Journal of Geodesy (2000) 74: 581±589

On outlier-hiding e€ects in speci®c Gauss±Markov models: geodetic examples W. Pr oszynÄski Institute of Applied Geodesy, Warsaw University of Technology, Pl. Politechniki 1, PL-00-661, Warszawa, Poland e-mail: [email protected]; Tel.: +48 22 660 7299; Fax: +48 22 625 1527 Received: 23 June 1998 / Accepted: 5 July 2000

Abstract. A speci®c subclass of Gauss±Markov models has been de®ned as containing the models for which the disturbance/response matrix, determined under the assumption of uncorrelated observations, consists of independent diagonal blocks. A proposed modi®cation of reliability assessment procedure for such models is presented By the appropriate reduction of a given full covariance matrix for the observations, the proposal allows the assessment to be made in the resulting model which, in contrast to the initial model, is free from outlier-hiding e€ects of the type not occurring in ordinary models. The theoretical ®ndings are demonstrated using simple numerical examples. All the proofs supporting the proposal are gathered in Appendixes. The proposal, which is not without its own weak points, is an attempt to associate the reliability assessment in speci®c Gauss±Markov models with e€ective outlier detection. Key words: Gauss±Markov model ± Least-squares estimation ± Diagonal pattern ± Outlier-hiding e€ects

1 Introduction When making a comparison between several types of reliability measures for systems with correlated observations (Wang and Chen 1994; Scha€rin 1997) and, therefore, studying the actual responses of the systems to observational outliers (Pr oszynÄski 1998), it became necessary to investigate more thoroughly the possible distorting e€ect of the speci®c algebraic structure of a Gauss±Markov model (GMM) upon the distribution of the outlier e€ects within the least-squares (LS) residuals. The elementary examples studied in the above publications proved that such an e€ect occurs with models having a speci®c algebraic structure, i.e. models for which the disturbance/response matrix, determined

under the assumption of uncorrelated observations, consists of independent diagonal blocks. Some introductory statements follow, which explain the philosophy of the approach presented in this paper. The correlations between the observations, contained in a given covariance matrix, basically do not apply to observational gross errors. In the LS estimation of parameters in GMMs having outliers, such correlations are an additional source of outlier-smearing e€ects, which result in the hiding of outliers. In GMMs with uncorrelated observations the elements in the i-th column of the reliability matrix (termed more generally the disturbance/response matrix) can be interpreted as distortions in the estimated standardized errors caused by the unit standardized gross error in the i-th observation. The i-th estimated error is distorted by a fraction of the gross error mentioned above. The distortion maintains the sign of the gross error. There is a symmetry in the disturbance/response relationships in such models. In addition to a lack of symmetry in the disturbance/response matrices, the correlations may cause distortions in the estimated errors, being of greater magnitude than the actual observational gross errors and even having the opposite sign. One of the possibilities for raising the e€ectiveness of outlier detection in GMMs with correlated observations could be to carry out a testing procedure starting with a given covariance matrix reduced to its diagonal and ending up with its full form. It should be emphasized that the full covariance matrix must be used for the ®nal parameter estimation after all the observational disturbances have been detected and removed. The reliability measures should correspond with the chances of the e€ective detection of outliers, or more precisely, with the extent to which a model is able to expose the outliers. In GMMs with uncorrelated observations the reliability measures are strictly associated with the degree of concentration of the model response in the actual location of the outlier. It would be dicult to extend this approach to GMMs with correlated observations as the disturbance/response relationships are rather complicated and often misleading about

582

the actual location of the outliers. Hence, for the reliability assessment in such GMMs it seems justi®able to sacri®ce a certain part of the stochastic information in favour of removing the excessive outlier-hiding e€ects. Such is the case, described herein, with GMMs which have a speci®c algebraic structure. Although not frequent in practice, they are important cases as far as the theory of reliability is concerned. 2 Basic concepts and notations Let us consider a (linearized) Gauss±Markov model de®ned by   2 e  0; ro Q …1† yˆ A x‡e nm

nn

where: y = the n  1 vector of observational increments A = the n  m coecient matrix of full column rank x = the unknown m  1 vector of parameter increments e = the unknown n  1 vector of random errors r2o = the variance component (known or unknown) Q = the symmetric, positive-de®nite n  n matrix of cofactors. The covariance matrix for random errors, being also the covariance matrix for the observations, will be written equivalently as Ce ˆ Cy ˆ r2o Q ˆ r2o P

1

ˆC

For further analysis we will introduce the following concepts. The LS algebraic correlation pattern of a GMM, denoted by dpat …HI † (`dpat' stands for diagonal pattern, see Appendix A), where HI ˆ A…AT A† 1 AT which is a hat matrix (see Chatterjee and Hadi 1988) de®ned for the ordinary LS procedure with C ˆ I, such that ^ yI ˆ HI y. This pattern shows the LS correlation blocks for a vector of adjusted observational increments ^yI , the correlations resulting entirely from the algebraic structure of A. The matrix RI ˆ I HI , being the redundancy matrix (see Chatterjee and Hadi 1988) de®ned with C ˆ I, such that ^eI ˆ RI y, does the same for a vector of estimated errors ^eI . The LS stochastic correlation pattern of a GMM, denoted by dpat…HC †, where HC ˆ A…AT C 1 A† 1 AT C

1

or using a weight matrix P instead of C T

1

T

HP ˆ HC ˆ A…A PA† A P

1

from the structure of both A and C. The matrix RC ˆ I HC , being a redundancy matrix such that ^e ˆ RC y, does the same for a vector of estimated errors ^e. A GMM of regular algebraic structure (further on termed an ordinary GMM), being a GMM in which dpat …HI † is complete (see Appendix A) and so is dpat …RI †. A GMM of irregular algebraic structure (further on termed a speci®c GMM), being a GMM in which dpat…HI † is incomplete (see Appendix A) and so is dpat…RI †; in a speci®c GMM we can write, for example      ^eI;1 RI;1 0 y1 ˆ ^eI;2 0 RI;2 y2 which means that, when applying ordinary LS, the vector ^eI;1 does not depend on y2 and ^eI;2 does not depend on y1 . In terms of reliability it means that the disturbances (gross errors) in y1 do not a€ect the residuals for y2 and vice versa. An outlier-smearing area, being the set of those observations in a GMM which are a€ected in the LS estimation by a gross error residing in one of them. 3 The outlier-hiding e€ects It is a well-known phenomenon that, in the process of LS estimation in a GMM with uncorrelated observations, the response to a gross error residing in one of the observations is smeared onto LS residuals of other observations due to functional relationships between the observed quantities. The distribution of the smearing e€ects is also a result of di€erences in a priori variances of observations which are valid only for random observational errors. In a GMM with correlated observations the distribution is additionally in¯uenced by taking into account the observational covariances, which, similarly to variances, are valid only for random errors. The following properties can be easily observed. (1) In an ordinary GMM dpat…HI † is complete, and hence for any covariance matrix C (pos. def.) we have dpat…HC †  dpat…HI †

…3†

which means that the outlier-smearing area for correlated observations cannot exceed the area resulting from the functional relationships between the observed quantities. (2) For a speci®c GMM [i.e. dpat…HI † is incomplete] there exist such covariance matrices C that dpat…HC †  dpat…HI †

…2†

^ ˆ HC y (to simplify the is a hat matrix, such that y notation HP has been used instead of HP 1 ). This pattern shows the LS correlation blocks for a vector of adjusted observational increments ^ y, the correlations resulting

which means that the outlier-smearing area in a speci®c GMM with correlated observations can be bigger than the area (or areas) resulting from the functional relationships between the observed quantities.

583

Such covariance matrices may thus be the cause of the response to an outlier, residing in a certain group of functionally interrelated observations, being smeared also upon a group having no functional links with the group containing the outlier. The extension of the outliersmearing area will additionally obscure the actual location of the outlier and decrease the chances of its detection. These speci®c outlier-hiding e€ects do not occur in ordinary GMMs. Following the introductory statements (see Sect. 2), in order to obtain consistency with reliability assessment in ordinary GMMs, it seems reasonable to transform a speci®c GMM so that it is free of additional `hiding e€ects' not occurring in ordinary GMMs. From the ®ndings given in the Appendix D it follows that a sucient condition to satisfy Eq. (3) is pat…Cin †  dpat…HI †

…4†

i.e. the input correlation pattern should be contained in the algebraic correlation pattern of a GMM. Note that Cin used in Eq. (4) should correspond to the same order of elements in y as does dcomp…HI † (`dcomp' stands for diagonal compact form; see Appendix A). The condition of Eq. (4) allows us to formulate the following general operational directives for the setting of the input correlation pattern to be used in the LS-based reliability assessment procedure:

(1) if we have an ordinary GMM [i.e. dpat(HI ) is then always complete; dcomp…HI † ˆ HI ] pat…Cin †  dpat…HI †. Thus Cin ˆ C; (2) if we have a speci®c GMM [i.e. dpat(HI ) is incomplete] but pat…Co †  dpat…HI †, where Co is the initial C transformed by the row and column permutations (see Appendix E) in order to obtain the correspondence with dcomp…HI †, then Cin ˆ C; (3) if we have a speci®c GMM and pat…Co † is not contained in dpat…HI † we have to reduce Co to Co;r , such that pat…Co;r † ˆ pat…Co † \ dpat…HI † and Cin ˆ Cr , where Cr is Co;r transformed to correspond with HI . The setting of the input correlation pattern explained above is shown in Fig. 1. The above proposal of adapting a speci®c GMM to the needs of internal reliability assessment also has a certain disadvantage. By reducing the initial covariance matrix we neglect some stochastic information as to the behaviour of random observational errors and the model thus obtained is not fully equivalent to the initial one. Considering the omitted covariances as reliable values we may conclude that such a reduced model yields a less accurate description of the true random observational errors. On the other hand, however, we eliminate the smearing of the outlier e€ects outside the area delimited by functional relationships in the model. This smearing is induced by random error correlations which do not apply to gross errors. 4 Numerical examples For each of the simple levelling schemes (see Fig. 2), we shall construct a GMM and check whether the given covariance matrices of observations, i.e. 2 3 2 3 1 0:5 1 0:5 2 0 3 6 0:5 1 0:5 1:5 7 7 C1 ˆ 4 0 1 0:5 5 C2 ˆ 6 4 1 0:5 4 1 5 3 0:5 5 0:5 1:5 1 5 are suitable for carrying out the assessment of internal reliability according to the procedure proposed in Sect. 3. 2 3 2 3 2 1 1 1 0 1 Scheme (a) A ˆ 4 1 1 5 HI ˆ 4 1 2 1 5 3 1 1 2 0 1 2

1 dpat…HI † ˆ 4 1 1

Fig. 1. Diagram for the setting of the input correlation pattern

1 1 1

3 1 15 1

As dpat(H1 ) is complete it is an ordinary GMM. Hence, a full covariance matrix C1 is a recommended input, i.e. Cin ˆ C1 . 2 3 2 3 1 0 0 0 1 Scheme (b) A ˆ 4 1 0 5 HI ˆ 4 0 0:5 0:5 5 0 0:5 0:5 1 0

584

Fig. 2. The test levelling schemes

2

1 dpat…HI † ˆ 4 0 0

0 1 1

3 0 15 1

The dpat(HI ) is incomplete and hence it is a speci®c GMM. The recommended correlation input is a matrix C1;r obtained from C1 so that pat…C1;r † ˆ pat…C1 † \ dpat…HI † i.e. 2

1 pat…C1;r † ˆ 4 0 1 and hence 2 2 0 C1;r ˆ 4 0 1 0 0:5

0 1 1

3 2 1 1 15 \ 40 1 0

0 1 1

3 2 0 1 15 ˆ 40 1 0

0 1 1

3 0 15 1

3

0 0:5 5 5

We shall now compute the matrix RP ˆ I HP showing the `disturbance/response' behaviour of the model, denoted in Scha€rin (1997) by Q^e P, i.e. RP  Q^e P ˆ I

A…AT PA† 1 AT P

With P ˆ C1;r1 …ro ˆ 1† we obtain 2 3 0 0 0 0:1 5 Q^e P ˆ 4 0 0:1 0 0:9 0:9 whereas with P ˆ C1 1 …ro ˆ 1† as in Scha€rin (1997) we have 2 3 0 0:6 0:6 0:1 5 Q^e P ˆ 4 0 0:1 0 0:9 0:9 Although the redundancy numbers (i.e. the diagonal elements) in both the matrices are the same, the latter discloses the smearing of a gross error in y2 or in y3 onto y1 (strictly speaking, onto the estimated value of e1 ).

Proceeding to PQ^e P taken in Wang and Chen (1994) and Scha€rin (1997) as the basis for ®nding the reliability measures for correlated observations, we obtain for both P ˆ C1;r1 and P ˆ C1 1 2 3 0 0 0 PQ^e P ˆ 4 0 0:2 0:2 5 0 0:2 0:2 The resulting values of reliability measures r; r, de®ned by ri ˆ fQgii fPQ^e Pgii

…Wang and Chen†

ri ˆ ‰fQ 1 gii Š 1 fPQ^e Pgii

…Schaffrin†

where: Q ˆ C (with ro ˆ 1), i ˆ 1; . . . ; n, are shown in Table 1. The signi®cant di€erence is in the value of r3 . A possible explanation may be that, as a consequence of neglecting the correlation of y3 and y1 , y3 being several times less accurate than y2 comes out considerably more reliable. 2 3 2 3 0 0 0 0 Scheme (c) A ˆ 4 1 5 HI ˆ 4 0 0:5 0:5 5 0 0:5 0:5 1 2

1 dpat…HI † ˆ 4 0 0

0 1 1

3 0 15 1

The dpat…HI † is incomplete and is the same as in example (b), and hence the recommended correlation input is the matrix C1;r as in (b). We shall compute the matrices Q^e P and PQ^e P for P ˆ C1;r1 and P ˆ C1 1 …ro ˆ 1† respectively, and ®nally the values of the reliability measures r, r. 2 3 1 0 0 0:1 5 For P ˆ C1;r1 : Q^e P ˆ 4 0 0:1 0 0:9 0:9

585 Table 1. The values of reliability measures for scheme (b) Cin ˆ C1;r

Cin ˆ C1

r1 r2 r3

0.00 0.20 1.00

0.00 0.20 1.00

r1 r2 r3

0.00 0.19 0.95

0.00 0.10 0.05

2

for P ˆ C1 1

1 : Q^e P ˆ 4 1:5 1:5

0 1 0

for P ˆ C1 1

5 : PQ^e P ˆ 4 3 3

1 61 dpat…HI † ˆ 6 40 0

1 1 0 0

0 0 1 1

2

61 6 pat…C2;r † ˆ 6 41 2

0 15 0

3 2 2

1

1 1

61 6 ˆ6 40 0

3 3 25 2

Table 2 shows the values of reliability measures r; r. The reliability measures  r (which are not normalized quantities) in the second column are proportional to the values in the ®rst column, whereas the normalized measures r show a di€erent ranking. 2 3 2 3 0:5 0:5 0 0 1 0 6 0:5 0:5 0 6 1 07 0 7 6 7 7 Scheme (d) A ˆ 6 4 1 1 5 HI ˆ 4 0 0 0:5 0:5 5 0 0 0:5 0:5 1 1 2

i.e.

3

Here we observe substantial di€erences both in the redundancy numbers for y2 and y3 and in the `disturbance/response' coecients for y1 ! e^2 and y1 ! e^3 . With the initial covariance matrix (C1 ) a gross error in y1 will be smeared onto y2 and y3 (strictly speaking, onto the estimated values of e2 and e3 ). 2 3 0:5 0 0 0:2 0:2 5 For P ˆ C1;r1 : PQ^e P ˆ 4 0 0 0:2 0:2 2

pat…C2;r † ˆ pat…C2 † \ dpat…HI †

3 0 07 7 15 1

As dpat…HI † is incomplete, the covariance matrix C2 should be reduced to C2;r such that

Table 2. The values of reliability measures for scheme (c) Cin ˆ C1;r

Cin ˆ C1

r1 r2 r3

1.00 0.20 1.00

10.00 2.00 10.00

r1 r2 r3

1.00 0.19 0.95

0.53 1.00 0.50

2

C2;r

1 6 0:5 6 ˆ4 0 0

1

1

1

1

1

1 1

1 0

1

0

0

1

1 3 0 07 7 7 15

0

1

1

0:5 1 0 0

1

3

1

2

1

6 17 7 61 7\6 15 40

0 0 4 1

0

0

3

1

0

1

0

0

1

07 7 7 15

0

1

1

3 0 0 7 7 15 5

We shall only show the matrix Q^e P for the full covariance matrix C2 , where we can observe the extension of the outlier-smearing area from one of the two separate levelling loops upon the other loop and vice versa. 2 3 0:56 0:56 0:12 0:12 6 0:44 0:44 0:12 0:12 7 7 Q^e P ˆ 6 4 0:42 0:42 0:51 0:51 5 0:42 0:42 0:49 0:49 It is easy to check that with the reduced covariance matrix C2;r the model is freed from the e€ect mentioned above. 5 Concluding remarks Taking into account the possibility of encountering in practice Gauss±Markov models having (at least in some of their segments) a speci®c algebraic structure, it is proposed that prior to starting the LS-based assessment of model internal reliability one should check the consistency between the pattern of a given covariance matrix and the GMM's algebraic correlation pattern. In the case of inconsistency a proper reduction of a given covariance matrix is recommended to eliminate the model's susceptibility to `outlier-hiding e€ects' which do not occur in ordinary GMMs. This paper does not o€er a complete treatment of the problem. For instance, it is necessary to investigate more thoroughly the in¯uence of the covariance matrix C upon the pattern of the projector HP . A separate but very important issue is the construction of algorithms for detecting the irregularities in the algebraic structure of larger systems. It should be emphasized that it was only due to the problems encountered in the reliability analysis of the systems with correlated observations (Wang and Chen 1994; Scha€rin 1997) that the present research could be initiated.

586

Appendix A: Diagonal pattern and ordinary pattern of a square matrix The concepts of diagonal pattern and ordinary pattern of a square matrix have been introduced to facilitate the notation of the requirements derived for covariance matrices in speci®c Gauss±Markov models. Before de®ning them, let us present the preparatory concepts. Diagonal compact form of a square matrix X [notation: dcomp (X)]: a diagonal-block form of X, which consists of disjoint diagonal square blocks, being nonzero matrices, and the complementary area of zero elements. A diagonal block is de®ned here as a (k  k) submatrix having: for k ˆ 1: a single element (a real number) for k  2: at least `k 1' non-zero elements such that their indices, either i or j, contain all the numbers 2; . . . ; k. The diagonal block cannot be decomposed into smaller-order blocks. Examples: 2 3 2 3   2 3 0 1 0 3 1 0 X3 ˆ 4 0 5 0 5 X1 ˆ 4 0 0 0 5 X2 ˆ 3 2 6 0 8 0 0 0 dcomp…X1 † ˆ X1

dcomp…X2 † ˆ X2 ;

dcomp…X3 † 6ˆ X3

diagonal compact form of X, such that each element of every diagonal block is assigned 1, whereas the nondiagonal elements outside the blocks remain zeroes. 2 3 2 3 1 0 0 0 1 0 0 0 60 2 6 7 1 07 7 60 1 1 07 dpat6 40 3 5 05 ˆ 40 1 1 05 0 0 0 0 0 0 0 1  1 dpat 0

  2 1 ˆ 6 1

1 1



 0 dpat 0

  0 1 ˆ 0 0

0 1



Since in general there can exist several diagonal compact forms of X, there can also be several equivalent diagonal patterns of X. For a block matrix   X1 0 Xˆ 0 X2 where X1 and X2 are square matrices, we have:   dpat…X1 † 0 dpat…X† ˆ 0 dpat…X2 † Assuming the following interpretation of the elements 0, 1 in diagonal patterns and the ordinary patterns: 0: the null set  1: the universal set U

The solid lines in X1 indicate the diagonal blocks. A diagonal compact form of X, if di€erent from the initial matrix X [see dcomp(X3 )], can be obtained by the symmetrical row and column permutations of X exposing its diagonal blocks, i.e.

we can introduce the relation of inclusion written

dcomp…X† ˆ UXUT

fdpat…X†gij  fdpat…Y†gij

where U is a permutation matrix (orthonormal) 2 32 32 3 0 1 0 1 0 3 0 1 0 6 76 76 7 dcomp…X3 † ˆ 4 1 0 0 54 0 5 0 54 1 0 0 5 0 0 1 6 0 8 0 0 1 2 3 5 0 0 6 7 ˆ 40 1 35 0 6 8 The form as de®ned above is not unique because in general there are several possible arrangements of blocks along the diagonal. Ordinary pattern of a square matrix X [notation: pat(X)]: a representation of X, such that each non-zero element is assigned 1, whereas all the zero elements remain unchanged, e.g.   1 0 pat…X2 † ˆ 1 1 Now we can de®ne the diagonal pattern of a square matrix X [notation dpat(X)] as a representation of the

dpat…X†  dpat…Y†

where X; Y 2 Rnn

and de®ned by i; j ˆ 1; . . . ; n

The de®nition applies also to ordinary patterns. For the following patterns:       1 1 1 1 1 1 pat…Y2 † ˆ dpat…X† ˆ dpat…Y1 † ˆ 1 1 0 1 1 1 we obtain dpat…Y1 †  dpat…X†; pat…Y2 †  dpat…X† Similarly, we can introduce the intersection of the patterns. We shall con®ne the presentation to intersecting the ordinary patterns and the diagonal patterns, as this case is used in the main part of this paper. The intersection of pat(X) and dpat(Y) written pat…X† \ dpat…Y† ˆ pat…Xr † where: X; Y; Xr 2 Rnn ; Xr is a reduced form of X, is de®ned by fpat…Xr †gij ˆ fpat…X†gij \ fdpat…Y†gij

i; j ˆ 1; . . . ; n

587

For 2

1 X ˆ 44 2

0 3 1

3

2

3 25 0

2 Y ˆ 40 0 2

1

0

1 3 0 1

which can easily be extended to

3

0 05 0 3

2

…AT1 A1 ‡ AT2 A2 †

1 Xr ˆ 4 4 0

0 3 0

ˆ …AT2 A2 †

…B5†

The superscript … † denotes g-inverse. 1

6 7 6 pat…X† \ dpat…Y† ˆ 4 1 1 1 5 \ 4 1 1 1 0 0 2 3 1 0 0 6 7 ˆ 41 1 05 0 0 0 Hence 2

1

1 1 0

0

Thus H11 ˆ A1 …AT1 A1 † AT1 ;

3

7 05 1

H22 ˆ A2 …AT2 A2 † AT2

The non-diagonal term H21 ˆ HT12 can be written equivalently as H21 ˆ A2 …AT1 A1 † AT1

…B6†

On the basis of Eq. (B4) we can write …AT1 A1 † …AT1 A1 ‡ AT2 A2 † ˆ I

3

0 05 0

…B7†

Premultiplying both sides of Eq. (B7) by A2 and making use of …AT1 A1 † ˆ …AT2 A2 † [see Eqs. (B4) and (B5)], we obtain A2 …AT1 A1 † AT1 A1 ‡ A2 …AT2 A2 † AT2 A2 ˆ A2

Appendix B: The condition for incomplete diagonal pattern of the orthogonal projector

which can be reduced to

Let a matrix A…A 2 Rnm ; rank…A† ˆ m† be partitioned as shown below:   A1 Aˆ …B1† A2

Since M…AT1 A1 † ˆ M…AT1 †, we obtain ®nally

where A1 …n1  m†; rank…A1 † ˆ r A2 …n2  m†; rank…A2 † ˆ m

r;

0rm

Property 1. If the spaces spanned by the rows of A1 and A2 are disjoint, i.e. M…AT1 † \ M…AT2 † ˆ f0g

…B2†

then H ˆ A…AT A† 1 AT has an incomplete diagonal pattern, i.e.     0 H11 H11 H12 ˆ …B3† Hˆ 0 H22 H21 H22 where H11 …n1  n1 †, H22 …n2  n2 †. Proof. For A partitioned as in Eq. (B1) we obtain   A1 …AT1 A1 ‡ AT2 A2 † 1 AT1 A1 …AT1 A1 ‡ AT2 A2 † 1 AT2 Hˆ A2 …AT1 A1 ‡ AT2 A2 † 1 AT1 A2 …AT1 A1 ‡ AT2 A2 † 1 AT2

A2 …AT1 A1 † AT1 A1 ˆ 0 A2 …AT1 A1 † AT1 ˆ 0 which yields H21 ˆ HT12 ˆ 0. We shall now check the reverse property. Given A 2 Rnm ; rank…A† ˆ m. Property 2. If the projector H ˆ A…AT A† 1 AT has an incomplete diagonal pattern [see Eq. (B3)], then M…AT1 † \ M…AT2 † ˆ f0g [see Eq. (B2)] where A1 and A2 are submatrices of A as in Eq. (B1). Proof. Since H12 ˆ HT21 ˆ 0, we have A1 …AT1 A1 ‡ AT2 A2 † 1 AT2 ˆ 0 which, besides the cases A1 ˆ 0 or A2 ˆ 0, implies that ‰A1 …AT1 A1 ‡ AT2 A2 † 1 Š
AT2 ˆ 0

…B8†

or A1
‰…AT1 A1 ‡ AT2 A2 † 1 AT2 Š ˆ 0

…B9†

Denoting the expressions in square brackets in Eqs. (B8) and (B9) by A1
and AT2
, respectively, we obtain A1 AT2 ˆ 0

…B10†

The cases r ˆ 0 or r ˆ m can be proved immediately, as

A1 AT2 ˆ 0

…B11†

A1 ˆ 0 or A2 ˆ 0 and H21 ˆ HT12 ˆ 0

Since rank…A1
† ˆ rank…A1 † and rank…A2
† ˆ rank…A2 †, it follows from Eq. (B10) and (B11) that

We also have either H11 ˆ 0 or H22 ˆ 0 respectively. Hence, we shall provide the proof for 0 < r < m. According to Rao (1973), with the condition of Eq. (B2) being satis®ed, the following equality holds true: …AT1 A1 ‡ AT2 A2 †

1

ˆ …AT1 A1 †

…B4†

rank…A1 † ‡ rank…A2 † ˆ m : Therefore, adding the cases A1 ˆ 0 or A2 ˆ 0, we obtain: rank…A1 † ˆ r; rank…A2 † ˆ m: r, where 0  r  m.

588

Considering the case 0 < r < m we shall assume that M…AT1 † \ M…AT2 † 6ˆ f0g Then, there must exist a matrix D…n2  m† such that A2 ˆ DA1 . Substituting this relationship into Eq. (B8), we obtain after simple transformations A1 ‰AT1 …I ‡ DT D†A1 Š 1 AT1 DT ˆ 0

…B12†

Since rank…A1 † < m the expression in square brackets will be singular for any matrix D. So we may conclude that there is no matrix D for which the relationship (B12) can be de®ned. Hence, it must be M…AT1 † \ M…AT2 † ˆ f0g, which completes the proof. From properties 1 and 2 it follows that the relationship of Eq. (B2) is a necessary and sucient condition for the incomplete diagonal pattern of the orthogonal projector. A detailed analysis, carried out on a di€erent basis, shows that the condition is satis®ed by the models which can be decomposed into the subsystems, which can be solved independently of one another or in a certain sequence. Appendix C: Property of the single-block orthogonal projector We shall consider an orthogonal projector H of the order n  2 (rank H ˆ r; r  n). Before we formulate and prove the main property we shall quote a wellknown auxiliary property. Auxiliary property. In an orthogonal projector H with a diagonal element fHgii being equal to 0 or 1, each nondiagonal element in the i-th row (fHgik ; k 6ˆ i) and in the i-th column (fHgki ; k 6ˆ i) is equal to 0. The proof follows immediately from the idempotency and symmetry of H, i.e. n n X X fHgii ˆ …fHgik fHgki † ˆ fHg2ii ‡ fHg2ik …k 6ˆ i† kˆ1

kˆ1

Property. In an orthogonal projector H with each of its diagonal elements fHgii (i ˆ 1; . . . ; n) satisfying the inequality 0 < fHgii < 1 each of its non-diagonal elements fHgij (j 6ˆ i) is di€erent from 0. Proof. We shall make use of the eigenvalue decomposition of H in the form   Ir 0 T …C1† WT H ˆ WKW ˆ W 0 0 where W is the n  n matrix of eigenvectors and K is the n  n matrix of the corresponding eigenvalues. Due to the idempotency of K, we can rewrite Eq. (C1) as follows: H ˆ WK…WK†T ˆ Wr WTr

…C2†

The rows of Wr are the orthogonal projections of the corresponding rows in W onto the r-dimensional

subspace spanned by the e1 ; e2 ; . . . ; er unit vectors of the n-dimensional orthocartesian coordinate system. In this representation we have for H of the order n2 fHgij ˆ …Wr †i …WTr †j ˆ …Wr †i ‰…Wr †j ŠT

…C3†

which is a scalar product of the row vectors …Wr †i
and …Wr †j
of the matrix Wr . Considering the case r < n as a general case, we notice the following. (1) With none of the n orthonormal row vectors of W coinciding with any one of the unit vectors e1 ; e2 ; . . . ; er there can be no pair of projections …Wr †i
, …Wr †j
such that …Wr †i
‰…Wr †j
ŠT ˆ 0 …j 6ˆ i†; …Wr †i
‰…Wr †i
ŠT ˆ 1 or 0; …Wr †j
‰…Wr †j
ŠT ˆ 1 or 0

…C4†

(2) With one row vector of W (e.g. Wi
) coinciding with one of the unit vectors e1 ; e2 ; . . . ; er or with one of the unit vectors er‡1 ; er‡2 ; . . . ; en , its projection …Wr †i
is the corresponding e or 0 respectively. In either case it is orthogonal not only to the j-th row vector of Wr but also to all the remaining row vectors of Wr . According to Eq. (C2), we obtain zero values in all the non-diagonal elements of the i-th row and the i-th column of H. The diagonal element fHgii will be 1 or 0 respectively. This result con®rms the auxiliary property. From the above two partial conclusions it follows that a zero value in any non-diagonal element of H implies 0 or 1 in at least one of the corresponding diagonal elements of H, which, by contradicting the assumptions, proves the property. The property can be restated in terms of patterns (for patterns see Appendix A). It would then read as follows. For an orthogonal projector H forming a single block of the size n  2 the following equality holds: dpat…H† ˆ pat…H† Appendix D: Examining the diagonal pattern of the weight modi®cation of the orthogonal projector Let A (A 2 Rnm ; rank…A† ˆ m) be partitioned as in Eq. (B1) and let H ˆ A…AT A† 1 AT have an incomplete diagonal pattern as in Eq. (B3). Property. With the weight matrix   P1 0 …pos. def.† Pˆ 0 P2 where P1 …n1  n1 †, P2 …n2  n2 †, the projector HP ˆ A…AT PA† 1 AT P has an incomplete diagonal pattern

589



HP…11† HP ˆ HP…21†

HP…12† HP…22†





HP…11† ˆ 0

0



matrix A with a new arrangement of rows and columns will take the form

HP…22†

where HP…11† …n1  n1 †, HP…22† …n2  n2 †. Proof. With A and P as in the assumptions, the projector HP can be written in the following form:   A1 GAT1 P1 A1 GAT2 P1 …D1† HP ˆ A2 GAT1 P2 A2 GAT2 P2 …AT1 P1 A1

AT2 P2 A2 † 1 .

A ˆ UAV

…E1†

We will also rearrange the rows and columns of the weight matrix according to the row permutations of the matrix A, i.e. P ˆ UPUT

…E2†

where G ˆ ‡ Let us introduce the standardized matrix     AS;1 b1 A1 ˆ AS ˆ AS;2 b2 A2

We may ®nd out easily that the projectors H and HP will take the form H ˆ A …AT A † 1 AT ˆ UHUT

…E3†

where b1 ; b2 are such that bT1 b1 ˆ P1 and bT2 b2 ˆ P2 . Since M…ATS;1 † ˆ M…AT1 † and M…ATS;2 † ˆ M…AT2 †, we have

HP ˆ A …AT P A † 1 AT P ˆ UHP UT

…E4†

M…ATS;1 † \ M…ATS;2 † ˆ f0g

…D2†

Applying the equalities of Eqs. (B4) and (B5) (see Appendix B) we obtain G ˆ …AT1 P1 A1 † ˆ …AT2 P2 A2 † and hence HP…11† ˆ A1 …AT1 P1 A1 † AT1 P1 HP…22† ˆ

A2 …AT2 P2 A2 †

AT2 P2

On the basis of Eq. (D2) we can write …AT1 P1 A1 † …AT1 P1 A1 ‡ AT2 P2 A2 † ˆ I The further part of the proof, which runs analogously to that in Appendix B, leads to A1 …AT1 P1 A1 † AT2 P1 ˆ 0 and A2 …AT2 P2 A2 † AT1 P1 ˆ 0 and hence HP…12† ˆ 0: HP…21† ˆ 0 Therefore, the property provides a sucient condition for the incomplete diagonal pattern of the projector HP in a speci®c GMM. Appendix E: In¯uence of permutations in the matrix A upon the pattern of the orthogonal and non-orthogonal projector Let U 2 Rnn , V 2 Rmm be the orthonormal matrices of the row and the column permutations respectively. A

On the basis of Eqs. (E3) and (E4) we may draw the following conclusions. (1) The permutations of columns in A do not a€ect H…HP † and consequently do not change their ordinary patterns and diagonal patterns. (2) The permutations of rows in A and the corresponding permutations of rows and columns in P result in identical permutations of rows and columns in H…HP †. This means that dpat…H † ˆ dpat…H† and dpat…HP † ˆ dpat…HP † With dpat…HP † ˆ dpat…H† we have also dpat…HP † ˆ dpat…H † References Chatterjee S, Hadi AS (1988) Sensitivity analysis in linear regression. Wiley, New York Pr oszynÄski W (1994) Criteria for internal reliability of linear leastsquares models. Bull Geod 68: 162±167 Pr oszynÄski W (1998) The reliability measures for the systems with correlated observations ± a comparative study. Proc Int Symp Geodesy for Geotechnical and Structural Engineering. Eisenstadt, Austria, pp. 216±221 Rao CR (1973) Linear statistical inference and its applications. Wiley, New York Scha€rin B (1997) Reliability measures for correlated observations. J Surv Eng 123(3): 126±133 Wang J, Chen Y (1994) On the reliability measure of observations. Acta Geod Cartograph Sin (English edition): 42±51