Extended Kalman Filter

拡張カルマンフィルタ

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Ptt

 Ptt (Ptm:jaxtyping.Float[Array,'NN'], w:jaxtyping.Float[Array,'N'],
      x:jaxtyping.Float[Array,'N'])

\(\!\) 推定誤差共分散行列 \(\mathbf P_{t/t}\) \[\sigma_t=\sigma(\hat{\mathbf w}_{t/t-1}^T\mathbf x_t)\] \[\mathbf P_{t/t}=\mathbf P_{t/t-1}-\frac{\sigma_t(1-\sigma_t)}{1+\sigma_t(1-\sigma_t)\mathbf x_t^T\mathbf P_{t/t-1}\mathbf x_t}(\mathbf P_{t/t-1}\mathbf x_t)(\mathbf P_{t/t-1}\mathbf x_t)^T\] \(\!\)

	Type	Details
Ptm	Float[Array, ‘N N’]	\(\mathbf P_{t/t-1}\)
w	Float[Array, ‘N’]	\(\hat{\mathbf w}_{t/t-1}\)
x	Float[Array, ‘N’]	\(\mathbf x_t\)
Returns	Float[Array, ‘N N’]	\(\mathbf P_{t/t}\)

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wtt

 wtt (Ptm:jaxtyping.Float[Array,'NN'], w:jaxtyping.Float[Array,'N'],
      x:jaxtyping.Float[Array,'N'], y:jaxtyping.Float[Array,'N'])

\(\!\) 濾波推定値 \(\hat{\mathbf w}_{t/t}\) \[\hat{\mathbf w}_{t/t}=\hat{\mathbf w}_{t/t-1}+\frac{1}{1+\sigma_t(1-\sigma_t)\mathbf x_t^T\mathbf P_{t/t-1}\mathbf x_t}\mathbf P_{t/t-1}\mathbf x_t(y_t-\sigma_t)\] \(\!\)

	Type	Details
Ptm	Float[Array, ‘N N’]	\(\mathbf P_{t/t-1}\)
w	Float[Array, ‘N’]	\(\hat{\mathbf w}_{t/t-1}\)
x	Float[Array, ‘N’]	\(\mathbf x_t\)
y	Float[Array, ‘N’]	\(y_t\)
Returns	Float[Array, ‘N’]	\(\hat{\mathbf w}_{t/t}\)

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EKF_out

 EKF_out (W:jaxtyping.Float[Array,'TN'], P:jaxtyping.Float[Array,'TNN'])

\(\!\) EKF 関数の返り値

\(\!\)	Type	Details
W	Float[Array, ‘T N’]	\(\{\hat{\mathbf w}_{t/t}\}_{t=0,\ldots,T-1}\)
P	Float[Array, ‘T N N’]	\(\{\mathbf P_{t/t}\}_{t=0,\ldots,T-1}\)

\(\!\)

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EKF

 EKF (N:int, T:int, x:jaxtyping.Float[Array,'{T}{N}'],
      y:jaxtyping.Float[Array,'{T}{N}'],
      G:jaxtyping.Float[Array,'{N}{N}'], w0:jaxtyping.Float[Array,'{N}'],
      P0:jaxtyping.Float[Array,'{N}{N}'])

\(\!\) 拡張カルマンフィルタ \(\!\)

	Type	Details
N	int	\(N\)
T	int	\(T\)
x	Float[Array, ‘{T} {N}’]	\(\{ \mathbf x_t \}_{t=0,\ldots,T-1}\)
y	Float[Array, ‘{T} {N}’]	\(\{ y_t \}_{t=0,\ldots,T-1}\)
G	Float[Array, ‘{N} {N}’]	\(\boldsymbol\Gamma\)
w0	Float[Array, ‘{N}’]	\(\hat{\mathbf w}_{0/-1}\)
P0	Float[Array, ‘{N} {N}’]	\(\mathbf P_{0/-1}\)
Returns	EKF_out