文章目录
概率状态空间模型(Probabilistic State Space Models)概率状态空间模型的定义状态 x k \boldsymbol{x}_k xk的马尔可夫性质(*Markov property of states*)观测向量 y k \boldsymbol{ y}_k yk的条件独立性(*Conditional independence of measurements*)概率状态空间模型(Probabilistic State Space Models)
概率状态空间模型的定义
(Probabilistic state space model) A probabilistic state space model or non-linear filtering model consists of a sequence of conditional probability distributions:
x k ∼ p ( x k ∣ x k − 1 ) y k ∼ p ( y k ∣ x k ) (1) \begin{aligned} \boldsymbol x_k &\sim p(\boldsymbol x_k | \boldsymbol x_{k-1}) \\ \boldsymbol y_k &\sim p(\boldsymbol y_k | \boldsymbol x_k) \end{aligned} \tag{1} xkyk∼p(xk∣xk−1)∼p(yk∣xk)(1) for k = 1 , 2 , ⋯ k = 1,2,\cdots k=1,2,⋯，其中
x k ∈ R n \boldsymbol x_k \in \mathbb{R}^{n} xk∈Rn是系统在时刻 k k k的状态； y k ∈ R m \boldsymbol y_k \in \mathbb{R}^{m} yk∈Rm是时刻 k k k的观测向量； p ( x k ∣ x k − 1 ) p(\boldsymbol x_k | \boldsymbol x_{k-1}) p(xk∣xk−1)是一个动态系统模型，描述了系统的随机动态变化(stochasticc dynamics)； p ( y k ∣ x k ) p(\boldsymbol y_k|\boldsymbol x_k) p(yk∣xk)是测量模型，描述了给定状态条件下的测量模型。
并且认为该模型为马尔可夫模型，如下图所示。这意味着它有两个重要性质。
状态 x k \boldsymbol{x}_k xk的马尔可夫性质(Markov property of states)
The states { x k : k = 0 , 1 , 2 , ⋯ } \{ \boldsymbol x_k: k=0,1,2,\cdots \} {xk:k=0,1,2,⋯} form a Markov sequence (or Markov chain if the state is discrete.). This Markov property means that x k \boldsymbol x_k xk (and actually the whole future x k + 1 , x k + 2 , ⋯ \boldsymbol x_{k+1}, \boldsymbol x_{k+2}, \cdots xk+1,xk+2,⋯) given x k − 1 \boldsymbol x_{k-1} xk−1 isindependent of anythingthat has happened before the time step k − 1 k-1 k−1：
p ( x k ∣ x 1 : k − 1 , y 1 : k − 1 ) = p ( x k ∣ x k − 1 ) (2) p(\boldsymbol x_k | \boldsymbol x_{1:k-1}, \boldsymbol y_{1:k-1}) = p(\boldsymbol x_k|\boldsymbol x_{k-1}) \tag{2} p(xk∣x1:k−1,y1:k−1)=p(xk∣xk−1)(2)
这里我们需要注意性质里边没有提到 x k \boldsymbol x_k xk之后的状态(future state)，由上图可知，我们知道 x k \boldsymbol x_k xk与之后的状态/观测向量是有联系的，比如说： ∃ T > k \exist T > k ∃T>k
p ( x k ∣ x 1 : k − 1 , y 1 : T ) = p ( x k ∣ x k − 1 , y k : T ) p(\boldsymbol x_k | \boldsymbol x_{1:k-1}, \boldsymbol y_{1:T}) = p(\boldsymbol x_k|\boldsymbol x_{k-1}, \boldsymbol y_{k:T}) p(xk∣x1:k−1,y1:T)=p(xk∣xk−1,yk:T)
Also, the past isindependent of the futuregiven the present:
p ( x k − 1 ∣ x k : T , y k : T ) = p ( x k − 1 ∣ x k ) (3) p(\boldsymbol x_{k-1} | \boldsymbol x_{k:T}, \boldsymbol y_{k:T}) = p(\boldsymbol x_{k-1}|\boldsymbol x_{k}) \tag{3} p(xk−1∣xk:T,yk:T)=p(xk−1∣xk)(3)
类似地，我们也需要注意， x k \boldsymbol x_k xk与之前的状态/观测向量是有联系的，比如说：
p ( x k − 1 ∣ x k : T , y 1 : T ) = p ( x k − 1 ∣ x k , y 1 : k − 1 ) p(\boldsymbol x_{k-1} | \boldsymbol x_{k:T}, \boldsymbol y_{1:T}) = p(\boldsymbol x_{k-1}|\boldsymbol x_{k}, \boldsymbol y_{1:k-1}) p(xk−1∣xk:T,y1:T)=p(xk−1∣xk,y1:k−1)
观测向量 y k \boldsymbol{ y}_k yk的条件独立性(Conditional independence of measurements)
The current measurement y k \boldsymbol{ y}_k yk given the current state x k \boldsymbol{ x}_k xk is conditionally independent of themeasurement and state histories:
p ( y k ∣ x 1 : k , y 1 : k − 1 ) = p ( y k ∣ x k ) p(\boldsymbol y_k|\boldsymbol x_{1:k}, \boldsymbol y_{1:k-1}) = p(\boldsymbol y_k | \boldsymbol x_k) p(yk∣x1:k,y1:k−1)=p(yk∣xk)
