Wang Haihua
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设系统有 $N$ 个指标变量, 对应的参考序列分别为 $$ x_{i}^{(0)}=\left(x_{i}^{(0)}(1), x_{i}^{(0)}(2), \cdots, x_{i}^{(0)}(n)\right), \quad i=1,2, \cdots, N, $$ 作累加运算 $x_{i}^{(1)}(k)=\sum_{j=1}^{k} x_{i}^{(0)}(j)$, 可得累加生成数列 $$ x_{i}^{(1)}=\left(x_{i}^{(1)}(1), x_{i}^{(1)}(2), \cdots, x_{i}^{(1)}(n)\right), \quad i=1,2, \cdots, N . $$
微分方程 $$ \frac{d x_{1}^{(1)}(t)}{d t}+a_{1} x_{1}^{(1)}(t)=a_{2} x_{2}^{(1)}(t)+a_{3} x_{3}^{(1)}(t)+\cdots+a_{N} x_{N}^{(1)}(t) $$ 是 1 阶 $N$ 个变量的微分方程模型, 记为 $\mathrm{GM}(1, N)$ 。 类似地, 当 $k-1<t \leq k$ 时, 令 $$ \begin{aligned} &x_{1}^{(1)}(t)=z_{1}^{(1)}(k)=\frac{1}{2}\left(x_{1}^{(1)}(k-1)+x_{1}^{(1)}(k)\right), \ldots, \\ &\frac{d x_{1}^{(1)}(t)}{d t}=x_{1}^{(1)}(k)-x_{1}^{(1)}(k-1)=x_{1}^{(0)}(k), \quad k=2,3, \cdots, n, \\ &x_{i}^{(1)}(t)=z_{i}^{(1)}(k)=\frac{1}{2}\left(x_{i}^{(1)}(k-1)+x_{i}^{(1)}(k)\right), \quad i=2,3, \cdots, N ; k=2,3, \cdots, n . \end{aligned} $$
将式化成离散模型 $$ x_{1}^{(0)}(k)+a_{1} z_{1}^{(1)}(k)=a_{2} z_{2}^{(1)}(k)+a_{3} z_{3}^{(1)}(k)+\cdots+a_{N} z_{N}^{(1)}(k), \quad k=2,3, \cdots, n . $$ 可以证明第二个式子是第一个式子的2 阶精度数值模型。
记 $$ \boldsymbol{u}=\left[\begin{array}{c} a_{1} \\ a_{2} \\ \vdots \\ a_{N} \end{array}\right], \quad Y=\left[\begin{array}{c} x^{(0)}(2) \\ x^{(0)}(3) \\ \vdots \\ x^{(0)}(n) \end{array}\right], \quad B=\left[\begin{array}{cccc} -z_{1}^{(1)}(2) & z_{2}^{(1)}(2) & \cdots & z_{N}^{(1)}(2) \\ -z_{1}^{(1)}(3) & z_{2}^{(1)}(3) & \cdots & z_{N}^{(1)}(3) \\ \vdots & \vdots & & \vdots \\ -z_{1}^{(1)}(n) & z_{2}^{(1)}(n) & \cdots & z_{N}^{(1)}(n) \end{array}\right], $$ 可将原式化为 $B u=Y$ ，则可以得到 $u$ 的最小二乘估计值 $$ \hat{\boldsymbol{u}}=\left[\begin{array}{c} \hat{\boldsymbol{a}}_{1} \\ \hat{\boldsymbol{a}}_{2} \\ \vdots \\ \hat{\boldsymbol{a}}_{N} \end{array}\right]=\left(\boldsymbol{B}^{T} \boldsymbol{B}\right)^{-1} \boldsymbol{B}^{T} \boldsymbol{Y} . $$
就得到 $x_{1}^{(1)}(k)(k=1,2, \cdots,$, 的预测值 $\hat{x}_{1}^{(1)}(k)(k=1,2, \cdots,$,还原到 $\hat{x}_{1}^{(0)}(k)(k=1,2, \cdots)$, 就得到 $x_{0}^{(1)}(k)(k=1,2, \cdots,$, 的预测值。