﻿ 扩展的Helmert型方差分量估计的通用公式
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 大地测量与地球动力学  2020, Vol. 40 Issue (12): 1313-1316  DOI: 10.14075/j.jgg.2020.12.021

引用本文

WANG Jielong, CHEN Yi. General Formulae of Extended Helmert Type for Estimating Variance Components[J]. Journal of Geodesy and Geodynamics, 2020, 40(12): 1313-1316.

第一作者简介

WANG Jielong, postgraduate, majors in modern geodetic and GNSS data processing, E-mail:wangjielong@tongji.edu.cn.

文章历史

1. 同济大学测绘与地理信息学院，上海市四平路1239号，200092

1 概括函数模型

 $\left\{ {\begin{array}{*{20}{l}} {\mathop F\limits_{c \times 1} (\mathit{\boldsymbol{\hat L}},\mathit{\boldsymbol{\hat X}}) = 0}\\ {\mathop \varPhi \limits_{s \times 1} (\mathit{\boldsymbol{\hat X}}) = 0} \end{array}} \right.$ (1)

 $\underset{c\times n}{\mathit{\boldsymbol{A}}}\,\underset{n\times 1}{\mathit{\boldsymbol{V}}}\,+\underset{c\times u}{\mathit{\boldsymbol{B}}}\,\underset{u\times 1}{\mathit{\boldsymbol{x}}}\,+\underset{c\times 1}{\mathit{\boldsymbol{W}}}\,=\underset{c\times 1}{\mathop{\mathbf{0}}}\,$ (2)
 $\underset{s\times u}{\mathit{\boldsymbol{C}}}\,\underset{u\times 1}{\mathit{\boldsymbol{\hat{x}}}}\,+\underset{s\times 1}{\mathop{{{\mathit{\boldsymbol{W}}}_{x}}}}\,=\underset{s\times 1}{\mathop{\mathbf{0}}}\,$ (3)

 $\underset{n\times n}{\mathop{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}}}\,=\sigma _{0}^{2}\mathit{\boldsymbol{Q}}=\sigma _{0}^{2}{{\mathit{\boldsymbol{P}}}^{-1}}$ (4)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} = {\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}} - 2{\mathit{\boldsymbol{K}}^{\rm{T}}}(\mathit{\boldsymbol{AV}} + \mathit{\boldsymbol{B\hat x}} + \mathit{\boldsymbol{W}}) - }\\ {2\mathit{\boldsymbol{K}}_s^{\rm{T}}(\mathit{\boldsymbol{C\hat x}} + {\mathit{\boldsymbol{W}}_x})} \end{array}$ (5)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat x}} = - (\mathit{\boldsymbol{N}}_{BB}^{ - 1} - \mathit{\boldsymbol{N}}_{BB}^{ - 1}{\mathit{\boldsymbol{C}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{CC}^{ - 1}\mathit{\boldsymbol{CN}}_{BB}^{ - 1}){\mathit{\boldsymbol{W}}_e} - }\\ {\mathit{\boldsymbol{N}}_{BB}^{ - 1}{\mathit{\boldsymbol{C}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{CC}^{ - 1}{\mathit{\boldsymbol{W}}_x}} \end{array}$ (6)
 $\mathit{\boldsymbol{V}} = - \mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{AA}^{ - 1}(\mathit{\boldsymbol{W}} + \mathit{\boldsymbol{B\hat x}})$ (7)

 $\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{N}}_{AA}} = \mathit{\boldsymbol{AQ}}{\mathit{\boldsymbol{A}}^{\rm{T}}},{\mathit{\boldsymbol{N}}_{BB}} = {\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{AA}^{ - 1}\mathit{\boldsymbol{B}},{\mathit{\boldsymbol{W}}_e} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{AA}^{ - 1}\mathit{\boldsymbol{W}},{\mathit{\boldsymbol{N}}_{CC}} = \mathit{\boldsymbol{CN}}_{BB}^{ - 1}{\mathit{\boldsymbol{C}}^{\rm{T}}}} \end{array}$ (8)

 ${{\mathit{\boldsymbol{Q}}_K} = \mathit{\boldsymbol{N}}_{AA}^{ - 1} - \mathit{\boldsymbol{N}}_{AA}^{ - 1}\mathit{\boldsymbol{B}}{\mathit{\boldsymbol{Q}}_{\hat x\hat x}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{AA}^{ - 1}}$ (9)
 ${{\mathit{\boldsymbol{Q}}_{\hat x\hat x}} = \mathit{\boldsymbol{N}}_{BB}^{ - 1} - \mathit{\boldsymbol{N}}_{BB}^{ - 1}{\mathit{\boldsymbol{C}}^{\rm{T}}}\mathit{\boldsymbol{N}}_{CC}^{ - 1}\mathit{\boldsymbol{CN}}_{BB}^{ - 1}}$ (10)
 ${\mathit{\boldsymbol{Q}}_W} = \mathit{\boldsymbol{AQ}}{\mathit{\boldsymbol{A}}^{\rm{T}}} = {\mathit{\boldsymbol{N}}_{AA}}$ (11)

 ${\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_V} = \mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{A}}^{\rm{T}}}{\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_w}{\mathit{\boldsymbol{Q}}_K}\mathit{\boldsymbol{AQ}}$ (12)

 ${\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_W} = \sigma _0^2{\mathit{\boldsymbol{Q}}_W} = \sigma _0^2\mathit{\boldsymbol{AQ}}{\mathit{\boldsymbol{A}}^{\rm{T}}} = \mathit{\boldsymbol{A}}\sigma _0^2\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{A}}^{\rm{T}}} = \mathit{\boldsymbol{A \boldsymbol{\varSigma} }}{\mathit{\boldsymbol{A}}^{\rm{T}}}$ (13)

 $\begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} E({\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}}) = {\rm{tr }}(\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_V}) = }\\ {{\rm{ tr}} ({\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{AA}}{\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_W}) = {\rm{tr}} ({\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{AA}}{\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{AA}})\sigma _0^2} \end{array}$ (14)
2 扩展的Helmert型方差分量通用公式

 $\mathit{\boldsymbol{ \boldsymbol{\varSigma} }} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_1}}&{}&{}&{}\\ {}&{{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_2}}&{}&{}\\ {}&{}& \ddots &{}\\ {}&{}&{}&{{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_k}} \end{array}} \right]$ (15)

 $\mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} {\mathop {{\mathit{\boldsymbol{L}}_1}}\limits_{{n_1} \times 1} }\\ {\mathop {{\mathit{\boldsymbol{L}}_2}}\limits_{{n_2} \times 1} }\\ \vdots \\ {\mathop {{\mathit{\boldsymbol{L}}_k}}\limits_{{n_k} \times 1} } \end{array}} \right],\mathit{\boldsymbol{V}} = \left[ {\begin{array}{*{20}{c}} {\mathop {{\mathit{\boldsymbol{V}}_1}}\limits_{{n_1} \times 1} }\\ {\mathop {{\mathit{\boldsymbol{V}}_2}}\limits_{{n_2} \times 1} }\\ \vdots \\ {\mathop {{\mathit{\boldsymbol{V}}_k}}\limits_{{n_k} \times 1} } \end{array}} \right],\mathit{\boldsymbol{A}} = \left[ {\mathop {{\mathit{\boldsymbol{A}}_1}}\limits_{c \times {n_1}} ,\mathop {{\mathit{\boldsymbol{A}}_2}}\limits_{c \times {n_2}} , \cdots ,\mathop {{\mathit{\boldsymbol{A}}_k}}\limits_{c \times {n_k}} } \right]$ (16)

Σ1, Σ2, …Σk为各类观测值的方差阵，且假设每类观测值含有多个方差分量，即

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_1} = \sigma _{11}^2{\mathit{\boldsymbol{Q}}_{11}} + \sigma _{12}^2{\mathit{\boldsymbol{Q}}_{12}} + \cdots + \sigma _{1{n_1}}^2{\mathit{\boldsymbol{Q}}_{1{n_1}}}}\\ {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_2} = \sigma _{21}^2{\mathit{\boldsymbol{Q}}_{21}} + \sigma _{22}^2{\mathit{\boldsymbol{Q}}_{22}} + \cdots + \sigma _{2{n_2}}^2{\mathit{\boldsymbol{Q}}_{2{n_2}}}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ldots }\\ {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_k} = \sigma _{k1}^2{\mathit{\boldsymbol{Q}}_{k1}} + \sigma _{k2}^2{\mathit{\boldsymbol{Q}}_{k2}} + \cdots + \sigma _{k{n_k}}^2{\mathit{\boldsymbol{Q}}_{k{n_k}}}} \end{array}} \right.$ (17)

 ${\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{V}}_1} + {\mathit{\boldsymbol{A}}_2}{\mathit{\boldsymbol{V}}_2} + \cdots + {\mathit{\boldsymbol{A}}_k}{\mathit{\boldsymbol{V}}_k} + \mathit{\boldsymbol{B\hat x}} + \mathit{\boldsymbol{W}} = 0$ (18)

 $\begin{array}{l} {\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_0} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{10}}}&{}&{}&{}\\ {}&{{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{20}}}&{}&{}\\ {}&{}& \ddots &{}\\ {}&{}&{}&{{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{k0}}} \end{array}} \right] = \mathit{\boldsymbol{Q}} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_1}}&{}&{}&{}\\ {}&{{\mathit{\boldsymbol{Q}}_2}}&{}&{}\\ {}&{}& \ddots &{}\\ {}&{}&{}&{{\mathit{\boldsymbol{Q}}_k}} \end{array}} \right] \end{array}$ (19)

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_1} = {\mathit{\boldsymbol{Q}}_{11}} + {\mathit{\boldsymbol{Q}}_{12}} + \cdots + {\mathit{\boldsymbol{Q}}_{1{n_1}}}}\\ {{\mathit{\boldsymbol{Q}}_2} = {\mathit{\boldsymbol{Q}}_{21}} + {\mathit{\boldsymbol{Q}}_{22}} + \cdots + {\mathit{\boldsymbol{Q}}_{2{n_2}}}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots }\\ {{\mathit{\boldsymbol{Q}}_k} = {\mathit{\boldsymbol{Q}}_{k1}} + {\mathit{\boldsymbol{Q}}_{k2}} + \cdots + {\mathit{\boldsymbol{Q}}_{k{n_k}}}} \end{array}} \right.$ (20)

 $\mathit{\boldsymbol{P}} = {\mathit{\boldsymbol{Q}}^{ - 1}} = \left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{P}}_1}}&{}&{}&{}\\ {}&{{\mathit{\boldsymbol{P}}_2}}&{}&{}\\ {}&{}& \ddots &{}\\ {}&{}&{}&{{\mathit{\boldsymbol{P}}_k}} \end{array}} \right]$ (21)

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{P}}_{i1}} = {\mathit{\boldsymbol{P}}_i}{\mathit{\boldsymbol{Q}}_{i1}}{\mathit{\boldsymbol{P}}_i}}\\ {{\mathit{\boldsymbol{P}}_{i2}} = {\mathit{\boldsymbol{P}}_i}{\mathit{\boldsymbol{Q}}_{i2}}{\mathit{\boldsymbol{P}}_i}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ldots }\\ {{\mathit{\boldsymbol{P}}_{i{n_i}}} = {\mathit{\boldsymbol{P}}_i}{\mathit{\boldsymbol{Q}}_{i{n_i}}}{\mathit{\boldsymbol{P}}_i}} \end{array},i = 1,2, \cdots k} \right.$ (22)

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{P}}_1} = {\mathit{\boldsymbol{P}}_{11}} + {\mathit{\boldsymbol{P}}_{12}} + \cdots + {\mathit{\boldsymbol{P}}_{1{n_1}}}}\\ {{\mathit{\boldsymbol{P}}_2} = {\mathit{\boldsymbol{P}}_{21}} + {\mathit{\boldsymbol{P}}_{22}} + \cdots + {\mathit{\boldsymbol{P}}_{2{n_2}}}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots }\\ {{\mathit{\boldsymbol{P}}_k} = {\mathit{\boldsymbol{P}}_{k1}} + {\mathit{\boldsymbol{P}}_{k2}} + \cdots + {\mathit{\boldsymbol{P}}_{k{n_k}}}} \end{array}} \right.$ (23)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{N}}_{AA}} = \sum\limits_{i = 1}^{{n_1}} {{\mathit{\boldsymbol{A}}_1}} {\mathit{\boldsymbol{Q}}_{1i}}\mathit{\boldsymbol{A}}_1^{\rm{T}} + \sum\limits_{j = 1}^{{n_2}} {{\mathit{\boldsymbol{A}}_2}} {\mathit{\boldsymbol{Q}}_{2j}}\mathit{\boldsymbol{A}}_2^{\rm{T}} + \cdots }\\ { + \sum\limits_{h = 1}^{{n_k}} {{\mathit{\boldsymbol{A}}_k}} {\mathit{\boldsymbol{Q}}_{kh}}\mathit{\boldsymbol{A}}_k^{\rm{T}}} \end{array}$ (24)

ΣW=AΣAT，顾及式(16)、(17)有：

 $\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_W} = \sum\limits_{i = 1}^{{n_1}} {{\mathit{\boldsymbol{A}}_1}} \sigma _{1i}^2{\mathit{\boldsymbol{Q}}_{1i}}\mathit{\boldsymbol{A}}_1^{\rm{T}} + \sum\limits_{j = 1}^{{n_2}} {{\mathit{\boldsymbol{A}}_2}} \sigma _{2j}^2{\mathit{\boldsymbol{Q}}_{2j}}\mathit{\boldsymbol{A}}_2^{\rm{T}} + \cdots }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \sum\limits_{h = 1}^{{n_k}} {{\mathit{\boldsymbol{A}}_k}} \sigma _{kh}^2{\mathit{\boldsymbol{Q}}_{kh}}\mathit{\boldsymbol{A}}_k^{\rm{T}}} \end{array}$ (25)

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{Q}}_{1i}}\mathit{\boldsymbol{A}}_1^{\rm{T}} = {\mathit{\boldsymbol{N}}_{1i}},i = 1,2, \cdots ,{n_1}}\\ {{\mathit{\boldsymbol{A}}_2}{\mathit{\boldsymbol{Q}}_{2j}}\mathit{\boldsymbol{A}}_2^{\rm{T}} = {\mathit{\boldsymbol{N}}_{2j}},j = 1,2, \cdots ,{n_2}}\\ {{\kern 1pt} {\kern 1pt} \cdots }\\ {{\mathit{\boldsymbol{A}}_k}{\mathit{\boldsymbol{Q}}_{kh}}\mathit{\boldsymbol{A}}_k^{\rm{T}} = {\mathit{\boldsymbol{N}}_{kh}},h = 1,2, \cdots ,{n_k}} \end{array}} \right.$ (26)
 $\begin{array}{*{20}{c}} {\sum\limits_{i = 1}^{{n_1}} {{\mathit{\boldsymbol{A}}_1}} {\mathit{\boldsymbol{Q}}_{1i}}\mathit{\boldsymbol{A}}_1^{\rm{T}} = {\mathit{\boldsymbol{N}}_1},\sum\limits_{j = 1}^{{n_2}} {{\mathit{\boldsymbol{A}}_2}} {\mathit{\boldsymbol{Q}}_{2j}}\mathit{\boldsymbol{A}}_2^{\rm{T}} = }\\ {{\mathit{\boldsymbol{N}}_2}, \cdots ,\sum\limits_{h = 1}^{{n_k}} {{\mathit{\boldsymbol{A}}_k}} {\mathit{\boldsymbol{Q}}_{kh}}\mathit{\boldsymbol{A}}_k^{\rm{T}} = {\mathit{\boldsymbol{N}}_k}} \end{array}$ (27)

 ${\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_W} = \sum\limits_{m = 1}^{{n_1}} {\sigma _{1m}^2} {\mathit{\boldsymbol{N}}_{1m}} + \sum\limits_{p = 1}^{{n_2}} {\sigma _{2p}^2} {\mathit{\boldsymbol{N}}_{2p}} + \cdots + \sum\limits_{q = 1}^{{n_k}} {\sigma _{kq}^2} {\mathit{\boldsymbol{N}}_{kq}}$ (28)
 ${\mathit{\boldsymbol{N}}_{AA}} = \sum\limits_{i = 1}^{{n_1}} {{\mathit{\boldsymbol{N}}_{1i}}} + \sum\limits_{j = 1}^{{n_2}} {{\mathit{\boldsymbol{N}}_{2j}}} + \cdots + \sum\limits_{h = 1}^{{n_k}} {{\mathit{\boldsymbol{N}}_{kh}}}$ (29)

 $\begin{array}{l} {\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}} = \sum\limits_{l = 1}^k {\mathit{\boldsymbol{V}}_l^{\rm{T}}} {\mathit{\boldsymbol{P}}_l}{\mathit{\boldsymbol{V}}_l} = \sum\limits_{i = 1}^{{n_i}} {\mathit{\boldsymbol{V}}_1^{\rm{T}}} {\mathit{\boldsymbol{P}}_{1i}}{\mathit{\boldsymbol{V}}_1} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{j = 1}^{{n_2}} {\mathit{\boldsymbol{V}}_2^{\rm{T}}} {\mathit{\boldsymbol{P}}_{2j}}{\mathit{\boldsymbol{V}}_2} + \cdots + \sum\limits_{h = 1}^{{n_k}} {\mathit{\boldsymbol{V}}_k^{\rm{T}}} {\mathit{\boldsymbol{P}}_{kh}}{\mathit{\boldsymbol{V}}_k} \end{array}$ (30)

 $\begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{tr}} ({\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{AA}}{\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_W}) = \\ {\rm{tr}} ({\mathit{\boldsymbol{Q}}_K}(\sum\limits_{i = 1}^{{n_1}} {{\mathit{\boldsymbol{N}}_{1i}}} + \cdots + \sum\limits_{h = 1}^{{n_k}} {{\mathit{\boldsymbol{N}}_{kh}}} ){\mathit{\boldsymbol{Q}}_K}(\sum\limits_{m = 1}^{{n_1}} {\sigma _{1m}^2} {\mathit{\boldsymbol{N}}_{1m}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {\sum\limits_{p = 1}^{{n_2}} {\sigma _{2p}^2} {\mathit{\boldsymbol{N}}_{2p}} + \cdots + \sum\limits_{q = 1}^{{n_k}} {\sigma _{kq}^2} {\mathit{\boldsymbol{N}}_{kq}})} \right) \end{array}$ (31)

 $\begin{array}{*{20}{c}} {\mathop {\hat \theta }\limits_{({n_1} + {n_2} + \cdots + {n_k}) \times 1} = }\\ {{{[\hat \sigma _{11}^2,\hat \sigma _{12}^2, \cdots ,\hat \sigma _{1{n_1}}^2, \cdots \cdots ,\hat \sigma _{k1}^2,\hat \sigma _{k2}^2, \cdots ,\hat \sigma _{k{n_k}}^2]}^{\rm{T}}}} \end{array}$ (32)
 $\begin{array}{*{20}{c}} {\mathop f\limits_{({n_1} + {n_2} + \cdots + {n_k}) \times 1} = [\mathit{\boldsymbol{V}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_{11}}{\mathit{\boldsymbol{V}}_1},\mathit{\boldsymbol{V}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_{12}}{\mathit{\boldsymbol{V}}_1}, \cdots ,}\\ {\mathit{\boldsymbol{V}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_{1{n_1}}}{\mathit{\boldsymbol{V}}_1}, \cdots ,\mathit{\boldsymbol{V}}_k^{\rm{T}}{\mathit{\boldsymbol{P}}_{k1}}{\mathit{\boldsymbol{V}}_k},\mathit{\boldsymbol{V}}_k^{\rm{T}}{\mathit{\boldsymbol{P}}_{k2}}{\mathit{\boldsymbol{V}}_k}, \cdots ,\mathit{\boldsymbol{V}}_k^{\rm{T}}{\mathit{\boldsymbol{P}}_{k{n_k}}}{\mathit{\boldsymbol{V}}_k}{]^{\rm{T}}}} \end{array}$ (33)
 $\begin{array}{*{35}{l}} \underset{({{n}_{1}}+{{n}_{2}}+\cdots +{{n}_{k}})\times 1}{\mathop{{{\mathit{\boldsymbol{H}}}_{ij}}}}\,=\left[ \begin{matrix} \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{11}} \right) \\ \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{12}} \right) \\ \cdots \\ \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{1{{n}_{1}}}} \right) \\ \cdots \\ \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{k1}} \right) \\ \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{k2}} \right) \\ \ldots \\ \cdots \\ \text{tr}\left( {{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{ij}}{{\mathit{\boldsymbol{Q}}}_{K}}{{\mathit{\boldsymbol{N}}}_{k{{n}_{k}}}} \right) \\ \end{matrix} \right], \\ \underset{({{n}_{1}}+{{n}_{2}}+\cdots +{{n}_{k}})\times ({{n}_{1}}+{{n}_{2}}+\cdots +{{n}_{k}})}{\mathop{\mathit{\boldsymbol{H}}}}\,= \\ \mathit{\boldsymbol{H}}_{ij}^{\text{T}},i=1,2,\cdots ,k;j=1,2,\cdots ,{{n}_{j}} \\ \end{array}$ (34)

 $\mathit{\boldsymbol{H\hat \theta }} = \mathit{\boldsymbol{f}}$ (35)

 $\begin{array}{l} \sigma _{ij}^2 = \frac{{\mathit{\boldsymbol{V}}_i^{\rm{T}}{\mathit{\boldsymbol{P}}_{ij}}{\mathit{\boldsymbol{V}}_i}}}{{ {\rm{tr}} ({\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{ij}}{\mathit{\boldsymbol{Q}}_K}{\mathit{\boldsymbol{N}}_{AA}})}},\\ i = 1,2, \cdots ,k;j = 1,2, \cdots ,{n_j} \end{array}$ (36)

3 特殊情况

A=-IC= 0时，式(2)、(3)可变为：

 $\left\{ {\begin{array}{*{20}{l}} {\mathop {{\mathit{\boldsymbol{V}}_1}}\limits_{{n_1} \times 1} = {\mathit{\boldsymbol{B}}_1}\mathit{\boldsymbol{\hat x}} - {\mathit{\boldsymbol{W}}_1}}\\ {\mathop {{\mathit{\boldsymbol{V}}_2}}\limits_{{n_2} \times 1} = {\mathit{\boldsymbol{B}}_2}\mathit{\boldsymbol{\hat x}} - {\mathit{\boldsymbol{W}}_2}}\\ \cdots \\ {\mathop {{\mathit{\boldsymbol{V}}_{k - 1}}}\limits_{{n_{k - 1}} \times 1} = {\mathit{\boldsymbol{B}}_{k - 1}}\mathit{\boldsymbol{\hat x}} - {\mathit{\boldsymbol{W}}_{k - 1}}}\\ {\mathop {{\mathit{\boldsymbol{V}}_k}}\limits_{{n_k} \times 1} = {\mathit{\boldsymbol{B}}_k}\mathit{\boldsymbol{\hat x}} - {\mathit{\boldsymbol{W}}_k}}\\ \end{array}} \right.$ (37)

 ${{\mathit{\boldsymbol{Q}}_{\hat x}} = {{({\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}})}^{ - 1}} = \mathit{\boldsymbol{N}}_{BB}^{ - 1}}$ (38)
 ${{\mathit{\boldsymbol{Q}}_K} = \mathit{\boldsymbol{P}} - \mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{Q}}_{\hat x}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P}}}$ (39)

 $E({\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}}) = {\rm{tr}} ({\mathit{\boldsymbol{Q}}_K}\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{Q}}_K}\mathit{\boldsymbol{Q}})\sigma _0^2$ (40)

 ${\mathit{\boldsymbol{Q}}_K}\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{Q}}_K}\mathit{\boldsymbol{Q}} = \mathit{\boldsymbol{PQ}} - 2\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{Q}}_{\hat x}}{\mathit{\boldsymbol{B}}^{\rm{T}}} + \mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{Q}}_{\hat x}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{Q}}_{\hat x}}{\mathit{\boldsymbol{B}}^{\rm{T}}}$ (41)

4 结语

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General Formulae of Extended Helmert Type for Estimating Variance Components
WANG Jielong1     CHEN Yi1
1. College of Surveying and Geo-Informatics, Tongji University, 1239 Siping Road, Shanghai 200092, China
Abstract: General formulae, based on a general function model and the theory of estimating variance components of Helmert type, are derived in this paper in order to estimate different variance components from same kind of observation. The simplified formulae from general formulae and some special cases are also given.
Key words: variance and co-variance components estimation; general function model; general formulae; Helmert variance components estimation