非线性状态空间模型与非线性自回归模型的联系
文章目录
- 非线性状态空间模型
- 非线性自回归模型
- 两者的联系
非线性状态空间模型
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\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}
rtvt=(1−α)rt−1+αf(Art−1+Winut)=Woutrt
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r∈RN 表示状态，
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u∈Rd 表示输入，
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A\in R^{N\times N}, W_{in} \in R^{N\times d} , f (\cdot)= tanh(\cdot)
A∈RN×N,Win∈RN×d,f(⋅)=tanh(⋅) 组成了非线性的状态转移方程。
非线性自回归模型
v t = g ( u t , u t − 1 , … , u t − q + 1 ) v_t = g(u_t, u_{t-1}, \ldots, u_{t-q+1}) vt=g(ut,ut−1,…,ut−q+1)
两者的联系
若具有如下特殊结构：
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A = \left[ \begin{array}{ll} O & O \\ I_{N-p} & O \end{array} \right] _{N\times N}
A=[OIN−pOO]N×N
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W_{in} = \left[ \begin{array}{ll} W \\ O \end{array} \right]_{N\times d} ,\quad W \in R^{p \times d}
Win=[WO]N×d,W∈Rp×d
假设
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\alpha = 1, N/p = q
α=1,N/p=q
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\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}
rt=f([OIN−pOO]rt−1+[WO]ut)=f([Wutrt−1,1:N−p])=⎣⎢⎢⎢⎢⎢⎡f(Wut)f∘f(Wut−1))⋮q
f∘⋯∘f(Wut−q+1)⎦⎥⎥⎥⎥⎥⎤
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\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}
rt=f⎝⎜⎛[OIN−pOO]rt−1+⎣⎢⎡W1⋮Wq⎦⎥⎤ut⎠⎟⎞=⎣⎢⎡f(W1ut)f(f(W1ut−1)+W2ut)⋮⎦⎥⎤=F(ut,ut−1,…,ut−q+1)
因此
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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})
vt=WoutF(ut,ut−1,…,ut−q+1)≜g(ut,ut−1,…,ut−q+1)
即把状态空间模型转化成了自回归模型。