## 定义

$o(t)=C_1*o(t-1)+c_2*o(t-2)+...+c_f(t-f) \tag{1}$

$s_0(t) = \{o(t),o(t-1),...,o(t-f+1)\} \tag{2}$

$s_0(t)=K_0*s_0(t-1) \tag{3}$
K_0是维度为$$(d*f)*(d*f)$$维度的矩阵。d是数据点的维度，对于二维路径点d = 2。

$\left[ \begin{matrix} o(t).x_1 \\ o(t).x_2 \\ o(t-1).x_1 \\ o(t-1).x_2 \end{matrix} \right] = \left[ \begin{matrix} k_{11} & k_{12} & k_{13} & k_{14} \\ k_{21} & k_{22} & k_{23} & k_{24} \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{matrix} \right] * \left[ \begin{matrix} o(t-1).x_1 \\ o(t-1).x_2 \\ o(t-2).x_1 \\ o(t-2).x_2 \end{matrix} \right]$

$\begin{cases} k_{ij} = 0 \space \text{for i>=d+1 and i!=j+d} \\ \tag{4} k_{ij} = 1 \space \text{for i>=d+1 and i=j+d} \end{cases}$

$s_0(t-1)^T*K_{i*} = o(t).x_i \tag{5}$

## 参数估计

h需要大于f。

$\left[ \begin{matrix} s(T_c-1)^T \\ s(T_c-2)^T \\ ... \\ s(T_c-h+f)^T \end{matrix} \right] * k_{i*} = \left[ \begin{matrix} l(T_c).x_i \\ l(T_c-1).x_i \\ ... \\ l(T_c-h+f+1).x_i \end{matrix} \right] \tag{6}$

$S = \left[ \begin{matrix} s(T_c-1)^T \\ s(T_c-2)^T \\ ... \\ s(T_c-h+f)^T \end{matrix} \right] \space l = \left[ \begin{matrix} l(T_c).x_i \\ l(T_c-1).x_i \\ ... \\ l(T_c-h+f+1).x_i \end{matrix} \right]$

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