Bias in least-squares adjustment of implicit functional models

Abstract

To evaluate the benefit of a measurement procedure onto the estimated parameters, the dispersion of the parameters is usually used. To draw objective conclusions, unbiased or at least almost unbiased estimates are required. In geodesy, most of the functional relations are nonlinear but the statistical properties of the estimates are usually obtained by a linearised substitute-problem. Since the statistical properties of linear models cannot be passed to the nonlinear case, the estimates are biased. In this contribution, the bias of the parameters as well as the bias of the dispersion in nonlinear implicit models is investigated, using a second-order Taylor expansion. Nonlinear implicit models are general models and are used, for instance, in the framework of surface-fitting or coordinate transformation, which considers errors for the coordinates in source and target system. The bias is introduced as a further indicator to validate the benefit of an adapted measurement process using more precise measuring instruments. Since some parametrisations yield an ill-posed problem, also the case of a singular equation system is investigated. To demonstrate the second-order effect onto the estimates, a best-fitting plane is adjusted under varying configurations. Such a configuration is recommended in evaluating uncertainties of optical 3D measuring systems, e.g. in the framework of the VDI/VDE 2634 guideline. The estimated bias is used as an indicator whether a large number of poor observations provides better results than a small but precise sample.

Full text

Bias in Least-Squares Adjustment of Implicit Functional Models
M. L¨oslera, R. Lehmannb, F. Neitzelcand C. Eschelbacha
aFrankfurt University of Applied Sciences, Faculty 1: Architecture - Civil Engineering -
Geomatics, Laboratory for Industrial Metrology, Nibelungenplatz 1, 60318 Frankfurt am
Main, Germany
bUniversity of Applied Sciences Dresden, Faculty of Spatial Information, Friedrich-List-Platz
1, 01069 Dresden, Germany
cTechnische Universit¨at Berlin, Institute of Geodesy and Geoinformation Science, Chair of
Geodesy and Adjustment Theory, Straße des 17. Juni 135, 10623 Berlin, Germany
ACCEPTED MANUSCRIPT
This is a preprint of an accepted manuscript of an article published by Taylor &
Francis Group in Survey Review on 29/01/2020, available online: https://doi.org/
10.1080/00396265.2020.1715680.
ABSTRACT
To evaluate the benefit of a measurement procedure onto the estimated parameters,
the dispersion of the parameters is usually used. To draw ob jective conclusions,
unbiased or at least almost unbiased estimates are required. In geodesy, most of the
functional relations are nonlinear but the statistical properties of the estimates are
usually obtained by a linearized substitute-problem. Since the statistical properties
of linear models cannot be passed to the nonlinear case, the estimates are biased.
In this contribution, the bias of the parameters as well as the bias of the dis-
persion in nonlinear implicit models is investigated, using a second-order Taylor
expansion. Nonlinear implicit models are general models and are used, for instance,
in the framework of surface-fitting or coordinate transformation, which considers
errors for the coordinates in source and target system. The bias is introduced as a
further indicator to validate the benefit of an adapted measurement process using
more precise measuring instruments. Since some parametrizations yield an ill-posed
problem, also the case of a singular equation system is investigated.
To demonstrate the second-order effect onto the estimates, a best-fitting plane
is adjusted under varying configurations. Such a configuration is recommended in
evaluating uncertainties of optical 3D measuring systems, e. g., in the framework of
the VDI/VDE 2634 guideline. The estimated bias is used as an indicator whether a
large number of poor observations provides better results than a small but precise
sample.
KEYWORDS
Bias; Nonlinear model; Taylor expansion; Least-squares; Surface-fitting
1. Introduction
The estimation of parameters using least-squares techniques has a wider range of ap-
plications, not only in geodesy. The used techniques such as Gauß-Markov model were
derived for linear functional relations. However, most of the functional relations in
geodesy are nonlinear. By applying a linear solver to a nonlinear problem, linearized
CONTACT M. L¨osler Email: michael.loesler@fb1.fra-uas.de

Bias in Least-Squares Adjustment of Implicit Functional Models
substitute-problems are solved iteratively instead of the original underlying nonlin-
ear problem. Even if a non-iterative solver might exist, the statistical properties of
the estimates are usually obtained by a linearized substitute-problem, derived by a
first-order Taylor expansion (TE1). Consequently, the statistical properties of the es-
timates are only obtained for the linear substituted problem and cannot be passed
to the nonlinear problem, as emphasised by Teunissen and Knickmeyer (1988). If the
nonlinearity is moderate, the bias of the estimated parameters and the dispersion may
be neglectable, and, thus, the derived substituted results are sufficient approximations
of the nonlinear problem. In most applications, the bias is assumed to be small and
the substituted results are equated with the nonlinear results without proof. However,
Xue and Yang (2017) studied the linearization error in short-distance positioning ap-
plications, such as indoor positioning or laser scanning, and emphasise that the bias
in nonlinear least-squares can become significant. As shown by Lehmann and L¨osler
(2018), the nonlinearity of the least-squares problem depends on the functional model,
but the stochastic model controls the impact of the nonlinearity onto the estimates.
An illustrative example to demonstrate the nonlinearity of the function and the
influence of the stochastic model onto the estimated parameters is the non-iterative
problem of converting polar coordinates into Cartesian coordinates, see Fig. 1. If the
distance sis assumed to be error-free, i. e., s= ˜s, but the angle φis a normally-
distributed quantity, the Cartesian coordinates depend only on the angle by x=f(φ).
By providing repeated measurements of the angle φl, the corresponding Cartesian
Figure 1. Nonlinear conversion of polar observations into Cartesian coordinates. Whereas the distance sis
assumed to be error-free, the angle measurement is a random quantity. Converting repeated angle measurements
φlyields the corresponding Cartesian Coordinates, i. e., xl=f(φl), which are denoted by small, light-grey
dots forming a circle arc. The red dot is obtained by ˆxTE1 =f(E {φ}). The expectation ˆxMCM = E {f(φ)}is
symbolised by a black triangle and approximated by a Monte-Carlo simulation. In contrast to ˆxTE1 , which lies
on the circle arc, ˆxMCM lies inside the circle. Error ellipses indicate the dispersion of the Cartesian coordinates
ˆxTE1 and ˆxMCM obtained by a first-order Taylor expansion and the Monte-Carlo method, respectively.
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Bias in Least-Squares Adjustment of Implicit Functional Models
coordinates xlform a circle arc with radius being equal to s. Obviously, the arc-
length depends only on the dispersion of φ. Averaging the coordinates yields an almost
unbiased estimator ˆx, if the sample size tends to infinity. Since ˆx is derived by averaging
points xllying on a circle arc, ˆx must lie inside the circle, i. e., ˆxTˆx <˜s2. Moreover,
the dispersion derived by a first-order Taylor expansion is underestimated. Since sis
assumed to be error-free, the error ellipse derived by the first-order Taylor expansion
becomes a straight line, lying orthogonal to the line of sight. It should be noted that if
sis also a normally-distributed measured quantity, the dispersion of the coordinates
derived by the first-order Taylor expansion becomes an ellipse, too. However, the semi-
axes of such a derived ellipse are smaller w.r. t. the ellipse derived by the Monte-Carlo
simulation. Converting polar coordinates into Cartesian coordinates and vice versa
yields biased parameters, if the nonlinearity of fis neglected (e. g. Manolakis 2011,
L¨osler et al. 2016). More generally, the linear property of the expectation, i. e.,
E{f(y)}=f(E {y}) (1)
is in general invalid for nonlinear functions (Carlton and Devore 2017, p. 87).
According to the International Vocabulary of Metrology (VIM), a measurement
result is composed of a measured quantity value and a measurement uncertainty
(JCGM200 2012, p. 19). The measured quantity value is the estimate of the value of
the measurand and derived by, e. g., measurement or calculation processes (JCGM200
2012, p. 19). The measurement uncertainty characterises the dispersion of the quan-
tity value (JCGM200 2012, p. 25). In terms of least-squares adjustments, observations
as well as estimated parameters are measured quantities, and the dispersion of such
quantities is expressed by the second moment, i. e., the related dispersion matrices also
known as variance-covariance matrices. In general, such dispersion of the estimated
parameters is used to evaluate the intended measurement process, e. g., in the frame-
work of surface-fitting, but the impact of the nonlinearity on the parameters is not
scrutinised. From this point of view, the estimated bias provides a further indicator to
validate, e.g., the benefit of an adapted measurement process including more precise
measuring instruments.
By approximating the nonlinear function by a second-order Taylor expansion (TE2),
B¨ahr (1988) analysed the bias of nonlinear functions in non-redundant models, i. e.,
variance-covariance propagation. Second-order improved estimates of a multivariate
least-squares adjustment having an explicit functional model, i. e.,
y=f(x) (2)
was derived in detail by Box (1971). In geodesy, similar expressions can be found in,
e. g., the contributions by Teunissen and Knickmeyer (1988), Teunissen (1989a,b,c,
1990), Xu and Grafarend (1996) or recently in, e. g., Wang and Zhao (2019).
By introducing direct observations as a further part of the parameter vector, these
derivations can also be applied to more general models. In the framework of planar
coordinate transformations, Teunissen and Knickmeyer (1988) estimated the bias of
the parameters of a similarity transformation by introducing the coordinates in the
source system as further observations. Wang and Zhao (2019) introduced direct ob-
servations to transform the implicit functional model of a straight line into its explicit
representation and studied the bias of the estimated parameters derived by the total
least-squares approach. By explicitly introducing direct observations, each observation
is expressed as self-contained function of the unknown parameters and the implicit
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Bias in Least-Squares Adjustment of Implicit Functional Models
more general model is transformed to an equivalent Gauß-Markov model, which re-
quires an explicit functional model. Mihajlovi´c and Cvijetinovi´c (2016) exemplified
the equivalence of both approaches in the framework of coordinate transformations,
wherein the coordinates are weighted in source and target systems.
In this investigation, the bias of the parameter is derived by a second-order Taylor
expansion. A third-order Taylor expansion (TE3) for variance-covariance propagation
in nonlinear explicit models was derived by Jeudy (1988). Even if TE3 may provide
improved adjustment results in the sense of a smaller bias, the computational effort
increases significantly. Thus, the possible accuracy benefit of higher-order Taylor ex-
pansions have to be balanced with respect to the computational costs. Therefore, the
Taylor expansion is restricted to the second order in this investigation. Due to the
wide range of applications in geodesy, and in contrast to prior investigations done by
(e. g. Box 1971, Teunissen and Knickmeyer 1988, Wang and Zhao 2019), who derived
the bias of the estimates for an explicit representation of the functional model, we will
focus on the implicit representation f(x,e) that yields the most general formulation of
least-squares adjustment problems. A common approach replaces the original nonlinear
functional model by a sequence of linearized equations, known as the Gauss-Helmert
model (e. g. Neitzel 2010). Instead of introducing direct observations, which trans-
form the most general case of the least-squares adjustment with implicit functional
model into an explicit representation, second-order improved estimates are derived for
the most general case of the least-squares adjustment with implicit functional model.
Also the case of an ill-posed problem is investigated, to provide a general solution for
deriving the bias of the estimates.
According to Julier et al. (2000), two strategies for investigating on nonlinear prob-
lems exist. Whereas the first strategy approximates the probability distribution, the
second strategy transforms the original underlying nonlinear problem into a substitute-
problem. The Unscented Transformation derived by Julier and Uhlmann (1997) is a
well-known method that approximates the probability distribution and is related to
the first strategy. For details on the Unscented Transformation, the interested reader is
referred to (e. g. Wan and van der Merwe 2000, Zhao et al. 2008, Wang and Zhao 2017,
2018). In this investigation, the second strategy is applied, and the original nonlinear
functional model is transformed to and analysed by its Taylor expansion.
In Section 2, the approach of Box (1971) is recapitulated, which estimates the bias
in nonlinear explicit functional models. The bias of the parameters as well as the bias
of the dispersion in nonlinear implicit functional models are derived by a second-order
Taylor expansion in Sec. 3. In Section 4, the estimation of the bias is exemplified in
the framework of surface-fitting. The bias of the parameters as well as the bias of
the dispersion of the implicit, nonlinear function of a best-fitting plane are estimated,
due to its wide range of applications for surface-fitting. Quadric surfaces like planes,
spheres or cylinders are frequently used to evaluate the uncertainty in measurement
of optical 3D instruments, e. g. in the framework of the VDI/VDE-2634 (2012, 2008)
guideline. For instance, Wujanz et al. (2017) verified their intensity-based stochastic
model for laser scanners using planes. Koch (2010) repeatedly measured a plane with
a terrestrial laser scanner to evaluate systematic effects, introducing a fully populated
dispersion matrix. A similar measurement configuration was used by Heinz et al. (2019)
to evaluate the benefit of a high discretisation but a poor single point measurement
against a high-precision single point measurement but a reduced number of observed
points. We will revisit the same topic, using the bias of the estimates as indicator.
The parameters of the best-fitting plane are estimated considering the coordinates
of the points as observations. Usually, such points are derived by indirect methods e. g.
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Bias in Least-Squares Adjustment of Implicit Functional Models
polar measurements or photogrammetric survey systems and generally depend on each
other, expressed by a fully populated dispersion matrix of the observational errors. In
some applications, the dispersion matrix is simplified by sparse matrices, e. g., a block
diagonal matrix, a diagonal matrix or a unified scaled identity matrix. However, since
we address the general case, a Taylor-Karman structured criterion matrix is introduced
(Grafarend and Schaffrin 1979) to define the a-priori dispersion of the observational
errors. To validate the results derived by the TE2 approach, the Monte-Carlo method
(MCM) is used, which provides asymptotically unbiased estimates.
The advantage of the MCM is that partial derivatives are not required for deriving
statistical moments (JCGM102 2011). Since the MCM estimates the parameters by
approximating the probability distribution of the nonlinear problem using the law of
large numbers (Julier et al. 2000), the computational costs of the sampling can become
expensive, if the nonlinear model is complex or the number of observations is large
(e. g. Teunissen 1990, Alkhatib and Schuh 2007, Schweitzer and Schwieger 2015). On
the one hand, in resent years, the computational power continually increases, which
allows for efficient processing of large sample sizes, but on the other hand, Wirth’s law
restricts the benefit, because the complexity of applications may be increasing faster
than the computational power (Wirth 1995). Due to this disadvantage, the MCM
cannot be fully recommended without knowing the application.
Finally, Sect. 5 concludes the paper.
2. Bias in explicit functional model
According to Box (1971), the derivations of the bias of the parameters as well as the
bias of the dispersion in explicit functional models are recapitulated and transferred
to implicit function models in Sec. 3. Similar expressions can also be found in e. g.
Teunissen and Knickmeyer (1988), Teunissen (1989a,b,c, 1990), Xu and Grafarend
(1996) or recently in the contribution by Wang and Zhao (2019).
Considering the nonlinear model
y−˜e =f(˜x) (3)
where the vector of the true parameters ˜x is mapped onto the observation space of yby
the vector-valued function f, the vector of true errors distributed as ˜e ∼N0,W−1
e
and Weis a known positive definite weight matrix. Estimating the parameters ˆx =
˜x +exin the least-squares sense yields the loss function (e. g. Koch 2007, p. 65)
Ω= (y−f(ˆx))TWe(y−f(ˆx)) = min .(4)
If fis expanded by its second-order Taylor series (e. g. Mana and Pennecchi 2007),
i. e.,
f(ˆx) = f(˜x) + Jxex+1
2eT
xHx,iexi.(5)
the estimated observational errors can be written as
ˆe =y−f(ˆx) = ˜e −Jxex−1
2eT
xHx,iexi.(6)
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Bias in Least-Squares Adjustment of Implicit Functional Models
The matrix Jxdenotes the Jacobian of fevaluated at the parameter vector ˜x, whose
(i, j)-th element reads ∂fi
∂ˆxj, and Hx,i is the Hessian of the i-th function in fevaluated
at ˜x, whose (j, k)-th element reads ∂2fi
∂ˆxj∂ˆxk. To denote a vector where the i-th element
is vi, the notation [vi]iis used. Similarly, the notation [Mij ]ij represents a matrix in
which element (i, j) is Mij .
Every true observational error ˜e generates a certain true estimation error ex. With-
out explicitly specifying this functional relationship, we expand it by a Taylor series.
With restriction to the second order, exbecomes
ex=J˜e +1
2˜eTH,i˜ei(7)
where Jand Hare the Jacobian and Hessian of this relationship, respectively. Note
that there cannot be a zero order term in Eq. 7 because ˜e =0yields ˆ
x=˜
x.
Taking the expectation E {·} in Eq. 7 into account yields for the linear term (e. g.
Teunissen 2003, p. 47)
E{J˜e}=JE{˜e}=0(8)
and for the quadratic term (e. g. Teunissen 2003, p. 49)
E˜eTH,i˜e = tr H,iW−1
e(9)
where tr(·) denotes the trace of the matrix (e. g. Bronshtein et al. 2007, p. 252). In
mind with Eq. 7, the expected value, i. e., the bias of the parameters, reads (Box 1971,
Wang and Zhao 2019)
E{ex}=1
2tr H,iW−1
ei.(10)
Since the observational errors are assumed to be normally-distributed, the second-
order improved dispersion becomes (e. g. Mana and Pennecchi 2007, Wang and Zhao
2019)
EexeT
x=JW−1
eJT
+1
2tr H,iW−1
eH,j W−1
eij .(11)
The aim is now to find expressions for the presently unknown matrices Jand H.
The necessary condition of the minimum in Eq. 4 yields
ˆ
JT
xWe(y−f(ˆx)) = 0(12)
where ˆ
Jxis evaluated at ˆx (Box 1971), i. e.,
ˆ
Jx=Jx+eT
xHx,ii.(13)
With Eqs. 6, 7, 12 in mind, the Jacobian Jis obtained neglecting the quadratic term
by
JT
xWe(˜e −JxJ˜e) = 0,
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Bias in Least-Squares Adjustment of Implicit Functional Models
which must hold for every ˜e and, consequently,
J=JT
xWeJx−1JT
xWe,(14)
where well-posedness of the least-squares problem is assumed. Taking the expectation
operator into account, after some algebraic effort, the quadratic term yields (Box 1971)
JT
xWe−1
2JxE˜eTH,i˜ei−1
2E˜eTJT
Hx,iJ˜ei=0
and with Eq. 14 in mind, one obtains
tr H,iW−1
ei=−Jtr JW−1
eJT
Hx,ii.(15)
Having Eqs. 14, 15, the bias of the parameters and the related dispersion are given by
E{ex}=−1
2Jtr JWe−1JT
Hx,ii=−1
2J[tr (ΣxHx,i)]i,(16a)
EexeT
x=Σx+1
2J[tr (ΣxHx,iΣxHx,j )]ij JT
(16b)
where Σx=JW−1
eJT
=JT
xWeJx−1is the first-order dispersion matrix of ˆx.
As shown by Wang and Zhao (2019), the Hessian Hin Eq. 7 can be approximated
by
H,i =X
j(J)ij JT
Hx,j J,
where (J)ij is (i, j)-th element in J. However, to estimate the bias of the parameters
and the bias of the dispersion, His not necessary in explicit representation.
It should be noted that Eq. 16 is evaluated at the true values, which are usually un-
known. For practical application, the true values have to be replaced by the estimated
values. In case of a linear function f, i. e., Hx=0, the estimated observational errors
ˆe =y−Jxˆx, the estimated parameters ˆx =JT
yas well as the resulting dispersion
matrix Σx=JW−1
eJT
of the Gauß-Markov model are unbiased (Koch 2007, p. 94).
If fis nonlinear, i. e., Hx6=0, Eq. 16 provides a second-order improved solution (e. g.
Box 1971, Wang and Zhao 2019).
3. Bias in implicit functional model
Having an explicit representation of the nonlinear function, i.e., each observation
is expressed as self-contained function of the unknown parameters, the bias of the
parameters as well as the bias of the dispersion can be estimated by Eq. 16. However,
in some applications like e. g. coordinate transformation with random errors in the
coordinates in the source and target frame, respectively, or curve and surface-fitting,
most of the nonlinear functions are given by an implicit representation. For that reason,
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Bias in Least-Squares Adjustment of Implicit Functional Models
the derivation of the bias in the explicit nonlinear functional model given in Sec. 2 is
transferred to the most general case, in this section.
An implicit representation of a nonlinear differentiable function is considered by
fc(˜u) = fc(˜x,˜e) = 0.(17)
Such a problem requires an adjustment with condition equations between the unknown
parameters and the observational errors. As shown in Eq. 16, the bias of the parameters
as well as the bias of the dispersion are derived from the Jacobian Jand the Hessian
H. Moreover, Halso depends on Jvia Eq. 15. For that reason, we focus on the
derivation of J. Since, each observation cannot be expressed as self-contained function,
the parameter vector is extended by the observational errors and denoted by uT=
xTeT.
To estimate the observational errors, the linear observation equations
fe(˜e) = y−˜y (18)
are introduced and the condition equations fcare considered as further pseudo-
observation equations, having weights Wcclose to infinity. According to Eq. 14, the
Jacobian of this model is obtained by
J=JT
uWfJu−1JT
uWf,(19)
where
Ju=0 I
JxJe
is the Jacobian of the extended functional model fT=fT
efT
cevaluated at the
extended parameters ˜u and the extended weight matrix is given by
Wf=We0
0 Wc.
The Jacobian Jtransforms the observational residuals into the parameter space
subjected to the condition equations by a linear transformation, i. e.,
eu=J˜e
fc.(20)
According to Eqs. 7, 15, a second-order improvement is obtained by introducing the
Hessian Huof the extended functional model fevaluated at ˜u, i. e.,
eu=J˜e
fc−1
2J[tr (ΣuHu,i)]i,(21)
where the first-order dispersion Σuis given by
Σu=JW−1
e0
0 0 JT
,(22)
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Bias in Least-Squares Adjustment of Implicit Functional Models
because W−1
ctends to zero.
Due to febeing linear, the related Hessian matrices become zeros and second-
order derivatives only exist for the condition equations fc. Even if the second-order
derivatives are partially zero, xand eare usually biased because Jis dense. Inserting
Σu,Huand Jinto Eq. 16 yields the bias and the related dispersion, introduced by
Box (1971).
Constrained adjustment problems with implicit functional models are rarely solved
by assuming the constraint equations as further pseudo-observations having weights
close to infinity. A more common way in numerical optimisations is to derive the
parameters by solving the nonlinear constrained problem (e. g. Nocedal and Wright
2006, Ch. 12)
min Ω(23a)
subject to
fc(ˆx,ˆe) = 0,(23b)
by introducing the Lagrangian function. In geodesy, such a solver is known as Gauß-
Helmert model, which is described in detail by e. g. Neitzel (2010). The Lagrangian
function combines the loss function Ωand the constraint function fcby
L=1
2Ω−λTfc,(24)
where λis the vector of Lagrangian multipliers.
The well-known necessary condition for a solution of the optimisation problem, also
known as Karush-Kuhn-Trucker (KKT) conditions, reads (e. g. Nocedal and Wright
2006, Ch. 12.3)


Weˆe −JT
eˆ
λ
−JT
xˆ
λ
fc(˜x,˜e) + Jxex+Je(ˆe −˜e)
=0,(25)
where second and higher-order terms are neglected.
By applying the Schur complement method (Nocedal and Wright 2006, p. 456),
the unknowns ˆx,ˆe and ˆ
λcan be expressed as a linear function of the observational
residuals ˜e. Since we are only interested in u, the linear transformation is obtained by
ˆx
ˆe =˜x
0+J(fc−Je˜e),(26)
where the Jacobian reads
J=ΣxJT
xW
W−1
eJT
eΣλ,(27)
the first-order dispersion matrices of xand λare given by
Σx=JT
xWJx−1,(28a)
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Bias in Least-Squares Adjustment of Implicit Functional Models
Σλ=W−WJxΣxJT
xW,(28b)
respectively, and W=JeW−1
eJT
e−1. In Eq. 26, the term (fc−Je˜e) is known as
vector of misclosures (e. g. Neitzel 2010).
Rearranging Eq. 27 yields the desired Jacobian given in Eq. 19 but without intro-
duced pseudo-observations, i.e.,
J=−ΣxJT
xWJeΣxJT
xW
I−W−1
eJT
eΣλJeW−1
eJT
eΣλ.(29)
Hence, having an implicit nonlinear functional model, the bias of the parameters as
well as the related dispersion are obtained by
E{eu}=−1
2J[tr (ΣuHu,i)]i,(30a)
EeueT
u=Σu+1
2J[tr (ΣuHu,iΣuHu,j )]ij JT
.(30b)
Teunissen (1990) showed that the nonlinearity of the model is caused by two types
of nonlinearity, the nonlinearity of the manifold itself and the nonlinearity of the
parameter curves in the manifold. Whereas the first type is intrinsic, the second type
depends on the parametrization (see also Lehmann and L¨osler 2018). Corresponding
explicit and implicit functional models differ in the parametrization but are related
to the same manifold. For that reason, the bias of the least-squares residuals are only
depending on the intrinsic normal curvature of the manifold and are invariant under
the choice of the parametrization (Teunissen 1990). Thus, Eq. 30 for the implicit
functional model corresponds to Eq. 16 for its explicit representation.
In some applications, further constraints fr(˜x) = 0on the parameters are required
to solve an ill-posed problem. In geodesy, a well-known ill-posed problem occurs in the
framework of so-called free network adjustments, where the observations describe the
internal network geometry but are insensitive about the absolute position of the points,
i. e., the definition of the geodetic datum. Also in surface-fitting, ill-posed problems
arise. For instance, the four parameters of a best-fitting plane are interdependent,
see Sec. 4. Hence, Eq. 28a becomes singular, i. e., nullity JT
xWJx=k > 0. By
introducing kindependent constraints fr, the defect of the ill-posed problem is solved
and Σxis obtained by (e. g. P´azman and Denis 1999)
Σx=U−UJT
rS(31)
where Jris the Jacobian of frevaluated at ˜x and
U=JT
xWJx+JT
rJr−1,(32a)
ST=JrUJT
r−1JrU.(32b)
Since fris known as a special case of a zero-variance computational base, see e.g.
the discussion by Neitzel (2004) in the framework of free network adjustments, only the
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Bias in Least-Squares Adjustment of Implicit Functional Models
estimated parameters ˆx are affected by the constraints but not the estimated obser-
vational errors ˆe. The extended Jacobian Jthat considers the additional constraints
fris given by
J=−ΣxJT
xWJeΣxJT
xW ST
I−W−1
eJT
eΣλJeW−1
eJT
eΣλ0.(33)
Since Jrepresents the Jacobian of fT=fT
efT
cfT
r, the Hessian Huof f
evaluated at ˜u must be introduced to derive the bias of the estimates via Eq. 30.
4. Example: Surface-fitting
In the framework of reverse engineering, model-specific geometric parameters, e. g.
dimension, curvature or orientation of an object are derived metrologically and mod-
elled by primitive surfaces. Such surfaces are also used, for instance, to evaluate the
uncertainty in measurement of optical 3D instruments and to optimise measurement
procedures, e. g. in the framework of the VDI/VDE-2634 (2012, 2008) guideline. Since
surface-fitting has a wide range of applications, we will focus on a best-fitting plane.
To demonstrate the impact of the bias on the estimated parameters as well as the
estimated dispersion, the parameters of the implicit model of a best-fitting plane are
derived under varying conditions. The estimated bias is used as an indicator whether a
large number of poor observations provides better results than a small but precise sam-
ple. All estimations are done introducing fully populated dispersion matrices because
in most applications, coordinates of observed points are derived by indirect methods
e. g. by polar observations or photogrammetric measurements. By converting indirect
measurements into Cartesian coordinates, correlations occur – even if the observations
of the measurement method are independent.
Due to different measurement methods and varying observation configurations, the
a-priori dispersion of the observed points is derived synthetically, using a Taylor-
Karman structured matrix. The Taylor-Karman structured matrix is frequently used
in optimisation of geodetic networks (e. g. Schmitt 1980, Roese-Koerner and Schuh
2014, Eschelbach et al. 2019), because the resulting confidences of the points offer
a desirable structure and are characterised by homogeneous and isotropic shapes. A
similar synthetic dispersion matrix, a Baarda-Aiberda structure matrix, was used by
Teunissen and Knickmeyer (1988) investigating on the bias in the planar Helmert
transformation.
The rotational-invariant Taylor-Karman structured matrix reads (Grafarend and
Schaffrin 1979)
σ2
0Kij =σ2
0 φt(sij )I+φt(sij )−φl(sij )
s2
ij
Dij !.(34)
Here, the transversal and longitudinal correlation functions are φt(s) and φl(s), re-
spectively, the variance of unit weight is denoted by σ2
0,sij =q∆x2
ij + ∆y2
ij + ∆z2
ij
is the spatial distance between the i-th and j-th point, and the symmetric matrix Dij
11 https://doi.org/10.1080/00396265.2020.1715680

Bias in Least-Squares Adjustment of Implicit Functional Models
reads
Dij =

∆x2∆x∆y∆x∆z
∆y∆x∆y2∆y∆z
∆z∆x∆z∆y∆z2
ij
.(35)
Using the modified Bessel function of the second kind, the longitudinal and transversal
correlation functions yield
φl(s) = 4d2
s2−2K0s
d−4d
sK1s
d,(36a)
φt(s) = 2s
dK1s
d−φl(s),(36b)
where K0and K1denote the modified Bessel function of zero and first order, respec-
tively, and d=√2 min (s), s > 0, is the characteristic distance as suggested by Yazji
(1998).
In recent years, terrestrial laser scanners are well established in the framework of en-
gineering geodesy, building information modelling, architecture, archaeology or reverse
engineering. Laser scanners are polar measurement systems, which allow for target-
less recording of high-resolution spatial point clouds. In laser scanning applications,
primitive surfaces like planes, spheres or cylinders are derived. Such primitive surfaces
are used, for instance, to combine point clouds observed by several instrument sta-
tions (e. g. Wujanz et al. 2018). Moreover, Koch (2010) investigated systematic effects
of a terrestrial laser scanner by repeatedly measuring a plane. Wujanz et al. (2017)
derived an intensity-based stochastic model for terrestrial laser scanners, which they
verified using planes. Furthermore, quadric surfaces are a common modelling strategy
in deformation analysis. A review of methods can be found in Neuner et al. (2016).
Among others, the quality of the point cloud depends on the discretisation, i. e.,
the spatial sampling rate, as well as the signal-noise level, i.e., the number of distance
measurement repetitions per point. Both parameters must be specified by the user. The
measurement effort increases for both, a higher discretisation and an improved signal-
noise level. The benefit of a high discretisation but a poor single point measurement
against a high-precision single point measurement but a reduced number of observed
points was studied by Heinz et al. (2019). In that contribution, a 25 cm ×25 cm
planar target was observed by several laser scanners using different settings for the
discretisation of the target and for the standard deviation of a single point. Based
on the estimated dispersion of the output quantities, i. e., the distance between the
instrument and the planar target as well as the inclination angles of the estimated
plane, Heinz et al. (2019) concluded that a higher discretisation but larger numerical
value for the standard deviation of a single point provides smaller dispersion and,
therefore, is more recommendable than a smaller numerical value for the standard
deviation of a single point but a lower discretisation w.r. t. the measurement effort. It
should be noted that Heinz et al. (2019) based their conclusions only on the standard
deviation of the estimated parameters. As an extension, the bias effects are now to be
investigated using the same configuration.
Following the measurement configuration used by Heinz et al. (2019), points lying
on a regular vertical grid are modelled. The edge length of the grid is 25 cm ×25 cm
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Bias in Least-Squares Adjustment of Implicit Functional Models
and the distance between the origin and the centre-point of the grid is 10m. Since
the grid is regular, the number of points per row is equal to the number of points
per column. To evaluate the impact of varying point uncertainties, a grid consisting
of 100 points is used, see Fig. 2. The dispersion of these points is derived by a Taylor-
Karman structured matrix given by Eq. 34 and scaled by eight different σ0, given in
Table 1. Please note, in the following examples, the parameters of TE1 as well as TE2
are derived by introducing unmodified grid points, In virtue of Eq. 9, the impact of
the nonlinearity depends on the stochastic model, i. e., σ0in Eq. 34, thus, no random
noise is added to the grid points.
-10 -5 0 5 10
-10
-5
0
5
10
Figure 2. Considered regular plane having an edge length lg= 25cm. The normalised normal vector of the
plane is n=100Tand the distance between the origin and the plane is d= 10 m.The ng= 100
points are arranged in a regular 10 ×10 grid. The centre point of the grid is located at 10 0 0 T, i. e.,
the Xcomponents of all grid points are 10 m. The distances between neighbour points lying in the same row
or column are 25
9≈2.78 cm.
The nonlinear implicit functional relation of a plane reads (e. g. Bronshtein et al.
2007, Ch. 3.5.3.4.)
nTPi−d= 0.(37a)
Here, PT
i=xiyizidenotes an arbitrary point of the plane, nis the normalised
normal vector and ddescribes the shortest distance between the origin and the plane.
The parameters of the plane xT=nxnynzdand the 3ngresiduals of the
points eT=exieyiezi·· · are estimated by Eq. 23, applying the parameter
13 https://doi.org/10.1080/00396265.2020.1715680

Bias in Least-Squares Adjustment of Implicit Functional Models
constraint equation
nTn−1=0.(37b)
Eq. 28a yields the first-order dispersion of the estimated parameters. Moreover, the
second-order improved solution of the parameters and the dispersion is derived by
Eq. 30, using the Jacobian matrices
Jx= xi−exiyi−eyizi−ezi−1
.
.
..
.
..
.
..
.
.!,
Je=−


nxnynz0 0 0 ···
0 0 0 nxnynz···
.
.
..
.
..
.
..
.
..
.
..
.
....

,
as well as the Jacobian of the parameter constraint equation fr, cf. Eq. 37b, i.e.,
Jr=2nx2ny2nz0.
Due to Eq. 18 being linear, the Hessians of the 3ngfunctions in febecome zero
matrices. Since the functional model of the plane in Eq. 37a is bi-linear, the Hessian
matrix of the i-th function of the ngfunctions in fcreads
Hu,i =0 Hxe,i
HT
xe,i 0,
where the off-diagonal matrix Hxe,i contains the non-zero second-order partial deriva-
tives, i. e.,
Hxe,i =−



·· · 0 1 0 0 0 ···
·· · 0 0 1 0 0 ···
·· · 0 0 0 1 0 ···
·· · 0 0 0 0 0 ···




,
and, furthermore, the Hessian of the k= 1 additional constraint equation fr, cf.
Eq. 37b, is given by
Hu,k =






2 0 0 0 ·· ·
0 2 0 0 ·· ·
0 0 2 0 ·· ·
0 0 0 0 ·· ·
.
.
..
.
..
.
..
.
....







.
To validate the estimated results, the Monte-Carlo method is applied, using a sample
size of m= 500 000 samples. Since MCM provides asymptotically unbiased estimates,
the derived parameters are used as reference solution. The parameters as well as the
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Bias in Least-Squares Adjustment of Implicit Functional Models
dispersion of the MCM are given by (e. g. Koch 2007, Ch. 6.3.3)
ˆxMCM =1
m
m
X
l=1
ˆxl,(39a)
Σx,MCM =1
m−1
m
X
l=1
(ˆxMCM −ˆxl) (ˆxMCM −ˆxl)T,(39b)
where ˆxlis derived by the l-th random sample using Eq. 23. Figure 3 depicts the
analysis procedure by a flow chart diagram.
Select number of points ng, edge length lg, variance of the
unit weight σ2
0, number of Monte-Carlo trials m,l= 1
Select nggrid points ˜
yand derive Weby Taylor-Karman matrix
Taylor series
Derive least-squares estimates
ˆxTE1 , Σx,TE1
Estimate second-order improved solution
ˆxTE2 , Σx,TE2
Monte-Carlo method
l=l+ 1
Prepare trial
yl∼N˜y,W−1
e
Derive parameters from sample yl
ˆxl
l < m
Estimate almost unbiased solution
ˆxMCM , Σx,MCM
Yes
No
Figure 3. Flow chart of analysis procedure. The grid points ˜y are derived by specifying the number of
grid points ngand the edge length lgof the grid. Weis derived by Eq. 34 using σ0. The parameters of the
best-fitting plane and the related dispersion are estimated by approximating the nonlinear function by its
first-order Taylor expansion (TE1) as well as its second-order Taylor expansion (TE2), see left part of the flow
chart. Moreover, the Monte-Carlo method (MCM) is applied to derive the parameters by approximating the
probability distribution using the law of large numbers, see right part of the flow chart.
Table 1 summarises the derived bias bdof the estimated parameter d, i. e., bd,TE1 =
ˆ
dTE1 −ˆ
dMCM and bd,TE2 =ˆ
dTE2 −ˆ
dMCM derived by TE1 and TE2 w.r. t. MCM,
respectively. The corresponding standard deviations for TE1 and TE2 are denoted
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Bias in Least-Squares Adjustment of Implicit Functional Models
Table 1. Derived bias bdof the estimated distance
dand related standard deviations w. r. t. the MCM.
The TE1 and TE2 solution are denoted by bd,TE1 =
ˆ
dTE1 −ˆ
dMCM and bd,TE2 =ˆ
dTE2 −ˆ
dMCM, respec-
tively. The derived standard derivation of the MCM,
TE1 and TE2 solution are given by ˆσMCM, ˆσTE1 and
ˆσTE2, respectively. All quantities are given in mm.
σ0ˆσMCM bd,TE1 ˆσTE1 bd,TE2 ˆσTE2
0.01 0.0 0.0 0.0 0.0 0.0
1.0 0.5 0.1 0.5 0.0 0.5
2.0 1.1 0.5 1.0 0.0 1.1
3.0 1.9 1.1 1.5 0.0 1.8
4.0 2.8 2.0 2.0 0.1 2.8
5.0 4.0 3.2 2.5 0.2 3.9
6.0 5.5 4.7 3.0 0.3 5.3
7.0 7.4 6.5 3.5 0.6 6.9
by ˆσTE1 and ˆσTE2, respectively. For comparison, the estimated standard deviation
of the MCM derived by Eq. 39b is denoted by ˆσMCM. As expected, the estimated
standard deviation ˆσdepends on σ0. Largest values can be found for σ0= 7 mm. For
σ0≤2 mm the estimated standard deviations are about 1 mm and are similar to each
other. Whereas for larger values of σ0the first-order solution ˆσTE1 is underestimated
w. r. t. the reference ˆσMCM, the second-order improved ˆσTE2 is quite similar to ˆσMCM.
For σ0= 7 mm the second-order improved ˆσTE2 is almost twice the size of ˆσTE1.
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
-2.0
0.0
2.0
4.0
6.0
8.0
Figure 4. Derived bias bdof the estimated distance dagainst σ0using a regular grid containing 10 ×10
grid points. The bias bd,TE1 =ˆ
dTE1 −ˆ
dMCM between TE1 and MCM are symbolised by red dots. The bias
bd,TE2 =ˆ
dTE2 −ˆ
dMCM of the second-order improved solution TE2 w.r. t. MCM are depicted by dark-grey
triangles. Error-bars indicate the estimated standard deviations and are given at the ratio of 1:4, to avoid
overlapping effects.
As shown in Figure 4, the bias bdof the estimated parameter dincreases as σ0
increases, i. e., the expectation E {d}becomes smaller while the impact of the non-
linearity increases. This behaviour can be explained geometrically. The normal unit
vector of the modelled plane is nT=100and, with Eq. 37 in mind, the dis-
tance ddepends only on nx. This dependency of nxon dholds, even if the plane is
slightly tilted. Since nis a normalised vector, i.e., nTn= 1, the range of the vector
components is restricted to values from 0 to 1. For that reason, nxcan be ≤1 but
never larger than 1. Whereas the expectation of each x-component of a grid point
is E {xi}= 10 m, the expectation of nxis E {nx}<1. Since the parameters of the
best-fitting plane are interdependent, the expectation of the parameter dbecomes
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Bias in Least-Squares Adjustment of Implicit Functional Models
E{d}<E{xi}. As already mentioned, no random noise is added to the grid points,
i. e., the observations are the expectations of the coordinates, but in general the linear
property of the expectation E {f(y)}=f(E {y}) is false, if the model is nonlinear
(e. g. Carlton and Devore 2017, p. 87). If Σuin Eq. 22 is factorized by Σu=σ2
0Ku
and substituted into Eq. 30, the dependency of the bias onto σ0becomes more com-
prehensible, i. e.,
E{eu}=−σ2
0
2J[tr (KuHu,i)]i,
EeueT
u=σ2
0Ku+σ4
0
2J[tr (KuHu,iKuHu,j )]ij JT
.
For σ0= 4 mm, the bias bd,TE1 and ˆσTE1 are about 2 mm and, thus, on the same or-
der of magnitude. As σ0gets larger, the estimated bias exceeds the ordinary confidence
interval (1σ). For comparison, the bias of the second-order solution is always <1 mm,
even if the influence of the nonlinearity is increased by a poor stochastic model like
for e. g. σ0= 7 mm. If almost unbiased estimates are expected for TE1, high-precision
measurements are recommended. Moreover, the differences between TE2 and MCM
are quite small and, therefore, the numerical effort of the Monte-Carlo method is
disproportionate to the accuracy benefit.
To reduce the leverage caused by the small edge length of the grid onto the bias of
the distance bd, larger edge lengths are recommended. Figure 5 depicts the estimated
bias bdfor several edge lengths lgof the grid, each consisting of 100 grid points.
The estimated bias decreases rapidly and tends to zero. Increasing the grid length by
10 cm halves the estimated bias. Further 10cm yields bd<1 mm. However, in practical
applications, the size of the object, e. g. the size of planar targets, is invariant and the
observation configuration, i.e., the distance between the instrument and the object, is
restricted e. g. by topography. Thus, leverage effects are difficult to compensate.
0.25 0.50 0.75 1.00 1.25 1.50
0.0
1.0
2.0
3.0
4.0
Figure 5. Derived bias bdof the estimated distance dagainst the edge length lgof the grid. Each grid consists
of 100 grid points and σ0= 5mm. Error-bars indicate the second-order improved standard deviations and are
given at the ratio of 1:4.
To evaluate a possible benefit of a larger number of observations onto the estimated
parameters, the number of grid points is increased, while σ0keeps fixed. The parame-
ters of the plane are adjusted for σ0= 1 mm, σ0= 2 mm and σ0= 5 mm, respectively,
using regular grids consisting of 100 to 625 grid points. Due to the large numerical
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Bias in Least-Squares Adjustment of Implicit Functional Models
effort of the MCM but negligible error between TE2 and MCM for σ0≤7 mm, the
MCM is omitted and the bias is derived by Eq. 30.
Table 2. Derived bias bd,TE1 =ˆ
dTE1 −ˆ
dTE2 of the distance
das well as the second-order improved standard deviation of d
using different numbers of grid points ng. The parameter are
estimated applying σ0= 1 mm, σ0= 2 mm and σ0= 5 mm,
respectively. Estimates are given in mm.
σ0= 1 σ0= 2 σ0= 5
ngbd,TE1 ˆσTE2 bd,TE1 ˆσTE2 bd,TE1 ˆσTE2
100 0.1 0.5 0.5 1.1 3.0 3.9
225 0.1 0.4 0.4 0.9 2.5 3.2
289 0.1 0.4 0.4 0.8 2.4 3.0
400 0.1 0.3 0.3 0.8 2.1 2.7
484 0.1 0.3 0.3 0.7 2.0 2.5
625 0.1 0.3 0.3 0.7 1.8 2.3
In Figure 6, the estimated bias and the second-order improved standard deviation
are depicted. Numerical values are given in Table 2. Due to the central limit theorem,
the standard deviation decreases while the number of grid points increases. Moreover,
the estimated bias becomes slightly smaller as the number of grid points gets larger.
As exemplified by σ0= 1 mm, the benefit of a large sample size becomes negligible
for precise measurements. Even for poor measurements, the benefit of larger sample
sizes is disputable. For instance, for σ0= 5 mm and 484 grid points, the estimated
bias is about 2 mm. Using Fig. 2, a similar value can be found for 100 grid points and
a slightly smaller σ0of about 4 mm. In virtue of the variance of the sample mean, i. e.,
σ2=σ2
m,(41)
where σ2denotes the variance of a single observation and mis the number of indepen-
dent repetitions, one obtains m= 2 ≈52
42. Instead of measuring ng= 484 grid points,
a bias of about 1.5 mm occurs, if only 100 grid points are used but measured twice.
Even for ng= 625 and σ0= 5 mm, the bias of about 1.8 mm is larger than for the
doubled measurement. The resulting second-order improved standard deviation of the
distance dderived by the doubled measurement is about 2.3 mm and similar to the
standard derivation derived by 625 individual points, cf. Table 2. For that reason, it
is more economic to improve the signal-noise level than to increase the number of ob-
servations. Considering Figure 1, this result was expectable, because here, the bias of
the Cartesian coordinates depends only on the dispersion of φbut not on the number
of observations.
In most applications, the statistical properties of the estimates of the nonlinear prob-
lem are derived by its linearized substitute-problem. Based on the first-order dispersion
of the estimated parameters of the best-fitting plane, Heinz et al. (2019) recommended
to increase the discretisation of the observed plane instead of improving the dispersion
of the observations. As demonstrated, the stochastic model controls the impact of the
nonlinearity onto the estimates (see also Lehmann and L¨osler 2018). If almost unbiased
estimates are desired, the user is advised to improve the dispersion of the observations
instead of increasing the discretisation. However, if high-precision observations are not
available, second-order improved results can be obtained by Eqs. 16, 30 in explicit and
implicit models, respectively.
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Bias in Least-Squares Adjustment of Implicit Functional Models
100 200 300 400 500 600
0.0
1.0
2.0
3.0
4.0
Figure 6. Dependencies between the number of grid points ngand the estimated bias bdof the distance
parameter d. Dark-red squares, dark-grey circles and light-red triangles symbolise the derived bias for σ0=
1 mm, σ0= 2 mm and σ0= 5 mm, respectively. Error-bars indicate the second-order improved standard
deviations and are given at the ratio of 1:4, to avoid overlapping effects.
5. Conclusion
In geodesy, most of the functional relations are nonlinear but the used models such
as Gauß-Markov model or Gauß-Helmert model were derived for linear models. By
applying such a solver, linearized substitute-problems are solved iteratively instead
of the original underlying nonlinear problem. Even if a non-iterative solver might
exist, the statistical properties of the estimates are usually obtained by a linearized
substitute-problem. Thus, the statistical properties of the derived estimates are related
to the linearized substituted problem but not necessarily to the original nonlinear
model. Without proof, in most applications the nonlinearity of the function is assumed
to be moderate and the bias of the estimated parameters as well as the bias of the
dispersion is neglected. Even if the functional model is moderate nonlinear e. g. bilinear,
the resulting least-squares problem may be strongly nonlinear, because the stochastic
model controls the impact of the nonlinearity onto the estimates.
In general, the estimated dispersion is used to evaluate the reliability of the adjusted
parameters or to validate measurement configurations. Following the line of reasoning
worked out by Box (1971) for bias estimation in nonlinear least-squares model with
explicit functional model, the second-order bias for a nonlinear implicit functional
model that yields the most general formulation of least-squares adjustment problems
was derived in Sec. 3. The estimated bias exceeds the analysis toolbox in adjustment
computations and provides a further indicator to validate the benefit of adapted mea-
surement procedures, varying observation configurations or the use of more precise
measuring instruments.
In Section 4, a similar observation configuration to Heinz et al. (2019) is used and
the bias of the distance parameter of the best-fitting plane is studied under varying
conditions. The results derived by a first-order and second-order Taylor expansion
are compared to the results derived by the Monte-Carlo method. The estimates of
the Monte-Carlo method are almost unbiased, but the computational costs of the
sampling can become expensive, if the nonlinear model is complex or the number of
observations is large (e.g. Teunissen 1990, Alkhatib and Schuh 2007, Schweitzer and
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Bias in Least-Squares Adjustment of Implicit Functional Models
Schwieger 2015). Due to Wirth’s law, the benefit of modern computational power is
limited by the complexity of the investigated problem (Wirth 1995), and, therefore,
the MCM cannot be fully recommended without knowing the application. Expanding
the nonlinear function by its Taylor series reduces the numerical effort. It has been
shown that the first-order solution is biased, even if the model is moderately nonlinear
like for the best-fitting plane. For instance, introducing a measurement uncertainty
σ0= 4 mm of the coordinate components of the grid points, the bias of the distance
parameter is about 2 mm for TE1 but only 0.1 mm for TE2. Whereas the estimated
dispersion of TE1 is always underestimated, i. e., the derived standard derivations
are too optimistic, the dispersion of TE2 is comparable to the dispersion of MCM.
For instance, having a measurement uncertainty σ0= 7mm, the dispersion of TE2
is nearly twice the derived value of TE1. For comparison, the difference between the
dispersion derived by MCM and TE2, respectively, is about 0.5 mm.
As demonstrated in Sec. 4, the nonlinearity of the least-squares problem depends on
the functional model, but the stochastic model controls the impact of the nonlinearity
onto the estimates. Therefore, smallest biases can be found for high-precision measure-
ments. Using the configuration suggested by Heinz et al. (2019), the estimated bias
decreases slightly but cannot be eliminated completely, if the number of grid points
gets larger. Thus, from an economical point of view, it is more recommended to im-
prove the signal-noise level than to increase the number of observations. For instance,
measuring the grid points twice reduces the bias much more than doubling the number
of points.
Whereas the linearized substitute-problem of the nonlinear function can be easily
obtained, the computational costs increase by deriving higher-order terms of the Tay-
lor series. In this investigation, the Taylor expansion is restricted to the second order,
because the accuracy benefits have to be balanced with respect to the computational
effort. As demonstrated by the analysis of the best-fitting plane, the bias of the es-
timates depends on the stochastic model of the observations, the number of points,
and the dimension of the measured plane. A general recommendation, whether the
bias becomes significant in a specific application, is not straightforward and should be
analysed individually using Eqs. 16, 30 (see also Teunissen 1990).
If almost unbiased estimates are desired, high-precision observations are recom-
mended. If such observations are not available, second-order improved results can be
obtained by Eqs. 16, 30 in explicit and implicit models, respectively.
References
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Strategy into Tailored Solution Procedures for Large-Scale Least Squares Problems. Journal
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Box, M.J., 1971. Bias in Nonlinear Estimation. Journal of the royal statistical society, series
b (methodological), 33 (2), 171–201.
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