﻿ 不等式约束Partial EIV模型的WHP拟牛顿修正解法及其精度评定的SUT法
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 大地测量与地球动力学  2019, Vol. 39 Issue (6): 648-653  DOI: 10.14075/j.jgg.2019.06.019

### 引用本文

WANG Leyang, ZOU Chuanyi, WU Lulu. The WHP Quasi Newton Correction Method for Inequality Constrained Partial EIV Model and the SUT Method for Its Precision Estimation[J]. Journal of Geodesy and Geodynamics, 2019, 39(6): 648-653.

### Foundation support

National Natural Science Foundation of China, No. 41874001, 41664001; Support Program for Outstanding Youth Talents of Jiangxi Province, No. 20162BCB23050; National Key Research and Development Program of China, No. 2016YFB0501405; Science and Technology Project of the Education Department of Jiangxi Province, No. GJJ171368.

### 第一作者简介

WANG Leyang, PhD, associate professor, majors in geodetic inversion and geodetic data processing, E-mail: wleyang@163.com.

### 文章历史

1. 东华理工大学测绘工程学院，南昌市广兰大道418号，330013;
2. 国家自然资源部流域生态与地理环境监测重点实验室，南昌市广兰大道418号，330013;
3. 江西省数字国土重点实验室，南昌市广兰大道418号，330013;
4. 江西水利职业学院建筑工程系，南昌市北山路99号，330013

1 附有不等式约束的Partial EIV模型的序列二次规划模型 1.1 附有不等式约束的Partial EIV模型

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{y}} = \left({{\mathit{\boldsymbol{\beta }}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)(\mathit{\boldsymbol{h}} + \mathit{\boldsymbol{B\bar a}}) + {\mathit{\boldsymbol{e}}_y}\\ \mathit{\boldsymbol{a}} = \mathit{\boldsymbol{\bar a}} + {\mathit{\boldsymbol{e}}_a}\\ {\mathop{\rm vec}\nolimits}(\mathit{\boldsymbol{\overline A}}) = \mathit{\boldsymbol{h}} + \mathit{\boldsymbol{B}}\mathit{\boldsymbol{\overline \alpha }} \\ \mathit{\boldsymbol{G\beta }}-\mathit{\boldsymbol{w}} \ge 0 \end{array} \right.$ (1)
 $\mathit{\boldsymbol{e}} = \left[{\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{e}}_y}}\\ {{\mathit{\boldsymbol{e}}_a}} \end{array}} \right]\sim \left({\left[{\begin{array}{*{20}{l}} 0\\ 0 \end{array}} \right], \sigma _0^2{\mathit{\boldsymbol{Q}}_e} = \sigma _0^2{\mathit{\boldsymbol{P}}^{-1}}} \right)$ (2)

 $\left\{ {\begin{array}{*{20}{l}} {\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{y}}\\ \mathit{\boldsymbol{a}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\left({{\mathit{\boldsymbol{\beta }}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)(\mathit{\boldsymbol{h}} + \mathit{\boldsymbol{B\bar a}})}\\ {\mathit{\boldsymbol{\bar a}}} \end{array}} \right]-\mathit{\boldsymbol{e}}}\\ {\mathit{\boldsymbol{G\beta }}-\mathit{\boldsymbol{w}} \ge 0} \end{array}} \right.$ (3)
1.2 附有不等式约束Partial EIV模型的序列二次规划模型

 $\left\{ {\begin{array}{*{20}{l}} {\min {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{Pe}}}\\ {{\rm{ s}}{\rm{.t}}{\rm{. }}\mathit{\boldsymbol{y}}-\left({{\mathit{\boldsymbol{\beta }}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)(\mathit{\boldsymbol{h}} + \mathit{\boldsymbol{Ba}}) + \mathit{\boldsymbol{Ce}} = 0} \end{array}} \right.$ (4)

 $\begin{array}{l} \mathit{\Phi }(\mathit{\boldsymbol{e}}, \mathit{\boldsymbol{\beta }}, \mathit{\boldsymbol{\lambda }}) = {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{Pe}} + 2{\mathit{\boldsymbol{\mu }}^{\rm{T}}}[\mathit{\boldsymbol{y}}-\\ \;\;\;\left({{\mathit{\boldsymbol{\beta }}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)(\mathit{\boldsymbol{h}} + \mathit{\boldsymbol{Ba}}) + \mathit{\boldsymbol{Ce}}] \end{array}$ (5)

 $\frac{{\partial \mathit{\Phi }}}{{\partial \mathit{\boldsymbol{e}}}} = 2\left({\mathit{\boldsymbol{Pe}} + {\mathit{\boldsymbol{C}}^{\rm{T}}}\mathit{\boldsymbol{\mu }}} \right) = 0$ (6)
 $\frac{{\partial \mathit{\Phi }}}{{\partial \mathit{\boldsymbol{\beta }}}} = -2{\mathit{\boldsymbol{\overline A}} ^{\rm{T}}}\mathit{\boldsymbol{\mu }} = 0$ (7)
 $\frac{{\partial \mathit{\Phi }}}{{\partial \mathit{\boldsymbol{\mu }}}} = 2\left[ {\mathit{\boldsymbol{y}}-\left({{\mathit{\boldsymbol{\beta }}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)(\mathit{\boldsymbol{h}} + \mathit{\boldsymbol{Ba}}) + \mathit{\boldsymbol{Ce}}} \right] = 0$ (8)

 $\mathit{\boldsymbol{e}} = -{(\mathit{\boldsymbol{P}})^{-1}}{\mathit{\boldsymbol{C}}^{\rm{T}}}\mathit{\boldsymbol{\mu }}$ (9)

 $\mathit{\boldsymbol{\mu }} = {\left({\mathit{\boldsymbol{C}}{{(\mathit{\boldsymbol{P}})}^{-1}}{\mathit{\boldsymbol{C}}^{\rm{T}}}} \right)^{-1}}\left[ {\mathit{\boldsymbol{y}}-\left({{\mathit{\boldsymbol{\beta }}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)(\mathit{\boldsymbol{h}} + \mathit{\boldsymbol{Ba}})} \right]$ (10)

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\hat e}}}^{\rm{T}}}\mathit{\boldsymbol{P\hat e}} = \left[ {\mathit{\boldsymbol{y}} - \left( {{\mathit{\boldsymbol{\beta }}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)(\mathit{\boldsymbol{h}} + \mathit{\boldsymbol{Ba}})} \right]{{\left( {\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{Q}}_\mathit{\boldsymbol{e}}}{\mathit{\boldsymbol{C}}^{\rm{T}}}} \right)}^{ - 1}} \bullet }\\ {\;\;\;\;\;\;\left[ {\mathit{\boldsymbol{y}} - \left( {{\mathit{\boldsymbol{\beta }}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)} \right.\left. {(\mathit{\boldsymbol{h}} + \mathit{\boldsymbol{Ba}})} \right]}\\ {\mathit{\boldsymbol{f}}\left( \mathit{\boldsymbol{\beta }} \right) = {{\mathit{\boldsymbol{\hat e}}}^{\rm{T}}}\mathit{\boldsymbol{P\hat e}}} \end{array}} \right.$ (11)

 $\left\{ \begin{array}{l} \min f(\mathit{\boldsymbol{\beta }}) = \left[ {\mathit{\boldsymbol{y}}-\left({{\mathit{\boldsymbol{\beta }}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)(\mathit{\boldsymbol{h}} + \mathit{\boldsymbol{Ba}})} \right] \bullet \\ \;\;\;\;\;{\left({\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{Q}}_\mathit{\boldsymbol{e}}}{\mathit{\boldsymbol{C}}^{\rm{T}}}} \right)^{-1}}\left[ {\mathit{\boldsymbol{y}}-\left({{\mathit{\boldsymbol{\beta }}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right)(\mathit{\boldsymbol{h}} + \mathit{\boldsymbol{Ba}})} \right]\\ \mathit{\boldsymbol{G}}\mathit{\boldsymbol{\widehat \beta }}-\mathit{\boldsymbol{w}} \ge 0 \end{array} \right.$ (12)

2 基于拟牛顿修正的SQP(序列二次规划)方法

 $L(\mathit{\boldsymbol{\beta }}, \mathit{\boldsymbol{\lambda }}) = f(\mathit{\boldsymbol{\beta }})-{\mathit{\boldsymbol{\lambda }}^{\rm{T}}}(\mathit{\boldsymbol{G\beta }}-\mathit{\boldsymbol{w}})$ (13)

 $\left\{ \begin{array}{l} \min \frac{1}{2}\mathit{\boldsymbol{d}}_j^{\rm{T}}\mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{\beta }}_j}} \right){\mathit{\boldsymbol{d}}_j} + \nabla f{\left( {{\mathit{\boldsymbol{\beta }}_j}} \right)^{\rm{T}}}{\mathit{\boldsymbol{d}}_j}\\ {\rm{s}}.{\rm{t}}.\;G\left( {{\mathit{\boldsymbol{\beta }}_j}} \right) + \nabla G\left( {{\mathit{\boldsymbol{\beta }}_j}} \right){\mathit{\boldsymbol{d}}_j} \ge 0 \end{array} \right.$ (14)

 $\mathit{\Psi }(\mathit{\boldsymbol{\beta }}) = f(\mathit{\boldsymbol{\beta }}) + \frac{{{{\left\| {{g_i}(\mathit{\boldsymbol{\beta }})} \right\|}_1}}}{\sigma }$ (15)

1) 给定初始参数解β0、初始拉格朗日乘子向量λ0、初始矩阵B0、不等式方程的雅戈比矩阵。β0为最小二乘解，λ0为元素全为0的向量，B0为单位阵，约束方程的雅戈比矩阵A0=$\nabla \mathit{\boldsymbol{C}}{\left({{\mathit{\boldsymbol{\beta }}_0}} \right)^{\rm{T}}}$

2) 求解序列二次规划子问题(14)，得到dj

3) 对于价值函数Ψ(β, σ)，选取价值函数的参数σj，使dj是该函数在Bj处的下降方向。

4) Armijo搜索，令mj是使下列不等式成立的最小非负整数m

 $\mathit{\Psi }\left( {{\mathit{\boldsymbol{B}}_j} + {p^m}{\mathit{\boldsymbol{d}}_j}, {\sigma _j}} \right) - \mathit{\Psi }\left( {{\mathit{\boldsymbol{B}}_j}, {\sigma _j}} \right) \le {\eta ^m}{\mathit{\Psi }^\prime }\left( {{\mathit{\boldsymbol{B}}_j}, {\sigma _j};{\mathit{\boldsymbol{d}}_j}} \right)$ (16)

${a_j} = {p^m}^{_j}, {\mathit{\boldsymbol{\beta }}_{j + 1}} = {\mathit{\boldsymbol{\beta }}_j} + {a_j}{\mathit{\boldsymbol{d}}_j}, \eta \in \left({0, \frac{1}{2}} \right), {\mathit{\Psi }^\prime }\left({{\mathit{\boldsymbol{B}}_j}, {\sigma _j}; {\mathit{\boldsymbol{d}}_j}} \right)$Ψ(Bj, σj)的方向导数。

5) 计算Aj+1=$\nabla \mathit{\boldsymbol{C}}{\left({{\mathit{\boldsymbol{\beta }}_{j + 1}}} \right)^{\rm{T}}}$和拉格朗日乘子向量${\mathit{\boldsymbol{\lambda }}_{j + 1}} = {\left[{{\mathit{\boldsymbol{A}}_{j + 1}}\mathit{\boldsymbol{A}}_{j + 1}^{\rm{T}}} \right]^{ - 1}}{\mathit{\boldsymbol{A}}_{j + 1}}\nabla {\mathit{\boldsymbol{f}}_{j + 1}}$

6) 校正矩阵BjBj+1。令

 ${\mathit{\boldsymbol{h}}_j} = {\mathit{\boldsymbol{\widehat \beta }}_{j + 1}} - {\mathit{\boldsymbol{\widehat \beta }}_j}$ (17)
 ${\mathit{\boldsymbol{l}}_j} = {\nabla _\beta }\mathit{\boldsymbol{L}}\left( {{{\mathit{\boldsymbol{\widehat \beta }}}_{j + 1}}, {{\mathit{\boldsymbol{\hat \lambda }}}_{j + 1}}} \right) - {\nabla _\beta }\mathit{\boldsymbol{L}}\left( {{{\mathit{\boldsymbol{\widehat \beta }}}_j}, {{\mathit{\boldsymbol{\hat \lambda }}}_j}} \right)$ (18)

WHP拟牛顿校正要求向量hjIj满足曲率条件，即$\mathit{\boldsymbol{h}}_j^{\rm{T}}{\mathit{\boldsymbol{I}}_j}$大于0，但是由式(17)、式(18)确定的向量hjIj可能不满足这一条件。为此，需要用如下步骤对hj进行修正[14]

 ${\mathit{\boldsymbol{Z}}_j} = {\theta _j}{\mathit{\boldsymbol{h}}_j} + {\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{l}}_j} - {\theta _j}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{l}}_j}$ (19)

 ${\theta _j} = \left\{ \begin{array}{l} 1, \mathit{\boldsymbol{I}}_j^{\rm{T}}\mathit{\boldsymbol{h}} \ge 0.2\mathit{\boldsymbol{I}}_j^{\rm{T}}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{I}}_j}\\ \frac{{0.8\mathit{\boldsymbol{I}}_j^{\rm{T}}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{I}}_j}}}{{\mathit{\boldsymbol{I}}_j^{\rm{T}}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{I}}_j} - \mathit{\boldsymbol{I}}_j^{\rm{T}}{\mathit{\boldsymbol{h}}_j}}}, \mathit{\boldsymbol{I}}_j^{\rm{T}}\mathit{\boldsymbol{h}} < 0.2\mathit{\boldsymbol{I}}_j^{\rm{T}}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{I}}_j} \end{array} \right.$ (20)

 ${\mathit{\boldsymbol{B}}_{j + 1}} = {\mathit{\boldsymbol{B}}_j} - \frac{{{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{I}}_j}\mathit{\boldsymbol{I}}_j^{\rm{T}}{\mathit{\boldsymbol{B}}_j}}}{{\mathit{\boldsymbol{I}}_j^{\rm{T}}{\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{I}}_j^{\rm{T}}}} + \frac{{{\mathit{\boldsymbol{Z}}_j}\mathit{\boldsymbol{Z}}_j^{\rm{T}}}}{{\mathit{\boldsymbol{I}}_j^{\rm{T}}{\mathit{\boldsymbol{Z}}_j}}}$ (21)

7) 替代矩阵Bj更新为Bj+1，令j=j+1，回到步骤2)，当最优解满足条件时停止循环。

3 不等式约束相关观测Partial EIV模型精度评定

 $\mathit{\boldsymbol{\widehat \beta }} = f(\mathit{\boldsymbol{l}})$ (22)

1) 计算不含约束条件时的参数估值$\mathit{\boldsymbol{\widehat \beta }}$、改正数$\mathit{\boldsymbol{\hat V}}$、单位权方差$\hat \sigma _0^2$

2) 根据输入量I=vec(y, A)、$\mathit{\boldsymbol{\hat V}}$Ql和单位权方差估值$\hat \sigma _0^2$构建观测值向量I的均值和协方差阵$\mathit{\boldsymbol{\bar l}} = \mathit{\boldsymbol{l}} + \mathit{\boldsymbol{\hat V}}, {\mathit{\boldsymbol{D}}_l} = \hat \sigma _0^2{\mathit{\boldsymbol{Q}}_l}$

3) 通过比例对称采样策略构建2t+1个sigma点列向量Ii

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{I}}_0} = \mathit{\boldsymbol{\bar l}}\\ {\mathit{\boldsymbol{I}}_i} = \mathit{\boldsymbol{\bar l}} + {\left( {\sqrt {t + \gamma } \sqrt {{\mathit{\boldsymbol{D}}_l}} } \right)_i}, i = 1, 2, \cdots , t\\ {\mathit{\boldsymbol{I}}_i} = \mathit{\boldsymbol{\bar l}} - {\left( {\sqrt {t + \gamma } \sqrt {{\mathit{\boldsymbol{D}}_l}} } \right)_i}, i = t + 1, t + 2, \cdots , 2t \end{array} \right.$ (23)

4) 计算sigma点列向量经过非线性变换后的样本：

 $\mathit{\boldsymbol{v}}_i^{\hat \beta } = f\left( {{\mathit{\boldsymbol{l}}_i}} \right), i = 0, 1, \cdots , 2t$ (24)

5) 确定样本$\mathit{\boldsymbol{v}}_i^{\hat \beta }$的权值：

 $\left\{ \begin{array}{l} W_0^m = \gamma /(t + \gamma )\\ W_0^c = \gamma /(t + \gamma ) + 1 - {a^2} + b\\ W_i^m = W_i^c = 1/(2(t + \gamma )), i = 1, 2, \cdots , 2t \end{array} \right.$ (25)

6) 加权计算参数估值的均值E($\mathit{\boldsymbol{\hat \beta }}$)、偏差${\mathit{\boldsymbol{b}}_{\hat \beta }}$、偏差改正后的参数估值$\mathit{\boldsymbol{\tilde \beta }}$、参数估值近似协方差阵${\mathit{\boldsymbol{D}}_{\hat \beta \hat \beta }}$

 $E(\mathit{\boldsymbol{\widehat \beta }}) = \sum\limits_{i = 0}^{2t} {W_i^m} \mathit{\boldsymbol{v}}_i^{\hat \beta }$ (26)
 ${\mathit{\boldsymbol{b}}_{\hat \beta }} = E(\mathit{\boldsymbol{\widehat \beta }}) - \mathit{\boldsymbol{\widehat \beta }}$ (27)
 $\mathit{\boldsymbol{\widetilde \beta }} = \mathit{\boldsymbol{\widehat \beta }} - {\mathit{\boldsymbol{b}}_{\hat \beta }}$ (28)
 ${\mathit{\boldsymbol{D}}_{\hat \beta \hat \beta }} = \sum\limits_{i = 0}^{2t} {W_i^c} \left( {\mathit{\boldsymbol{v}}_i^{\hat \beta } - \mathit{\boldsymbol{\widetilde \beta }}} \right){\left( {\mathit{\boldsymbol{v}}_i^{\hat \beta } - \mathit{\boldsymbol{\widetilde \beta }}} \right)^{\rm{T}}}$ (29)
4 算例及分析

4.1 算例1(不等式约束的平面拟合模型)

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}} {{y_1}}\\ {{y_2}}\\ \vdots \\ {{y_n}} \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} {{e_{{y_1}}}}\\ {{e_{{y_2}}}}\\ \vdots \\ {{e_{{y_n}}}} \end{array}} \right] = \\ \left( {\left[ {\begin{array}{*{20}{c}} 1&{{c_1}}&{{b_1}}\\ 1&{{c_2}}&{{b_2}}\\ \vdots & \vdots & \vdots \\ 1&{{c_n}}&{{b_n}} \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 0&{{e_{{c_1}}}}&{{e_{{b_1}}}}\\ 0&{{e_{{c_2}}}}&{{e_{{b_2}}}}\\ \vdots & \vdots & \vdots \\ 0&{{e_{{c_n}}}}&{{e_{{b_n}}}} \end{array}} \right]} \right)\left[ {\begin{array}{*{20}{c}} {{\beta _1}}\\ {{\beta _2}}\\ {{\beta _3}} \end{array}} \right] \end{array}$ (30)

 $\mathit{\boldsymbol{\bar a}} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{c}}_{n \times 1}}}\\ {{\mathit{\boldsymbol{b}}_{n \times 1}}} \end{array}} \right], \mathit{\boldsymbol{h}} = \left[ {\begin{array}{*{20}{c}} {{{\bf{1}}_{n \times 1}}}\\ {{{\bf{0}}_{2n \times 1}}} \end{array}} \right], \mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {{{\bf{0}}_{n \times 2n}}}\\ {{\mathit{\boldsymbol{I}}_{2n \times 2n}}} \end{array}} \right]$ (31)

4.2 算例2(不等式约束的坐标转换模型)

 $\begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{X_1}}\\ {{Y_1}}\\ \vdots \\ {{X_n}}\\ {{X_n}} \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} {{e_{{X_1}}}}\\ {{e_{{Y_1}}}}\\ \vdots \\ {{e_{{X_n}}}}\\ {{e_{{Y_n}}}} \end{array}} \right] = \left( {\left[ {\begin{array}{*{20}{c}} {{x_1}}&{ - {y_1}}&1&0\\ {{y_1}}&{{x_1}}&0&1\\ \vdots & \vdots & \vdots & \vdots \\ {{x_n}}&{ - {y_n}}&1&0\\ {{y_n}}&{{x_n}}&0&1 \end{array}} \right] - } \right.\\ \;\;\;\;\;\;\;\;\;\left. {\left[ {\begin{array}{*{20}{c}} {{e_{{x_1}}}}&{ - {y_1}}&0&0\\ {{e_{{y_1}}}}&{{x_1}}&0&0\\ \vdots & \vdots & \vdots & \vdots \\ {{e_{{x_n}}}}&{ - {e_{{y_n}}}}&0&0\\ {{e_{{y_n}}}}&{{e_{{x_n}}}}&0&0 \end{array}} \right]} \right)\left[ {\begin{array}{*{20}{l}} {{\beta _1}}\\ {{\beta _2}}\\ {{\beta _3}}\\ {{\beta _4}} \end{array}} \right] \end{array}$ (32)

 $\begin{array}{*{20}{l}} {\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{P}}_{XY}} = {\rm{diag}}([1, 2, 3, 1, 5, }\\ {4, 2, 7, 2, 1, 1, 2, 3, 1, 5, 4, 2, 7, 2, 1{]^{ - 1}})} \end{array}$ (33)
 $\begin{array}{*{20}{l}} {\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{P}}_{xy}} = {\rm{diag}}([1, 3, 6, 1, 1, }\\ {8, 4, 3, 6, 5, 1, 3, 6, 1, 1, 8, 4, 3, 6, 5{]^{ - 1}})} \end{array}$ (34)

 ${\mathit{\boldsymbol{D}}_{XY}} = \sigma _0^2\mathit{\boldsymbol{P}}_{XY}^{ - 1}, {\mathit{\boldsymbol{D}}_{xy}} = \sigma _0^2\mathit{\boldsymbol{P}}_{xy}^{ - 1}$ (35)

5 结语

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The WHP Quasi Newton Correction Method for Inequality Constrained Partial EIV Model and the SUT Method for Its Precision Estimation
WANG Leyang1,2,3     ZOU Chuanyi1,2     WU Lulu4
1. Faculty of Geomatics, East China University of Technology, 418 Guanglan Road, Nanchang 330013, China;
2. Key Laboratory of Watershed Ecology and Geographical Environment Monitoring, MNR, 418 Guanglan Road, Nanchang 330013, China;
3. Key Laboratory for Digital Land and Resources of Jiangxi Province, 418 Guanglan Road, Nanchang 330013, China;
4. Department of Architecture and Civil Engineering, Jiangxi Water Resources Institute, 99 Beishan Road, Nanchang 330013, China
Abstract: We propose a new method to determine solution and precision evaluation of the inequality constrained partial errors-in-variables(Partial EIV) model. Under the total least square rule, the inequality constrained Partial EIV model is converted to standard optimization problems. Using the WHP(Wilson-Han-Powell) quasi Newton correction sequential quadratic programming(SQP) to solve the problem, the SUT method is used to evaluate the accuracy of parameter estimation. Simulation results show that this method can reduce iterations and increase convergence rate, the precision evaluation is simple and effective.
Key words: total least square; inequality constraint; Partial EIV model; quasi Newton correction; SUT method