Impact of mathematical correlations: Exemplified in industrial applications

Michael Lösler; Cornelia Eschelbach; Rüdiger Lehmann

doi:10.1515/jag-2025-0040

Artikel Open Access

Impact of mathematical correlations

Exemplified in industrial applications

Michael Lösler , Cornelia Eschelbach und Rüdiger Lehmann

Veröffentlicht/Copyright: 6. Juni 2025

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Aus der Zeitschrift Journal of Applied Geodesy Band 19 Heft 3

Abstract

Close-range photogrammetry offers a wide range of industrial applications in the field of large volume metrology. The object coordinates are derived from captured images using a bundle adjustment. Even if the observations are assumed to be stochastically independent within the adjustment procedure, the estimated object coordinates are correlated. In subsequent applications such as surface fitting or deformation analysis, these estimated object coordinates are usually treated as independent and even identically distributed observations, neglecting stochastic information of the prior bundle adjustment. However, simplifications in stochastic modelling lead to misinterpretations of the adjustment results in terms of precision and reliability. Based on the estimates of a bundle adjustment, the impact of neglected correlations in subsequent applications is investigated. It is demonstrated that the chosen stochastic model affects the resulting standard deviations significantly. In surface fitting the derived standard deviations of datum-independent form parameters are two to five times overestimated when neglecting stochastic dependencies. Applying hypothesis tests to the estimates as part of quality assurance, for instance, lead to incorrect decisions, because the test statistics are biased. Analogously, in deformation analysis the risk of type I decision errors increases when in fact stable networks are falsely detected as deformed. This contribution indicates the advantage of the fully-populated dispersion matrix because the identified discrepancies cannot be compensated by introducing simple stochastic models, such as a diagonal variance matrix or a point-based block-diagonal matrix.

Keywords: correlation; dispersion; surface fitting; hypothesis test; close-range photogrammetry; industrial metrology

1 Introduction

Determining the positions and dimensions of certain objects is a common task in engineering surveying. Regardless of the instruments involved, a multi-station configuration is generally used, to capture the whole object redundantly. In order to obtain the observed coordinates in a consistent reference frame, the multi-station measurements are evaluated using a suitable network or bundle adjustment approach. The adjustment procedure yields the estimated coordinates of the observed object points and quantities for evaluating the precision and the reliability of the estimates.

Even if the original observations of a multi-station network are assumed to be identically distributed and even stochastically independent, the estimated coordinates are mutually correlated. As shown by Lehmann [1], p. 218], this kind of correlations is known as mathematical correlations and results from functional relationships that depend entirely or at least partially on the same parameters. In contrast to mathematical correlations, physical correlations result from unconsidered and unknown, non-estimable systematic errors, as discussed by Jäger et al. [2], p. 63]. However, considering such physical correlations requires assumptions that are difficult to validate. Due to the lack of proven knowledge, this kind of correlation is often neglected [1], p. 217f].

In most applications such as surface fitting or deformation analysis, the estimates of the prior adjustment procedure are only an intermediate result that does not correspond to the desired result of the final product. In order to achieve the desired results, these estimates are treated as incoming data in dedicated applications. It is common practice to consider only the parameters but not the related dispersion parameters, i.e., the estimated coordinates are treated, for instance, as independent and even identically distributed observations by eliminating their scattering. Apart from computational costs and memory aspects, the main reason for simplified stochastic modelling is the assumption of a negligible influence of correlations on the estimates as mentioned by Gotthardt [3]. However, the order of magnitude for small correlations remains unsettled. Reillyand et al. [4], p. 59] state that many analysts agree that correlation coefficients of 0.6 or less do not cause significant changes in a solution, and are negligible.

On the other hand, more recent case studies indicate a considerable impact of the stochastic model on the estimates. Based on simulated laser scan data, Zhao et al. [5] investigate the effect of correlations resulting from the conversion of polar measurements into Cartesian coordinates on B-spline surface approximation. Jurek et al. [6] model spatial correlations and analyse the influence of the resulting stochastic model on parameter estimation using a simulated laser scan of a plane. In the framework of reference point determination of laser telescopes used for satellite laser ranging, Lösler et al. [7], 8] investigate the impact of oversimplified stochastic models in photogrammetric applications. Kerekes [9], p. 95ff] derives an elementary error model for terrestrial laser scanning to generate fully-populated synthetic variance-covariance matrices, and evaluates the relevance of such a matrix in surface fitting exemplified by a sphere estimation. Similarly, Raschhofer et al. [10] apply an elementary error model to point clouds obtained from a panoramic scanner to investigate simplifications in stochastic modelling in the field of freeform determination. All last named studies emphasise that stripping (parts of) these mutual correlations within the stochastic modelling leads to misinterpretations of the adjustment results in terms of accuracy as well as reliability and yields biased test statistics.

Starting from the Gauss–Markov model commonly used in parameter estimation, Section 2 recapitulates essential mathematical fundamentals when evaluating neglected correlations independent of case and causation. In order to demonstrate these relations in practice, neglecting correlations is studied in industrial applications exemplified in surface fitting in Section 3. For that purpose, geometric objects such as a sphere and a paraboloid were observed by means of close-range photogrammetry. The object coordinates as well as the related fully-populated dispersion matrix were derived from captured images using a bundle adjustment. By stripping stochastic information off the fully-populated dispersion matrix, the effect of simplified stochastic models is illustrated. Due to the limited number of datasets, supplementary Monte-Carlo simulations (MCS) were carried out to investigate the effect of simplified stochastic modelling on hypothesis tests. The results of the analysis are discussed in Section 4. Finally, Section 5 concludes this investigation.

2 Mathematical background

The following subsections describe essential mathematical principles for evaluating the effects of the stochastic model on the estimates. All derivations are independent of the use case analysed.

2.1 Impact of stochastic model

Let x ̃ be the vector of true parameters connected to the vector of true observations y ̃ via

(1) E y − e = E y = y ̃ = A x ̃ ,

where A is the coefficient matrix of the linear transformation, e ∼ N 0 , Σ y denotes the vector of normal distributed observational errors, and D y = D e ≕ Σ y is the positive definite dispersion matrix defining the stochastic model in parameter estimation. The least-squares solution for x results from the well-known linear transformation

(2) x ̂ = J y = A T Σ y − 1 A − 1 A T Σ y − 1 y ,

where J ≔ A T Σ y − 1 A − 1 A T Σ y − 1 [11]. Therefrom it is easy to verify that the parameters to be estimated are unbiased, i.e.,

(3) E x ̂ = E J y − e = J E y = J A x ̃ = x ̃ .

Applying Steiner’s theorem [12], p. 37] yields the related dispersion, i.e.,

(4) D x ̂ = Σ x ̂ = E J e e T J T − E J e E J e = J D e J T = J Σ y J T = A T Σ y − 1 A − 1 ,

which is identical to the least-squares solution.

In order to investigate the effect of a modified stochastic model Σ ̄ y on x, Jäger et al. [2], p. 213ff] propose to decompose the true stochastic model defined by Σ_y as

(5) Σ y = Σ ̄ y + Δ Σ y ,

where matrix Σ ̄ y is still positive definite. According to the Woodbury matrix identity, the inverse is [13]

(6) Σ y − 1 = Σ ̄ y + Δ Σ y − 1 = Σ ̄ y − 1 − Σ ̄ y − 1 I + Δ Σ y Σ ̄ y − 1 − 1 Δ Σ y Σ ̄ y − 1 = Σ ̄ y − 1 + Δ Σ ̄ y − 1 .

Similar to Eq. (5), the matrix J can be expressed as

(7) J = J ̄ + Δ J ,

where J ̄ ≔ A T Σ ̄ y − 1 A − 1 A T Σ ̄ y − 1 . The linear transformation introducing J ̄ instead of J reads

(8) x ̄ ̂ = J ̄ y = J − Δ J y .

Analogous to Eq. (3), the expectation is defined as

(9) E x ̄ ̂ = J ̄ E y = J − Δ J E y = J − Δ J A x ̃ = I − Δ J A x ̃ .

Observing that the term

(10) Δ J A = J A − J ̄ A = 0

vanishes, the parameter vector x ̄ ̂ is also an unbiased estimation of x, i.e.,

(11) E x ̄ ̂ = E x ̂ = x ̃ .

Therefore, the parameters to be estimated are unbiased regardless of the stochastic model under consideration. Obviously, the estimates will differ in individual samples but x ̂ and x ̄ ̂ will scatter about the identical value x ̃ if the experiment is repeated several times. Hence, the expectation of the difference reads E x ̂ − x ̄ ̂ = 0 .

For the sake of completeness, the consideration of corrections for a discrete sample as derived by Wolf [14] is outside the scope of this study, as we are interested in the effects of neglected mathematical correlations and simplified stochastic modelling.

Analogous to Eq. (4), the corresponding dispersion of x ̄ ̂ results from the variance-covariance propagation and can be expressed in terms of ΔJ and ΔΣ_y, respectively. In mind with Eq. (7), the dispersion w.r.t. ΔJ is

(12) D x ̄ ̂ = Σ x ̄ ̂ = J ̄ Σ y J ̄ T = J − Δ J Σ y J − Δ J T = J Σ y J T + Δ J Σ y Δ J T = Σ x ̂ + Δ J Σ y Δ J T ,

where Eq. (10) was taken into account. On the other hand, substituting Eq. (5) yields an expression of the dispersion w.r.t. ΔΣ_y, which reads

(13) D x ̄ ̂ = Σ x ̄ ̂ = J ̄ Σ y J ̄ T = J ̄ Σ ̄ y + Δ Σ y J ̄ T = J ̄ Σ ̄ y J ̄ T + J ̄ Δ Σ y J ̄ T .

From Eq. (12) follows that the dispersion of x ̄ ̂ exceeds the dispersion of x ̂ by the term ΔJΣ_yΔJ^T, because this additional term is at least positive semi-definite. Moreover, this result is independent of the sign of ΔJ used in Eq. (7). Thus, introducing an insufficient stochastic model leads to unbiased parameters but does not provide the minimum (best) dispersion. In contrast to Eq. (12), which allows a direct comparison between the true dispersion of x ̂ and the true dispersion of x ̄ ̂ , Eq. (13) shows the discrepancy between the true dispersion of x ̄ ̂ and the distorted dispersion derived from the inverted normal equation matrix J ̄ Σ ̄ y J ̄ T = A T Σ ̄ y − 1 A − 1 , which is generally used to evaluate the precision of the parameters to be estimated. In case of a positive semi-definite matrix ΔΣ_y, the true dispersion Σ x ̄ ̂ exceeds J ̄ Σ ̄ y J ̄ T and Σ ̄ y reflects an overly optimistic stochastic model. In case of a negative semi-definite matrix, J ̄ Σ ̄ y J ̄ T overestimates the true dispersion, and Σ ̄ y indicates an overly pessimistic stochastic model. For indefinite matrices ΔΣ_y the dispersion of some parameters is underestimated, while the dispersion of other parameters is overestimated. Hereinafter, the distorted variance derived from the inverted normal equation matrix is denoted by ς ̂ 2 .

2.2 Impact on arbitrary functions

The sensitivity analysis of a non-detectable model error on an arbitrary function f x of the estimated parameters x ̂ introduced by Baarda [15] is a measure of the exterior reliability of geodetic networks. This analysis principle can readily be adapted to evaluate the effect of an incorrectly specified stochastic model. Let F be the Jacobian of f evaluated at x ̂ , the variance of the function value reads

(14) σ f 2 = F Σ x ̂ F T .

Moreover, the (first-order) distortion ∇f of the function value w.r.t. a discrete vector ∇ x = x ̂ − x ̄ ̂ is given by

(15) ∇ f = F ∇ x .

Applying the Cauchy–Schwarz inequality yields a measure of the maximum distortion of the (linearized) function as an upper boundary [12], p. 135],

(16) ∇ f = F ∇ x ≤ F Σ x ̂ F T ∇ x T Σ x ̂ − 1 ∇ x = σ f ∇ x T Σ x ̂ − 1 ∇ x = σ f δ ′ ,

where the Mahalanobis distance δ′ is known as sensitivity parameter [16]. From Eq. (15) and in mind with Eq. (11) it is easy to verify that the expectation of the distortion is E ∇ f = 0 . However, the corresponding variance σ ∇ f 2 is usually not zero. As shown by Jäger et al. [2], p. 218], a measure of the sensitivity of the variance is obtained by squaring Eq. (16) and applying the expectation operator. In mind with Eq. (12), the expectation of the (squared) sensitivity parameter reads

(17) δ 2 = E ∇ x T Σ x ̂ − 1 ∇ x = tr Σ x ̂ − 1 E ∇ x ∇ x T = tr Σ x ̂ − 1 Δ J Σ y Δ J T = tr Σ x ̂ − 1 Σ x ̄ ̂ − Σ x ̂ ,

and the maximum distortion is obtained from

(18) σ ∇ f 2 ≤ σ f 2 δ 2 .

Thus, the estimated variance σ f 2 is almost undisturbed if σ ∇ f 2 ≈ 0 , which implies δ² ≈ 0.

Analogously, the variance is potentially disturbed when using Σ ̄ y instead of Σ_y, cf. Eq. (13). Replacing δ² with

(19) δ ̄ 2 = tr J ̄ Σ ̄ y J ̄ T − 1 Σ x ̄ ̂ − J ̄ Σ ̄ y J ̄ T

in Eq. (18) yields

(20) σ ̄ ∇ f 2 ≤ σ f 2 δ ̄ 2 .

Equation (20) provides a measure of the sensitivity of the variance of an arbitrary function f x in most practical applications, where the inverted normal equation is equated with the parameter dispersion, and the estimates are introduced into further analysis procedures.

2.3 Impact on hypothesis testing

In addition to disturbed results, the interpretation and the analysis of the estimates are also impeded. For instance, the risk of a type I decision error increases in hypothesis tests. Hypothesis tests are often used in quality assurance to optimise production and reduce costs [17], p. 221ff]. Moreover, conformance tests are important in industrial metrology to evaluate predefined geometric specifications on workpieces [18]. In geodetic sciences, changes in shape or position of an object are evaluated using hypothesis tests [19], 20]. However, the test statistics are biased, if unsuitable stochastic models are used. Moreover, reasonable critical values cannot be derived from standard statistical functions.

Let y ∼ N y ̃ , Σ y be a realised sample, and ∇ = y − y ̃ be the vector of deviations. According to Wilks [21], a commonly applied test statistic to evaluate the null hypothesis H 0 : ∇ ∼ N 0 , Σ y is given by

(21) T = ∇ T Σ y − 1 ∇ ∼ χ v 2 | H 0 ,

where v = rg Σ y is the degrees of freedom. The expectation of the test statistic is E T = v and the dispersion reads D T = 2 v [22], p. 107].

Substituting Eq. (6) into (21) gives the biased test statistic

(22) T ̄ = ∇ T Σ ̄ y − 1 ∇ = ∇ T Σ y − Δ Σ y − 1 ∇ = ∇ T Σ y − 1 ∇ − ∇ T Δ Σ ̄ y − 1 ∇ = T − Δ T .

The expectation of this biased test statistic is defined as

(23) E T ̄ = E ∇ T Σ y − 1 ∇ − E ∇ T Δ Σ ̄ y − 1 ∇ = tr Σ y − 1 E ∇ ∇ T − tr Δ Σ ̄ y − 1 E ∇ ∇ T = v − tr Δ Σ ̄ y − 1 Σ y = tr Σ ̄ y − 1 Σ y .

Depending on the definiteness of Δ Σ ̄ y − 1 , the test statistic T ̄ is underestimated or overestimated, and the expectation of the additional bias term E Δ T is generally not zero and E T ̄ ≠ E T . An exception for an unbiased test statistic arises, for instance, when Σ ̄ y ≔ Σ D is a diagonal variance matrix, whose elements are identical to the diagonal of Σ_y. The dispersion of the biased test statistic is given by

(24) D T ̄ = D T − Δ T = 2 v − 4 tr Δ Σ ̄ y − 1 Σ y + 2 tr Δ Σ ̄ y − 1 Σ y Δ Σ ̄ y − 1 Σ y = 2 tr Σ ̄ y − 1 Σ y Σ ̄ y − 1 Σ y .

Even if the expectation of T ̄ is identical with T, the dispersion differs. Therefrom, T ̄ does not follow a χ v 2 distribution. Following the same line of reasoning worked out for the test statistic, it is easy to verify that the variance of the unit weight is also disturbed when the stochastic model is simplified. For a detailed investigation of the effect of incorrect weights on the estimated variance of the unit weight, the interested reader is referred to Xu [23].

Based on numerical investigations Kermarrec et al. [24], 25] identify the Γ distribution as a suitable first-order approximation of the distribution of T ̄ . The Γ k , θ distribution is defined by a shape k and a scale θ, and relates to the χ v 2 distribution via k = v/2 and θ = 2. The expectation of a Γ distributed value γ is E γ = k θ and the dispersion reads D γ = k θ 2 [22], p. 106f]. In mind with Eqs. (23), (24), the shape and the scale given by

(25) k = tr Σ ̄ y − 1 Σ y 2 2 tr Σ ̄ y − 1 Σ y Σ ̄ y − 1 Σ y ,

(26) θ = 2 tr Σ ̄ y − 1 Σ y Σ ̄ y − 1 Σ y tr Σ ̄ y − 1 Σ y ,

respectively, define an approximation of the test statistic’s distribution. In contrast to numerical approaches for estimating k and θ, Eqs. (25), (26) are known as method of matching moments [22], p. 206], and ensure that the expectation and the dispersion of the adopted Γ distribution are identical to the expectation and the dispersion of T ̄ .

3 Industrial applications

Surface fitting is a common application in today’s industrial metrology for assessing and optimising the quality of manufactured products [26], 27]. Test specimens such as planes, spheres, or cylinders are fitted by means of least-squares to evaluate the uncertainties of measuring instruments within the set of Geometrical Product Specification and Verification ISO standards [28]. In reverse engineering, surface fitting is used to reconstruct geometric properties such as dimension, shape, orientation, or location of geometric objects [29]. In order to analyse the deformations of objects, Kermarrec et al. [25] extracted small planes from a measured bridge structure and evaluated their changes under artificial load. Surface fitting is also used to investigate the impact of gravity-induced deformations of the receiving unit of radio telescopes [30], 31].

3.1 Surface fitting

Due to the wide range of applications for surface fitting, a reference sphere originally used in laser scanner applications and the dish of a antenna designed as a rotational symmetric paraboloid were chosen to investigate the impact of neglected correlations. The diameter of the sphere and the dish of the antenna is about 14.5 cm and 40 cm, respectively.

The functional model of a sphere is defined as [29]

(27) S : 0 = x − x 0 2 + y − y 0 2 + z − z 0 2 − r 2 ,

where the vector x 0 y 0 z 0 T is the centre of the sphere and x y z T are the coordinates of an arbitrary point lying on the surface.

The canonical equation of an upwardly open paraboloid having its apex at the origin reads [32]

(28) P : 0 = u 2 + v 2 4 F − w .

The surface point u v w T relates to the canonical form and is connected to the arbitrarily oriented reference frame of the measurement system by

(29) u v w = R x − x 0 y − y 0 z − z 0 ,

where vector x 0 y 0 z 0 T is the apex of the paraboloid and R is an orthogonal matrix defining the rotation sequence via the angles ω_x and ω_y by

R = cos ω y 0 sin ω y sin ω x ⁡ sin ω y cos ω x − cos ω y ⁡ sin ω x − cos ω x ⁡ sin ω y sin ω x cos ω x ⁡ cos ω y .

Both, the sphere and the paraboloid, consist of datum-dependent isometric parameters defining the position and – if applicable – the orientation of the object w.r.t. the measurement frame, and datum-independent surface parameters. According to the equivalence theorem of estimable and invariant quantities derived by Grafarend & Schaffrin [33] only the datum-independent surface parameters are examined in this contribution, i.e., the sphere radius r and the focal length F of the antenna.

The sphere and the dish of the antenna were observed by means of close-range photogrammetry using Hexagon’s Aicon DPA Industrial measurement system. The maximum permissible error (MPE) of the system, defined as length deviation between two signalised points, is specified by 15 µm + 15 μm m⁻¹ [34]. Following the recommendations given by Mason [35], multi-station configurations were realised enclosing the object surface. Fraser [36] pointes out the advantages of such a configuration when investigating several multi-station configurations for observing the main dish of a radio telescope photogrammetrically and indicated small point-to-point correlations.

The object under investigation was embedded in a local reference frame as shown in Figure 1. Four certified scale-bars were established to trace the scaleless photogrammetric measurements to the SI metre. The lengths of the two long scale-bars are about 1.4 m. The two short scale-bars are about 17.5 cm.

Figure 1:

Prepared 40 cm antenna embedded in a stable reference frame, realised with coded targets and equipped with two 1.4 m scale-bars (black) and two 17.5 cm scale-bars (yellow).

3.2 Case studies

The analysis is based on one basic dataset for each object under investigation. Derived datasets only differ in their characteristics, i.e., the consideration of the reference frame and the length of the scale-bars used. Introducing the reference frame improves the multi-station configuration by additional observations and, thus, increases the precision and the reliability of the interior and exterior orientation parameters as well as the derived object coordinates. Without the external reference frame, the position and orientation of the camera stations are only determined from the object frame. A dataset that contains the external reference frame is labelled by an R. If only the object frame is taken into account, the dataset is labelled by an O. To reduce scaling extrapolation errors, long scale-bars exceeding the object diameter are usually preferred [37], and are indicated by an L. Short scale-bars labelled by an S tag the more practical situation in large volume metrology, where the object dimensions correspond to or exceed the established scale-bars, and long scales cannot be realised for various reasons. The following datasets were investigated: RL, RS, OL, and OS.

The captured images of the multi-station configuration were pre-analysed by Aicon 3D Studio [38]. The software provides information about the correlations of the estimated interior orientation parameters of the camera, which is useful for evaluating the in situ calibration of the instrument, but does not determine the fully-populated dispersion matrix, which is mandatory for investigating the effects of simplified stochastic models in subsequent applications. In order to obtain the object coordinates and the fully-populated dispersion Σ_y, the final bundle adjustments were carried out as a free-network adjustment using the in-house developed software package JAiCov [39].

To investigate the effect of neglected mathematical correlations onto the estimates, three commonly used simplified stochastic models were derived from the fully-populated dispersion Σ_y of the bundle adjustment using Eq. (5). In the simplest form, the coordinates of the n points are considered to be independent and identically distributed with variance σ 0 2 , and a scaled identity matrix

(30) Σ I = σ 0 2 I

forms the stochastic model. This model is frequently used in surface fitting and corresponds to an orthogonal distance fit [40], 41]. Treating the observations as independent but with individual variances yields a diagonal dispersion matrix

(31) Σ D = diag σ x 1 2 σ y 1 2 … σ z n 2 .

Such a model is the most complex model when using software packages for pre-analysing, which do not provide the fully-populated dispersion but the standard deviations of the estimated coordinates. Maintaining the correlation of the coordinates of each point results in a block-diagonal matrix

(32) Σ B = blkdiag Σ x y z 1 … Σ x y z n ,

where each block is defined as

Σ x y z i = σ x i 2 σ x y i σ x z i σ x y i σ y i 2 σ y z i σ x z i σ y z i σ z i 2 .

Such a model is frequently used in laser scanner applications, when polar observations are converted into Cartesian coordinates [5].

4 Analysis results

The results of the analysed surfaces are presented and discussed in this section. Due to the limited number of datasets, the effects of the simplified stochastic modelling on hypothesis tests are studied by means of Monte-Carlo simulations.

4.1 Focal length of paraboloid

Table 1 summarises the results for the focal length F. Regardless of the dataset analysed, the consideration of an external reference frame has only a marginal impact on the estimates. Largest differences result from the scale-bars used. For datasets with short scale-bars, the standard deviation σ ̂ is almost three times as large as for corresponding datasets with long scale-bars.

Table 1:

Standard deviations of the focal length F for various stochastic models: ς ̂ and σ ̄ ̂ are derived from J ̄ Σ ̄ y J ̄ T and J ̄ Σ y J ̄ T , respectively. All quantities are given in µm. The unitless sensitivity coefficients are δ² and δ ̄ 2 .

Paraboloid		RL	RS	OL	OS
Σ _y	σ ̂	61.0	172.2	67.4	170.9
Σ _I	ς ̂	47.6	92.6	46.1	75.4
	σ ̄ ̂	61.7	172.4	68.2	171.1
	δ ²	0.09	0.08	0.08	0.08
	δ ̄ 2	6.42	8.33	4.31	35.51
Σ _D	ς ̂	54.6	97.7	52.9	75.7
	σ ̄ ̂	61.1	172.8	67.6	171.5
	δ ²	0.04	0.21	0.06	0.29
	δ ̄ 2	3.73	11.60	2.16	39.26
Σ _B	ς ̂	58.1	65.8	56.7	75.3
	σ ̄ ̂	61.0	172.5	67.5	171.1
	δ ²	0.00	0.09	0.03	0.13
	δ ̄ 2	2.45	19.49	0.92	34.85

The biased standard deviation ς ̂ , which is derived from the inverted normal equation, is overestimated and only about half the size of σ ̄ ̂ . However, the potential distortion could be significantly larger, as indicated by the sensitivity parameter δ ̄ 2 . In case of Σ_D, a potential additional distortion δ ̄ 2 of the variance of an arbitrary function of the estimated parameters becomes about 40 ς ̂ 2 . Applying Eq. (13) almost completely reduces the discrepancy between ς ̂ and σ ̂ . This finding is confirmed by δ², which is close to zero.

Simplifications of the stochastic model lead to overoptimistic results, especially if short scale-bars are used. Figure 2 compares the correlation coefficients of the datasets. The behaviour for RL and RS is similar for OL and OS, respectively. The upper histogram and the upper triangular matrix refer to datasets with short scale-bars, while the lower histogram and the lower triangular matrix refer to datasets with long scale-bars.

Figure 2:

Correlation coefficients ρ of the surface points of the paraboloid derived from Σ_y. The upper histogram and the upper triangular matrix refer to datasets with short scale-bars, while the lower histogram and the lower triangular matrix refer to datasets with long scale-bars. (a) RS versus RL. (b) OS versus OL.

In case of long scale-bars, the point-to-point correlations are close to zero. About 97 % and 98 % of the correlation coefficients are less than 0.1 for OL and RL, respectively. For that reason, the results obtained from the simplified models Σ_I, Σ_D and Σ_B are almost identical and close to the results derived from Σ_y. In case of short scale-bars, almost the whole range from −1 to +1 is covered by the determined correlation coefficients. About 79 % and 92 % of the correlation coefficients exceed 0.1 for OS and RS, respectively. Obviously, considering the variances and the correlations of the coordinates of each point but neglecting mutual point-to-point correlations as in Σ_B is by no means a suitable substitute for an appropriate uncertainty modelling.

4.2 Radius of sphere

The results for the sphere are summarised in Table 2. Similar to the paraboloid results, the standard deviation σ ̂ depends on the length of the scale-bars and is three to five times greater for datasets with short scale-bars. Moreover, simplifications in stochastic modelling lead to a disturbed dispersion ς ̂ , especially when short scale-bars are used. Note that ς ̂ derived from the inverted normal equations is greater than the true standard deviation σ ̂ for the RL dataset. However, applying Eq. (13) yields standard deviations σ ̄ ̂ close to σ ̂ . Introducing σ ̄ ̂ instead of σ ̂ has practically no effect on an arbitrary function as indicated by δ² ≈ 0. On the other hand, δ ̄ 2 reveals the large impact of an unconsidered correction. In particular, datasets with an external reference frame react sensitively to an oversimplified stochastic model such as Σ_I, Σ_D, or Σ_B.

Table 2:

Standard deviations of the sphere radius r for various stochastic models: ς ̂ and σ ̄ ̂ are derived from J ̄ Σ ̄ y J ̄ T and J ̄ Σ y J ̄ T , respectively. All quantities are given in µm. The unitless sensitivity coefficients are δ² and δ ̄ 2 .

Sphere		RL	RS	OL	OS
Σ _y	σ ̂	7.2	33.3	14.3	32.2
Σ _I	ς ̂	9.3	21.8	5.1	5.6
	σ ̄ ̂	7.4	33.3	14.7	32.4
	δ ²	0.08	0.05	0.44	0.43
	δ ̄ 2	45.29	60.78	8.42	36.03
Σ _D	ς ̂	9.3	17.3	5.4	6.1
	σ ̄ ̂	7.5	33.6	14.6	32.4
	δ ²	0.10	0.24	0.17	0.14
	δ ̄ 2	45.32	60.30	8.84	34.79
Σ _B	ς ̂	9.5	15.1	5.8	6.9
	σ ̄ ̂	7.4	33.8	14.6	32.4
	δ ²	0.09	0.32	0.17	0.11
	δ ̄ 2	42.61	54.83	7.13	25.79

It is interesting to note that the distortion δ ̄ 2 affects the variance of the radius almost completely for the OS dataset. Let the function of the estimated parameters simple be f r = r , the boundary value reads ς 2 1 + δ ̄ 2 ≈ 34 μm , which is close to the standard deviation given by σ ̄ ̂ ≈ 32 μm .

The correlation structure and distribution for the datasets under investigation are depicted in Figure 3. Similar to the paraboloid results shown in Figure 2, the correlations decrease when the scale-bars exceed the object dimension. However, the consideration of the reference frame surrounding the sphere causes stronger correlations than the length of the scale-bars. In case of long scale-bars, only 44 % of the correlations are less than 0.1 if the reference frame is taken into account, while 78 % are less than 0.1 when the reference frame is omitted. The same applies to datasets with short scale-bars. Here, 95 % of the correlations exceed 0.1 if the reference frame is considered. Excluding the reference frame decreases the number of correlations greater than 0.1 to 59 %.

Figure 3:

Correlations ρ of the surface points of the sphere derived from Σ_y. The upper histogram and the upper triangular matrix refer to datasets with short scale-bars, while the lower histogram and the lower triangular matrix refer to datasets with long scale-bars. (a) RS versus RL. (b) OS versus OL.

Figures 2 and 3 show that the size and distribution of mutual correlations are not comparable, even if the application is almost identical. The mathematical correlations depend on various factors such as the selected datum and the configuration of the network and can only be predicted to a certain degree. Thus, the only serious recommendation is to consider the fully-populated dispersion instead of evaluating different configurations for specific tasks to be solved.

4.3 Hypothesis testing

In order to illustrate the effects of neglected correlations in hypothesis testing, an overall congruence test of the surface points of the antenna was evaluated by means of Monte-Carlo simulations following the procedure proposed by Lehmann & Lösler [42]. For that purpose, the surface points y obtained from the bundle adjustment were simulated m = 500,000 times w.r.t. Σ_y.

According to Eq. (21) the χ v 2 test statistic of the overall congruence test reads

T = y i − y T Σ y + y i − y .

Due to the free-network adjustment, the dispersion Σ_y is rank deficient, and the equivalent decomposition given in Eq. (6) becomes [13]

Σ y + = Σ ̄ y + Δ Σ y + = Σ ̄ y + − Σ ̄ y + I + Δ Σ y Σ ̄ y + − 1 Δ Σ y Σ ̄ y + .

Considering the simplified stochastic model Σ ̄ y = Σ I , Σ D , Σ B yields the biased test statistic

T ̄ = y i − y T Σ ̄ y + y i − y .

Table 3 compares the 5 % quantiles c derived from the MCS, and taken from the inverse cumulative distribution function (CDF) of the χ² and the adapted Γ distribution for various stochastic models and configurations. The χ² quantile depends only on the degrees of freedom v = 282 and reads c χ 2 = 322.17 . It is identical for all investigated datasets. The type I decision error α derived by the MCS indicates the probability of a wrongly rejected null hypothesis.

Table 3:

5 % quantiles c derived from the MCS and taken from the inverse cumulative distribution function of the χ² and the adapted Γ distribution. The type I decision error derived by the MCS is denoted by α.

		RL	RS	OL	OS
Σ _I	c _MCS	333.35	882.51	331.04	793.35
	c _Γ	332.54	875.90	330.45	785.01
	α _Γ	0.052	0.051	0.052	0.051
	c χ 2	322.17	322.17	322.17	322.17
	α χ 2	0.091	0.275	0.084	0.270
Σ _D	c _MCS	330.07	751.21	329.38	633.78
	c _Γ	329.07	740.31	328.68	619.63
	α _Γ	0.053	0.052	0.052	0.054
	c χ 2	322.17	322.17	322.17	322.17
	α χ 2	0.079	0.265	0.076	0.251
Σ _B	c _MCS	330.54	513.96	329.37	501.01
	c _Γ	329.57	496.93	328.47	485.09
	α _Γ	0.053	0.056	0.053	0.057
	c χ 2	322.17	322.17	322.17	322.17
	α χ 2	0.079	0.226	0.075	0.221

Regardless of the simplification of the stochastic model, the χ² quantile is not a reliable quantity. Especially for datasets with short scale-bars, the type I decision error α χ 2 is four to five times greater than desired, and about 25 % of the simulated samples are wrongly rejected. The parameters of the adapted Γ distribution were derived from Eqs. (25), (26) and the corresponding quantiles are close to the numerically evaluated quantiles of the MCS, as shown in Figure 4. The differences are less than 1 %.

Figure 4:

Cumulative distribution functions (CDF) and distribution quantiles for the stochastic model Σ_B derived from the OS dataset of the paraboloid.

Figure 4 depicts for the OS dataset the cumulative distribution functions of T ̄ for the stochastic model Σ_B. Particularly for small type I decision errors, the Γ distribution provides an appropriate approximation of the true distribution, and is preferable to χ². However, irrespective of a possible approximation of the underlying distribution, it is more advisable to specify a reliable stochastic model.

5 Conclusions

Parameter estimation provides a wide part of applications and is part of the daily business in industrial metrology and surveying engineering, as desired quantities are rarely observed directly and non-redundantly. The adjustment procedure yields the estimated parameters and quantities for evaluating the accuracy and the reliability of the estimates. For that purpose, an appropriate functional relation, which transforms the observations to the parameter space, as well as a reliable stochastic model must be specified. However, the impact of correlations in adjustment calculus is often underestimated in particular in subsequent applications where results of a prior adjustment are treated as incoming data.

Based on an introduced sensitivity measure, the effect of a simplified stochastic model was studied for the case of neglected correlations as well as for the case of a subsequent correction of the stochastic model. Both cases refer to the practical case of using applications that do not support a complex stochastic model. The analysis of photogrammetric data used in the framework of surface fitting demonstrated the strong impact of the stochastic model onto the estimates. The standard deviations obtained from the inverted system of normal equations, which is commonly used to evaluate the reliability of the estimated parameters, are clearly disturbed when neglecting stochastic dependencies. Therefore, a subsequent correction is advisable as the remaining deviations are practically insignificant.

Simplifications in stochastic modelling affect the derived results and lead to misinterpretations and incorrect conclusions. In hypothesis testing, test statistics are biased and do not follow the expected distribution. Based on the empirically evaluated distribution of the biased test statistic, the type I decision error was studied using quantiles of the χ² distribution and an adapted Γ distribution. Regardless of the simplification of the stochastic model under investigation, the χ² quantile did not reflect a reliable quantity. In contrast to the χ² distribution, the quantiles of the adapted Γ distribution were comparable with the empirical results of the Monte-Carlo simulation. Particularly for small type I decision errors, the Γ distribution provides an appropriate approximation of the true distribution and is preferable to χ².

Subsequent corrections, optimisation of the measurement configuration, or adaption of limit values only treat the symptoms of incorrect stochastic modelling. From this point of view, it is advisable to derive reliable stochastic models and to develop efficient algorithms. Even if the effects of simplifications may be negligible in some individual cases, we strongly advise against their use.

Even though this contribution focused on neglected mathematical correlations, all derived quantities are fully valid for physical correlations, too. However, considering physical correlations requires assumptions that are difficult to validate. Due to the lack of proven knowledge, this kind of correlation is often neglected. In contrast to physical correlations, mathematical correlations result from the functional model, which transforms the observations to the parameter space. Neglecting this kind of correlation is a conscious decision on the part of the user.

Corresponding author: Michael Lösler, Faculty 1: Architecture, Civil Engineering, Geomatics, Laboratory for Industrial Metrology, Frankfurt University of Applied Sciences, D-60318 Frankfurt am Main, Germany, E-mail: michael.loesler@fb1.fra-uas.de

Acknowledgments

The first author would like to sincerely thank Frank Neitzel for his personal encouragement and unwavering support over the last few years and dedicates this article to him.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The authors state no conflict of interest.
Research funding: None declared.
Data availability: The raw data can be obtained on request from the corresponding author.

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Received: 2025-03-04

Accepted: 2025-03-16

Published Online: 2025-06-06

Published in Print: 2025-07-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/jag-2025-0040

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correlation; dispersion; surface fitting; hypothesis test; close-range photogrammetry; industrial metrology

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