期刊菜单

Application of Optimization Algorithm for Unequal Time Interval MGM (1, N) in Engineering Survey

Abstract: The coefficients matrix of the errors in variables model with both fixed column elements, random elements and non-random elements was considered by introducing the structure matrix, the op-timization algorithm was used to solve the parameters of the non-equidistant multivariable grey model. It was applied to the fitting and prediction of the deformation monitoring of the building site above the old goaf. The experimental results show that the proposed algorithm is effective and feasible. In addition, the accuracy of multivariate grey model is higher than that of univariate grey model, because the multivariate grey model takes into account the temporal and spatial correlation of deformation data series.

1. 引言

2. 非线性高斯赫尔默特模型的混合结构总体最小二乘算法

$L+{e}_{L}=\left(A+{E}_{A}\right)X$ (1)

$\left[\begin{array}{c}{e}_{L}\\ vec\left({E}_{A}\right)\end{array}\right]~N\left\{\left[\begin{array}{c}0\\ 0\end{array}\right],{\sigma }_{0}^{2}\left[\begin{array}{cc}{I}_{m}& 0\\ 0& {I}_{n}\otimes {I}_{m}\end{array}\right]\right\}$ (2.1)

$\left[\begin{array}{c}{e}_{L}\\ vec\left({E}_{A}\right)\end{array}\right]~N\left\{\left[\begin{array}{c}0\\ 0\end{array}\right],{\sigma }_{0}^{2}\left[\begin{array}{cc}{Q}_{LL}& 0\\ 0& {Q}_{AA}\end{array}\right]\right\}$ (2.2)

${e}_{L}^{\text{T}}{e}_{L}+{e}_{A}^{\text{T}}{e}_{A}=min$ (3)

$AX-L=\left[\begin{array}{cc}{I}_{n}& -\left({X}_{p}\otimes {I}_{n}\right)\cdot D\end{array}\right]\left[\begin{array}{c}{e}_{L}\\ {e}_{a}\end{array}\right]$ (4)

$M=\left[\begin{array}{cc}{I}_{n}& -\left({X}_{p}\otimes {I}_{n}\right)\cdot D\end{array}\right]$ , $e=\left[\begin{array}{c}{e}_{L}\\ {e}_{a}\end{array}\right]$ (5)

$\Phi \left(e,\lambda ,X\right)={e}^{\text{T}}Pe+2{\lambda }^{T}\left(L-AX+M\cdot e\right)$ (6)

$X={\left({\left(A+{E}_{A}\right)}^{\text{T}}{\left(MQ{M}^{\text{T}}\right)}^{-1}A\right)}^{-1}{\left(A+{E}_{A}\right)}^{\text{T}}{\left(MQ{M}^{\text{T}}\right)}^{-1}L$ (7)

${\left(A+{E}_{A}\right)}^{\text{T}}{\left(MQ{M}^{\text{T}}\right)}^{-1}A$ 之后，再同时加上 ${\left(A+{E}_{A}\right)}^{\text{T}}{\left(MQ{M}^{\text{T}}\right)}^{-1}{E}_{A}X$

$X={\left[{\left(A+{E}_{A}\right)}^{\text{T}}{\left(MQ{M}^{\text{T}}\right)}^{-1}\left(A+{E}_{A}\right)\right]}^{-1}{\left(A+{E}_{A}\right)}^{\text{T}}{\left(MQ{M}^{\text{T}}\right)}^{-1}\left(L+{E}_{A}X\right)$(8)

3. 非等时距序列MGM模型优化

3.1. 非等时距序列GM (1, 1)模型建模机理 [13] [14]

$\Delta {t}_{k}={t}_{k}-{t}_{k-1},k=2,3,\cdots ,n$ (9)

$\Delta {t}_{k}$ 不为常数时，即两相邻时刻的时间差不固定，此时原始数据序列 ${x}^{\left(0\right)}$ 为非等间隔观测序列，一次累加生成序列如下式：

${x}^{\left(1\right)}\left({t}_{k}\right)=\underset{i=1}{\overset{k}{\sum }}\Delta {t}_{i}{x}^{\left(0\right)}\left({t}_{i}\right),k=1,2,\cdots ,n,\Delta {t}_{1}=1$ (10)

${x}^{\left(0\right)}\left({t}_{k}\right)=\frac{{x}^{\left(1\right)}\left({t}_{k}\right)-{x}^{\left(1\right)}\left({t}_{k-1}\right)}{\Delta {t}_{k}},k=2,3,\cdots ,n$ (11)

${x}^{\left(0\right)}\left({t}_{k}\right)+{a}_{1}{z}^{\left(1\right)}\left({t}_{k}\right)={b}_{1}$ (12)

${x}^{\left(1\right)}\left({t}_{k}\right)=\frac{{b}_{1}}{{a}_{1}}+\left({x}^{\left(9\right)}\left({t}_{1}\right)-\frac{{b}_{1}}{{a}_{1}}\right)\cdot {\text{e}}^{-{a}_{1}\left({t}_{k}-{t}_{1}\right)}$ (13)

$X={\left({A}^{\text{T}}A\right)}^{-1}{A}^{\text{T}}L$(14)

3.2. 非等时距多变量灰色模型MGM (1, N)建模机理 [13] [14]

MGM (1, N)模型考虑了多个变量相互影响和发展的变形情况，完全不同于GM (1, N)，也不是GM (1, 1)模型的简单叠加，而是GM (1, 1)模型在N元变量情况下的拓展，通过建立N个N元微分方程，联立求解MGM (1, N)模型中的灰参数，而多个变量之间的相互影响程度便反映在所求灰参数中，灰参数主导模型的整个拟合和后续的预测过程。

${X}_{j}^{\left(0\right)}={\left[\begin{array}{cccc}{x}_{j}^{\left(0\right)}\left({k}_{1}\right)& {x}_{j}^{\left(0\right)}\left({k}_{2}\right)& \cdots & {x}_{j}^{\left(0\right)}\left({k}_{m}\right)\end{array}\right]}^{\text{T}}$ (15)

${X}^{\left(0\right)}={\left[\begin{array}{cccc}{x}_{1}^{\left(0\right)}& {x}_{2}^{\left(0\right)}& \cdots & {x}_{n}^{\left(0\right)}\end{array}\right]}^{\text{T}}$ (16)

$\frac{\text{d}{X}^{\left(1\right)}\left(t\right)}{\text{d}t}={C}_{1}{X}^{\left(1\right)}\left(t\right)+{C}_{2}$ (17)

${X}^{\left(1\right)}\left({t}_{k}\right)=\left({C}_{1}^{-1}{C}_{2}+{X}^{\left(0\right)}\left({t}_{1}\right)\right)\cdot {\text{e}}^{{C}_{1}\left({t}_{k}-{t}_{1}\right)}-{C}_{1}^{-1}{C}_{2}$ (18)

${\text{e}}^{{C}_{1}\Delta }=I+{C}_{1}\Delta +\frac{{C}_{1}^{2}}{2!}{\Delta }^{2}+\cdots =I+\underset{i=1}{\overset{\infty }{\sum }}\frac{{C}_{1}^{i}}{i!}{\Delta }^{i}$ (19)

${x}_{j}^{\left(0\right)}\left({t}_{k}\right)=\underset{i=1}{\overset{n}{\sum }}{a}_{ji}{z}_{i}^{\left(1\right)}\left({t}_{k}\right)+{b}_{j}\text{\hspace{0.17em}}\left(j=1,2,\cdots ,n;k=2,3,\cdots ,m\right)$，写成矩阵形式为

$Y=L\cdot H$ (20)

$H={\left({L}^{\text{T}}L\right)}^{-1}{L}^{\text{T}}Y$(21)

3.3. 建模优化

4. 案例分析(Examples Analysis)

Table 1. Original data series

A方案：对选取的J89、J90、J91三个水准监测点的9期监测数据建立各自的非等间距GM (1, 1)模型，分别采用最小二乘法和上文中提到的总体最小二乘法进行模型参数求取，并建立相应的时间响应函数，进行拟合和预测；

B方案：对选取的J89、J90、J91三个水准监测点的9期监测数据建立非等间距MGM (1, 3)模型，分别采用最小二乘法和上文中提到的总体最小二乘法进行模型参数求取，并建立相应的时间响应函数，进行拟合和预测。

Table 2. Calculation results of GM (1, 1) model parameters in scheme A

Table 3. Fitting error of point J90 of two algorithms under scheme A

Table 4. Calculation results of MGM (1, 3) model parameters in scheme B

Table 5. Residual error of least squares parameter fitting in scheme B

Table 6. Total least squares fitting residuals in scheme B

Figure 1. Fitting/prediction error of two models of J89

Figure 2. Fitting/prediction error of two models of J90

Figure 3. Fitting/prediction error of two models of J91

5. 结论

1) 将文中算法应用于高速公路建设场地残余沉降的拟合和预报，实验结果表明：混合结构总体最小二乘平差方法对于非等时距GM (1, 1)模型参数的优化效果并不明显，对非等时距MGM (1, N)模型参数的优化效果比较明显。

2) 由于非等时距多变量灰色模型MGM (1, N)兼顾了变形数据序列的时间、空间相关性，建立的模型更能贴切的反映变形数据之间相互影响的发展态势，故此，该模型的拟合和预报精度均明显高于非等时距单变量灰色模型GM (1, 1)。

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