16.4.1: 予測部分の計算:
$z_s
=
\Phi^{s-1,s}_* (\widetilde{z}_{s-1})$
in (16.9)(in $\S$16.3)
次の補題を用意しておく.
$(B_1):$
$
\int_{- \infty}^{\infty}
\frac{1}{\sqrt{2 \pi} A}
\exp[
-\frac{(x -By )^2}{2A^2}
]
\frac{1}{\sqrt{2 \pi} C}
\exp[
-\frac{(y- D)^2}{2C^2}
]
dy
=
\frac{1}{\sqrt{2 \pi} \sqrt{A^2 + B^2C^2}}
\exp[
-\frac{(x- BD)^2}{2(A^2 + B^2C^2)}
]
$
ここで, 記号"$\approx${\rq\rq}の意味は,以下の通りとする.
\begin{align*}
"f(\omega) \approx g(\omega )"
\Longleftrightarrow
\mbox{"$\frac{f(\omega)}{g(\omega )}
= $定数"
}
\end{align*}
$(B_2):$
$
\exp[
-\frac{(A\omega -B )^2}{2E^2}
]
\exp[
-\frac{(C \omega- D)^2}{2F^2}
]
\approx
\exp
[
-\frac{1}{2}
(
\frac{A^2 F^2+ C^2 E^2}{E^2 F^2}
)
\Big(\omega -
\frac{(
{AB}{F^2}
+
{CD}{E^2}
)}{({A^2}{F^2} + {C^2}{E^2})}
\Big)^2
]
$
証明
証明は,初等的で簡単なので,省略する.
さて,
(16.3)と(D)から, 次を得る.
\begin{align}
\widetilde{z}_0(\omega_0)=&\lim_{\Xi_0 \to x_0} \frac{F(\Xi_0) z_0}{\int_{\mathbb R} F(\Xi_0) z_0 d \omega_0}
\nonumber \\
\approx
&
\frac{1}{\sqrt{2 \pi} {q}_0}
\exp[
-\frac{(x_0 -c_0 \omega_0 -d_0 )^2}{2{q}_0^2}
]
\frac{1}{\sqrt{2 \pi} \sigma_0}
\exp[-
\frac{(\omega_0-\mu_0)^2}{2 \sigma_0^2}
]
\nonumber \\
\approx
&
\frac{1}{\sqrt{2 \pi} \widetilde{\sigma}_0}
\exp[-
\frac{(\omega_0-\widetilde{\mu}_0)^2}{2 \widetilde{\sigma}_0^2}
]
\tag{16.10}
\end{align}
ここで,
\begin{align}
\widetilde{\sigma}_0^2=\frac{q_0^2 \sigma_0^2}{q_0^2 + c_0^2 \sigma_0^2},
\quad
\widetilde{\mu}_0
=
\mu_0 + \widetilde{\sigma}_0^2 (
\frac{c_0}{q_0^2})(x_0-d_0- c_0 \mu_0)
\tag{16.11}
\end{align}
さらに,
補題16.3の
(B$_1$)と(16.6)
から,
次が言える.
\begin{align}
z_1(\omega_1)
&
=
[\Phi_*^{0,1}
\widetilde{z}_0](\omega_1)
\nonumber \\
&
=
\int_{- \infty}^{\infty}
\frac{1}{\sqrt{2 \pi} r_1}
\exp[
-\frac{(\omega_1 - a_1 \omega_{0} -b_1 )^2}{2r_1^2}
]
\frac{1}{\sqrt{2 \pi} \widetilde{\sigma}_0}
\exp[-
\frac{(\omega_0-\widetilde{\mu}_0)^2}{2 \widetilde{\sigma}_0^2}
]
d \omega_0
\nonumber \\
&
=
\frac{1}{\sqrt{2 \pi}{\sigma}_1}
\exp[-
\frac{(\omega_1-{\mu}_1)^2}{2 {\sigma_1}^2}
]
\tag{16.12}
\end{align}
ここで,
\begin{align}
{\sigma}_1^2=\ a_1^2 \widetilde{\sigma}_0^2 + r_1^2,
\quad
{\mu}_1
=
a_1 \widetilde{\mu}_0 + b_1
\tag{16.13}
\end{align}
したがって,
補題16.3の(B$_2$)から,次を得る.
\begin{align}
\widetilde{z}_{t-1}(\omega_{t-1})
=
&
\lim_{\Xi_{t-1} \to x_{t-1}} \frac{F(\Xi_{t-1}) z_{t-1}}{\int_{\mathbb R} F(\Xi_{t-1}) z_{t-1} d \omega_{t-1}}
\nonumber \\
\approx
&
\frac{1}{\sqrt{2 \pi} {q}_{t-1}}
\exp[
-\frac{(x_{t-1} -c_{t-1} \omega_{t-1} - d_{t-1})^2}{2{q}_{t-1}^2}
]
\frac{1}{\sqrt{2 \pi} \sigma_{t-1}}
\exp[-
\frac{(\omega_{t-1}-\mu_{t-1})^2}{2 \sigma_{t-1}^2}
]
\nonumber \\
\approx
&
\frac{1}{\sqrt{2 \pi} \widetilde{\sigma}_{t-1}}
\exp[-
\frac{(\omega_{t-1}-\widetilde{\mu}_{t-1})^2}{2 \widetilde{\sigma}_{t-1}^2}
]
\tag{16.14}
\end{align}
ここで,
\begin{align}
\widetilde{\sigma}_{t-1}^2 &=\frac{q_{t-1}^2 \sigma_{t-1}^2}{q_{t-1}^2 + c_{t-1}^2 \sigma_{t-1}^2}
=
\sigma_{t-1}^2
\frac{q_{t-1}^2 + c_{t-1}^2 \sigma_{t-1}^2 +q_{t-1}^2 -q_{t-1}^2 - c_{t-1}^2 \sigma_{t-1}^2}{q_{t-1}^2 + c_{t-1}^2 \sigma_{t-1}^2}
\nonumber \\
&
=
\sigma_{t-1}^2
(1-
\frac{ c_{t-1}^2 \sigma_{t-1}^2}{q_{t-1}^2 + c_{t-1}^2 \sigma_{t-1}^2}
)
\quad
\nonumber \\
\widetilde{\mu}_{t-1}
&
=
\mu_{t-1} + \widetilde{\sigma}_{t-1}^2 (
\frac{c_{t-1}}{q_{t-1}^2})(x_{t-1}- c_{t-1} \mu_{t-1})
\tag{16.15}
\end{align}
さらに,補題16.3の(E$_1$)により,
\begin{align}
z_t(\omega_{t})
&
=
[\Phi_*^{{t-1},{t}}
\widetilde{z}_{t-1}](\omega_{t})
\nonumber \\
&
\approx
\int_{- \infty}^{\infty}
\frac{1}{\sqrt{2 \pi} r_{t}}
\exp[
-\frac{(\omega_{t} - a_{t} \omega_{{t-1}} -b_{t} )^2}{2r_{t}^2}
]
\frac{1}{\sqrt{2 \pi} \widetilde{\sigma}_{t-1}}
\exp[-
\frac{(\omega_{t-1}-\widetilde{\mu}_{t-1})^2}{2 \widetilde{\sigma}_{t-1}^2}
]
d \omega_{t-1}
\nonumber \\
&
\approx
\frac{1}{\sqrt{2 \pi} {\sigma}_{t}}
\exp[-
\frac{(\omega_{t}-{\mu}_{t})^2}{2 {\sigma_{t}}^2}
]
\tag{16.16}
\end{align}
where
\begin{align}
{\sigma}_t^2=\ a_t^2 \widetilde{\sigma}_{t-1}^2 + r_t^2,
\quad
{\mu}_t
=
a_t \widetilde{\mu}_{t-1} + b_t
\tag{16.17}
\end{align}
以上の(16.10)--(16.17)をまとめると,
\begin{align*}
{\small
\overset{\mbox{}}{\underset{\mu_0, \sigma_0}{\fbox{$z_0$}}}
\xrightarrow[(16.11)]{\mbox{$x_0$}}
\underset{\widetilde{\mu}_0, \widetilde{\sigma}_0}{\fbox{$\widetilde{z}_0$}}
\xrightarrow[(16.13)]{\Phi_*^{0,1}}
\overset{\mbox{}}{\underset{\mu_1, \sigma_1}{\fbox{$z_1$}}}
\xrightarrow[]{\mbox{$x_1$}}
\cdots
\xrightarrow[]{\Phi_*^{t-2,t-1}}
\overset{\mbox{}}{\underset{\mu_{t-1}, \sigma_{t-1}}{\fbox{$z_{t-1}$}}}
\xrightarrow[(16.15)]{\mbox{$x_{t-1}$}}
\underset{\widetilde{\mu}_{t-1}, \widetilde{\sigma}_{t-1}}{\fbox{$\widetilde{z}_{t-1}$}}
\xrightarrow[(16.17)]{\Phi_*^{t-1,t}}
\overset{\mbox{}}{\underset{\mu_t, \sigma_t}{\fbox{$z_t$}}}
\xrightarrow[]{\mbox{$x_{t+1}$}}
\cdots
\xrightarrow[]{\Phi_*^{s-1,s}}
\overset{\mbox{}}{\underset{\mu_s, \sigma_s}{\fbox{$z_s$}}}
}
\end{align*}
したがって,次を得る:
\begin{align}
z_s
=
\Phi^{s-1,s}_* (\widetilde{z}_{s-1})
\tag{16.18}
\end{align}
in (16.9) (in $\S$16.3).
16.4: カルマンフィルタ:予測の部分
This web-site is the html version of "Linguistic Copehagen interpretation of quantum mechanics; Quantum language [Ver. 4]" (by Shiro Ishikawa; [home page] )
PDF download : KSTS/RR-18/002 (Research Report in Dept. Math, Keio Univ. 2018, 464 pages)
補題16.3
次が成立する.