large \displaystyle F(\omega) =A(\omega)e^{j\phi(\omega)}$
$\Large \displaystyle E(\omega) =A^2(\omega)$
$\Large \displaystyle E(\omega) =A^2(\omega)$
$\Large \displaystyle g(t) =\int_{-\infty}^{\infty}f(\tau)h(t-\tau)d\tau=f(t)*g(t)$
$\Large \displaystyle \rho_g(t) =\rho_f(t)*h(t)*h(-t)$
$\Large \displaystyle g(t)\leftrightarrow F(\omega)*H(\omega)$
以下の記事は、 「 自己相関関数と周期性解析」を引用しています。
$\Large \displaystyle a_{n+2}=c_{1}a_{n+1}+c_{2}a_{n}\ \ \ (n=0,1,2,\cdots)$
$\Large \displaystyle e_1=(1,0,\cdots)$
$\Large \displaystyle e_2=(0,1,\cdots)$
$\Large \displaystyle a=(a_0,a_1,\cdots)$
$\Large \displaystyle a=a_0 e_1+a_1 e_2$
$\Large \displaystyle y(k)=\sum_{l=0}^{\infty}g(l)u(k-l)$
$\Large \displaystyle y_t=\int_0^t g(\tau)u(t-\tau)d\tau=g(t)*u(t)$
$\Large \displaystyle y(z)=G(z)u(z)$
$\Large \displaystyle u(z)=\sum_{l=0}^{\infty}u(k)z^{-k}$
$\Large \displaystyle y(z)=\sum_{l=0}^{\infty}y(k)z^{-k}$
$\Large \displaystyle G(z)=\sum_{l=0}^{\infty}g(k)z^{-k}$
$\Large \displaystyle y(k)+a_1 y(k-1)+\cdots+a_n y(k-n)=b_1 u(k-1)+b_2 u(k-2)+\cdots+b_mu(k-m)$
$\Large \displaystyle (1+a_1 z^{-1}+\cdots+a_n z^{-n})y(z)=(b_1 z^{-1}+b_2 z^{-2}+\cdots+b_m z^{-m})u(z)$
$\Large \displaystyle G(z)=\frac{y(z)}{u(z)}=\frac{b_1 z^{-1}+b_2 z^{-2}+\cdots+b_m z^{-m}}{1+a_1 z^{-1}+\cdots+a_n z^{-n}}$
$\Large \displaystyle qy(k)=y(k+1)$
$\Large \displaystyle q^{-1}y(k)=y(k-1)$
$\Large \displaystyle A(q)y(k)=B(q)u(k)+w(k)$
以上
$\Large \displaystyle x(k+1)=x(k)+v(k)$
$\Large \displaystyle y(k)=x(k)+w(k)$
$\Large \displaystyle x(0)\sim N(0,\sigma_0^2)$
$\Large \displaystyle v(k)\sim N(0,\sigma_v^2)$
$\Large \displaystyle w(k)\sim N(0,\sigma_w^2)$
$\Large \displaystyle y=cx +w$
$\Large \displaystyle \hat{x}=f(y)=\alpha y + \beta$
$\Large \displaystyle e=x-\hat{x}$
$\Large \displaystyle E[e]=E[x-\hat{x}]=E[x-\alpha y - \beta]$
$\Large \displaystyle =E[x-\alpha (cx+w) - \beta]$
$\Large \displaystyle =(1-\alpha c)\bar{x}-\alpha\bar{w}-\beta=0$
$\Large \displaystyle \hat{\beta}=(1-\alpha c)\bar{x}-\alpha\bar{w}=\bar{x}-\alpha(c\bar{x}+\bar{w})$
$\Large \displaystyle E[\{e-E(e)\}^2]=E[[(x-\alpha y-\beta)-\{(1-\alpha c)\bar{x}-\alpha \bar{w}-\beta \}]^2]$
$\Large \displaystyle =E[\{ x-\alpha(cx+w)-(1-\alpha c)\bar{x}+\alpha \bar{w}\}^2]$
$\Large \displaystyle =E[\{ (1-\alpha c)(x-\bar{x})-\alpha(w-\bar{w}) \}^2]$
$\Large \displaystyle =E[(1-\alpha c)^2(x-\bar{x})^2+\alpha^2(w-\bar{w})^2 -2\alpha(1-\alpha c)(x-\bar{x})(w-\bar{w}) ]$
$\Large \displaystyle =(1-\alpha c)^2E[(x-\bar{x})^2]+\alpha^2E[(w-\bar{w})^2] -2\alpha(1-\alpha c)E[(x-\bar{x})(w-\bar{w})] $
$\Large \displaystyle =(1-\alpha c)^2\sigma_x^2+\alpha^2\sigma_w^2$
$\Large \displaystyle E[\{e-E(e)\}^2]=(1-2\alpha c+\alpha^2 c^2)\sigma_x^2+\alpha^2\sigma_w^2$
$\Large \displaystyle =(c^2 \sigma_x^2 +\sigma_w^2)\alpha^2 - 2c\sigma_x^2 \alpha + \sigma_x^2$
$\Large \displaystyle =(c^2 \sigma_x^2 +\sigma_w^2)[\alpha^2 -\frac{2c\sigma_x^2}{c^2 \sigma_x^2 +\sigma_w^2}\alpha+\frac{\sigma_w^2}{c^2 \sigma_x^2 +\sigma_w^2} ]$
$\Large \displaystyle =(c^2 \sigma_x^2 +\sigma_w^2)(\alpha -\frac{c\sigma_x^2}{c^2 \sigma_x^2 + \sigma_w^2})^2 - \frac{c^2 \sigma_x^4}{c^2 \sigma_x^2 + \sigma_w^2} + \sigma_x^2$
$\Large \displaystyle =(c^2 \sigma_x^2 +\sigma_w^2)(\alpha -\frac{c\sigma_x^2}{c^2 \sigma_x^2 + \sigma_w^2})^2 - \frac{\sigma_x^2 \sigma_w^2}{c^2 \sigma_x^2 + \sigma_w^2}$
$\Large \displaystyle =(c^2 \sigma_x^2 +\sigma_w^2)(\alpha -\frac{c\sigma_w^{-2}}{c^2 \sigma_w^{-2} + \sigma_x^{-2}})^2 - \frac{1}{c^2 \sigma_w^{-2} + \sigma_x^{-2}}$
$\Large \displaystyle \sigma^2=\frac{1}{c^2 \sigma_w^{-2} + \sigma_x^{-2}}$
$\Large \displaystyle E[\{e-E(e)\}^2]=(c^2 \sigma_x^2+\sigma_w^2)(\alpha - c\sigma_w^{-2}\sigma^2)^2+\sigma^2$
$\Large \displaystyle \alpha=c\frac{\sigma^2}{\sigma_w^2}$
$\Large \displaystyle E[\{e-E(e)\}^2]=\sigma^2$
$\Large \displaystyle x(k+1)=Ax(k)+bv(k)$
$\Large \displaystyle y(k)=c^Tx(k)+w(k)$
$\Large \displaystyle x(1)=x(0)+v(0)$
$\Large \displaystyle y(1)=x(1)+w(1)=x(0)+v(0)+w(1)$
$\Large \displaystyle \begin{pmatrix} y(1) \\ x(1) \end{pmatrix} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} w(1) \\ v(0) \\ x(0) \end{pmatrix} $
$\Large \displaystyle x(k+1)=Ax(k)+bv(k)$
$\Large \displaystyle y(k)=c^Tx(k)+w(k)$