Nonlinear Gaussian State Space with Linear Observation Matrix Derivation
$begingroup$
For the following State Space Model:
$x_k = f(x_{k-1}) + v_k ~~ sim ~ Bbb N(0_{n_{v times 1}}, sum_v) \
y_k = Cx_{k} + w_k ~~ sim ~ Bbb N(0_{n_{w times 1}}, sum_w)$
where $f: Bbb R^{n_{x}} rightarrow R^{n_{x}} $ is a real valued nonlinear function, $ C ~epsilon R^{n_{y} small times normalsize n_x} ~$ is an observation matrix, $v_k$ and $w_k$ are mutually independent i.i.d Gaussian sequences with $sum_v$ > 0 and $sum_w$ > 0, $C$, $sum_v$ and $sum_w$ are assumed known.
Let $x_k ~|~ x_{k-1}, y_{k} sim Bbb N(m_k, sum)$.
Then $m_k$, $sum$ is given by
$m_k = sum(sum^{-1}_vf(x_{k-1})+ C^tsum^{-1}_wy_k)
\ sum^{-1} = sum^{-1}_v + C^tsum^{-1}_wC$
Can somebody please help me how this relation $x_k~|~x_{k-1}, y_k$ has been derived.
estimation-theory non-linear-dynamics
asked Dec 5 '18 at 13:41
lathelathe
12
$endgroup$
add a comment |
$begingroup$
For the following State Space Model:
$x_k = f(x_{k-1}) + v_k ~~ sim ~ Bbb N(0_{n_{v times 1}}, sum_v) \
y_k = Cx_{k} + w_k ~~ sim ~ Bbb N(0_{n_{w times 1}}, sum_w)$
where $f: Bbb R^{n_{x}} rightarrow R^{n_{x}} $ is a real valued nonlinear function, $ C ~epsilon R^{n_{y} small times normalsize n_x} ~$ is an observation matrix, $v_k$ and $w_k$ are mutually independent i.i.d Gaussian sequences with $sum_v$ > 0 and $sum_w$ > 0, $C$, $sum_v$ and $sum_w$ are assumed known.
Let $x_k ~|~ x_{k-1}, y_{k} sim Bbb N(m_k, sum)$.
Then $m_k$, $sum$ is given by
$m_k = sum(sum^{-1}_vf(x_{k-1})+ C^tsum^{-1}_wy_k)
\ sum^{-1} = sum^{-1}_v + C^tsum^{-1}_wC$
Can somebody please help me how this relation $x_k~|~x_{k-1}, y_k$ has been derived.
estimation-theory non-linear-dynamics
asked Dec 5 '18 at 13:41
lathelathe
12
$endgroup$
add a comment |
$begingroup$
For the following State Space Model:
$x_k = f(x_{k-1}) + v_k ~~ sim ~ Bbb N(0_{n_{v times 1}}, sum_v) \
y_k = Cx_{k} + w_k ~~ sim ~ Bbb N(0_{n_{w times 1}}, sum_w)$
where $f: Bbb R^{n_{x}} rightarrow R^{n_{x}} $ is a real valued nonlinear function, $ C ~epsilon R^{n_{y} small times normalsize n_x} ~$ is an observation matrix, $v_k$ and $w_k$ are mutually independent i.i.d Gaussian sequences with $sum_v$ > 0 and $sum_w$ > 0, $C$, $sum_v$ and $sum_w$ are assumed known.
Let $x_k ~|~ x_{k-1}, y_{k} sim Bbb N(m_k, sum)$.
Then $m_k$, $sum$ is given by
$m_k = sum(sum^{-1}_vf(x_{k-1})+ C^tsum^{-1}_wy_k)
\ sum^{-1} = sum^{-1}_v + C^tsum^{-1}_wC$
Can somebody please help me how this relation $x_k~|~x_{k-1}, y_k$ has been derived.
estimation-theory non-linear-dynamics
asked Dec 5 '18 at 13:41
lathelathe
12
$endgroup$
For the following State Space Model:
$x_k = f(x_{k-1}) + v_k ~~ sim ~ Bbb N(0_{n_{v times 1}}, sum_v) \
y_k = Cx_{k} + w_k ~~ sim ~ Bbb N(0_{n_{w times 1}}, sum_w)$
where $f: Bbb R^{n_{x}} rightarrow R^{n_{x}} $ is a real valued nonlinear function, $ C ~epsilon R^{n_{y} small times normalsize n_x} ~$ is an observation matrix, $v_k$ and $w_k$ are mutually independent i.i.d Gaussian sequences with $sum_v$ > 0 and $sum_w$ > 0, $C$, $sum_v$ and $sum_w$ are assumed known.
Let $x_k ~|~ x_{k-1}, y_{k} sim Bbb N(m_k, sum)$.
Then $m_k$, $sum$ is given by
$m_k = sum(sum^{-1}_vf(x_{k-1})+ C^tsum^{-1}_wy_k)
\ sum^{-1} = sum^{-1}_v + C^tsum^{-1}_wC$
Can somebody please help me how this relation $x_k~|~x_{k-1}, y_k$ has been derived.
estimation-theory non-linear-dynamics
estimation-theory non-linear-dynamics
asked Dec 5 '18 at 13:41
lathelathe
12
asked Dec 5 '18 at 13:41
lathelathe
12
asked Dec 5 '18 at 13:41
lathelathe
12
asked Dec 5 '18 at 13:41
lathelathe
12
asked Dec 5 '18 at 13:41
lathelathe
12
12
add a comment |
add a comment |
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