Probabilistische Systemmodelle Mit Additivem Rauschen Allgemein:
z βΎ = a βΎ ( x βΎ ) + v βΎ
\underline{z} = \underline{a}(\underline{x}) + \underline{v}
z β = a β ( x β ) + v β β \Rightarrow β
f ( z βΎ β£ x βΎ ) = f v ( z βΎ β a βΎ ( x βΎ ) )
f(\underline{z} \mid \underline{x})=f_v(\underline{z}-\underline{a}(\underline{x}))
f ( z β β£ x β ) = f v β ( z β β a β ( x β )) Beispiel:
z = x 2 + v v βΌ f v ( v )
z = x^2 + v \qquad v \sim f_v(v)
z = x 2 + v v βΌ f v β ( v ) Gesucht: f ( z β£ x ) f(z|x) f ( z β£ x )
f ( z β£ x , v ) = Ξ΄ ( z β x 2 β v ) , f ( z , v β£ x ) = f ( z β£ x , v ) β
f v ( v )
f(z \mid x, v)=\delta\left(z-x^{2}-v\right), \quad f(z, v \mid x)=f(z \mid x, v) \cdot f_v(v)
f ( z β£ x , v ) = Ξ΄ ( z β x 2 β v ) , f ( z , v β£ x ) = f ( z β£ x , v ) β
f v β ( v ) f ( z β£ x ) = Marginalisierung β« R f ( z , v β£ x ) d v = β« R f ( z β£ x , v ) β
f v ( v ) d v = β« R Ξ΄ ( z β x 2 β v ) β
f v ( v ) d v = f v ( z β x 2 )
\begin{aligned}
f(z \mid x) &\overset{\text{Marginalisierung}}{=}\int_{\mathbb{R}} f(z, v \mid x) d v\\
&=\int_{\mathbb{R}} f(z \mid x, v) \cdot f_v(v) d v \\
&=\int_{\mathbb{R}} \delta\left(z-x^{2}-v\right) \cdot f_v(v) d v \\
&=f_{v}\left(z-x^{2}\right)
\end{aligned}
f ( z β£ x ) β = Marginalisierung β« R β f ( z , v β£ x ) d v = β« R β f ( z β£ x , v ) β
f v β ( v ) d v = β« R β Ξ΄ ( z β x 2 β v ) β
f v β ( v ) d v = f v β ( z β x 2 ) β In dem Fall
z = x k + 1 x = x k ,
z = x_{k + 1} \quad x = x_{k},
z = x k + 1 β x = x k β , heiΓt
f v ( z β£ x ) = f v ( x k + 1 β£ x k ) = f v ( x k + 1 β a ( x k ) ) (additive)
f_v(z \mid x) = f_v(x_{k+1} \mid x_k) = f_v(x_{k+1} - a(x_k))
\tag{additive}
f v β ( z β£ x ) = f v β ( x k + 1 β β£ x k β ) = f v β ( x k + 1 β β a ( x k β )) ( additive ) Transitionsdichte (Engl. transition density).
Mit Multiplikativem Rauschen Abbildung
z = x β
v v βΌ N ( v , 0 , Ο v )
z = x \cdot v \quad v \sim \mathcal{N}(v, 0, \sigma_v)
z = x β
v v βΌ N ( v , 0 , Ο v β ) Annahme: z , x , v z, x, v z , x , v
Gesucht: f ( z β£ x ) f(z \mid x) f ( z β£ x )
RΓΌckfΓΌhrung auf additiven Fall mit log β‘ ( β
) \log(\cdot) log ( β
)
log β‘ ( z ) β z Λ = log β‘ ( x β
v ) = log β‘ ( x ) β x Λ + log β‘ ( v ) β v Λ β z Λ = x Λ + v Λ
\underbrace{\log (z)}_{\bar{z}}=\log (x \cdot v)=\underbrace{\log (x)}_{\bar{x}}+\underbrace{\log (v)}_{\bar{v}} \Leftrightarrow \bar{z}=\bar{x}+\bar{v}
z Λ log ( z ) β β = log ( x β
v ) = x Λ log ( x ) β β + v Λ log ( v ) β β β z Λ = x Λ + v Λ Dichte von v Λ = log β‘ ( v ) \bar{v} = \log(v) v Λ = log ( v )
f ( v Λ β£ v ) = Ξ΄ ( v Λ β log β‘ ( v ) ) = exp β‘ ( v Λ ) Ξ΄ ( v β exp β‘ ( v Λ ) )
f(\bar{v} \mid v) = \delta(\bar{v} - \log(v)) = \exp(\bar{v})\delta(v - \exp(\bar{v}))
f ( v Λ β£ v ) = Ξ΄ ( v Λ β log ( v )) = exp ( v Λ ) Ξ΄ ( v β exp ( v Λ ))
\begin{aligned}
f_\bar{v}(\bar{v}) &= \int_{\mathbb{R}} f(\bar{v} \mid v) f_v(v) dv \\\\
&= \int_{\mathbb{R}} \exp(\bar{v})\delta(v - \exp(\bar{v})) f_v(v) dv \\\\
&= \exp(\bar{v}) f_v(\exp(\bar{v})) \\\\
&= \frac{1}{\sqrt{2 \pi} \sigma_{v}} \exp (\bar{v}) \exp\left\{-\frac{1}{2} \frac{[\exp(\bar{v})]^{2}}{\sigma_{v}^{2}}\right\}
\end{aligned}
Dann
\begin{aligned}
f(\bar{z} \mid \bar{x}) &= f_\bar{v}(\bar{z} - \bar{x}) \\
&= \frac{1}{\sqrt{2 \pi} \sigma_{v}} \exp \{\bar{z} - \bar{x}\} \exp\left\{-\frac{1}{2} \frac{[\exp(\bar{z} - \bar{x})]^{2}}{\sigma_{v}^{2}}\right\}
\end{aligned}
z = exp β‘ { z Λ } β g ( z Λ ) = z β exp β‘ ( z Λ ) β g β² ( z Λ ) = β exp β‘ ( z Λ ) Nullstelle : z Λ = log β‘ ( z )
\begin{aligned}
z = \exp\{\bar{z}\} &\Rightarrow g(\bar{z}) = z - \exp(\bar{z}) \\
&\Rightarrow g^{\prime}(\bar{z}) = -\exp(\bar{z}) \quad \text{Nullstelle}: \bar{z} = \log(z)
\end{aligned}
z = exp { z Λ } β β g ( z Λ ) = z β exp ( z Λ ) β g β² ( z Λ ) = β exp ( z Λ ) Nullstelle : z Λ = log ( z ) β f ( z β£ x Λ ) = 1 β£ z β£ f ( log β‘ ( z ) β£ x Λ )
f(z \mid \bar{x}) = \frac{1}{|z|} f(\log(z) \mid \bar{x})
f ( z β£ x Λ ) = β£ z β£ 1 β f ( log ( z ) β£ x Λ ) x = exp β‘ ( x Λ ) β x = \exp(\bar{x}) \Rightarrow x = exp ( x Λ ) β
f ( z β£ x ) = 1 2 Ο Ο v 1 β£ x β£ exp β‘ { β 1 2 z 2 Ο v 2 x 2 }
f(z \mid x)=\frac{1}{\sqrt{2 \pi} \sigma_{v}} \frac{1}{|x|} \exp \left\{-\frac{1}{2} \frac{z^{2}}{\sigma_{v}^{2} x^{2}}\right\}
f ( z β£ x ) = 2 Ο β Ο v β 1 β β£ x β£ 1 β exp { β 2 1 β Ο v 2 β x 2 z 2 β } Direkte LΓΆsung:
f ( z β£ x , v ) = Ξ΄ ( z β x β
v )
f(z \mid x, v) = \delta(z - x \cdot v)
f ( z β£ x , v ) = Ξ΄ ( z β x β
v ) f ( z , v β£ x ) = f ( z β£ x , v ) β
f v ( v ) = Ξ΄ ( z β x β
v ) f v ( v )
f(z, v \mid x) = f(z \mid x, v) \cdot f_v(v) = \delta(z - x \cdot v) f_v(v)
f ( z , v β£ x ) = f ( z β£ x , v ) β
f v β ( v ) = Ξ΄ ( z β x β
v ) f v β ( v ) f ( z β£ x ) = β« R f ( z , v β£ x ) d v = β« R Ξ΄ ( z β x β
v ) f v ( v ) d v
f(z \mid x) = \int_{\mathbb{R}} f(z, v \mid x) dv = \int_{\mathbb{R}}\delta(z - x \cdot v) f_v(v) dv
f ( z β£ x ) = β« R β f ( z , v β£ x ) d v = β« R β Ξ΄ ( z β x β
v ) f v β ( v ) d v Setze
g ( v ) : = z β x v β g β² ( v ) = β x , Nullstelle v = z x
\begin{aligned}
g(v) := z - xv &\Rightarrow g^\prime(v) = -x, \quad \text{Nullstelle } v = \frac{z}{x}
\end{aligned}
g ( v ) := z β xv β β g β² ( v ) = β x , Nullstelle v = x z β β Daher
f ( z β£ x ) = β« R 1 β£ x β£ Ξ΄ ( v β z x ) β
f v ( v ) d v = 1 β£ x β£ β
f v ( z x ) ( multiplicative )
\begin{aligned}
f(z \mid x)&=\int_{\mathbb{R}} \frac{1}{|x|} \delta\left(v-\frac{z}{x}\right) \cdot f_v(v) d v \\
&=\frac{1}{|x|} \cdot f_v\left(\frac{z}{x}\right) \qquad \qquad (\text{multiplicative})
\end{aligned}
f ( z β£ x ) β = β« R β β£ x β£ 1 β Ξ΄ ( v β x z β ) β
f v β ( v ) d v = β£ x β£ 1 β β
f v β ( x z β ) ( multiplicative ) β Mixed Additive and Multiplicative Noise (Script Chp. 9.2.2) System equation
x k + 1 = x k v k + w k
x_{k+1} = x_k v_k + w_k
x k + 1 β = x k β v k β + w k β with additive noise w k w_k w k β v k v_k v k β f k v w ( v k , w k ) f_{k}^{vw}(v_k, w_k) f k v w β ( v k β , w k β )
The joint density of the state at time step k + 1 k+1 k + 1
f ( x k + 1 , v k , w k β£ x k ) = f ( x k + 1 β£ x k , v k , w k ) f k v w ( v k , w k ) ,
f\left(x_{k+1}, v_{k}, w_{k} \mid x_{k}\right)=f\left(x_{k+1} \mid x_{k}, v_{k}, w_{k}\right) f_{k}^{v w}\left(v_{k}, w_{k}\right),
f ( x k + 1 β , v k β , w k β β£ x k β ) = f ( x k + 1 β β£ x k β , v k β , w k β ) f k v w β ( v k β , w k β ) , where according to the system equation the density of the state at time step k + 1 k + 1 k + 1 k k k v k v_k v k β w k w_k w k β
f ( x k + 1 β£ x k , v k , w k ) = Ξ΄ ( x k + 1 β x k v k β w k ) .
f(x_{k+1} \mid x_{k}, v_{k}, w_{k}) = \delta(x_{k+1} - x_{k}v_{k} - w_{k}).
f ( x k + 1 β β£ x k β , v k β , w k β ) = Ξ΄ ( x k + 1 β β x k β v k β β w k β ) . The desired transition density is now given by
f ( x k + 1 β£ x k ) = β« R β« R f ( x k + 1 , v k , w k β£ x k ) d w k d v k = β« R β« R Ξ΄ ( x k + 1 β x k v k β w k ) f k v w ( v k , w k ) d w k d v k = additive f k ( x k + 1 β£ x k ) = β« R f k v w ( v k , x k + 1 β x k v k ) d v k β£ v k , w k independent = β« R f k v ( v k ) f k w ( x k + 1 β x k v k ) d v k
\begin{aligned}
f\left(x_{k+1} \mid x_{k}\right) &=\int_{\mathbb{R}} \int_{\mathbb{R}} f\left(x_{k+1}, v_{k}, w_{k} \mid x_{k}\right) d w_{k} d v_{k} \\
&=\int_{\mathbb{R}} \int_{\mathbb{R}} \delta\left(x_{k+1}-x_{k} v_{k}-w_{k}\right) f_{k}^{v w}\left(v_{k}, w_{k}\right) \mathrm{d} w_{k} \mathrm{~d} v_{k}\\
&\overset{\text{additive}}{=} f_{k}\left(x_{k+1} \mid x_{k}\right)=\int_{\mathbb{R}} f_{k}^{v w}\left(v_{k}, x_{k+1}-x_{k} v_{k}\right) \mathrm{d} v_{k} \mid v_k, w_k \text{ independent}\\
&=\int_{\mathbb{R}} f_{k}^{v}\left(v_{k}\right) f_{k}^{w}\left(x_{k+1}-x_{k} v_{k}\right) \mathrm{d} v_{k}
\end{aligned}
f ( x k + 1 β β£ x k β ) β = β« R β β« R β f ( x k + 1 β , v k β , w k β β£ x k β ) d w k β d v k β = β« R β β« R β Ξ΄ ( x k + 1 β β x k β v k β β w k β ) f k v w β ( v k β , w k β ) d w k β d v k β = additive f k β ( x k + 1 β β£ x k β ) = β« R β f k v w β ( v k β , x k + 1 β β x k β v k β ) d v k β β£ v k β , w k β independent = β« R β f k v β ( v k β ) f k w β ( x k + 1 β β x k β v k β ) d v k β β These expressions cannot in general be solved analytically.