The effect of observation correlations upon the basic characteristics of reliability matrix as oblique projection operator
Abstract
The reliability matrix, being an oblique projection operator, transforms correlated observations into the least squares residuals in Gauss–Markov models. It also allows to study model responses in individual observations to the assumed configurations of gross errors. The variability of the basic characteristics of the operator due to the increase in observation correlations is investigated by means of numerical tests and theoretical derivations. The characteristics such as diagonal elements and asymmetry indices have not that long ago been introduced as the responsebased measures of internal reliability and subjected to the analysis. Here, additionally, the relationship between the asymmetry indices and the angles of nonorthogonality of projection is derived. The measures are compared in terms of the effect of observation correlations with the commonly used reliability measures obtained on the basis of statistical tests for detection and identification of outliers, such as generalized reliability numbers and minimal detectable biases. For the purposes of the present paper, the latter are termed the testingbased measures. The comparative analysis shows that both the types, when taken together, provide complete information about the behaviour of a GMM with correlated observations in the presence of a gross error in a particular observation and about its detectability. Hence, the conclusion is that the responsebased measures can be a useful supplementation of the testingbased measures for the phase of network design.
Keywords
Internal reliability Responsebased measures Testingbased measures Oblique projection operator Angles of nonorthogonality1 Introduction
The covariance matrices for observations are a basis for constructing the stochastic models for satellite and ground positioning systems. The covariances have an effect upon the elements of reliability matrices. It was already found in Wang and Chen (1994) and Schaffrin (1997) that at some level of observation correlations there may appear on a main diagonal of a reliability matrix the negative elements as well as the elements greater than 1. It was also noticed that the diagonal elements being onedimensional quantities do not fully describe the responses of a model to gross errors in the observations. So, the proposal of a generalized reliability number appeared. Further study of this problem led to the socalled responsebased reliability measures (Prószyński 2010) being a pair of quantities for each observation, i.e. a reliability number and an asymmetry index. They make ranking of observations with respect to internal reliability more correct than when using the reliability numbers alone. The measures will here be subject to further analyses.

confirmed by publications on applied mathematics and numerical methods (Davies and Higham 2000; Budden et al. 2008; Numpacharoen and Atsawarungruangkit 2012), the difficulty in generating positive definite correlation matrices in a way that ensures a purely random character of the matrix elements throughout all the levels of correlation;

the lack of injectivity of the function f: \( {\mathbf{C}}_{\text{s}} \to { \det }{\mathbf{C}}_{\text{s}} \) (\( {\mathbf{C}}_{\text{s}} \)—a positive definite correlation matrix), where the degree of departure from injectivity increases with the number of observations in a system. Due to the above drawback, in order to make a graph presenting a correlationdependent quantity as a function of the correlation level expressed in terms of \( { \det }{\mathbf{C}}_{\text{s}} \), we have a dispersion band instead of getting an unambiguous curve.
The above problems appear also in the present paper but are approached in a way that raises effectiveness of a matrix generating method and minimizes the effect of the lack of injectivity. The approach is explained briefly in “Appendix A”.

to formulate the properties of an oblique projection operator that are connected with the effect of observation correlations;

to provide a more detailed knowledge on variability of the responsebased reliability measures with respect to the increase in observation correlations;

to compare the behaviour of the responsebased reliability measures and of the socalled testingbased measures (i.e. generalized reliability numbers and MDBs) with the increase in observation correlations;

to create a theoretical basis for operating with reliability matrices in designing the positioning systems.
2 Recalling basic formulas and properties
Let us remind that the form above is obtained by premultiplying both sides of the original GMM by \( {\mathbf{S}}^{  1} \), where \( {\mathbf{S}} = ( {\text{diag }}{\mathbf{C}} )^{1/2} \), and transforming the covariance matrix C (positive definite) accordingly.
The function f: \( {\mathbf{C}}_{\text{s}} \to { \det }{\mathbf{C}}_{\text{s}} \) (or f:\( {\mathbf{C}}_{\text{s}} \to \rho_{\text{G}} \)) is not injective. This implies some ambiguity of the graphs for correlationdependent quantities with \( \rho_{\text{G}} \) being a variable. That is why the use will also be made of constant correlation matrices (Tiit and Helemae 1997; Wied 2017), i.e. the positive definite matrices with all offdiagonal elements being set to one value of correlation. They are denoted here as \( {\mathbf{C}}_{{{\text{s,}}a}} \), where the function f:\( {\mathbf{C}}_{{{\text{s,}}a}} \to \rho_{{{\text{G,}}a}} \) is injective, separately for \( a \in (  \frac{1}{n  1}, \, 0) \) and \( a \in (0, 1) \). We can readily come to this conclusion by analysing the sign of derivative \( [\rho_{{{\text{G,}}a}} ]^{\prime } \).
 (a)
H—the reliability matrix, being an oblique projection operator (HH = H, \( {\mathbf{H}}^{\text{T}} \ne {\mathbf{H}} \))
 (b)
h_{i}, w_{i} (i =1,…, n)—the responsebased measures of internal reliability for the ith observation; h_{i} is a reliability number, w_{i} is an asymmetry index. They are basic characteristics of the operator H and are defined as
 (c)
G_{(i)}—a global response to gross error in the ith observation (Prószyński 2010), defined by
 (d)
r_{i} (i =1,…, n)—the generalized reliability numbers, given by
The quantities r_{i} and \( \sqrt {r_{i} } \) are noncentrality parameters for unit gross error, in a global model test statistic and the local outlier test statistics, respectively. For \( {\mathbf{C}}_{\text{s}} = {\mathbf{I}} \), we have \( r_{i} = h_{{{\mathbf{I}}_{i} }} \).
Instead of the measures h_{i}, w_{i}, we can equivalently use a pair h_{i}, k_{i} . The parameter k is a ratio of the quasiglobal response and a local response to a gross error in a particular observation (Prószyński 2010). It characterizes the relative strength of the local response (i.e. in that observation). The reliability criteria as expressed in terms of h_{i}, k_{i} are met in the region \( 0.5 < h_{i} < 1 \) and \( 0 < k_{i} < 1 \), termed in the abovementioned reference the outlierexposing area.
3 Methodology of research
The effect of observation correlations upon the reliability matrix always depends on the design matrix used. Also, the relationships linking the investigated indices h_{i}, w_{i} and G_{(i)} of the reliability matrix and the observation correlations contained in the matrices \( {\mathbf{C}}_{\text{s}} \) are fairly complex. These two reasons call for the use of the specially planned methodology of research. So, for investigating the correlation effect numerical testing based on simulated networks will have to be the main research tool. For better determining of the effect itself, several networks with different levels of internal reliability will be used in the tests. Obviously, wherever feasible the purely theoretical derivations will be carried out to support the analyses of the test results.
Let us consider a correlationdependent quantity x given as f(\( {\mathbf{A}}_{\text{s}} ,{\mathbf{C}}_{\text{s}} \)), where \( {\mathbf{A}}_{\text{s}} \), \( {\mathbf{C}}_{\text{s}} \) define the model (1). The correlation effect will be contained in a graph characterizing, for a given \( {\mathbf{A}}_{\text{s}} \), the variability of x due to the increase in observation correlations. Assuming the use of the global correlation index \( \rho_{\text{G}} \), the function x = f(\( {\mathbf{A}}_{\text{s}} ,{\mathbf{C}}_{\text{s}} \)) will have a form \( x = f(\rho_{\text{G}} {\mathbf{A}}_{\text{s}} ) \), as expressed in a notation borrowed from conditional distributions. Since \( {\mathbf{A}}_{\text{s}} \) can be obtained with different values of observational standard deviations, we shall reduce the task to a single case of unit standard deviations (\( \sigma_{1} = \sigma_{2} = \cdots = \sigma_{n} = 1 \)) where \( {\mathbf{A}}_{\text{s}} = {\mathbf{A}} \), and hence, we can use the notation \( x = f(\rho_{\text{G}} {\mathbf{A}}) \). The effect of the increase in observation correlations will be determined on the basis of the values of x computed for different design matrices using specially created auxiliary indices.
Due to numerical problems with generating matrices of greater sizes (n ≥ 5), we decided to use also the simplest possible representation of correlation matrix \( {\mathbf{C}}_{\text{s}} \), i.e. the already mentioned constant correlation matrix, denoted as \( {\mathbf{C}}_{{{\text{s,}}a}} \). Although it yields specific matrix configurations, it represents injective function (see Sect. 2) and is readily created.
4 Auxiliary indices and their properties supporting analyses
 (a)
d_{h}, \( d_{{h_{{\mathbf{I}}} }} \)—the indices of mutual differentiation of h_{i} values and \( h_{{{\mathbf{I}}_{i} }} \) values
\( \Delta d_{h} \), \( [h_{{\rm max} } \, , \, h_{{\rm min} } ] \)—difference of indices as given above and the interval of h_{i} values.
 (b)
\( \bar{w} \), \( [w_{{\rm max} } \, , \, w_{{\rm min} } ] \)—average asymmetry index and the interval of w_{i} values.
In all the above formulas, i = 1,…, n, n being the number of observations in a network.
5 Nonorthogonality versus asymmetry of projection H
To operate with one common term when referring to projection H and to the matrix H itself, we shall use in the former case the term nullspace instead of a more appropriate term the kernel.
The angles between spaces and their different definitions have been widely investigated in the mathematical literature (e.g. Golub and Van Loan 1989). A specific concept of minimal angle between complementary subspaces was introduced in (Ipsen and Meyer 1995). In the present paper, using basic definition of the angle between a pair of vectors, we start from standard methods of computing the angles between the range and the nullspace of projection operator H (see formula (4)) and on this basis derive a relationship between the angles of nonorthogonality and the asymmetry indices w.
Let U and V denote the range and the nullspace of projection H. Since H is an oblique projection, the subspaces U and V are not orthogonal. To determine the angles between U and V, we first find the vectors \( {\mathbf{u}} \in {\text{U}} \) such that Hu = u and \( {\mathbf{v}} \in {\text{V}} \) such that Hv = 0. The vectors u and v can be expressed in terms of \( {\mathbf{A}}_{\text{s}} \),\( {\mathbf{C}}_{\text{s}} \) (case 1) or can be determined directly on the basis of SVD (Singular Value Decomposition) of the matrix H (case 2). Both cases are presented below, i.e.
Case 1
The dimensions of the subspaces are dimU = n − (u − d) and dimV = u − d, what yields dimU + dimV = n.
Case 2
Using the column vectors \( {\mathbf{u}}^{ \bot } = \{ {\mathbf{P}}_{\text{s}} {\mathbf{A}}_{\text{s}} \}_{ \bullet i} \) and \( {\mathbf{v}} = \{ {\mathbf{A}}_{\text{s}} \}_{ \bullet i} \) as in (19) and (20), we can find the angles of nonorthogonality \( \alpha \), defined as \( \alpha = \pi /2  \beta \). The angles \( \alpha \) obtained on the basis of these vectors will obviously not be consistent (both in number and value) with those that result from \( \beta_{j,k} \), i.e. \( \alpha_{j,k} =\uppi/2  \beta_{j,k} \).
As could be expected the increase in observation correlations exerts a distorting effect on the range of projection H. This can be explained by the character of variability of the ratio (condition number) \( k^{ * } = \frac{{\lambda_{\rm max}^{ * } }}{{\lambda_{\rm min}^{ * } }} \) for the matrix \( {\mathbf{P}}_{\text{s}} \) with the decrease in det \( {\mathbf{C}}_{\text{s}} \).
On the basis of spectral decomposition of \( {\mathbf{C}}_{\text{s}} \), i.e. \( {\mathbf{C}}_{\text{s}} \) = WΛW^{T}, and the definition (3), i.e. \( {\mathbf{C}}_{\text{s}} = q{\mathbf{P}}_{\text{s}}^{  1} \), we get \( {\mathbf{P}}_{\text{s}} = {\mathbf{W}}(q{\varvec{\Lambda}}^{  1} ){\mathbf{W}}^{\text{T}} \) and \( \lambda_{i}^{ * } = q\lambda_{i}^{  1} \) (i = 1, …, n). We can check that \( \, \mathop \Pi \nolimits_{1}^{n} \lambda_{i}^{ * } = { \det }{\mathbf{P}}_{\text{s}} = 1 \).
Furthermore, we get \( \lambda_{\rm max}^{ * } = q\lambda_{\rm min}^{  1} \), \( \lambda_{\rm min}^{ * } = q\lambda_{\rm max}^{  1} \), and finally, \( k^{ * } = \frac{{\lambda_{\rm max}^{ * } }}{{\lambda_{\rm min}^{ * } }} = \frac{{\lambda_{\rm max} }}{{\lambda_{\rm min} }} = k \).
With the decrease in det \( {\mathbf{C}}_{\text{s}} \) (and thus, with the increase in \( \rho_{\text{G}} \)), the ratio k, and hence \( k^{ * } \), increases. With \( \mathop \Pi \nolimits_{1}^{n} \lambda_{i} \to 0 \), both these ratios tend to infinity. This shows that the increase in observation correlations distorts more and more the angles between the vectors of the range of the operator H.
6 Results of numerical tests and their discussion
6.1 Test networks

the range of internal reliability indices for \( {\mathbf{C}}_{\text{s}} = {\mathbf{I}} \) [in brackets];

number of observations n and redundancy f.
Point coordinates for a horizontal network
Point no.  X _{[m]}  Y _{[m]} 

1  150  650 
2  200  100 
3  400  400 
4  800  700 
5  350  950 
6  950  350 
6.2 Results of the tests
The graphs represent a greater number of graphs that have been carried out in the research. Their reduction, aimed at saving space of the present paper, was possible thanks to the previously derived theoretical relationships linking some of the above quantities (Sect. 4).
6.3 Discussion of the results

as shown in Fig. 3 the values of \( \tfrac{1}{n}\sum\nolimits_{n} {h_{i}^{2} } \) and \( \Delta d_{h} = d_{h}  d_{{h_{I} }} \) (interrelated by (10) and (11)) are increasing. Their increase is considerably smaller for \( {\mathbf{C}}_{{{\text{s,}}a}} \) graphs. The differentiation of \( h_{i} \) values due to observation correlations rises clearly for \( \rho_{\text{G}} \) > 0.6;

in compliance with the increase in \( \Delta d_{h} \) (as in Fig. 3), we observe in Fig. 4 an increase in the width of the interval \( (h_{\rm min} ,h_{\rm max} ) \). This effect is much smaller for \( {\mathbf{C}}_{{{\text{s,}}a}} \) graphs;

for the network V1 of small redundancy (f = 2) \( h_{{\rm min} } \) comes down to 0 at \( \rho_{\text{G}} \) = 0.8 and assumes negative values for \( \rho_{\text{G}} \) > 0.8, while for V2 with f = 5 \( h_{{\rm min} } \) reaches 0 only at \( \rho_{\text{G}} \) = 0.95). For the network H with f = 18, the increase in the width of the interval \( (h_{\rm min} ,h_{\rm max} ) \) as computed for \( {\mathbf{C}}_{{{\text{s,}}a}} \) configurations is smaller than for networks with f = 2 and f = 5. We may then conclude that the higher the network redundancy, the smaller is its reaction to the increase in observation correlations;

the values of \( \bar{w} \) are decreasing (Fig. 5) and the width of the interval \( (w_{{\rm min} } ,w_{{\rm max} } ) \) is increasing (Fig. 6). The lower limit moves towards high negative values. The results are consistent with formulas (12), (13) and the growth of \( \, \tfrac{1}{n}\sum\nolimits_{n} {h_{i}^{ 2} } \) (Fig. 3) and \( \, \bar{g} \) (Fig. 7);

as shown in Fig. 7 the increase in \( \bar{r} \) is of similar character as that in \( \bar{g} \). However, for higher values of \( \rho_{\text{G}} \) and especially for \( \rho_{\text{G}} \) approaching 1 the increase in \( \bar{r} \) is greater than that in \( \bar{g} \). This can be due to the influence of the scale factor q (see formula (3)) being in \( \bar{r} \) a magnifying factor \( q^{  1} \), while \( \bar{g} \) is independent of q. The differentiation of the \( r_{i} \) values increases faster than that of \( g_{i} \) values (Fig. 8), which can be explained by the similar reason with that in this case a magnifying factor in \( d_{r} \) is \( q^{  2} \).
We can see that despite the lack of injectivity of the function f :\( {\mathbf{C}}_{\text{s}} \to { \det }{\mathbf{C}}_{\text{s}} \), the graphs \( x = f(\rho_{\text{G}} {\mathbf{A}}) \) are consistent between themselves and also with the graphs obtained on the basis of \( {\mathbf{C}}_{{{\text{s,}}a}} \) configurations. So, there are reasonable grounds to consider that they yield a general but correct description of the effects of increase in observation correlations upon the measures of internal reliability. Such a description is for use in designing networks with correlated observations, the networks with low redundancies in particular.
7 Analysis of suitability of responsebased measures as compared to testingbased measures
It should be noted first of all that the responsebased measures \( h_{i} ,w_{i} \) are applicable only for a priori analyses of networks, while the testingbased measures \( r_{i} \) and \( {\text{MDB}}_{{{\text{s,}}i}} \) can be used both for a priori analyses (\( r_{i} ,{\text{ MDB}}_{{{\text{s,}}i}} \)) and for outlier detection (\( r_{i} \)) and identification (\( \sqrt {r_{i} } \)).

the pairs (\( h_{i} ,w_{i} \)) are the unambiguous measures of internal reliability of networks with correlated observations, as they are based on two basic characteristics of the oblique projection operator H, i.e. the elements on a main diagonal (\( h_{i} \)) and asymmetry indices (\( w_{i} \)) linked with the angles of nonorthogonality;

\( h_{i} \) is a direct response of a model in the ith observation to a single gross error residing in this observation. The \( r_{i} \) and \( {\text{MDB}}_{{{\text{s,}}i}} \) measures characterize the sensitivity of a global model test to a single gross error residing in the ith observation. You can see the difference between the two types of measures particularly clearly in the case when f = 1 (at any level of observation correlations), where the \( h_{i} \) values are mutually differentiated but the \( {\text{MDB}}_{{{\text{s,}}i}} \) ones are all identical. It is obviously an extreme and purely theoretical example, since such networks should be avoided in practice;

the criteria for (\( h_{i} ,w_{i} \)) are clearly interpretable since they are formulated in terms of reactions to a gross error (i.e. magnitude and sign of the error compensation and relative strength of the response);

the values of \( h_{i} \) are mutually related and, as shown in the present paper, so are the values of \( w_{i} \). Both types are interrelated by the inequality. Unlike \( r_{i} \), they do not depend on the scale factor q;

with the increase in observation correlations the differentiation of \( h_{i} \) values increases and the average value of \( w_{i} \) decreases. It means a gradual worsening of the model responses in individual observations. The smaller the redundancy f of the model, the worsening becomes greater. In the extreme case, i.e. when f = 1, at larger values of \( \rho_{\text{G}} \) (0.7 and more) the measures h for some observations may reach negative values as well as values greater than 1. The responses in such observations do not properly compensate the effect of a gross error in terms of both magnitude and sign. There may as well be the responses that are too small (compared to other responses), for the error to be detected and identified;

since the MDB only describes the sensitivity of a global model test to the presence of a single gross error in the observations, it should be supplemented with minimal identifiable error MIB (Teunissen 2017; Zaminpardaz and Teunissen 2018; Imparato et al. 2018) or the identifiability index ID (Prószyński 2015), to be aware of the magnitudes of the type III errors. Therefore, for the purposes of a priori analyses, MDBs cannot be considered as autonomous measures. The \( h_{i} ,w_{i} \) measures are only to a certain extent burdened with this disadvantage, since the pairs (\( h_{i} ,w_{i} \)) falling into the outlierexposing area, except the pairs falling outside, show satisfactory values of ID indices. The conclusion can be that the direct response \( (h_{i} ,w_{i} ) \) in a particular observation contains also some information as to the level of detectability and identifiability of a gross error in that observation.
8 Concluding remarks
Basic characteristics of the reliability matrix, which is an oblique projection operator, describe the responses of a network (or more generally—an observation system) to a single gross error and are sensitive to the increase in observation correlations. This justifies their use as socalled responsebased measures of internal reliability (\( h_{i} ,w_{i} \)). The measures can be a supplementation of the testingbased measures \( r_{i} ,{\text{ MDB}}_{{{\text{s,}}i}} \) for a priori analyses of networks carried out at the stage of their design. Although both types of measures show compliance in responding to the increase in observation correlations, they have specific features of their own. So, while taken together they can provide more complete information about the behaviour of networks in the presence of gross errors and about the chances for their detection and identification. For the systems with small redundancy, as may occur in engineering surveys, the responsebased measures seem to be specially useful.
The graphs \( x = f(\rho_{\text{G}} {\mathbf{A}}) \) based on generated \( {\mathbf{C}}_{\text{s}} \) configurations are limited to small network sizes (\( n \le 5 \)) and despite the applied minimizing procedure are affected by the lack of injectivity of the function f :\( {\mathbf{C}}_{\text{s}} \to \rho_{\text{G}} \). The graphs for networks of greater sizes and less affected by this drawback could be obtained by means of more powerful computers and more advanced software than those commonly available used in the present research. Having in mind the need to improve the design methods of systems with correlated observations, it seems purposeful to undertake such studies.
For further studies of correlation effect on the internal reliability of networks, it seems useful to take into account other relevant contributions as for instance those in Wang and Knight (2012) and Yang et al. (2013).
Notes
Acknowledgements
The work was created as part of statutory research on internal reliability of systems with correlated observations, in the Chair of Engineering Surveys and Measurement and Control Systems, Faculty of Geodesy and Cartography, Warsaw University of Technology.
References
 Budden M, Hadavas P, Hoffman L (2008) On the generation of correlation matrices. Appl Math E Notes 8:279–282Google Scholar
 Davies PI, Higham NJ (2000) Numerically stable generation of correlation matrices and their factors. BIT Numer Math 40:640. https://doi.org/10.1023/A:1022384216930 CrossRefGoogle Scholar
 Golub G, Van Loan C (1989) Matrix computations, 2nd edn. The Johns Hopkins University Press, BaltimoreGoogle Scholar
 Imparato D, Teunissen PJG, Tiberius CCJM (2018) Minimal detectable and identifiable biases for quality control. Surv Rev. https://doi.org/10.1080/00396265.2018.1437947 Google Scholar
 Ipsen EC, Meyer CD (1995) The angle between complementary subspaces. NCSU technical report #NA–019501 Series 4.24.667Google Scholar
 Numpacharoen K, Atsawarungruangkit A (2012) Generating correlation matrices based on the boundaries of their coefficients. PLoS ONE 7(11):e48902. https://doi.org/10.1371/journal.pone.0048902 CrossRefGoogle Scholar
 Prószyński W (2010) Another approach to reliability measures for systems with correlated observations. J Geod 84:547–556CrossRefGoogle Scholar
 Prószyński W (2015) Revisiting Baarda’s concept of minimal detectable bias with reference to outlier identifiability. J Geod 89:993–1003CrossRefGoogle Scholar
 Prószyński W, Kwaśniak M (2018) Analytic tools for investigating the structure of network reliability measures with regard to observation correlations. J Geod 92:321–332CrossRefGoogle Scholar
 Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New YorkGoogle Scholar
 Robert C, Casella G (1999) Monte Carlo statistical methods. Springer, New YorkCrossRefGoogle Scholar
 Schaffrin B (1997) Reliability measures for correlated observations. J Surv Eng 123(3):126–137CrossRefGoogle Scholar
 Teunissen PJG (1994) A new method for fast carrierphase ambiguity estimation. IEEE Plans 94:562–573Google Scholar
 Teunissen PJG (2017) Distributional theory for the DIA method. J Geod 122:122. https://doi.org/10.1007/s0019001710457 Google Scholar
 Tiit E, Helemae H (1997) Boundary distributions with fixed marginals. In: Distributions with given marginals and moment problems. Springer, Dordrecht. https://doi.org/10.1007/9789401155328
 Wang J, Chen Y (1994) On the reliability measure of observations. Acta Geodaet. et Cartograph. Sinica, English Edition 42–51Google Scholar
 Wang J, Knight NL (2012) New outlier separability test and its application in GNSS positioning. J Glob Position Syst 11(1):46–57CrossRefGoogle Scholar
 Wied D (2017) A nonparametric test for a constant correlation matrix. Econom Rev 36(10):1157–1172. https://doi.org/10.1080/07474938.2014.998152 CrossRefGoogle Scholar
 Yang L, Wang J, Knight NL, Yunzhong S (2013) Outlier separability analysis with a multiple alternative hypotheses test. J Geod 87(6):591–604. https://doi.org/10.1007/s0019001306290 CrossRefGoogle Scholar
 Zaminpardaz S, Teunissen PJG (2018) DIA datasnooping and identifiability. J Geod 122:122. https://doi.org/10.1007/s0019001811413 Google Scholar
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