MINQUE of Variance-Covariance Components in Linear Gauss-Markov Models Peng Junhuan1; Shi Yun2; Li Shuhui3; and Yang Honglei4
Abstract: For heterogeneous and correlated observations, the variance components and the covariance components sometimes must be estimated. The forms of best invariant quadratic unbiased estimate (BIQUE) and Helmert-type estimation of variance and covariance components have already been derived by Koch and Grafarend, respectively. After obtaining the minimum norm quadratic unbiased estimate (MINQUE) of variance components, Rao derived only the MINQUE of the variance and covariance components for a special case in which the error vector is composed of a linear combination of independent random effect vectors of zero mean and the same variance-covariance matrix whose variance and covariance components were to be determined. However, an explicit expression of the MINQUE suitable to more general situations has not been derived. This paper defines the natural estimation of covariance components from errors and derives the MINQUE of variance and covariance components. The BIQUE and MINQUE of variance components without covariance components have the same iteration solution; the Helmert solution is only a special case of the MINQUE. However, the three estimates of variance and covariance components are different. The two MINQUE methods obtained in this paper have the advantage independence of the error distribution and offer a reasonable alternative in estimating variance and covariance components, and they can be used in the most general case. Numeric results show that the two MINQUE methods obtained in this paper are feasible. DOI: 10.1061/(ASCE)SU.1943-5428 .0000050. © 2011 American Society of Civil Engineers. CE Database subject headings: Heterogeneity; Estimation; Markov process; Gaussian process. Author keywords: Variance components; Variance-covariance components; Minimum norm quadratic unbiased estimate (MINQUE); Least squares and minimum norm quadratic unbiased estimation (LSMINQUE); Best invariant quadratic unbiased estimate (BIQUE); Helmert-type estimation.
Introduction The estimation of variance components has been studied and applied in the science of surveying and mapping (Grafarend and Schaffrin 1976; Grafarend 1980, 1985; Helmert 1907, 1924; Koch 1986, 1999; Kubik 1970; Ou 1989; Peng 2009; Schaffrin 1981a, b, 1983; Xu et al. 2007; Yu 1992, 1996) and statistics (Azais et al. 1993; Drygas 1977; Henderson 1953; Lamotte 1973; Rao 1970, 1971a, b, 1972, 1973, 1997; Rao and Kleffe 1988; Searle 1992; Verdooren 1988; Volaufová and Witkovsky 1992; Volaufová 1993; Witkovsky 1996). In the case of heterogeneous and correlated observations, there are four different unbiased methods to estimate the variance components: the restricted maximum likelihood estimation (RMLE) (Rao 1997; Searle 1992); best invariant 1 School of Land Science and Technology, China Univ. of Geosciences (Beijing), Xueyuan Rd. 29, Haiding District, 100083, Beijing, China (corresponding author). E-mail:
[email protected] 2 Dept. of Surveying and Mapping, Xi’an Univ. of Technology, Yanta Middle Rd. 58, 710054, Xi’an, China. E-mail:
[email protected] 3 School of Land Science and Technology, China Univ. of Geosciences (Beijing), Xueyuan Rd. 29, Haiding District, 100083, Beijing, China. E-mail:
[email protected] 4 School of Land Science and Technology, China Univ. of Geosciences (Beijing), Xueyuan Rd. 29, Haiding District, 100083, Beijing, China. E-mail:
[email protected] Note. This manuscript was submitted on May 9, 2010; approved on November 24, 2010; published online on December 9, 2010. Discussion period open until April 1, 2012; separate discussions must be submitted for individual papers. This paper is part of the Journal of Surveying Engineering, Vol. 137, No. 4, November 1, 2011. ©ASCE, ISSN 0733-9453/ 2011/4-129–139/$25.00.
quadratic unbiased estimation (BIQUE) (Drygs 1977; Koch 1999; Lamotte 1973), which is the special case of minimum variance invariant quadratic unbiased estimation (MVIQUE) (Rao 1971b, 1997) with a normal assumption; the minimum norm quadratic unbiased estimation (MINQUE) (Rao 1970, 1971a, 1973); and the Helmert-type estimation (Helmert 1907, 1924). They have very close mathematical relations. The RMLE and the BIQUE depend on the normality assumption of errors; the MINQUE need not know the error distribution. For the same initial values of variance components, the iterative forms of the three methods are completely identical and lead to the same numeric result. Frequently, in practice, the three methods need not to be distinguished. The Helmert method is a special case of the former three methods as the stochastic model is block-diagonal-structured matrix. The MINQUE is the most general method because the error distribution need not be known, and the stochastic model need not be block-diagonal structured. However, when observations are not independent and the covariance components cannot be represented as a linear function of variance components, not only must the variance components be estimated, but also must the covariance components be estimated. For a stochastic model with the block-partitioned matrix, Grafarend (1980) generalized the Helmert method into an estimation of variance-covariance components and said that the Helmert-type unbiased estimation of variance-covariance components exists if and only if the test matrix is regular. The normal matrix for solving the Helmert-type estimation of variance-covariance components is not symmetric. Using the orthogonal complement likelihood function, Koch (1986), under a normality assumption, derived equally the REML of variance-covariance components and proved that it equals the BIQUE of variance-covariance components.
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Ou (1989) discussed the relation between Helmert-type estimation and BIQUE and gave an approximate iterative method of Helmerttype estimation; Crocetto et al. (2000) gave the simplified iterative procedures of BIQUE. Relating the squares and products of the residuals to the variance components to be determined and restructuring them as a linear model is called the dispersion-mean model (Searle 1992) and solved by the least-squares method, as first proposed by Pukelsheim (1974). It can also be viewed as the seminal work of Seely (1971, 1972). A variation of it is used by Malley (1986). For the ordinary and generalized least-squares criteria, the MINQUE and REML or BIQUE can be derived from the dispersion-mean model. The least-squares variance components estimation proposed by Teunissen and Amiri-Simkooei (2007) is in substance similar to the dispersion-mean model, and early in 1988, Verdooren (1988) also named the least-squares estimator of variance components. In Amiri-Simkooei et al. (2009), the leastsquares variance components estimation is practically the BIQUE or REML. Except for the MINQUE method, these methods for estimating the variance components without covariance components all have the only corresponding form for estimating variance-covariance components. However, for the MINQUE, Rao (1971a, 1972) only derived the MINQUE of variance-covariance components of two special cases and did not give a general formulation suitable for any situation (Rao 1971a, 1972, 1973). Different from the BIQUE and the Helmert-type estimation, the norm criterion of MINQUE is derived from the nature estimation of random effect components of the error vector, which requires not only that the variancecovariance matrix is clarified, but also the corresponding error vector structure must be defined clearly. For the BIQUE and the Helmert-type estimation, it is enough to clarify the structure of the variance-covariance matrix. In theory and practice, deriving the MINQUE of variance-covariance components suitable for the more general situation is interesting because the BIQUE and the Helmert-type estimation are respectively limited by error distribution and the block-partitioned structure of the stochastic model, and the two MINQUEs derived by Rao are only convenient in computation for two special cases (Rao 1971a, 1972; Rao and Kleffe 1988). This paper discusses the approach to derive the MINQUE of variance-covariance components suitable to more general situations and gives a comparison among different types of estimation, such as the BIQUE, MINQUE, and Helmert estimation. A numerical example demonstrating this method when errors are normally distributed is also given.
BIQUE and Helmert-Type Estimation of Variance-Covariance Components
2
σ21
6 6 σ21 DðeÞ ¼ 6 6 4
and
σn1
y ¼ Xβ þ e
ð1Þ
EðeÞ ¼ 0
ð2Þ
Stochastic model
θr ¼ and
~r ¼ Q
σ2i σij
σ22
7 σ2n 7 7 7 5
σn2
σ2n
e ¼ Hξ ¼ H1 ξ1 þ H2 ξ2 þ þ Hm ξm
ð4Þ
where ξi = ni × 1 random vector with Eðξi Þ ¼ 0; m = number of the independent random vectors; Dðξi Þ ¼ σ2i Ii and Dðξi ; ξj Þ ¼ 0 for i ≠ j; Ii = ni × ni identity matrix; σ2i = variance component; Hi = n × ni known design matrix (Rao 1973); H ¼ ðH1 ; H2 ; …; Hm Þ; and ξ ¼ ðξT1 ; ξT2 ; …; ξTm ÞT . The corresponding variance matrix of the error vector e is represented by DðeÞ ¼
m X
Hi Dðξi ÞHTi ¼
m X
i¼1
σ2i Hi HTi ¼
i¼1
m X
σ2i Ti
ð5Þ
i¼1
P ¼ m Ti ¼ HHT . The form in where Ti ¼ HTi Hi . Denote T i¼1 Eq. (4) and the form in Eq. (5) do not include the case with covariance components. To include covariance components, reassume that different random vectors ξi (i ¼ 1; 2; …; m) are correlated with Dðξi ; ξj Þ ¼ σij Iij , where σij = covariance component; and Iij ¼ ð1Þni ×nj = ni × nj matrix whose elements are all unity. The variancecovariance matrix of the error vector e is DðeÞ ¼
m X m X
Hi Dðξ i ; ξ j ÞHTj
i¼1 j¼1
¼
m X
σ2i Hi HTi þ
i¼1
¼
m X i¼1
m X m X
σij ðHi Iij HTj þ Hj Iji HTi Þ
i¼1 j¼iþ1
σ2i Ti þ
m X m X
σij ðTij þ Tji Þ
ð6Þ
i¼1 j¼iþ1
P Pm Pm where Tij ¼ Hi Iij HTj . Denote T ¼ m i¼1 Ti þ i¼1 j¼iþ1 ðTij þ Tji Þ. For convenience of derivation in subsequent sections, Eq. (6) is rewritten as DðeÞ ¼
l X
~r θr Q
r¼1
where l ¼ mðm þ 1Þ=2
r ¼ ð2m þ 2 iÞði 1Þ=2 þ 1; i ¼ j ¼ 1; 2; …; m r ¼ ð2m þ 2 iÞði 1Þ=2 þ j i þ 1 i ¼ 1; 2; …; m; j ¼ i þ 1; …; m
Ti Tij þ Tji
ð3Þ
where y = vector of n observations; β = vector of t unknown parameters; X = n × t design matrix; e = error vector; σ2i and σij = variance and covariance components, respectively, and the number of estimable unknown components must be less than or equal to rðr þ 1Þ=2 with r ¼ n t (Xu et al. 2007); E = mathematical expectation sign; and D = variance sign. For the estimation of variance components without covariance components, the error vector e can be represented as a linear function of different and independent random vectors ξi by (Rao 1973)
Consider the Gauss-Markov linear model: Function model
σ1n
3
σ12
r ¼ ð2m þ 2 iÞði 1Þ=2 þ 1; i ¼ j ¼ 1; 2; …; m r ¼ ð2m þ 2 iÞði 1Þ=2 þ j i þ 1 i ¼ 1; 2; …; m; j ¼ i þ 1; …; m
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ð7Þ
Denote θ ¼ ðθ1 ; θ2 ; θ3 ; …; θl1 ; θl ÞT 2 σ3 ; …; σðm1Þm ; σ2m ÞT .
¼ ðσ21 ; σ12 ; σ22 ; σ13 ; σ23 ;
~T θ ¼ α
m X
αi σ2i þ
i¼1
¼
Principle of BIQUE
l X
m X m X
αij σij
i¼1 j¼iþ1
~ 1 σ21 þ α ~ 2 σ12 þ α ~ 3 σ22 þ α ~ 4 σ13 ~ r θr ¼ α α
r¼1
For the BIQUE of variance-covariance components, the quadratic form yT My is defined as an estimation of the linear function:
~ 5 σ23 þ α ~ 6 σ23 þ þ α ~ l1 σðm1Þm þ α ~ l σ2m þα
ð8Þ
under the three conditions
1: MX ¼ 0 ~ r θr ¼ 2: EðyT MyÞ ¼ α
ð9Þ l X
~ r θr α
ð10Þ
r¼1
3: and
DðyT MyÞ ¼ min
ð11Þ
~ i are arbitrary; and where M = n × n unknown matrix to be determined; the coefficients α αi r ¼ ð2m þ 2 iÞði 1Þ=2 þ 1; i ¼ j ¼ 1; 2; …; m ~r ¼ α αij r ¼ ð2m þ 2 iÞði 1Þ=2 þ j i þ 1 i ¼ 1; 2; …; m; j ¼ i þ 1; …; m
Condition 1 implies that the quadratic form is translation-invariant: EðyT MyÞ ¼ E½ðXβ þ eÞT MðXβ þ eÞ ¼ E½ðXβ 0 þ
eÞT MðXβ
0
þ eÞ ¼
EðeT MeÞ
ð13Þ
Condition 2 means that the quadratic form is an unbiased estimation of the linear function under Condition 1; therefore, it follows that EðyT MyÞ ¼ E½ðXβ þ eÞT MðXβ þ eÞ ¼ tr½MDðeÞ ¼
l X
~ rÞ θr trðMQ
ð14Þ
Helmert-Type Estimation For Gauss-Markov linear models defined by Eq. (1), Grafarend (1980) extended the Helmert estimation of variance components into the estimation of variance-covariance components in the special case that the stochastic model is a block-partitioned matrix: 2 2 3 σ1 Q11 σ12 Q12 σ1m Q1m 6 7 l 6 σ12 Q21 σ22 Q22 σ2m Q2m 7 X 6 7 ~ DðeÞ ¼ 6 θQ ð19Þ ¼ 7 .. .. .. .. 6 7 r¼1 r r . . . . 4 5 σm1 Qm1
r¼1
Allowing for the form of Eqs. (10) and (14), the unbiased Condition 2 equals ~ r Þ; ~ r ¼ trðMQ α
r ¼ 1; 2; …; l
ð15Þ
Condition 3 indicates that the variance of the quadratic form has to be minimum, which equals, under the assumption of normally distributed errors (Koch 1999) Dðy MyÞ ¼ 2tr½MDðeÞMDðeÞ ¼ min T
ð16Þ
Let all the initial values of the variance components and covariance components be one. Eq. (16) reduces to DðyT MyÞ ¼ 2trðMTMTÞ ¼ min
ð17Þ
Minimizing trðMTMTÞ under the constraint of Eqs. (9) and (15) leads to the BIQUE of variance components and covariance components, as follows (Koch 1999): 1
^θ ¼ S W
ð18Þ
~ i CQ ~ j Þ; i; j ¼ 1; 2; …; l; W ¼ ðW i Þ; where S ¼ ðSij Þ; Sij ¼ trðCQ T 1 ~ W i ¼ Y CQi CY; and C ¼ T T1 XðXT T1 XÞ1 XT T1 . The equality DðyT MyÞ ¼ 2tr½MDðeÞMDðeÞ holds only under the normality assumption of errors, which means that the form of Eq. (18) cannot be derived if errors are not normally distributed, theoretically.
ð12Þ
σm2 Qm2
σ2m Qmm
where DðeÞ = block-partitioned matrix; Qij = block matrix; 2 3 2 0 Q12 Q11 0 0 6 7 6 6 0 6 Q21 0 0 07 7 ~2 ¼ 6 ~1 ¼ 6 Q Q 6 . 7; 6 . . . . .. . .. 7 .. .. 6 .. 6 .. .. . 4 5 4 0 0 0 0 0 2 3 0 0 0 6 7 6 0 Q22 0 7 6 7 ~ ; …; Q3 ¼ 6 . .. 7 .. .. 6 .. .7 . . 4 5
0
0
0 6. 6 .. ~ l1 ¼ 6 Q 6 60 4 0 2 0 6. 6 .. ~l ¼ 6 Q 6 60 4
.. .
0 .. .
0
Qmðm1Þ
2
.. .
0
0
3 0 .. 7 . 7 7 7 0 0 7 5 0 Qmm
0 .. .
3
7 7 7 7; Qðm1Þm 7 5 0
and 3 0 7 07 7 ; .. 7 .7 5 0
ð20Þ and
0 .. .
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Denote Q ¼ ðQij Þm×m . The weight matrix is P ¼ Q1 ¼ P ~ r , here ðPij Þm×m ¼ lr¼1 P 2 6 6 ~1 ¼ 6 P 6 6 4 2
P11
0
0 .. .
0 .. .
.. .
0
0
0 6 60 ~3 ¼ 6 P 6. 6 .. 4
0
0
P22 .. .
.. .
7 07 7 ; .. 7 .7 5 0 3 0 7 07 7 ; .. 7 .7 5
0
0
0
0 6. 6 .. 6 ¼6 60 4
.. .
0 .. .
0
0
~ mðm1Þ P
0 6. 6 .. ~l ¼ 6 P 6 60 4 0
.. .
0 .. .
0
0
2
~ l1 P
2
2
3
6 6 P21 ~2 ¼ 6 P 6 . 6 .. 4 0
3
P12
0 .. .
.. .
7 07 7 ; .. 7 .7 5
0
0
0
;
0 .. .
3 0 .. 7 . 7 7 7 0 7 5 ~ mm P
0
Derivation of MINQUE of Variance-Covariance Components ð21Þ
3
7 7 7 7; ~ ðm1Þm 7 P 5 0
and
The BIQUE of variance-covariance components holds only under the normality assumption of errors; the Helmert-type estimation depends on the least-squares estimation of unknown parameters and can only be used in the stochastic model with a block-partitioned matrix. Rao (1971a, b, 1972) only proposed the MINQUE of variance-covariance components for two special cases, which are not suited to the usual geodetic situation. Here, the MINQUE of variance and covariance components suitable for the more general situation will be discussed and solved.
MINQUE Based on e Hξ H1 ξ1 H2 ξ2 Hm ξ m
Helmert used the residuals based on the least-squares method to estimate the variance components. The principle can also be used for estimating variance components–covariance components. Let the initial value of variance components–covariance components all be unity. The least-squares estimation of the parameters is ^ ¼ ðXT PXÞ1 XT Py β
ð22Þ
The residuals are v ¼ ðXT ðXT PXÞ1 XP IÞy ~ ¼ Re ~ ¼ ðXT ðXT PXÞ1 XP IÞe ¼ Ry
ð23Þ
~ ¼ XðXT PXÞ1 XT P I, called the test matrix by where R Grafarend (1980); and I = n × n identity matrix. The variance matrix of residuals is ~ ~T ¼ DðvÞ ¼ RDðeÞ R
There are a few important points to consider: ~ j Þ ¼ ~Sji ¼ trðR ~ i Þ cannot always hold; ~iR ~Q ~jR ~Q ~TP ~TP 1. ~Sij ¼ trðR ~ is generally asymmetric (see Grafarend 1980, therefore, S p. 174); 2. The precondition of the derivation is that the stochastic model is a block-partitioned matrix; therefore, the method cannot be usually used when the stochastic model is not a blockpartitioned matrix; and 3. This method is dependent on the least-squares estimation of unknowns but independent of error distribution.
l X
~ rR ~Q ~T θr R
ð24Þ
Eq. (4) is the most general case in statistical textbooks and literature. Rao (1970) assumed that all the random vectors ξi are known and defined a natural estimate of variance component T σ2i as σ2i ¼ n1 i ξ i ξi , but Rao did not define the natural estimation of covariance components suitable to a general situation. Therefore, except for the two special cases (Rao 1971a, 1972), Rao’s MINQUE cannot be directly used for estimating covariance components. However, Rao’s ideas on natural estimation can also be helpful in estimating covariance components. Similarly, a natural estimation of covariance component σij is first defined in this paper as 1 1 1 T σij ¼ n1 ð27Þ i nj ½ξ i ½ξ j ¼ ni nj ξ i Iij ξ j Pni where ½ξi ¼ t¼1 ξit = sum of all the elements of the vector ξi. Inserting the natural estimation of σ2i and σij into the linear function ~ T θ leads to an estimate of the linear function: α
~T θ ¼ α
i¼1
r¼1
~ i v is The expectation of the quadratic form vT P
¼
¼
~iR ~Q ~TP ~ rÞ θr trðR
ð25Þ
r¼1
Canceling the expectation symbol of Eq. (25) leads to the Helmert estimation of variance-covariance components (Grafarend 1980; Cui 2001): ~θ ¼ S ~ 1 w ~
m X i¼1
~ i vÞ ¼ E½trðP ~ i vvT Þ ¼ tr½P ~ i EðvvT Þ ¼ tr½P ~ i DðvÞ EðvT P l X
m X
ð26Þ
~ j Þ; ~ ¼ ð~Sij Þ; ~Sij ¼ trðR ~iR ~Q ~TP where θ~ ¼ ð~θ1 ; ~θ2 ; ; ~θl ÞT ; S T~ ~ T T T ~ ~ ~ ~ ~ ~ w ¼ ð~ w1 ; w2 ; ; wl Þ ; and wi ¼ v Pi v ¼ y R Pi Ry.
αi σ2i þ
m X m X
αij σij
i¼1 j¼iþ1 T n1 i αi ξ i ξ i þ
m X n X
1 T T n1 i nj αij ξi Iij ξ j ¼ ξ Rξ ð28Þ
i¼1 j¼iþ1
where R ¼ ðRij Þ = m × m symmetric block-partitioned matrix; T 1 1 1 Rii ¼ n1 i αi Ii ; and Rij ¼ Rji ¼ 2 ni nj αij Iij . Subjecting the T quadratic form Y MY to the translation-invariance (Condition 1) ~ T θ: leads to another estimate of α eT Me ¼ ξT HT MHξ
ð29Þ
The difference between the two estimations ξT Rξ and ξT HT MHξ is ξT ðHT MH RÞξ. The Euclidean kBk2 of any a matrix B ¼ ðbij Þ is defined P P norm b2ij (Rao 1973). Therefore, the Euclidean norm as kBk2 ¼ kHT MH Rk2 is (Rao 1973)
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kHT MH Rk2 ¼ trðMHHT MHHT Þ 2trðHT MHRÞ þ trðR2 Þ ¼ trðMHHT MHHT Þ trðR2 Þ TÞ trðR2 Þ ¼ trðMTM
ð30Þ
where trðHT MHRÞ ¼ trðR2 Þ [referring to Eq. (65)] and trðR2 Þ ¼ Pm 1 2 Pm Pm 1 1 1 2 i¼1 ni αi þ i¼1 j¼iþ1 2 ni nj αij [referring to Eq. (63)]. In Eq. (30), the matrix M is to be determined, so minimizing TÞ under the translation-invariance (Condition 1) and the trðMTM unbiasedness (Condition 2) leads to the MINQUE of variancecovariance components. To find the MINQUE of variancecovariance components, construct with Eqs. (9) and (15) the Lagrangian function TÞ 4trðMXKT Þ 4 ϕðMÞ ¼ 2trðMTM
l X
ð31Þ where K = Lagrangian multiplier matrix of the constraint MX ¼ 0; and ~ηr = l Lagrangian multiplier coefficient of the constraint ~ rÞ α ~ r ¼ 0. trðMQ For ∂ϕðMÞ=∂M ¼ 0 l X
~r ¼ 0 ηr Q
ð32Þ
r¼1
and 1 KXT T 1 þ M¼T
l X
~ iÞ ¼ ~ i ¼ trðMQ α
~ rT 1 Q 1 ηr T
l X
Q ~ iÞ Q ~ rC ~ηr trðC
i ¼ 1; 2; …; l
ð40Þ
r¼1
Eq. (40) can be represented by the matrix form η and ~η ¼ S 1 α ~ ~ ¼ S~ α
ð41Þ
Q ~ j Þ; and ¼ ðSij Þ; Sij ¼ trðC Q ~ iC where ~η ¼ ð~η1 ; η~2 ; …; ~ηl ÞT ; S i; j ¼ 1; 2; …; l is symmetric. Inserting Eq. (39) into the quadratic form yT My leads to yT My ¼
~ rÞ α ~r ~ηr ½trðMQ
r¼1
T KXT TM
Inserting Eq. (39) into Eq. (15) leads to
l X
Q ~ i Cy ¼ ~ηT w ~ηi yT C
ð42Þ
i¼1
Q ~ i Cy. ¼ ð 2 ; …; w l ÞT ; and w i ¼ yT C where w w1 ; w T T ~ ~ Denoting α θ as the MINQUE of α θ, it follows that T θ ¼ yT My ¼ ~ηT w α
ð43Þ
1 α ~ for ~η in Eq. (43) leads to Substituting ~η ¼ S 1 w ~ T θ ¼ α ~T S α
ð44Þ
~ is arbitrary, the MINQUE of varianceBecause the coefficient α covariance components is θ ¼ S 1 w
ð33Þ
ð45Þ
r¼1
Postmultiplying the two sides of Eq. (33) by X leads to 1 X þ 1 KXT T MX ¼ T
l X
~ rT 1 Q 1 X ¼ 0 ηr T
ð34Þ
r¼1
¯ ξ¯ H ¯ 1 ξ¯ 1 H ¯ 2 ξ¯ 2 MINQUE Based on e H ¯ m ξ¯ m H In the science of surveying and mapping, a more general case is 2ξ2 þ þ H mξm ξ ¼ H 1ξ1 þ H e¼H
Solving Eq. (34) leads to K¼
l X
~ rT 1 XðXT T 1 XÞ1 ηr Q
ð35Þ
r¼1
Inserting K into the right side of Eq. (33) leads to " # l X 1 ~ M¼T ηr Qr C
ð36Þ
= n × N matrix; ξ = N-vector; and N ¼ n1 þ where H n2 þ þ nm . All ξi have similar properties to ξi in Eq. (4), except ii and Dðξi ; ξj Þ ¼ σ ij . The variance of the error Dðξi ; ξi Þ ¼ σ 2i Q ij Q vector e is DðeÞ ¼ ¼
¼T 1 T 1 XÞ1 XT T 1 . Premultiplying the 1 XðXT T where C T leads to two sides of Eq. (36) by C T M ¼ C
Q ~ rC ηr C
ð37Þ
r¼1
¼T 1 T 1 XÞ1 XT T 1 by TM 1 XðXT T Postmultiplying C leads to T 1 XÞ1 XT T 1 ÞTM M ¼ ðT 1 T 1 XðXT T C 1 XÞ1 XT M ¼ M 1 XðXT T ¼MT
ð38Þ
Allowing for Eqs. (37) and (38) M¼
l X r¼1
Q ~ rC ηr C
ð39Þ
m X m X
i Dðξi ; ξj ÞH T H j
i¼1 j¼1
r¼1
l X
ð46Þ
m X i¼1
ii H iQ T þ σ 2i H i
m X m X
ij H ji H T þ H TÞ iQ jQ σ ij ðH j i
i¼1 j¼iþ1
ð47Þ The corresponding BIQUE is derived by Koch (1999). It has ii HT and the same formulation as Eq. (18), except Tii ¼ Hi Q i T Tij ¼ Hi Qij Hj . In the case of Eq. (47), how to estimate DðeÞ, σ 2i , and σ ij from the MINQUE approach will be to some extent different from the case of Eq. (6). Three cases are discussed as follows. Case 1 In this case, if a transformation ξi ¼ Zi ξi exists such that the ii variance-covariance matrix satisfies Dðξi ; ξi Þ ¼ σ 2i Zi ZTi ¼ σ 2i Q ij , then the error vector and its and Dðξi ; ξj Þ ¼ σ ij Zi ZTj ¼ σ ij Q variance are respectively
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2
1 Z1 ξ 1 þ H 2 Z2 ξ 2 þ þ H m Zm ξ m e¼H ¼ H1 ξ1 þ H2 ξ2 þ þ Hm ξm
DðeÞ ¼
m X
σ 2i Ti þ
m X m X
i¼1
ð48Þ
σ ij ðTij þ Tji Þ
ð49Þ
i¼1 j¼iþ1
i . Therefore, the variance-covariance in Eq. (49) where Hi ¼ H can be computed from Eq. (45).
11 12 σ 12 Q σ 21 Q 6 2 22 6σ Q σ 22 Q 6 12 21 Dðξ; ξÞ ¼ 6 .. .. .. 6 . . . 4 2 σ 1m Qm1 σ 2m Qm2 2 2 3 σ1 σ12 σ1N 6 7 6 σ12 σ22 σ2N 7 6 7 ¼6 . .. 7 .. .. 6 . 7 . 5 . . 4 . σ1N
Case 2 If the transformation in the Case 1 does not hold, there will surely be the following transformation: ξi ¼ Zi1 ξ1 þ Zi2 ξ2 þ þ Zim ξm ;
i ¼ 1; 2; …; m
σ2N
DðeÞ ¼
ð50Þ
where Zij = transformation matrix. The corresponding variancecovariance matrices are respectively m X
σ2t Zit ZTit þ
t¼1
m X m X
σtk ðZit Itk ZTik þ Zik Ikt ZTit Þ
t¼1 k¼tþ1
ð51Þ
Dðξi ; ξj Þ ¼
m X t¼1
σ2t Zit ZTjt
þ
m X m X
σtk ðZit Itk ZTjk
N X
σ2i hi hTi þ
¼
N X N X
σij ðhi hTj þ hj hTi Þ
i¼1 j¼iþ1
N X
σ2i Ti þ
i¼1
N X
N X
σij ðTij þ Tji Þ
þ
Substituting Eq. (50) into (46) leads to
where Ti ¼ hi hTi ; and Tij ¼ hi hTj . Denote σ ¼ ðσ21 ; σ12 ; σ22 ; σ13 ; σ23 ; σ23 ; …; σðN1ÞN ; σ2N ÞT . The MINQUE of σ has a similar formulation to Eq. (45): ð57Þ
In fact, it is not necessary to solve Eq. (57), Step 2 can be applied directly. Step 2. Denote σ ¼ ð σ21 ; σ 12 ; σ 22 ; σ 13 ; σ 23 ; σ 23 ; …; σ ðm1Þm ; σ 2m ÞT . According to the identity of Eq. (55), σ can be represented by σ and is the linear function of σ:
1 ðZ11 ξ1 þ Z11 ξ2 þ þ Z1m ξm Þ e¼H
σ ¼ F σ
2 ðZ21 ξ1 þ Z22 ξ2 þ þ Z2m ξm Þ þ þH m ðZm1 ξ1 þ Zm1 ξ2 þ þ Zmm ξm Þ þH ¼ H1 ξ2 þ H2 ξ2 þ þ Hm ξm
ð53Þ
1 Z1i þ H 2 Z2i þ þ H m Zmi . After the transformawhere Hi ¼ H tion, the error vector e in Eq. (53) has the same variance as Eq. (6). The variance-covariance components σ2i and σij can also be solved by Eq. (45). Substituting the estimated variance-covariance components into Eqs. (51) and (52), respectively, leads to the MINQUE of Dðξi ; ξi Þ and Dðξi ; ξj Þ. However, in this case, the MINQUE of variance-covariance components σ 2i and σ ij cannot be directly obtained. Case 3 σ 2i
To obtain the MINQUE of variance-covariance components and σ ij , a two-step strategy is adopted as follows. ξ, and e are respectively rewritten as Step 1. For Eq. (46), H, ¼ ðh1 ; h2 ; …; hN Þ; H
e ¼ h1 ε1 þ h2 ε2 þ þ hN εN ¼ hε
ð56Þ
i¼1 j¼iþ1
Zjk Ikt ZTit Þ ð52Þ
There exists the variance equality
σ2N
¼w Sσ
t¼1 k¼tþ1
ξ ¼ ε ¼ ðε1 ; ε2 ; …; εN ÞT ;
ð55Þ
The variance of error vector e becomes
i¼1
Dðξi ; ξi Þ ¼
1m 3 σ 1m Q 2m 7 7 σ 22m Q 7 7 ¼ Dðε; εÞ .. 7 . 5 2 σ m Qmm
ð54Þ
ðRefer to Appendix IIIÞ
ð58Þ
where F = constant matrix determined by the cofactor matrix ii and Q ij . Substituting Eq. (58) into (57) and minimizing ðSF σ Q T ðSF leads to wÞ σ wÞ 1 ðSFÞ Tw T SF σ ¼ ½ðSFÞ
ð59Þ
where σ is called the least squares and minimum norm quadratic unbiased estimation (LSMINQUE).
Numeric Example Fig. 1 is a simulated hybrid distance and angle triangulation network; A, B, C, and D are points whose coordinates are known; and P1–P9 are points whose coordinates are to be determined. The horizontal angles are 1–39; the distances are 40–63; and the observations are assumed to be normally distributed. All the angles are assumed to have the same observational variance σ21 , their a priori value is 1 (arc s2 ). All the distances are assumed to have the same observational variance σ22 due to approximately equal distance lengths, their a priori value is 4 ðcm2 Þ. Any pair of contiguous angles is assumed to have the same covariance ρ, their a priori value is 0:5 ðarc s2 Þ. The variance matrix of observations is defined by
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Fig. 1. Hybrid distance-angle network
Table 1. Observational Equation Data 1
2
3
4
ν
Coefficients and unknowns
f
y
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
0:1569x2 0:2839y2 0:2555x2 0:0962y2 0:0986x2 þ 0:3801y2 0:3341x1 þ 0:1753y1 0:2555x2 þ 0:0962y2 0:4778x10:1596y10:1436x2 0:3349y2 0:1436x1 0:3349y1 þ 0:1118x2 þ 0:2387y2 0:1673x1 0:3923y1 0:1436x2 þ 0:3349y2 þ 0:3109x6 þ 0:0574y6 0:3109x1 þ 0:0574y1 0:1480x2 0:22602 0:1629x6 þ 0:1686y6 0:1436x1 þ 0:3349y1 þ 0:2917x2 0:1089y2 0:1480x6 0:2260y6 0:1480x2 þ 0:2260y2 þ 0:0845x5 0:2424y5 0:2326x6 þ 0:0164y6 0:2930 × 2 þ 0:0515y2 þ 0:2085x5 þ 0:1909y5 þ 0:0845x6 0:2424y6 0:1449x2 0:2774y2 0:2930x5 þ 0:0515y5 þ 0:1480x6 þ 0:2260y6 0:2930x2 0:0515y2 0:2296x3 0:2425y3 0:0634x50:2939y5 0:1366x2 þ 0:3008y2 þ 0:3662x3 0:0583y3 0:2296x5 0:2425y5 0:1564x2 0:2493y2 0:1366x3 þ 0:3008y3 þ 0:2930x5 0:0515y5 0:1366x2 0:3008y2 þ 0:2278x30:2931y3 0:1569x2 þ 0:2839y2 0:3644x3 þ 0:0077y3 0:0845x5 þ 0:2424y5 0:1875x6 0:1570y6 þ 0:2720x7 0:0854y7 0:1849x5 0:2353y5 þ 0:2720x6 0:0854y6 0:0872x7 þ 0:3207y7 0:2694x5 0:0071y5 0:0845x6 þ 0:2424y6 0:1849x7 0:2353y7 0:1849x5 þ 0:2353y5 0:2998x7 þ 0:0512y7 þ 0:1149x8 0:2865y8 0:3017x5 þ 0:0350y5 þ 0:1149x7 0:2865y7 þ 0:1868x8 þ 0:2515y8 0:1169x5 0:2703y5 þ 0:1849x7 þ 0:2353y7 0:3017x8 þ 0:0350y8 0:2282x4 0:2162y4 þ 0:3017x5 0:0350y5 0:0736x8 þ 0:2511y8 0:3561x4 0:1255y4 0:1279x5 þ 0:3417y5 0:2282x8 0:2162y8 0:1279x4 þ 0:3417y4 0:1738x5 0:3068y5 þ 0:3017x8 0:0350y8 0:3256x3 þ 0:0423y3 þ 0:1978x4 þ 0:2994y4 þ 0:1279x5 0:3417y5 0:0961x3 0:2847y3 0:3256x4 þ 0:0423y4 þ 0:2296x5 þ 0:2425y5 0:0987x4 0:3175y4 þ 0:2282x8 þ 0:2162y8 0:3269x90:1013y9 0:2282x4 þ 0:2162y4 0:2786x8 þ 0:1034y8 þ 0:0504x9 0:3196y9 0:3269x4 þ 0:1013y4 þ 0:0504x8 0:3196y8 þ 0:2765x9 þ 0:2183y9 0:0504x8 þ 0:3196y8 þ 0:2943x9 0:1224y9
0″.00 000 :01 0″.47 0 300 :45 000 :01 000 :01 300 :82 000 :01 0 100 :04 000 :01 0 1″.63 000 :01 500 :26 6″.04 000 :01 1″.23 000 :01 000 :01 9″.04 0 0 400 :51 0 3″.78 300 :91 0 000 :01 2″.18 0″.89
54°49′31″.50 43°28′18″.06 81°42′09″.96 48°19′03″.78 85°31′35″.45 46°09′24″.22 77°14′58″.98 46°18′21″.18 56°26′36″.02 52°27′12″.94 60°48′48″.26 66°43′59″.83 56°31′34″.45 67°51′49″.11 55°36′34″.81 64°22′05″.95 62°16′38″.58 53°21′12″.41 69°16′26″.4 57°22′19″.96 60°00′22″.16 61°31′58″.59 58°27′30″.21 50°03′43″.27 67°04′01″.07 62°52′20″.16 62°04′47″.27 53°57′44″.86 60°40′23″.62 55°30′29″.76 63°49′04″.44 60°00′26″.09
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σ2y1 ¼ σ2y2 ¼ ¼ σ2y39 ¼ σ21 ; σ2y40 ¼ σ2y41 ¼ ¼ σ2y63 ¼ σ22 ; 2 y5 y37 y33 y31 y1 y2 σ1 ¼D ¼D ¼D ¼D ¼D ¼ D y4 y7 y38 y35 y32 y17 ρ 02 31 02 31 02 31 02 3 1 02 2 31 y19 y27 y16 y8 σ1 ρ 0 D@4 y10 5A ¼ D@4 y21 5A ¼ D@4 y25 5A ¼ D@4 y14 5A ¼ @4 ρ σ21 ρ 5A y18 y39 y29 y28 0 ρ σ21
02
31 31 31 2 2 02 02 y3 y13 y22 σ1 B6 y6 7C B6 y11 7C B6 y24 7C 6 ρ 7C 7C 7C 6 B6 B6 B6 6 7C 7C 7C 6 B6 B6 DB B6 y9 7C ¼ DB6 y20 7C ¼ DB6 y30 7C ¼ 6 0 @4 y12 5A @4 y23 5A @4 y34 5A 4 0 y15 y26 y36 0
^ i ; Y^ i Þ denotes the adjusted coordinates of unknown point ðX Pi, their approximate coordinates are ðX 0i ; Y 0i Þ, and the increment or correction is ðxi; yiÞ. The unit of the former two is meter, the ^ i ; Y^ i Þ ¼ ðX 0i ; Y 0i Þ þ ðxi; yiÞ=100. latter is centimeter. There is ðX The linearized error equation is v ¼ Xdx þ f, dx ¼ ðx1; y1; x2; y2; …; x9; y9ÞT . Tables 1 and 2 show the data of linearized error equation of 63 obervations. Column 1 shows the residuals ν i ; Column 2 shows the unknowns and their coefficient in the error equation;
ρ σ21 ρ 0 0
0 ρ σ21 ρ 0
0 0 ρ σ21 ρ
ρ ; σ21
3 0 0 7 7 0 7 7 ρ 5 σ21
Column 3 is the constant vector f (the differences between the approximate observation computed from the approximate coordinates and the practical observation), their physical units are arc-second from 1–39 and centimeter from 40–63; Column 4 provides practical observations y, the unit of angle from 1–39 is degree, minute, and second, and that of the distance from 40–63 is meter. Table 3 shows the coordinates of known points (A, B, C, and D) and the approximate coordinates of unknown points (P1–P9).
Table 2. Observational Equation Data ν
Coefficients and unknowns
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
0:3001x8 0:1063y8 0:2439x9 0:1972y9 0:2497x8 0:2133y8 0:0504x9 þ 0:3196y9 0:3001x8 þ 0:1063y8 0:0164x8 0:2738y8 0:2837x8 þ 0:1675y8 0:2283x7 0:0875y7 0:2837x8 0:1675y8 0:1134x7 0:1990y7 0:1149x8 þ 0:2865y8 0:4646x1 0:8855y1 0:3523x2 0:9359y2 0:9190x1 þ 0:3942y1 0:9190x2 0:3942y2 0:1816x1 þ 0:9834y1 þ 0:1816x6 0:9834y6 0:8365x2 þ 0:5480y2 þ 0:8365x6 0:5480y6 0:1730x2 þ 0:9849y2 0:1730x5 0:9849y5 0:9442x5 0:3292y5 þ 0:9442x6 þ 0:3292y6 0:2994x6 þ 0:9541y6 0:2994x7 0:9541y7 0:7864x5 þ 0:6177y5 þ 0:7864x7 0:6177y7 0:1151x5 þ 0:9934y5 0:1151x8 0:9934y8 0:9281x7 þ 0:3723y7 0:9281x8 0:3723y8 0:3578x7 þ 0:9338y7 0:5084x8 þ 0:8611y8 0:3339x8 þ 0:9426y8 0:6287x9 þ 0:7777y9 0:9878 × 8 þ 0:1558y8 0:9878 × 9 0:1558y9 0:2961x4 þ 0:9552y4 0:2961x9 0:9552y9 0:6878x4 þ 0:7259y4 þ 0:6878x8 0:7259y8 0:9365x4 0:3505y4 þ 0:9365x5 þ 0:3505y5 0:1288x3 þ 0:9917y3 0:1288x4 0:9917y4 0:7261x3 þ 0:6876y3 þ 0:7261x5 0:6876y5 0:9105x2 þ 0:4135y2 0:9105x3 0:4135y3 0:0210x3 0:9998y3 0:919x2 0:3942y2
f 3″.05 500 :60 100 :46 4″.18 300 :67 3″.04 1″.70 15.58 2.14 14.06 7.62 10.09 14.63 15.74 9.54 9.24 9:43 7:77 24:17 7:25 13:14 19:13 4:40 1.33 4.11 2.52 14.98 15.11 12.18 3.83 4:07
y
cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm
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58°27′31″.06 61°32′04″.50 62°52′19″.24 50°03′39″.19 67°04′02″.52 51°31′14″.86 47°10′43″.57 5,466.094 m 7,555.869 m 5,660.181 m 6,523.628 m 7,635.247 m 6,934.021 m 8,034.367 m 7,234.704 m 6,892.307 m 6,790.312 m 6,681.955 m 8,437.411 m 6,260.640 m 6,478.554 m 6,575.864 m 6,375.089 m 6,026.734 m 6,561.906 m 5,653.177 m 6,281.210 m 6,177.309 m 6,244.185 m 5,659.608 m 5,660.247 m
Discussion and Conclusions
Table 3. Coordinates X (m)
Y (m)
Approximate coordinates of unknown points P1 P2 P3 P4 P5 P6 P7 P8 P9
2,804,774.012 2,799,571.889 2,793,886.472 2,793,077.517 2,798,372.034 2,805,958.609 2,803,792.216 2,797,590.637 2,791,293.424 Coordinates of known points 2,795,427.59 2,800,773.509 2,802,234.19 2,794,005.704
C D B A
19,432,985.849 19,4307,54.895 19,428,172.765 19,421,943.714 19,423,925.325 19,426,570.598 19,419,667.768 19,417,180.218 19,416,187.092 19,411,073.567 19,411,789.111 19,437,826.22 19,433,831.155
This example demonstrates how to determine variancecovariance components. Table 4 shows the computation results from the LSMINQUE form [Eq. (59)] and the BIQUE form [Eq. (18)]. According to the root of mean squared (RMS) differences between the simulated variance-covariance components and the estimated components, the BIQUE is slightly better than the MINQUE because the BIQUE is the optimal estimation with minimum variance under the normally simulating errors. However, the difference between the methods is very small according to the RMS, which means that the two methods have good consistency. The LSMINQUE can be used as a reasonable alternative when the error distribution is in ambiguity. Strictly speaking, the MINQUE form [Eq. (45)] cannot be used to compute this example because the observational errors e cannot be represented strictly in the form of Eq. (4). But for comparision, the approximate result listed in Table 4 is still computed from Eq. (45). The three results have good consistency. Additionally, the stochastic model, the variance-covariance matrix of the observational errors, is not a block-partitioned matrix; therefore, the Helmert-type estimation fails. If the covariance component is neglected, this example become a pure variance components estimation. The results in Table 5 show that they are completely identical.
For the estimation of variance components, the MINQUE, BIQUE, and Helmert-type estimation are all unbiased and have a completely equal mathematical form; therefore, their numerical results should be the same, as shown in Table 5. However, for variance-covariance components estimation, that conclusion does not hold. Comparing are symmetric, Eqs. (18), (26), and (45), it can be seen that S and S ~ is asymmetric, and S ≠ S ~ ≠ S, which implies that the three types S of estimation are not equal mathematically. The most general stochastic model should be represented by Eqs. (46) and (47). Eqs. (4) and (6) are their reduction or a special case. Eqs. (46) and (47) can also be reduced to Eqs. (4) and (6) by algebraic transformation. The variance-covariance components in Eqs. (6) and (47) can both be estimated from Eq. (18) if the observations are normally distributed. If the principle of MINQUE is used, it is not necessary to know the error distribution. The variance-covariance components in Eq. (6) can be estimated from Eq. (45). However, the components in Eq. (47) can only be estimated from Eq. (59) rather than Eq. (45). From the former discussion, it can be concluded that for the estimation of variancecovariance components, 1. Contrary to the estimation of variance components, the three types of estimations are not identical to one another, so they may have different numerical results, but the differences should usually be small; 2. The BIQUE can usually be used in the case that errors are normally distributed, and it has the minimum estimation variance; 3. The Helmert-type estimation can be only used in the case that the stochastic model is a block-partitioned matrix; and 4. The MINQUE, including LSMIQUE, is independent of error distribution and can be applied in the most general situation.
Appendix I. Derivation of Unbiased Conditions Inserting Eq. (6) into Eq. (14) leads to EðyT MyÞ ¼ tr½MDðeÞ ¼
m X
σ2i trðMTi Þ þ
i¼1
m X m X
σij tr½MðTij þ Tji Þ ð60Þ
i¼1 j¼iþ1
P Pm Pm 2 ~T θ ¼ m Because EðyT MyÞ ¼ α i¼1 αi σi þ i¼1 j¼iþ1 αij σij and allowing for Eq. (60), there are αi ¼ trðMTi Þ;
αij ¼ tr½MðTij þ Tji Þ
ð61Þ
Table 4. Estimation of Variance-Covariance Components LSMINQUE σ ^ 21 σ ^ 22
ðs2 Þ
ðcm2 Þ ρ ðs2 Þ RMS
2.7205 4.4080 1:3299 1.12
BIQUE 2.735 4.161 1:129 1.07
MINQUE 2.802 4.137 1:408 1.17
Appendix II. Relation between t r R2 and Unbiased Coefficients Vector α
trðR2 Þ ¼
i¼1 j¼1
Table 5. Estimation of Variance Components
σ ^ 21 ðs2 Þ σ ^ 22 ðcm2 Þ
m X m X
MINQUE
BIQUE
Helmert
2.778 4.215
2.778 4.215
2.778 4.215
trðRij Rji Þ ¼
m X
trðR2ii Þ þ 2
i¼1
m X m X
trðRij Rji Þ
i¼1 j¼iþ1
ð62Þ T 1 1 1 Inserting Rii ¼ n1 i αi Ii and Rij ¼ Rji ¼ 2 ni nj αij Iij into (62) leads to
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trðR Þ ¼ 2
m X
2 2 n2 i αi trðIi Þ
þ2
i¼1
¼
m X
m X
t ði; jÞ = cross between the ith line and jth column in the where Q t . Therefore, Eq. (67) leads to the linear function σ ¼ F matrix Q σ.
2 2 22 n2 i nj αij trðIij Iji Þ
i¼1 j¼iþ1 2 n2 i αi ni þ 2
i¼1
¼
m X m X
m X m X
2 2 22 n2 i nj αij ni nj
Acknowledgments
i¼1 j¼iþ1 2 n1 i αi þ
i¼1
m X m X
1 2 21 n1 i nj αij
ð63Þ
i¼1 j¼iþ1
References
Appendix III. Relation between t r HT MHR and Unbiased Coefficients Vector α trðHT MHRÞ ¼ trðMHRHT Þ ¼ tr M
m X m X
! Hi Rij HTj
i¼1 j¼1
¼
m X
trðMHi Rii HTi Þ þ 2
m X m X
i¼1
¼
m X
trðMHi Rij HTj Þ
i¼1 j¼iþ1 T n1 i αi trðMHi Hi Þ
i¼1
þ2
m X m X
1 T 21 n1 i nj αij trðMHi Iij Hj Þ
i¼1 j¼iþ1
¼
m X
n1 i αi trðMTi Þ
i¼1
þ2
m X m X
1 21 n1 i nj αij trðMTij Þ
ð64Þ
i¼1 j¼iþ1
Inserting Eq. (61) into Eq. (64) leads to trðHT MHRÞ ¼
m X
2 n1 i αi þ
i¼1
m X m X
1 2 2 21 n1 i ni αij ¼ trðR Þ
i¼1 j¼iþ1
ð65Þ
Appendix IV. Derivation of Variance Components Linear Model Denote Dðξ; ξÞ as 2 2 σ 1 Q11 6 2 6σ Q 6 12 21 Dðξ; ξÞ ¼ 6 .. 6 . 4 m1 σ 1m Q
12 σ 12 Q 22 σ 2 Q
1m 3 σ 1m Q 2m 7 7 σ 22m Q 7 7 .. 7 . 5 2 σ m Qmm
.. .. . . 2 σ 2m Qm2 2 3 þ σ 4 þ σ 5 þ σ 6 þ 12 Q2 þ σ 22 Q 13 Q 23 Q 23 Q ¼σ 1 Q1 þ σ mðm1Þ=2þ1 þ þ σ mðmþ1Þ=2 þσ 1m Q 2m Q ð66Þ 2
i has a similar formulation to Eq. (20). Any element of where Q Dðε; εÞ can be represented as follows because of the identity of Eq. (53): 1 ði; jÞ þ σ 2 ði; jÞ þ σ 3 ði; jÞ þ σ 4 þ σ 5 ði; jÞ σij ¼ σ 21 Q 12 Q 22 Q 13 Q 23 Q 6 ði; jÞ þ þ σ mðm1Þ=2þ1 ði; jÞ þ þσ 2 Q 1m Q 3
mðmþ1Þ=2 ði; jÞ þσ 2m Q
This project was sponsored by the “863” Project (2007AA12Z226) of the National Natural Science Foundation of China (NSFC41074009, NSFC-40674015).
ð67Þ
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