非線性狀態空間模型與非線性自迴歸模型的聯繫

非線性狀態空間模型

$\begin{array}{ll} r_{t} &= (1-\alpha) r_{t-1} + \alpha f(Ar_{t-1} + W_{in}u_{t}) \\ v_{t} &= W_{out} r_{t} \end{array}$
其中 $r\in R^N$ 表示狀態， $u \in R^d$ 表示輸入， $A\in R^{N\times N}, W_{in} \in R^{N\times d} , f (\cdot)= tanh(\cdot)$ 組成了非線性的狀態轉移方程。

非線性自迴歸模型

$v_t = g(u_t, u_{t-1}, \ldots, u_{t-q+1})$

兩者的聯繫

若具有如下特殊結構：
$A = \left[ \begin{array}{ll} O & O \\ I_{N-p} & O \end{array} \right] _{N\times N}$
$W_{in} = \left[ \begin{array}{ll} W \\ O \end{array} \right]_{N\times d} ,\quad W \in R^{p \times d}$
假設 $\alpha = 1, N/p = q$
$\begin{array}{ll} r_{t} &= f\left(\left[ \begin{array}{ll} O & O \\ I_{N-p} & O \end{array} \right]r_{t-1} + \left[ \begin{array}{ll} W \\ O \end{array} \right]u_{t}\right) \\\\ &=f\left( \left[ \begin{array}{ll} Wu_{t} \\ r_{t-1,1:N-p} \end{array} \right]\right) \\\\ &= \left[ \begin{array}{c} f(Wu_{t}) \\ f\circ f(Wu_{t-1})) \\ \vdots \\ \underbrace{f\circ \cdots\circ f}_{q}(Wu_{t-q+1}) \end{array} \right] \end{array}$

$\begin{array}{ll} r_{t} &= f\left(\left[ \begin{array}{ll} O & O \\ I_{N-p} & O \end{array} \right]r_{t-1} + \left[ \begin{array}{c} W_1 \\ \vdots \\ W_q \end{array} \right]u_{t}\right) \\\\ &= \left[ \begin{array}{c} f(W_1u_{t}) \\ f(f(W_1u_{t-1})+W_2u_t) \\ \vdots \end{array} \right] \\\\ &= F(u_t, u_{t-1}, \ldots, u_{t-q+1}) \end{array}$
因此
$v_t = W_{out} F(u_t, u_{t-1}, \ldots, u_{t-q+1}) \triangleq g(u_t, u_{t-1}, \ldots, u_{t-q+1})$
即把狀態空間模型轉化成了自迴歸模型。