An approach to response-based reliability analysis of quasi-linear Errors-in-Variables models
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Abstract
The paper presents an approach to internal reliability analysis of observation systems known as Errors-in-Variables (EIV) models with parameters estimated by the method of least squares. Such problems are routinely treated by total least squares adjustment, or orthogonal regression. To create a suitable environment for derivations in the analysis, a general nonlinear form of such EIV models is assumed, based on a traditional adjustment method of condition equations with unknowns, also known as the Gauss–Helmert model. However, in order to apply the method of reliability analysis based on the approach to response assessment in systems with correlated observations, presented in the earlier work of this author, it was necessary to confine the considerations to a quasi-linear form of the Gauss–Helmert model, representing quasi-linear EIV models. This made it possible to obtain a linear disturbance/response relationship needed in that approach. Several specific cases of quasi-linear EIV models are discussed. The derived formulas are consistent with those already functioning for standard least squares adjustment problems. The analysis shows that, as could be expected, the average level of response-based reliability for such EIV models under investigation is lower than that for the corresponding standard linear models. For EIV models with homoscedastic and uncorrelated observations, the relationship between the average reliability indices for the independent and the dependent variables is formulated for multiple regression and coordinate transformations. Numerical examples for these two applications are provided to illustrate this analysis.
Keywords
Errors-in-Variables Total least squares Disturbance/response relationship Oblique projector Response-based reliability1 Introduction
Total least squares (TLS) adjustment referring to Errors-in-Variables (EIV) models has a wide mathematical literature, e.g., Golub van Loan (1980), van Huffel and Vandewalle (1991), and Rao and Toutenburg (1999). It has also been extensively explored by researchers in the field of geodesy. There are a number of contributions analyzing the relationships between the EIV models and the standard iteratively linearized models, well established in geodesy, and simultaneously proposing suitable algorithms for the rigorous evaluation of parameters in nonlinear EIV models (e.g., Schaffrin and Wieser 2008; Schaffrin and Felus 2008; Neitzel 2010).
The present contribution is focussed entirely on the problem of response-based reliability analysis for TLS adjustment. It should be noted that analyses of this type are usually carried out at the design stage when one wants to evaluate the reliability properties of the originally nonlinear adjustment model under consideration. In such a priori analyses, the nonlinearity problems may be overcome by using approximate values of the parameters when observation results are lying sufficiently close to the true values, or, practically, by using nominal values of these quantities.
In an attempt to generalize the EIV model for the purpose of the response-based reliability analysis, the most reasonable approach, backed by an appropriate proof, appeared to this author to take, as a basis, a nonlinear stochastic model containing two types of quantities, namely, the error-free unknown parameters to be determined and the observations as random variables of well-known values and accuracy characteristics. This led to the use of the so-called combined case of least-squares adjustment (Krakiwsky 1975), being termed a method of condition equations with unknowns, also known as the Gauss–Helmert model. This equivalent approach to TLS adjustment as a specific least-squares problem turned out to be consistent with that discussed in Schaffrin et al. (2006), Schaffrin and Snow (2010), and Neitzel (2010), and it is followed here since it seems to be most suitable for the purpose of the response-based reliability analysis along the lines of the approach as in Prószyński (2010).
However, since such an approach requires the use of the linear relationship between the observations and residuals, restrictions to a general G–H model had to be made confining the considerations to its quasi-linear form only. Such a form means here a nonlinear G–H model that is linear with respect to the observation vector formed of both the dependent and the independent variables.
To establish a link between this paper and publications that do not use the term reliability, but are concerned with similar properties of over-determined linear models (e.g. Chatterjee and Hadi 1988), the domain of this paper could as well be expressed as the “sensitivity” analysis of orthogonal regression.
2 Generalized EIV model and its linearized form for the purpose of reliability analysis
We shall first show that the TLS adjustment problem referring to a nonlinear EIV model is, with respect to response-based reliability analysis, equivalent to the LS problem referring to a linearized form of this model.
The equivalence between the conditions (3) and (4) as applied to the EIV model (2) makes it possible to formulate the TLS problem for correlated observations, using a suitably modified condition (3).
- (a)\(\mathbf{y}_\mathrm{obs} +\mathbf{v}_{ y} =(\mathbf{G}_\mathrm{obs} +\mathbf{E}_\mathrm{G })\mathbf{x}+\mathbf{z}\), with x and z being the vectors of unknown parameters, the aggregated vectors are \(\mathbf{u}=\left[ {{\begin{array}{c} \mathbf{x} \\ \mathbf{z} \\ \end{array} }} \right], \mathbf{r}=\left[ {{\begin{array}{c} {\text{ vec} \mathbf{G}_\mathrm{obs} } \\ {\mathbf{y}_\mathrm{obs} } \\ \end{array} }} \right]\) The TLS condition and the equivalent LS condition will have the form as for the EIV model (2), i.e.$$\begin{aligned} \Vert {[{{\begin{array}{l@{\quad }l} {\mathbf{V}_\mathrm{G} }&{\mathbf{v}_{ y} } \\ \end{array} }}]} \Vert _\mathrm{F}^2 =\min \equiv \left\Vert {{\begin{array}{c} {\text{ vec}\mathbf{V}_\mathrm{G} } \\ {\mathbf{v}_{ y} } \\ \end{array} }} \right\Vert_2^2 =\min \end{aligned}$$
- (b)\(\mathbf{y}_\mathrm{obs} +\mathbf{v}_{ y} =\mathbf{G}(\mathbf{t})\cdot (\mathbf{x}_\mathrm{obs} +\mathbf{v}_{ x} )+\mathbf{z}\), with t and z being the vectors of unknown parameters, the aggregated vectors are \(\mathbf{u}=\left[ {{\begin{array}{c} \mathbf{t} \\ \mathbf{z} \\ \end{array} }} \right],\mathbf{r}=\left[ {{\begin{array}{c} {\mathbf{x}_\mathrm{obs} } \\ {\mathbf{y}_\mathrm{obs} } \\ \end{array} }} \right]\) The TLS condition and the equivalent LS condition will have the formwhich is consistent with the approach for the model (2), since \(\mathbf{v}_{ x} \) can be interpreted as a one-column matrix of residuals, i.e. \(\text{ vec}\mathbf{v}_{ x} =\mathbf{v}_{ x} \).$$\begin{aligned} \Vert {[ {{\begin{array}{l@{\quad }l} {\mathbf{v}_{ x} }&{\mathbf{v}_{ y} } \\ \end{array} }}]}\Vert _\mathrm{F}^2 =\min \equiv \left\Vert {{\begin{array}{c} {\mathbf{v}_{ x} } \\ {\mathbf{v}_{ y} } \\ \end{array} }} \right\Vert_2^2 =\min \end{aligned}$$
3 Derivation of disturbance/response relationship for quasi-linear EIV models
In contrast to Schaffrin (1997), the approach to “reliability analysis” for systems with correlated observations according to Prószyński (2010) requires the use of the observation model with random variables which are correlated, dimensionless variables of equal accuracy. We thus have to modify the model (8), rescaling the random errors so that instead of the vector v we operate with the vector \(\mathbf{v}_\mathrm{s} ={\varvec{\Sigma }}^{-1}\mathbf{v}\), where \({\varvec{\Sigma }}=(\text{ diag}\;\mathbf{C})^{1/2}\). This naturally results in that the covariance matrix of the rescaled random errors coincides with the original correlation matrix.
4 Indices for response-based reliability of quasi-linear EIV models
In the numerical examples that will follow, the results of such a response-based reliability analysis for EIV and GM models will be shown in a tabular and/or a graphic form. To distinguish the case of uncorrelated observations, we shall replace \(h_{ii} \) by the index \(\bar{{h}}_{ii} \), as in (Prószyński 2010).
The method of reliability analysis applied in the present paper does not follow the traditional approach of Baarda, since it does not lead to specifying the minimal detectable biases for individual observations. It is based entirely on the model responses to gross errors, and therefore is termed here a “response-based” reliability analysis. This approach offers “reliability criteria” interpretable in terms of model responses to observation disturbances.
the response in the individual observation (i.e. a local response) should compensate for at least half of the disturbance residing in that observation;
the local response with its absolute value should surpass the quasi-global response,
5 Formulas for reliability analysis of specific cases of quasi-linear EIV models
We shall discuss specific cases of quasi-linear EIV models assuming the systems with correlated observations with given positive-definite covariance matrix. The cases themselves are very important in geodetic technologies, since they represent the observation systems frequently met in practice that fall into the class of EIV models.
5.1 Multiple linear regression
We omit discussion of the structure of C, since it will depend on the properties of the observations used in a particular task.
5.2 Similarity transformation (2D)
5.3 Affine transformation (3D)
6 Specific properties of quasi-linear EIV models concerning the average reliability indices
- i.
the relationship between average reliability indices in quasi-linear EIV models versus those in GM models
- ii.
the relationship between average reliability indices for dependent and independent variables in quasi-linear EIV models with homoscedastic and uncorrelated observations
- ad i.Let us compare the average reliability indices \(\bar{{h}}_{ii} \) for the EIV and GM models. Introducing an auxiliary coefficient \(\gamma =n/r\), where due to \(r>n,\) it is always \(\gamma <1\), we shall writeand hence$$\begin{aligned} {\bar{{h}}}_\mathrm{{avr}} \text{(EIV)}&= \frac{\text{ Tr} \mathbf{H}}{\dim \mathbf{H}}=\frac{\text{ rank} \mathbf{H}}{\dim \mathbf{H}}=\frac{n-u}{r} =\gamma (1-\frac{u}{n})\nonumber \\&=\gamma \cdot {\bar{{h}}}_\mathrm{{avr}} (\text{ GM}) \end{aligned}$$(33)The values of the coefficient \(\gamma \) as in (33) for specific cases of quasi-linear EIV models will be as follows:$$\begin{aligned} {\bar{{h}}}_\mathrm{{avr}} (\text{ EIV})<{\bar{{h}}}_\mathrm{{avr}} (\text{ GM}) \end{aligned}$$As shown above, the value of \(\gamma \) reaches 0.5 for similarity and affine transformation and is smaller than that for multiple regression with \(s>1\). For instance, with \(s = 4\) we have \(\gamma = 0.2\), which implies a very low level of reliability. As could be expected, in terms of the response-based reliability the EIV models are weaker than the corresponding GM models. It follows from (33) that no matter how high the redundancy level of the EIV model is, we will have \({\bar{{h}}}_\mathrm{{avr}} \text{(EIV)}<0.5\). Thus, the reliability criteria proposed for GM models (see Sect. 4) are too rigorous for EIV models, and should be weakened. The decrease in average internal reliability between the GM and EIV models that have the same number of parameters and observation equations can be explained by a specific property of EIV models. The explanation of the property can be that the independent variables being treated as observed quantities do not cause the increase in the rank of the operator H, as it is the case when adding equations for the new observed dependent variables both in GM and EIV models. Hence, in EIV models the sum of reliability indices being equal to the rank of H depends upon the number (n) of condition equations, but not on the number (r) of observed variables \((r>n)\). Therefore, in EIV models the sum of reliability indices must be shared by a greater number of observed variables than in GM models.$$\begin{aligned}&\text{ multiple} \text{ regression}\quad \gamma =\frac{n}{r}=\frac{n}{ns+n}=\frac{1}{1+s}\\&\text{ similarity} \text{ transformation} \text{(2D,} \text{3D;} d \text{=} \text{2,} \text{3)} \\&\quad \gamma =\frac{n}{r}=\frac{{d}k}{{2d}k}=\frac{1}{2}\\&\text{ affine} \text{ transformation} \text{(2D,} \text{3D;} d \text{=} \text{2,} \text{3)}\\&\gamma =\frac{n}{r}=\frac{{d}k}{{2d}k}=\frac{1}{2} \end{aligned}$$
- ad ii.For such models the reliability matrix H as in (13) will take the formwhere \(\sigma ^{2}\) is the common variance and U is the \((n\times n)\) central matrix. Substituting \(\mathbf{B}=[{{\begin{array}{ll} \mathbf{K}&{-\mathbf{I}_{ n} } \\ \end{array} }}]\) (see (8)) into (34) and after simple manipulations we obtain$$\begin{aligned} \mathbf{H}=\mathbf{B}_\mathrm{s}^\mathrm{T} \mathbf{UB}_\mathrm{s} =\sigma ^{2}\mathbf{B}^\mathrm{{T}}\mathbf{UB} \end{aligned}$$(34)Denoting by \(\text{ Tr} \mathbf{H}_\mathrm{{ind}} \) and \(\text{ Tr} \mathbf{H}_\mathrm{{dep}} \) the traces for blocks of H corresponding to independent and dependent variables and by \({\bar{{h}}}_\mathrm{{avr}} (\text{ ind})\) and \({\bar{{h}}}_\mathrm{{avr}} (\text{ dep})\) the average reliability indices for independent and dependent variables, we shall introduce a coefficient \(\upeta \) defined as$$\begin{aligned} \mathbf{H}=\sigma ^{2}\left[ {{\begin{array}{l@{\quad }c} {\mathbf{K}^\mathrm{T}\mathbf{UK}}&{-\mathbf{K}^\mathrm{T}\mathbf{U}} \\ {-\mathbf{UK}}&\mathbf{U} \\ \end{array} }} \right] \end{aligned}$$(35)For multiple regression we have r = ns + n$$\begin{aligned} {\upeta }=\frac{{\bar{{h}}}_\mathrm{{avr}} (\text{ ind})}{{\bar{{h}}}_\mathrm{{avr}} (\text{ dep})}&= \frac{\text{ Tr} \mathbf{H}_\mathrm{{ind}} /({r-n)}}{\text{ Tr} \mathbf{H}_\mathrm{{dep}} /{n}}\nonumber \\&= \frac{{n}}{{r-n}}\cdot \frac{\text{ Tr} \mathbf{U}\mathbf{KK}^\mathrm{{T}}}{\text{ Tr} \mathbf{U}} \end{aligned}$$(36)and hence$$\begin{aligned} \mathbf{KK}^\mathrm{{T}}&= (\mathbf{I}_{ n} \otimes \mathbf{a}^\mathrm{{T}})(\mathbf{I}_{ n} \otimes \mathbf{a}^\mathrm{{T}})^\mathrm{{T}}=(\mathbf{I}_\mathrm{n} \otimes \mathbf{a}^\mathrm{{T}})(\mathbf{I}_\mathrm{n} \otimes \mathbf{a})\\&=\mathbf{I}_{ n} \otimes \mathbf{a}^\mathrm{{T}}\mathbf{a}=\left\Vert \mathbf{a} \right\Vert^{2}\cdot \mathbf{I}_{ n} \end{aligned}$$For similarity transformation (2D, 3D) we have: n = dk, r = 2dk, where \(d = 2\) or 3.$$\begin{aligned} {\upeta }=\frac{{n}}{{ns}}\cdot \frac{\Vert \mathbf{a} \Vert ^\mathrm{{2}}\text{ Tr} \mathbf{U}}{\text{ Tr} \mathbf{U}}=\frac{\Vert \mathbf{a}\Vert ^\mathrm{{2}}}{{s}} \end{aligned}$$(37)and hence$$\begin{aligned} \mathbf{KK}^\mathrm{{T}}&= (\mathbf{I}_{k} \otimes \upmu \mathbf{T}_\alpha )(\mathbf{I}_{k} \otimes \upmu \mathbf{T}_\alpha )^\mathrm{{T}}=(\mathbf{I}_{k} \otimes \upmu \mathbf{T}_\alpha )\\&\times \,(\mathbf{I}_{k} \otimes \upmu \mathbf{T}_\alpha ^\mathrm{T} )=\mathbf{I}_\mathrm{k} \otimes \upmu ^{2}\mathbf{I}_{d} =\upmu ^{2}\cdot \mathbf{I}_{{dk}} \end{aligned}$$For isometric transformation \((\upmu = 1)\) we get \(\upeta = 1\). For affine transformation it was not possible to reduce the formula (36) to a simple form as was done for the cases above.$$\begin{aligned} \upeta =\frac{{dk}}{{dk}}\cdot \frac{\upmu ^\mathrm{{2}}\text{ Tr} \mathbf{U}}{\text{ Tr} \mathbf{U}}=\upmu ^\mathrm{{2}} \end{aligned}$$(38)
7 Numerical examples of reliability analysis for EIV versus GM modelling
We will consider the models of similarity transformation and multiple regression. For each model we shall compare the reliability indices for EIV, resp. GM modelling.
Example 1
The matrices A and B will have the form as in (27) and the dimensions \((12\times 4)\) and \((12\times 24)\), respectively.
Observed coordinates and approximate transformation parameters
Point no. | Old system | New system | ||
---|---|---|---|---|
x | y | X | Y | |
1 | 167.23 | 62.58 | 167.62 | 165.13 |
2 | 149.66 | 119.71 | 123.55 | 213.92 |
3 | 50.24 | 108.88 | 29.47 | 156.90 |
4 | 39,03 | 38.31 | 51.10 | 81.33 |
5 | 106.68 | 19.26 | 127.39 | 93.80 |
6 | 104.81 | 72.29 | 100.88 | 145.79 |
Parameters | \(\upmu _\mathrm{o} = 1.10; \alpha _\mathrm{o} = 25^{\circ }; {a}_\mathrm{o} = 30; {b}_\mathrm{o} = 25\) |
uncorrelated observations : \(\mathbf{C}_{x, \mathrm{obs}} =\mathbf{C}_{{y},\mathrm{obs}} =\mathbf{C}_{X,\mathrm {obs}} =\mathbf{C}_{Y,\mathrm {obs}} =\sigma ^{2}\cdot \mathbf{I}; \sigma =0.005\)
correlated observations : \(\mathbf{C}_{x,\mathrm{obs}} =\sigma ^{2}\cdot \mathbf{C}_{\mathrm{{s}},x} ; \mathbf{C}_{y,\mathrm{obs}} =\sigma ^{2}\cdot \mathbf{C}_{\mathrm{{s}},y} ; \mathbf{C}_{X,\mathrm {obs}} =\sigma ^{2}\cdot \mathbf{C}_{\mathrm{{s,}}{X}} ; \mathbf{C}_{Y,\mathrm {obs}} =\sigma ^{2}\cdot \mathbf{C}_{\mathrm{{s,}}{Y}} ; \mathbf{C}_{\mathrm{{s}},{x}} ,\mathbf{C}_{\mathrm{{s}},{y}} ; \mathbf{C}_{\mathrm{{s}},{X}} ,\mathbf{C}_{\mathrm{{s}},{Y}}\) are independently generated correlation matrices, each such that \(\left| {\left\{ {\mathbf{C}_\mathrm{s} } \right\} _{ {ij}} } \right|\le 0.5\)\((j\ne i)\). There is no correlation between the vectors \(\mathbf{x}_\mathrm{obs} ,\mathbf{y}_\mathrm{obs} , \mathbf{X}_\mathrm{obs} ,\mathbf{Y}_\mathrm{obs}\).
Reliability indices for GM and EIV modelling—similarity transformation
Obs. No | GM | EIV | GM (cor) | EIV (cor) | ||||||
---|---|---|---|---|---|---|---|---|---|---|
\(\bar{h} _{ii} \) | \(k_i \) | \({\bar{h}}_{ii} \) | \(k_i \) | \(h_{ii} \) | \(w_{ii} \) | \(k_i \) | \(h_{ii} \) | \(w_{ii} \) | \(k_i \) | |
1(\(x,y\)) | – | – | 0.35 | 1.89 | – | – | – | 0.31 | \(-\)0.213 | 4.40 |
0.27 | \(-\)0.161 | 4.83 | ||||||||
2(\(x,y\)) | – | – | 0.33 | 1.99 | – | – | – | 0.26 | \(-\)0.283 | 7.04 |
0.27 | \(-\)0.139 | 4.52 | ||||||||
3(\(x,y\)) | – | – | 0.34 | 1.91 | – | – | – | 0.29 | \(-\)0.137 | 4.18 |
0.28 | \(-\)0.220 | 5.26 | ||||||||
4(\(x,y\)) | – | – | 0.32 | 2.10 | – | – | – | 0.34 | \(-\)0.044 | 2.31 |
0.22 | \(-\)0.205 | 7.57 | ||||||||
5(\(x,y\)) | – | – | 0.39 | 1.58 | – | – | – | 0.41 | \(-\)0.185 | 2.49 |
0.29 | \(-\)0.267 | 5.53 | ||||||||
6(\(x,y\)) | – | – | 0.46 | 1.19 | – | – | – | 0.46 | \(-\)0.109 | 1.72 |
0.44 | \(-\)0.249 | 2.51 | ||||||||
1(\(X,Y\)) | 0.63 | 0.58 | 0.29 | 2.50 | 0.64 | \(-\)0.085 | 0.76 | 0.22 | \(-\)0.200 | 7.96 |
0.57 | \(-\)0.087 | 1.01 | 0.32 | \(-\)0.055 | 2.63 | |||||
2(\(X,Y\)) | 0.61 | 0.64 | 0.28 | 2.62 | 0.61 | \(-\)0.049 | 0.76 | 0.38 | \(-\)0.012 | 1.71 |
0.57 | \(-\)0.180 | 1.30 | 0.29 | \(-\)0.109 | 3.82 | |||||
3(\(X,Y\)) | 0.63 | 0.59 | 0.28 | 2.52 | 0.63 | \(-\)0.029 | 0.66 | 0.36 | 0.009 | 1.72 |
0.61 | \(-\)0.049 | 0.77 | 0.39 | \(-\)0.025 | 1.73 | |||||
4(\(X,Y\)) | 0.59 | 0.70 | 0.27 | 2.76 | 0.68 | 0.028 | 0.42 | 0.34 | 0.004 | 1.90 |
0.52 | \(-\)0.113 | 1.34 | 0.34 | 0.001 | 1.95 | |||||
5(\(X,Y\)) | 0.71 | 0.41 | 0.32 | 2.12 | 0.72 | \(-\)0.103 | 0.59 | 0.43 | \(-\)0.099 | 1.83 |
0.67 | \(-\)0.152 | 0.82 | 0.29 | \(-\)0.134 | 3.97 | |||||
6(\(X,Y\)) | 0.83 | 0.20 | 0.38 | 1.65 | 0.85 | \(-\)0.030 | 0.21 | 0.39 | \(-\)0.077 | 2.07 |
0.91 | \(-\)0.105 | 0.22 | 0.39 | \(-\)0.181 | 2.72 |
The coefficient \(\upeta \) as defined in (36), is \({\upeta }={\upmu }^{2}=1.1^{2}=1.21\)
We can check that \(\upeta =\frac{{\bar{{h}}}_\mathrm{{avr}} (\text{ ind})}{{\bar{{h}}}_\mathrm{{avr}} (\text{ dep})}=\frac{0.365}{0.302}=1.21\)
Since \(\upeta > 1\) the average reliability index for independent variables (i.e. coordinates in the old system) is greater than that for dependent variables (i.e. coordinates in the new system). We can see it in Fig. 2, where the line EIV(x) \(\equiv \) EIV(y) runs above the line EIV(X) \(\equiv \) EIV(Y). The separation between both lines is not great, since the scale coefficient \({\upmu }\) does not differ much from 1.
Example 2
Multiple regression
C = \(\mathbf{C}_{\mathrm{s}} =\mathbf{I}\) and \(\mathbf{C}_{\mathrm{s}}\ne \mathbf{I}\), where \(| {\{{\mathbf{C}}_{s} \}_{ij} }|\le 0.5, j\ne i\)
\(\mathbf{a}_{\mathrm{o}}^{\mathrm{T}} (1)=[{{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 2&{-3}&1&4 \\ \end{array} }}]\) and \(\mathbf{a}_{\mathrm{o}}^{\mathrm{T}} (2) =[{{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {-0.43}&{-0.20}&{0.59}&{-0.49} \end{array}}}]\)
To save space in this article, the analysis results will be presented in graphical form only, i.e. for the variant \(\mathbf{a}_\mathrm{o} (1)\)—in Figs. 4 and 5, and for \(\mathbf{a}_\mathrm{o} (2)\)—in Figs. 6 and 7. In each case the two variants of the correlation matrix will be taken into consideration.
The coefficient \(\upeta \) denoted here by \(\upeta \)(2), is \(\upeta (2)=\frac{\Vert \mathbf{a}_\mathrm{o} (2)\Vert _{2}^{2} }{\mathrm{s}}=\frac{0.81}{4}=0.20\).
We can check that \(\upeta (2)=\frac{{\bar{{h}}}_\mathrm{{avr}} (\text{ ind})}{{\bar{{h}}}_\mathrm{{avr}} (\text{ dep})}=\frac{0.0420}{0.207}=0.20\).
The analysis shows (Figs. 5, 7) that the investigated GM model, except for one observation, does not satisfy the reliability criteria (\({h}_\mathrm{{ii}} >0.5)\). According to the theory we have \(\gamma =\frac{\bar{{h}}_{avr} (EIV)}{\bar{{h}}_{avr} (GM)}=\frac{0.075}{0.375}=0.2\), which means that the average value of the reliability index being in GM model equal to 0.375, drops down in the EIV model to 0.075. We observe significant differences in the values of the reliability indices \({h}_\mathrm{{ii}} \) both for uncorrelated and the correlated observations. Some values reach 0.03 or even 0.01.
For the EIV model with uncorrelated observations, in the variant \(\mathbf{a}_\mathrm{o} (1)\) (see Fig. 4) all the y-observations and in the variant \(\mathbf{a}_\mathrm{o} (2)\) (see Fig. 6) most of the x-observations are practically uncontrolled by the other observations in the model, and hence, potential gross errors residing in them are practically undetectable. This example of multiple regression confirms the theory that the distribution of the response-based reliability indices between the independent and dependent variables is dependent on the norm of the vector of regression coefficients (a).
In the case a\(_\mathrm{o}\)(1), the coefficient \({\upeta }\) is much greater than 1 and the independent variables x display better average reliability than the dependent variables y. For the case a\(_\mathrm{o}\)(2), where \({\upeta }\) is much smaller than 1, we have the opposite relation, i.e. the dependent variables \(y\) show better average reliability than the independent variables \(x\).
8 Conclusions
The response-based reliability of EIV models can be analyzed in an analogous way as for the corresponding GM models. The theoretical derivations showed that in terms of average reliability indices EIV models are at least two times weaker than the GM models. This can be simply explained by the fact that the coefficients are treated as error-free (deterministic) quantities in GM models, whereas they are considered as random variables in the EIV models. This confirms that the EIV models are subject to a greater number of sources of observation errors than GM models, which results in the lower level of their response-based reliability. Therefore, the reliability criteria for EIV models should be set at a lower level than for GM models. Such criteria are not proposed in this paper and require separate research.
Taking into account the empirically confirmed connection between the level of reliability indices and effectiveness of outlier detection in GM models, we have grounds to conclude that the relatively low response-based reliability of EIV models may indicate lower effectiveness of outlier detection than in GM models.
The a priori reliability analysis proposed within this paper is only one particular aspect of EIV models. Other aspects, obviously of greater importance when considering a full scope of practical problems, include numerical algorithms for parameter estimation and the associated outlier detection procedures (see e.g., Schaffrin 2011). It seems, however, that the revealed reliability properties of EIV models can be helpful in constructing the outlier detection procedures. For doing so, the research findings of geodesists in the area of hypotheses testing (eg. Teunissen 1996) can be a valuable theoretical basis. On the grounds of this theory, one might also undertake the task of deriving a generalized formula for minimal detectable biases (MDBs) of observed quantities in EIV models. The testing-based approach to reliability measures (Schaffrin 1997; Knight et al. 2010) might be helpful in carrying out that task.
The equality \(r=n\) as a specific case of EIV models being equivalent to GM models, has been proposed in this paper only for the needs of the response-based reliability analysis. Therefore, it does not have a general character. At any rate, it is commonly known that both EIV and GM models can be treated by the classical method of least-squares adjustment.
A more forward-looking approach to reliability analysis, however, has already been undertaken by Schaffrin and Uzun (2011) who applied the TLS-techniques within EIV models. It would be interesting to see any correspondence to the approach presented. However, despite differences in the assumptions, both the approaches are important to the development of geodetic technologies, as they are extending the methods of reliability analyses upon the observation systems that fall into the class of EIV models.
Notes
Acknowledgments
The research presented in this paper has been carried out under the Grant No. N N 526 135134 funded by the National Research Council in Poland. The author is greatly indebted to this institution for the financial support. Special thanks are due to one of the anonymous reviewers for very helpful and constructive comments and suggestions that substantially improved the manuscript.
Open Access
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References
- Chatterjee S, Hadi AS (1988) Sensitivity analysis in linear regression. Wiley, New YorkCrossRefGoogle Scholar
- Golub GH, van Loan CF (1980) An analysis of the total least-squares problem. SIAM J Numer Anal 17(6):883–893CrossRefGoogle Scholar
- Knight NL, Wang J, Rizos C (2010) Generalised measures of reliability for multiple outliers. J Geod 84:625–635CrossRefGoogle Scholar
- Krakiwsky EJ (1975) A synthesis of recent advances in the method of least squares. Lecture note No. 42, Dept. of Engineering Surveying, UNB, Fredericton, CanadaGoogle Scholar
- Neitzel F (2010) Generalization of total least-squares on example of unweighted and weighted similarity transformation. J Geod 84:751–762CrossRefGoogle Scholar
- Prószyński W (2010) Another approach to reliability measures for systems with correlated observations. J Geod 84:547–556CrossRefGoogle Scholar
- Rao CR (1973) Linear statistical inference and its applications. Wiley, New YorkCrossRefGoogle Scholar
- Rao CR, Toutenburg H (1999) Linear models: least squares and alternatives, 2nd edn. Springer, BerlinGoogle Scholar
- Schaffrin B (1997) Reliability measures for correlated observations. J Eng Surv 123:126–137Google Scholar
- Schaffrin B, Lee I, Felus Y, Choi Y (2006) Total least-squares (TLS) for geodetic straight-line and plain adjustment. Boll Geod Sci Affini 65(3):141–168Google Scholar
- Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geod 82(7):415–421CrossRefGoogle Scholar
- Schaffrin B, Felus YA (2008) On the multivariate total least-squares approach to empirical coordinate transformations. J Geod 82:373–383CrossRefGoogle Scholar
- Schaffrin B, Snow K (2010) Total least-squares regularization of Tykhonov type and an ancient racetrack in Corinth. Linear Algebra Appl 432(8):2061–2076CrossRefGoogle Scholar
- Schaffrin B (2011) Errors-in-Variables for mobile mapping algorithms in the presence of outliers. In: Proceedings of the international symposium on mobile mapping technology, Kraków, PolandGoogle Scholar
- Schaffrin B, Uzun S (2011) On the reliability of Errors-in-Variables models. In: Proceedings of the 9thTartu conference on multivariate statistics and the 20th international workshop on matrices and statistics, Tartu, Estonia.Google Scholar
- Teunissen PJG (1996) Testing theory, an introduction. Delft University Press, DelftGoogle Scholar
- van Huffel S, Vandewalle J (1991) The total least-squares problem. Computational aspects and analysis. Society for Industrial and Applied Mathematics, PhiladelphiaGoogle Scholar