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J. Geod. Sci. 2018; 8:98–114
Research Article Open Access
R. Lehmann* and M. Lösler
Hypothesis testing in non-linear models
exemplied by the planar coordinate
transformations
https://doi.org/10.1515/jogs-2018-0009
Received February 13, 2018; accepted June 25, 2018
Abstract: In geodesy, hypothesis testing is applied to a
wide area of applications e.g. outlier detection, defor-
mation analysis or, more generally, model optimisation.
Due to the possible far-reaching consequences of a deci-
sion, high statistical test power of such a hypothesis test
is needed. The Neyman–Pearson lemma states that un-
der strict assumptions the often-applied likelihood ratio
test has highest statistical test power and may thus fulll
the requirement. The application, however, is made more
dicult as most of the decision problems are non-linear
and, thus, the probability density function of the param-
eters does not belong to the well-known set of statistical
test distributions. Moreover, the statistical test power may
change, if linear approximations of the likelihood ratio test
are applied.
The inuence of the non-linearity on hypothesis testing
is investigated and exemplied by the planar coordinate
transformations. Whereas several mathematical equiva-
lent expressions are conceivable to evaluate the rotation
parameter of the transformation, the decisions and, thus,
the probabilities of type 1 and 2 decision errors of the re-
lated hypothesis testing are unequal to each other. Based
on Monte Carlo integration, the eective decision errors
are estimated and used as a basis of valuation for linear
and non-linear equivalents.
Keywords: Hypothesis testing; Likelihoodr atio test;M onte
Carlo integration; Non-linear model; Coordinate transfor-
mation
*Corresponding Author: R. Lehmann: University of Applied Sci-
ences Dresden, Faculty of Spatial Information, E-mail:
M. Lösler: Frankfurt University of Applied Sciences, Faculty of
Architecture, Civil Engineering and Geomatics
1Introduction
Hypothesis testing plays an important role in the frame-
work of parameter estimation. In the context of outlier de-
tection, hypothesis testing is used to detect and to iden-
tify implausible observations (e.g. Lehmann and Lösler
2016, Klein et al. 2017). In congruence analysis, hypoth-
esis testing is introduced to distinguish stable points or
areas from instable parts of an epochal observed network
(e.g. Velsink 2015, Lehmann and Lösler 2017). To nd an
adequate number of model parameters, e.g. in the frame-
work of reverse engineering, hypothesis testing indicates
the benet of a more complex model versus a simplied
model (e.g. Ahn 2005).
In geodesy, the likelihood ratio (LR) test is most of-
ten applied (Koch 1999, Teunissen 2000). It bases on the
Neyman–Pearson lemma, which demonstrates that under
various assumptions such a test has the highest statistical
test power (Neyman and Pearson 1933). In practice, most
of the decision problems are non-linear and the under-
lying likelihood function must be maximized iteratively,
e.g. by ordinary least-squares techniques with the risk of
nding only a local maximum. Moreover, the often-used
LR test in the linearized model deteriorates the decision
due to a potential loss of statistical test power. Finally,
the true probability density function of such a test does
not belong to well-known class of statistical test distribu-
tions, and therefore, critical values cannot be computed
with standard statistical functions. To derive the true prob-
ability density function as well as corresponding critical
values, a Monte Carlo integration can be carried out (see
e.g. Lehmann 2012).
Estimation in non-linear geodetic models has been
widely investigated. Teunissen (1985) found that two types
of non-linearity exist: The rst is inherent in the problem
and manifests itself in the non-linearity of the model oper-
ator. The second is perhaps introduced by a parametriza-
tion, which can even make an inherently linear problem
non-linear. This is the case when the planar four parame-
Hypothesis testing in non-linear models exemplied by the planar coordinate transformations |99
ter transformation is parameterized by rotation angle and
scale.
This investigation focuses on the inuence of the non-
linearity on hypothesis testing exemplied by the planar
coordinate transformations. Here, several mathematical
equivalent expressions are conceivable to evaluate the ro-
tation parameter of the transformation by hypothesis test-
ing. Depending on the degree of non-linearity, the eec-
tive αcan dier in comparison to its usually used χ2equiv-
alent. The planar geodetic coordinate transformation is a
good example to study non-linear eects in geodetic mod-
els, because under standard assumptions on the covari-
ance matrix it admits an analytical solution (Teunissen
1985, 1986). Moreover, the planar coordinate transforma-
tions have a wide range of applications in geodesy.
The paper is organized as follows: After briey in-
troducing the non-linear Gauss–Markov model, we focus
on the LR test as a general decision method. Then the
least squares solutions of planar coordinate transforma-
tions are introduced. As an example for hypothesis testing
in non-linear models, we set up a test problem for the rota-
tion angle and solve it by various dierent applications of
the LR test. Finally, we compare these dierent solutions
in terms of decision errors, for which the method of Monte
Carlo integration is used.
2Hypothesis test in the non-linear
Gauss–Markov model
Throughout this paper, true values of quantities will be de-
noted by tilde and estimates by hat.
We start from the non-linear Gauss–Markov model
(GMM)
Y=A˜
X+e(2.1)
where Yis a n-vector of observations and ˜
Xis a u-vector
of unknown true model parameters. Ais a known non-
linear operator mapping from the u-dimensional param-
eter space to the n-dimensional observation space. eis an
unknown random n-vector of normally distributed obser-
vation errors. The associated stochastic model reads:
e∼N(0,σ2P−1)(2.2)
Pis a known positive denite n×n-matrix of weights
(weight matrix). σ2is the a priori variance factor, which
may be either known or unknown. Estimates ^
X,^
Yof the
unknown true values of ˜
X,˜
Y=Y−eare desired.
In geodesy, a decision problem is generally posed as
a statistical hypothesis test. Opposing the special model
represented by the GMM Eqs. (2.1), (2.2) augmented by non-
linear equality constraints
B˜
X=b(2.3)
to a general model represented by the GMM without equal-
ity constraints is equivalent to opposing the null hypothe-
sis
H0:B˜
X=b(2.4)
to the alternative hypothesis
HA:B˜
X=b.(2.5)
The standard solution of the testing problem in classi-
cal statistics goes as follows (e.g. Tanizaki 2004 p. 49 ):
1. A test statistic T(Y)is introduced, which is known to
assume extreme values if H0does not hold true.
2. Under the condition that H0holds true, the probability
distribution of T(Y)is derived, represented by a cumu-
lative distribution function (CDF) F(T|H0).
3. A probability of type 1 decision error α(signicance
level) is suitably dened (say 0.01 or 0.05 or 0.10), see
Fig. 1.
4. For one-sided tests, a critical value cis derived by
c=F−1(1−α|H0), where F−1denotes the inverse CDF
(also known as quantile function) of T|H0. (For two-
sided tests two critical values are needed, but this case
does not show up in this investigation.)
5. The empirical value of the test statistic T(Y)is com-
puted from the given observations Y. If T(Y)>cthen
H0must be rejected, otherwise we fail to reject H0.
In principle, we are free to choose a test statistic. Even
heuristic choices like
T(Y):=
B^
X−b
(2.6)
with some suitable norm ‖·‖are conceivable. Although
the statistical test power (probability of rejection of H0
when it is false) of such a test might be non-optimal or even
poor.
3The likelihood ratio test
In geodesy, we most often apply the likelihood ratio
(LR) test (e.g. Tanizaki 2004 p. 54 ). The test statistic of
the LR test reads
TLR (Y):= max LX,σ2|Y:B(X)=b
max LX,σ2|Y (3.1)
100 |R. Lehmann and M. Lösler
Figure 1: Probability density functions fof test statistic Tunder H0
and HAand decision errors α,β.
where LX,σ2|Ydenotes the likelihood function of the
GMM to be maximized with no restriction (denominator)
and with the restriction B(X)=b(numerator). For the
GMM Eqs. (2.1), (2.2) the likelihood function reads
LX,σ2|Y=
det 2πσ2P−1−0.5
exp −1
2σ2(Y−A(X))TP(Y−A(X))
(3.2)
It is well known that maximizing LX,σ2|Yis equiv-
alent to minimizing the least squares error functional (e.g.
Koch 1999 p. 161f, Lösler et al. 2017)
Ω(X) := (Y−A(X))TP(Y−A(X)) (3.3)
either with constraints B(X)=bor without constraints. In
the rst case, Ωis augmented by the Lagrange term
Ω′(X,k)=Ω(X)+ 2kTB(X)−b(3.4)
where kis the vector of Lagrange multipliers, in geodesy
also known as correlates.
To simplify matters, we will restrict the derivation to
the case of a known a priori variance factor σ2. In this case,
(3.1) can be expressed as
TLR (Y)=
exp −min(Ω′(X,k))
2σ2
exp −min(Ω(X))
2σ2
= exp −min(Ω′(X,k))−min(Ω(X))
2σ2(3.5)
Moreover, we may replace TLR(Y)by the fully equiva-
lent test statistic
T(Y):= −2·log (TLR (Y)) =min(Ω′(X,k))−min(Ω(X))
σ2
(3.6)
If T(Y)>cwith a properly chosen critical value c,
then H0must be rejected, otherwise we fail to reject H0.
Note that all these derivations are fully valid even if Aor
Bare non-linear operators.
In the case that Aand Bare both linear operators, we
obtain the expression (e.g. Lehmann and Neitzel 2013)
T(Y)=^
wTΣ−1
^
w^
w(3.7)
where
^
w:= B^
X−b(3.8)
is the vector of estimated misclosures and Σ^
wis the re-
lated covariance matrix. ^
Xdenotes the minimizer of Ω(X)
in Eq. (3.3), known as least squares estimate of X. Equa-
tion (3.7) can be seen as a special case of Eq. (2.6). If σ2is
known and Eq. (2.2) holds true, the test statistic Eq. (3.7)
follows the distributions:
T(Y|H0)∼χ2(m)(3.9a)
T(Y|HA)∼χ′2m,Λ′(3.9b)
with the non-centrality parameter
Λ′=˜
wTΣ−1
^
w˜
w(3.10)
mdenotes the number of independent constraints. The
vector of true misclosures ˜
w=B˜
X−band hence also
Λ′are naturally unknown.
All established tests in geodesy belong to the class of
LR tests. The rationale of these tests is provided by the
famous Neyman–Pearson lemma (Neyman and Pearson
1933), which demonstrates that under various assump-
tions such a test has the highest statistical test power
among all competitors. It is often applied even if we can-
not exactly or only approximately make these assump-
tions in practice, because we know that the power is still
larger than for rival tests (Teunisssen 2000, Kargoll 2012,
Lehmann and Voß-Böhme 2017).
In truly non-linear models, we generally encounter
three special problems:
1. The likelihood function Eq. (3.2) can only be maxi-
mized iteratively with the danger of nding only a lo-
cal maximum. (Global optimization methods, which
promise to nd also the global maximum, are not yet
widely applied practically because of the considerable
computational workload for multidimensional prob-
lems.)
2. Test statistic Eq. (3.7) is only an approximation of the
true LR test statistic Eq. (3.6) because the likelihood
ratio is taken in the linearized GMM.
3. The probability density function (PDF) of T(Y)or
some equivalent of it does not belong to the well-
known set of statistical test distributions (t,χ2,Fetc.)
Hypothesis testing in non-linear models exemplied by the planar coordinate transformations |101
such that the critical values must be computed numer-
ically.
In the next sections, we will illustrate some consequences
of these problems. In the conclusions, we will return to
these points.
4The least squares solution of the
three-parameter transformation
Figure 2: Planar parameter transformation
In a plane consider two Cartesian reference frames x,yand
X,Y, which are related by translation and rotation, such
that an arbitrary point Phas coordinates xP,yP,XP,YP
satisfying the non-linear transformation equations
XP=X0+xP·cos ϵ−yP·sin ϵ
YP=Y0+xP·sin ϵ+yP·cos ϵ(4.1)
with transformation parameters X0,Y0,ϵ, see Fig. 2. Re-
lated equations can be formulated for the opposite trans-
formation direction.
We start from a set of Npoints having observed coordi-
nates
x1,y1,. . . ,xN,yN,X1,Y1,. . . ,XN,YN(4.2)
in both frames. The problem is to nd the best estimates for
X0,Y0,ϵin the least squares sense, also known as the least
squares solution of the three-parameter transformation.
In the following, we restrict ourselves to the case that
the coordinates of one system are non-stochastic (error-
free) xed quantities. Without restriction of generality, the
error-free coordinates are denoted as x1,y1,. . . ,xN,yN.
Moreover, we assume that for each pair of observations
Xi,Yi, both Xiand Yihave the same weight pi. This GMM
reads
Y=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
X1
Y1
.
.
.
XN
YN
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,˜
X=⎛
⎜
⎝
˜
ϵ
˜
X0
˜
Y0
⎞
⎟
⎠,
A˜
X=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
˜
X0+x1·cos ˜
ϵ−y1·sin ˜
ϵ
˜
Y0+x1·sin ˜
ϵ+y1·cos˜
ϵ
.
.
.
˜
X0+xN·cos ˜
ϵ−yN·sin ˜
ϵ
˜
Y0+xN·sin ˜
ϵ+yN·cos˜
ϵ
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
P=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
p10· · · 0 0
0p10 0
.
.
.....
.
.
0 0 pN0
0 0 · · · 0pN
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(4.3)
This setting is one of the rare cases, where an analytical
solution exists. For the sake of simplicity, we assume that
σ2is chosen such that the weights fulll Σpi= 1.
For the sake of compact notation, we introduce in ei-
ther coordinate system the following abbreviations:
1. the weighted barycentres are given by
X*:=
N
i=1
piXi,Y*:=
N
i=1
piYi,
x*:=
N
i=1
pixi,y*:=
N
i=1
piyi(4.4)
2. the coordinates related to the barycentres as origins
∆Xi:= Xi−X*,∆Yi:= Yi−Y*,
∆xi:= xi−x*,∆yi:= yi−y*(4.5)
3. the moments of inertia related to the barycentres
h:=
N
i=1
pi∆x2
i+∆y2
i=−x2
*−y2
*+
N
i=1
pix2
i+y2
i
(4.6a)
H:=
N
i=1
pi∆X2
i+∆Y 2
i=−X2
*−Y2
*+
N
i=1
piX2
i+Y2
i
(4.6b)
4. the auxiliary terms
c:=
N
i=1
pi(∆Xi∆xi+∆Yi∆yi)=
N
i=1
pi(Xi∆xi+Yi∆yi)
(4.7a)
102 |R. Lehmann and M. Lösler
s:=
N
i=1
pi(∆Y i∆xi−∆Xi∆yi)=
N
i=1
pi(Yi∆xi−Xi∆yi)
(4.7b)
In GMM Eq. (4.3) the non-linear least squares solution
for ϵ,X0,Y0reads (see appendix 1)
^
ϵ= arctan s
c(4.8a)
^
X0=X*−x*·cos ^
ϵ+y*·sin ^
ϵ=X*−x*·c−y*·s
√c2+s2
(4.8b)
^
Y0=Y*−x*·sin ^
ϵ−y*·cos ^
ϵ=Y*−x*·s+y*·c
√c2+s2
(4.8c)
Ω^
X=h+H−2c2+s2(4.8d)
These formulas do not directly contain the observa-
tions Y, but only the statistics c,s,X*,Y*. All other quanti-
ties are xed. Therefore, the vector (c,s,X*,Y*)Tis a su-
cient statistic of the problem. Moreover, it is normally dis-
tributed because of the linear relationship
Z:= ⎛
⎜
⎜
⎜
⎝
c
s
X*
Y*
⎞
⎟
⎟
⎟
⎠
=⎛
⎜
⎜
⎜
⎝
∆x1∆y1· · · ∆yN
−∆y1∆x1· · · ∆xN
1 0 · · · 0
0 1 · · · 1
⎞
⎟
⎟
⎟
⎠
PY (4.9)
The covariance matrix of Zcan be derived by covariance
propagation
ΣZ=⎛
⎜
⎜
⎜
⎝
∆x1∆y1· · · ∆yN
−∆y1∆x1· · · ∆xN
1 0 · · · 0
0 1 · · · 1
⎞
⎟
⎟
⎟
⎠
Pσ2P−1P⎛
⎜
⎜
⎜
⎝
∆x1∆y1· · · ∆yN
−∆y1∆x1· · · ∆xN
1 0 · · · 0
0 1 · · · 1
⎞
⎟
⎟
⎟
⎠
T
=σ2⎛
⎜
⎜
⎜
⎝
h0 0 0
0h0 0
0 0 1 0
0 0 0 1
⎞
⎟
⎟
⎟
⎠
(4.10)
Thus, c,s,X*,Y*are even independent random vari-
ables. For the expectations we obtain
E{c}=
N
i=1
pi(E{Xi}∆xi+E{Yi}∆yi)
=
N
i=1
pi˜
X0+xi·cos ˜
ϵ−yi·sin ˜
ϵ∆xi
+˜
Y0+xi·sin ˜
ϵ+yi·cos ˜
ϵ∆yi
=
N
i=1
pi((∆xi·cos ˜
ϵ−∆yi·sin ˜
ϵ)∆xi
+(∆xi·sin ˜
ϵ+∆yi·cos ˜
ϵ)∆yi)=h·cos ˜
ϵ(4.11a)
E{s}=
N
i=1
pi(E{Yi}∆xi−E{Xi}∆yi)=h·sin ˜
ϵ(4.11b)
E{X*}=
N
i=1
piE{Xi}=
N
i=1
pi˜
X0+xi·cos ˜
ϵ−yi·sin ˜
ϵ
=˜
X0+x*·cos ˜
ϵ−y*·sin ˜
ϵ(4.11c)
E{Y*}=
N
i=1
piE{Yi}=
N
i=1
pi˜
Y0+xi·sin ˜
ϵ+yi·cos ˜
ϵ
=˜
Y0+x*·sin ˜
ϵ+y*·cos ˜
ϵ(4.11d)
Starting from an initial guess for ϵ,X0,Y0, the solu-
tion Eq. (4.8) can also be obtained as the limit of a se-
quence of linearized GMM. Despite of the non-linearity of
the GMM, we obtain a unique solution for the parameter
estimation problem. Thus, there is no danger of nding
only a local minimum here.
5The least squares solution of the
four-parameter transformation
In extension of Eq. (4.1) we introduce a scale parame-
ter µsuch that the new observation equations read
XP=X0+µ·(xP·cos ϵ−yP·sin ϵ)
YP=Y0+µ·(xP·sin ϵ+yP·cos ϵ)(5.1)
By the substitution a:= µ·cos ϵ,o:= µ·sin ϵwe
obtain the linear representation
XP=X0+xP·a−yP·o
YP=Y0+xP·o+yP·a(5.2)
with the parameter vector
˜
X=⎛
⎜
⎜
⎜
⎝
˜
a
˜
o
˜
X0
˜
Y0
⎞
⎟
⎟
⎟
⎠
(5.3)
Hypothesis testing in non-linear models exemplied by the planar coordinate transformations |103
The least squares solution of this linear GMM is simple and well known:
^
a=c
h,^
o=s
h(5.4a)
^
X0=X*−x*·^
a+y*·^
o(5.4b)
^
Y0=Y*−x*·^
o−y*·^
a(5.4c)
Ω^
X=H−c2+s2
h(5.4d)
where h,H,c,sare as dened in Eqs. (4.6a,b), (4.7a,b). This solution permits an estimate of the rotation angle and scale
parameter:
^
ϵ= arctan ^
o
^
a= arctan s
c(5.5a)
^
µ=^
a2+^
o2=√c2+s2
h(5.5b)
See also appendix 3. Note that Eq. (5.5a) coincides with Eq. (4.8a).
6LR hypothesis testing in the three-parameter transformation
As an example of a hypothesis test in a planar transformation model, we want to test a hypothesis for the rotation angle
˜
ϵof the form
H0:˜
ϵ=ϵ0vs.HA:˜
ϵ=ϵ0(6.1)
which can be identied as a special case of Eqs. (2.4),(2.5) by
B˜
X=⎛
⎜
⎝
1
0
0⎞
⎟
⎠
T
˜
X,b=ϵ0,^
w=^
ϵ−ϵ0(6.2)
with m= 1. Obviously, Bis a linear operator, but Ais not. To apply test statistic Eq. (3.7), A(X)must be linearized by
Taylor expansion:
A(X)=A^
X+A·X−^
X+o
X−^
X
(6.3)
In the following, we investigate four dierent derivations of a test statistic for problem Eq. (6.1).
¬Starting from an initial guess for ϵ,X0,Y0, a sequence of linear GMM is computed, until the iteration converges. In
the nal step, the Jacobian matrix Aassumes the form
A=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−x1·sin ^
ϵ−y1·cos ^
ϵ1 0
x1·cos ^
ϵ−y1·sin ^
ϵ0 1
.
.
.
−xN·sin ^
ϵ−yN·cos ^
ϵ
xN·cos ^
ϵ−yN·sin ^
ϵ
.
.
.
1
0
.
.
.
0
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(6.4)
104 |R. Lehmann and M. Lösler
This gives an approximation of the covariance matrix of the estimated parameters (see appendix 4)
Σ^
X=σ2ATPA−1=σ2⎛
⎜
⎝
Σpix2
i+y2
isymm .
−x*·sin ^
ϵ−y*·cos ^
ϵ1
x*·cos ^
ϵ−y*·sin ^
ϵ0 1
⎞
⎟
⎠
−1
=σ2
h⎛
⎜
⎝
1symm .
x*·sin ^
ϵ+y*·cos ^
ϵ h + (x*·sin ^
ϵ+y*·cos ^
ϵ)2
−x*·cos ^
ϵ+y*·sin ^
ϵy2
*−x2
*
2sin 2^
ϵ−x*y*cos 2^
ϵ h + (x*·cos ^
ϵ−y*·sin ^
ϵ)2⎞
⎟
⎠(6.5a)
σ2
^
ϵ=σ2
h(6.5b)
When we perform the LR test of Eq. (4.3) using the linear approximation of A, we come up with Eq. (3.7), which
reads here
T3.1(Z)=^
ϵ−ϵ0TΣ−1
^
ϵ^
ϵ−ϵ0=^
ϵ−ϵ0
σ^
ϵ2
=h
σ2arctan s
c−ϵ02(6.6)
(‘’3.1” denotes here the 1st version of the three-parameter test statistic.)
A practically equivalent formulation of Eq. (6.1) is
H0: tan ˜
ϵ= tan ϵ0vs.HA : tan ˜
ϵ=tan ϵ0(6.7)
(disregarding the impractical non-issue that tan ϵ= tan ( ϵ+π)).
Here, B(X)is also non-linear and must be linearized by Taylor expansion:
B(X)=B^
X+BT·X−^
X+o
X−^
X
(6.8)
In the nal step of the iteration, the Jacobian matrix Bassumes the form
B=⎛
⎜
⎝
cos−2^
ϵ
0
0⎞
⎟
⎠(6.9)
In this case, Eq. (3.7) reads
T3.2(Z)=tan ^
ϵ−tan ϵ0TBTΣ^
XB−1tan ^
ϵ−tan ϵ0
=tan ^
ϵ−tan ϵ02
σ2
hcos−4^
ϵ=h
σ2sc −c2tan ϵ0
c2+s22
(6.10)
This result is obviously dierent from Eq. (6.6). One could argue that Eq. (6.10) should be less reliable than Eq. (6.6)
because also Bmust now be linearized too. But this argument is not conclusive, because we could have obtained this
result also by substituting t:= tan ϵin the transformation equations and solving and testing for the new parameter
tinstead of ϵ. In this case, Bwould be the same as in Eq. (6.2). The same line of reasoning would apply for other
trigonometric functions in Eq. (6.7).
The main reason why (6.6) and (6.10) are dierent is not the “non-issue” discussed above, but the fact that the
linearization errors by truncating the corresponding Taylor expansions are dierent. Proof: Use “cot” instead of “tan”
in (6.7). Although the same ˜
ϵ=ϵ0+kπ,k∈Zholds, we arrive at a dierent test statistic than in (6.10).
®Applying covariance propagation to Eq. (4.8a) and using the quotient rule and the chain rule, we obtain the expres-
sion:
σ2
^
ϵ=⎛
⎝
−s
1 + s2
c2c2
1
1 + s2
c2c
0 0⎞
⎠ΣZ⎛
⎝
−s
1 + s2
c2c2
1
1 + s2
c2c
0 0⎞
⎠
T
=c2+s2σ2h
1 + s2
c22c4
=σ2h
c2+s2(6.11)
Hypothesis testing in non-linear models exemplied by the planar coordinate transformations |105
This is dierent from Eq. (6.5b), because the linearization is applied at a later stage. Therefore, we can assume that
this is a better approximation than Eq. (6.5b). Using this expression in Eq. (3.7) yields
T3.3(Z)=^
ϵ−ϵ0
σ^
ϵ2
=c2+s2
σ2harctan s
c−ϵ02(6.12)
But still this test statistic is a linear approximation via Eq. (3.7).
¯To obtain a fully non-linear LR test statistic, we revert to Eq. (3.6):
T3.4(Z)=min(Ω′)−min(Ω)
σ2
=Ω(^
X0,^
Y0,ϵ0)−Ω(^
X0,^
Y0,^
ϵ)
σ2
=1
σ2[(h+H−2(c·cos ϵ0+s·sin ϵ0))
−h+H−2c2+s2
=2
σ2c2+s2−c·cos ϵ0−s·sin ϵ0(6.13)
where appendix 1 and Eq. (4.8d) have been used.
Note that this test statistic is as simple to compute as the three previous versions.
7LR hypothesis testing in the four-parameter transformation
We want to test the same hypothesis Eq. (6.1), but now for the four-parameter transformation. In terms of the substitution
model parameters Eq. (5.5a), it can be formulated as
H0: arctan ˜
o
˜
a=ϵ0vs.HA: arctan ˜
o
˜
a=ϵ0(7.1)
This can be identied as a special case of Eqs. (2.4), (2.5) by
B˜
X= arctan ˜
o
˜
a,b=ϵ0,^
w= arctan ^
o
^
a−ϵ0(7.2)
with m= 1.
In the following, we investigate four dierent derivations of a test statistic for problem Eq. (7.1).
¬Acting on a,o, operator Bis non-linear, but Ais linear here. Consequently, B(X)must be linearized as in Eq. (6.8):
B=⎛
⎝
−o
1 + o2
a2a2
1
1 + o2
a2a
0 0⎞
⎠
T
=1
o2+a2⎛
⎜
⎜
⎜
⎝
−o
a
0
0
⎞
⎟
⎟
⎟
⎠
(7.3)
In this case, Eq. (3.7) reads
T4.1(Z)=arctan ^
o
^
a−ϵ0TBTΣ^
XB−1arctan ^
o
^
a−ϵ0=c2+s2
σ2harctan s
c−ϵ02(7.4)
where Σ^
Xis the well-known covariance matrix (e.g. Wolf 1966, Somogyi and Kalmár 1988)
Σ^
X=σ2⎛
⎜
⎜
⎜
⎝
1/h0
0 1/h
0 0
0 0
0 0
0 0
1 0
0 1
⎞
⎟
⎟
⎟
⎠
(7.5)
106 |R. Lehmann and M. Lösler
It turns out that T4.1(Z)≡T3.3(Z). However, the corresponding models are dierent.
Alternatively, we can solve the non-linear four-parameter transformation with parameters µ,ϵinstead of a,oby iter-
ation. In the nal step, the Jacobian matrix Aassumes the form
A=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−x1·^
µ·sin ^
ϵ−y1·^
µ·cos ^
ϵ x1·cos ^
ϵ−y1·sin ^
ϵ
x1·^
µ·cos ^
ϵ−y1·^
µ·sin ^
ϵ x1·sin ^
ϵ+y1·cos^
ϵ
.
.
..
.
.
1 0
0 1
.
.
..
.
.
−xN·^
µ·sin ^
ϵ−yN·^
µ·cos ^
ϵ xN·cos ^
ϵ−yN·sin ^
ϵ
xN·^
µ·cos ^
ϵ−yN·^
µ·sin ^
ϵ xN·sin ^
ϵ+yN·cos^
ϵ
1 0
0 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(7.6)
This gives the covariance matrix of the estimated parameters (see appendix 5)
Σ^
X=σ2ATPA−1=σ2⎛
⎜
⎜
⎜
⎜
⎝
^
µ2Σpix2
i+y2
i
0Σpix2
i+y2
i
symm .
^
µ·−x*·sin ^
ϵ−y*·cos ^
ϵx*·cos ^
ϵ−y*·sin ^
ϵ
^
µ·x*·cos ^
ϵ−y*·sin ^
ϵx*·sin ^
ϵ+y*·cos ^
ϵ
1
0 1
⎞
⎟
⎟
⎟
⎟
⎠
−1
=σ2
h
⎛
⎜
⎜
⎜
⎜
⎝
^
µ−2
0 1
symm .
^
µ−1·x*·sin ^
ϵ+y*·cos ^
ϵy*·sin ^
ϵ−x*·cos ^
ϵ
^
µ−1·y*·sin ^
ϵ−x*·cos ^
ϵ−x*·sin ^
ϵ−y*·cos ^
ϵ
Σpix2
i+y2
i
0Σpix2
i+y2
i
⎞
⎟
⎟
⎟
⎟
⎠
(7.7a)
σ2
^
ϵ=σ2
h·^
µ2(7.7b)
The hypotheses are now formulated as in Eq. (6.1). Acting on µ,ϵ, operator Ais non-linear, but Bis linear here and
corresponds to Eq. (6.2).
When we perform the LR test using the linear approximation of A, we come up with Eq. (3.7), which reads here
T4.2(Z)=^
ϵ−ϵ0TΣ−1
^
ϵ^
ϵ−ϵ0=^
ϵ−ϵ0
σ^
ϵ2
=c2+s2
h2·h
σ2arctan s
c−ϵ02(7.8)
It turns out that T4.2(Z)≡T4.1(Z)≡T3.3(Z).
®Let us now study the special case ϵ0= 0.Here, the hypotheses can be written as
H0:˜
o= 0 vs.HA:˜
o=0(7.9)
In the four-parameter transformation, this can be identied as a special case of Eqs. (2.4), (2.5) by
B˜
X=⎛
⎜
⎜
⎜
⎝
0
1
0
0
⎞
⎟
⎟
⎟
⎠
T
˜
X,b= 0,^
w=^
o(7.10)
In this case, both Aand Bare linear operators and Eq. (3.7) reads
T4.3(Z)=^
o2
σ2
^
o
=h^
o2
σ2=s2
σ2h(7.11)
where Eq. (5.4) and Eq. (7.5) have been used.
Hypothesis testing in non-linear models exemplied by the planar coordinate transformations |107
¯To obtain a fully non-linear LR test statistic, we revert to
Eq. (3.6):
T4.4(Z)=min(Ω′)−min(Ω)
σ2
=Ω(^
X0,^
Y0,ϵ0,^
µ)−Ω(^
X0,^
Y0,^
ϵ,^
µ)
σ2
=1
σ2H−(c·cos ϵ0+s·sin ϵ0)2
h−H+c2+s2
h
=c2+s2−(c·cos ϵ0+s·sin ϵ0)2
hσ2
=(c·sin ϵ0−s·cos ϵ0)2
hσ2(7.12)
where appendix 2 and Eq. (5.4d) have been used.
8Distributions
Due to the coincidence with T3.3, the test statistics
T4.1,T4.2will not be further discussed.
Note that all derived test statistics Ti(Z)depend on
only two of the four elements of Z, i.e. cand s. This will
be highlighted by the notation Ti(c,s)used below:
T3.1(c,s)=h
σ2arctan s
c−ϵ02(8.1a)
T3.2(c,s)=h
σ2sc −c2tan ϵ0
c2+s22
(8.1b)
T3.3(c,s)=c2+s2
σ2harctan s
c−ϵ02(8.1c)
T3.4(c,s)=2
σ2c2+s2−c·cos ϵ0−s·sin ϵ0
(8.1d)
T4.3(c,s)=s2
σ2h(8.1e)
T4.4(c,s)=(c·sin ϵ0−s·cos ϵ0)2
σ2h(8.1f)
In the linear or linearized GMM, we obtain from
Eq. (3.9) the following distributions of the LR test statistics:
Ti(c,s)|H0∼χ2(1),i= 3.1,3.2,3.3,4.3,4.4(8.2a)
Ti(c,s)|HA∼χ′21,h
σ2(˜
ϵ−ϵ0)
2,i= 3.1,3.3(8.2b)
T3.2(c,s)|HA∼χ′21,h
σ2(tan ˜
ϵ−tan ϵ0)2cos4^
ϵ
(8.2c)
T4.3(c,s)|HA∼χ′21,˜
s2
σ2h(8.2d)
T4.4(c,s)|HA∼χ′21,(˜
c·sin ϵ0−˜
s·cos ϵ0)2
σ2h(8.2e)
However, observing that test statistics Ti(c,s),
i= 3.1,3.2,3.3,4.4in Eq. (8.2a) are obtained by lin-
earization of Aor Bor both, these distributions can be
no more than approximations of the true distributions of
Ti(c,s)in the vicinity of ^
ϵ. But oftentimes Ti(c,s)is eval-
uated far away from ^
ϵ, especially if αis small. Test statistic
T3.4,T4.4is dened in the fully non-linear model and test
statistic T4.3is dened in the fully linear model. Therefore,
no such approximation is made here.
Remark: In Eq. (8.2b) it would not be correct to apply Eq.
(6.11) instead of Eq. (6.5b). Equation (6.5b) must be used
even in case i= 3.3, because Eq. (6.5b) is derived from Σ^
X
in Eq. (6.5a), as it is required by Eq. (3.9).
Simplications:
¬If we rotate the source system (x,y) by ϵ0about the
barycentre (x*,y*)and solve the same transformation
problem with the rotated coordinates, c,sare replaced by
c′=c·cos ϵ0+s·sin ϵ0
s′=−c·sin ϵ0+s·cos ϵ0(8.3)
Note that the vector (c′,s′)Tis the result of the rota-
tion of (c,s)Tby angle −ϵ0about the origin (0,0) and has
therefore the same covariance matrix
Σc′,s′=σ2h0
0h(8.4)
The transformation problems with the rotated coordi-
nates have the solution
cos ^
ϵ′=c′
√c′2+s′2=c·cos ϵ0+s·sin ϵ0
√c2+s2
= cos ^
ϵ·cos ϵ0+ sin ^
ϵ·sin ϵ0= cos ^
ϵ−ϵ0(8.5)
Now, testing Eq. (6.1) is obviously identical to testing
H0:˜
ϵ′= 0 vs.HA:˜
ϵ′=0(8.6)
with the rotated coordinate system, i.e. with c′,s′. This ob-
viously results in the same test statistics T3.1,T3.3. Less
obvious, the same applies to T3.4and T4.4by virtue of
T3.4c′,s′=2
σ2c′2+s′2−c′
108 |R. Lehmann and M. Lösler
=2
σ2c2+s2−c·cos ϵ0−s·sin ϵ0
=T3.4(c,s)(8.7a)
T4.4c′,s′=s′2
hσ2=(−c·sin ϵ0+s·cos ϵ0)2
hσ2
=T4.4(c,s)(8.7b)
For T4.3no rotation is necessary, because it only ap-
plies to the special case ϵ0= 0. Moreover, note that for
ϵ0= 0 we nd a coincidence of T4.3and T4.4. This shows
that T4.4follows the χ2-distribution Eq. (8.2a,d,e). Hence,
we will further discuss only T4.4.
However, the situation for T3.2is dierent. Here, a dif-
ferent result is obtained. Note that the solution in terms of
the parameter t:= tan ϵis not rotational invariant. This
becomes obvious in the case of ˜
ϵ=±π/2, where ˜
tnot even
exists.
Disregarding this non-issue, we will continue with
ϵ0= 0, noting that almost no restriction of generality is
made.
If we scale both coordinate systems by the factor σ−1and
solve the transformation problem with the scaled coordi-
nates, c′,s′,h,are replaced by
c′′ =c′
σ2,s′′ =s′
σ2,h′′ =h
σ2,σ′′2=σ2
σ2= 1 (8.8)
The weights do not change, such that Σpi= 1 is re-
tained. Note that the new vector (c′′,s′′)Thas the covari-
ance matrix
Σc′′,s′′ =σ−2h0
0h=h′′ 0
0h′′ (8.9)
The new solution is ^
ϵ′′ =^
ϵ′.
Now testing Eq. (8.6) with the scaled coordinates, i.e.
with c′′,s′′, obviously results in
T3.1c′′,s′′=h′′ arctan s′′
c′′ 2
=T3.1c′,s′(8.10a)
T3.2c′′,s′′=h′′ s′′ c′′
c′′2+s′′ 22
=T3.2c′,s′(8.10b)
T3.3c′′,s′′=c′′ 2+s′′2
h′′ arctan s′′
c′′ 2
=T3.3c′,s′
(8.10c)
T3.4c′′,s′′= 2 c′′ 2+s′′2−c′′ =T3.4c′,s′
(8.10d)
T4.4c′′,s′′=s′′ 2
h′′ =T4.4c′,s′(8.10e)
Thus, all test statistics are scale invariant, too.
A special problem exists for T3,2, which can be written
as
T3.2c′′,s′′=h′′
4sin 22^
ϵ′′ ≤h′′
4(8.11)
The fact that the χ2density function is non-zero on the
whole positive real line again proves that T3.2has not the
χ2distribution.
Henceforth, we drop double-primes, such that
ϵ0:= 0,σ2:= 1 (8.12)
is assumed with almost no loss of generality. (Remember
that “almost” here concerns only T3.2, which is not rota-
tion invariant.)
The question, which test statistic is best, must be an-
swered by the resulting probabilities of decision error.
1. The probability of type 1 decision error αis usually se-
lected by the user. But if the 1−α-quantile of χ2(1) is
used for T3.1,T3.2,T3.3, the eective αcan be dier-
ent.
2. The probability of type 2 decision error βshould be
small.
Both probabilities αand βare linked via the critical value
c, see Fig. 1.
9Probability of type 1 decision
error
The idea is to compare the 1−α-quantiles of χ2(1) with
the quantiles of the true distribution of Ti|H0obtained
by Monte Carlo integration. This method has been suc-
cessfully used e.g. by Lehmann (2012) for the computation
of critical values of normalized and studentized residuals
employed in geodetic outlier detection. In principle, it re-
places
•random variates by computer generated pseudo ran-
dom numbers,
•probability distributions by histograms and
•statistical expectations by arithmetic means
computed from a large number of Monte Carlo experi-
ments, i.e. computations with pseudo random numbers in-
stead of noisy observations.
Hypothesis testing in non-linear models exemplied by the planar coordinate transformations |109
In the case that H0is true, we have ˜
ϵ= 0, such that
from Eq. (4.11) follows
E{c|H0}=h,E{s|H0}= 0.(9.1)
According to Eqs. (4.10), (8.12) we need to generate the
following pseudo random numbers:
c|H0∼N(h,h),s|H0∼N(0,h)(9.2)
We use M= 108Monte Carlo samples, which turns out
to be suciently high, because the results only insigni-
cantly change, when the computations are repeated with
dierent pseudo random numbers.
We use three stages of non-linearity, expressed by the
signal/noise ratio: h= 1000 means that the signal is 1000
times larger than the noise (σ= 1), which causes only
weak non-linear eects. Analogous, h= 100 and h= 10
cause medium and strong non-linear eects, respectively.
In Table 1, the 1−α-quantiles of χ2(1) and the quan-
tiles of the true distribution of Ti|H0,i= 3.1,3.2,3.3are
compared. For T4.3≡T4.4we can directly use the quan-
tiles of χ2(1). As expected, the largest dierences occur for
h= 10 and T3.2. Using the χ2-quantile as a critical value
can be both, an advantage and a disadvantage in terms of
α. Consider h= 10 and a desired α= 0.01 in T3.1, we erro-
neously select 6.63 as a critical value, instead of 8.92. The
true αfor T3.1is not 0.01, but even larger than 0.02. By
interpolation of the derived quantiles in Table 1, we obtain
an eective α= 0.021. In contrast to that, we nd from
Eq. (8.11) that |T3.2|<2.5always holds, such that 6.63 is
never exceeded, which corresponds to an eective α= 0.
The true quantiles of T3.4are given in Table 2, but
should not be compared to the χ2-quantiles, because they
are obtained in the non-linear model. It is perhaps un-
expected that T3.4follows the χ2distribution even better
than other test statistics, as can be seen from a comparison
of Table 1 and 2.
10 Probability of type 2 decision
error
The aim of this investigationis to nd out , whichtest statis-
tic has the highest statistical test power, i.e. the best abil-
ity to reject a false H0. For comparison, we plot the power
function of Ti|HA,i= 3.1,. . . ,4.4, denoted as
1−βi(|˜
ϵ|)(10.1)
Due to the symmetry of βi, all plots are produced
only for positive ˜
ϵ. Whenever Eq. (8.2b,c) hold only ap-
proximately, we again use Monte Carlo integration to
compute the true distribution of Ti|HA. According to
Eqs. (4.10), (4.11), (8.12) we need to generate the following
pseudo random numbers:
c|HA∼N(h·cos ˜
ϵ,h),s|HA∼N(h·sin ˜
ϵ,h)(10.2)
We nd
1−βi(|˜
ϵ|)= Pr(Ti>ci|HA),i= 3.1,. . . ,4.4(10.3)
where ciis the critical value, which equals the 1−α-
quantile of either the χ2(1) distribution or the true distri-
butions obtained in the preceding section, whenever this
is dierent. The rst case is practically applied. Below we
restrict ourselves to the choice of α= 0.05.
In Fig. 3, the power function Eq. (10.3) is plotted for
T4.4, which requires no Monte Carlo integration because
Eq. (8.2e) holds exactly. We see that the power is increas-
ing with |˜
ϵ|, which is expected, because H0and HAare
getting more and more dierent, cf. Fig. 1. Furthermore,
the statistical test power is worse when his small, which is
also expected. Remember that h= 10 means that the mo-
ment of inertia of the points are only 10 times larger than
the standard deviations σ= 1 of the target coordinates,
which makes testing hypotheses nearly hopeless.
Figure 3: Power functions for T4.4and various values of h.
In Fig. 4-6 the other power functions Eq. (10.3) are
plotted relative to that of T4.4. A ratio >1means that Ti
outperforms T4.4and vice versa. Test results with χ2(1)-
quantiles are displayed by dotted curves and are denoted
by Tχ2, while those using true distributions computed
by the Monte Carlo method are displayed by solid curves
and are denoted by T(α). For T3.4only the solid curve
makes sense.
In case of weak non-linearity, i.e. h= 1000, see
Fig. 4, practically no dierence is visible. All seven power
110 |R. Lehmann and M. Lösler
Table 1: Quantiles of χ2(1)(column 2) vs. quantiles of the true distribution of Ti|H0,i= 3.1,3.2,3.3(following columns)
T4.3≡T4.4T3.1T3.2T3.3≡T4.1≡T4.2
χ2(1 −α,1) h=10 h=100 h=1000 h=10 h=100 h=1000 h=10 h=100 h=1000
α=0.10 2.71 2.99 2.73 2.71 1.93 2.63 2.70 2.99 2.73 2.71
α=0.05 3.84 4.46 3.89 3.85 2.28 3.69 3.83 4.38 3.89 3.85
α=0.02 5.41 6.78 5.51 5.42 2.46 5.12 5.38 6.40 5.51 5.42
α=0.01 6.63 8.92 6.78 6.64 2.49 6.19 6.58 8.07 6.77 6.64
Table 2: Quantiles of the true distribution of T3.4|H0
T3.4h=10 h=100 h=1000
α=0.10 2.79 2.71 2.71
α=0.05 3.96 3.85 3.84
α=0.02 5.59 5.43 5.41
α=0.01 6.86 6.65 6.63
Figure 4: Power function ratios for h=1000 (weak non-linearity).
Dotted curves: using χ2(1)-quantiles and are denoted by Tχ2,
solid curves: using true quantiles for critical values and are denoted
by T(α). Black and red solid curves visually overlap.
functions behave equally well. In case of medium non-
linearity, i.e. h= 100, there is also no great dierence be-
tween the test statistics, except for T3.2, when the χ2(1)-
quantile is used (red dotted curve), see Fig. 5. The reason
is that this approximate quantile (c= 3.69) diers much
from the true value (c= 3.84). Otherwise, χ2(1)-quantiles
are outperforming the true quantiles.
The strong non-linear case, i.e. h= 10, is depicted in
Fig. 6. The dierences between the tests are even ampli-
ed. Note, the dierent vertical scales in Fig. 4-6. When
the χ2(1)-quantile c= 3.84 is used, T3.2is unable to reject
a false H0, no matter how large |˜
ϵ|is (red dotted curve).
This is a consequence of Eq. (8.11) and the price we have to
pay that α= 0 has been obtained in the preceding section.
Figure 5: Power function ratios for h=100 (medium non-linearity).
Dotted curves: using χ2(1)-quantiles and are denoted by Tχ2,
solid curves: using true quantiles for critical values and are denoted
by T(α). Black and red solid curves visually overlap.
Figure 6: Power function ratios for h=10 (strong non-linearity). Dot-
ted curves: using χ2(1)-quantiles and are denoted by Tχ2, solid
curves: using true quantiles for critical values and are denoted by
T(α).
Due to the strong non-linearity, the power is again worst, if
the true quantiles are applied. This behavior is expected,
because a shift of the critical value cchanges αand βin
opposite directions, see Fig. 1. It follows that the increase
Hypothesis testing in non-linear models exemplied by the planar coordinate transformations |111
of probability of type 2 decision error corresponds to the
loss of probability of type 1 decision error observed in the
preceding section.
All solid curves are free of this eect, because they
truly refer to α= 0.05. This can easily be validated be-
cause for ˜
ϵ= 0 the power is always equal. The only signi-
cant dierences between the powers of Tioccur for strong
non-linearity, so we will only focus on the case h= 10, see
Fig. 6.
In the interval 0<˜
ϵ<0.2the best power is obtained
for T3.3, where the covariance propagation has been ap-
plied to Eq. (4.8a). This is even better than for the full non-
linear test T3.4(green curve). But this advantage is very
small and could be accidental. Remember that there is no
mathematical proof that Eq. (3.6) outperforms Eq. (3.7).
This has been demonstrated here. However, for values of
˜
ϵ>0.4the situation changes, as is displayed in Fig. 7.
Figure 7: Same as Fig. 6, solid curves only, but larger range of ˜
ϵ
Note that a comparison of T3.ivs. T4.jis less instruc-
tive, because if the scale is unknown, one should always
use the four-parameter transformation, even though a test
in a three-parameter transformation model may be more
powerful.
Finally, note that the results in this section are not
obtained from a “numerical experiment”, but are strictly
valid for all planar coordinate transformations with error-
free coordinates in one coordinate system and the conven-
tional assumption on the weights Eq. (4.3).
11 Conclusions
We have presented an analysis of the decision errors, when
performing LR tests in planar coordinate transformation
models. Several mathematical equivalent expressions are
conceivable to apply the LR test to one specic hypothesis
test Eq. (6.1), but dierent results are obtained.
At the end of section 3, we named three problems,
which arise, if we apply the LR test to non-linear models
in the usual way, which we now want to further comment
on.
¬The likelihood function Eq. (3.2) can only be maximized
iteratively with the danger of nding only a local maxi-
mum. For problems like many transformations, which per-
mit a unique analytical non-linear least squares solution
like Eq. (4.8), this problem does not exist. The likelihood
function has a unique maximum.
Test statistic Eq. (3.7) gives an LR-Test only in the lin-
earized GMM, i.e. not in the truly non-linear GMM. While
Eq. (3.6) requires the minimization of Ωand Ω′, Eq. (3.7)
only relies on the minimization of Ω.min(Ω′)−min(Ω)is
computed only by linear approximation. The consequence
could be a small loss of statistical power of the test, de-
pending on the degree of non-linearity. For the planar co-
ordinate transformations with α= 0.05 this has not al-
ways been found, not even for strong non-linearity. How-
ever, if αis chosen smaller, the dierences between the
power functions amplify.
®The PDF of Eq. (3.6) or Eq. (3.7) does not belong to the
well-known set of test distributions (t,χ2,Fetc.) such that
the critical values must be computed numerically. This is
usually not done, because it requires numerical eort. But
using Monte Carlo integration it is simple, as has been
demonstrated in section 9. The advantage would be that
we eectively obtain the desired value of α. Otherwise, we
found a shift of some probability from type 1 to type 2 de-
cision error or back, which is undesired.
The same analytical computation can be done for
other problems, for which explicit non-linear analytical
least squares solutions exist. This encloses
•many other transformation problems, also 3D trans-
formations (e.g. Grafarend and Awange 2003), also
transformation where coordinates in both systems are
error-aected (e.g. Chang 2015)
•many curve and surface tting problems (e.g. Ahn
2005)
The four parameter transformation is an exceptional case,
because it is intrinsically linear, but can be made non-
linear by parameterization Eq. (5.1). The resulting non-
112 |R. Lehmann and M. Lösler
linear eects can be investigated easily by comparison
with the linear model Eq. (5.2).
Also, more complex hypothesis tests can be studied in this
way, e.g. in the framework of multiple outlier detection.
The same approach can be applied to study other deci-
sion methods like model selection by information crite-
ria, which has also been applied to transformations and
other geodetic models (Lehmann 2014, 2015, Lehmann and
Lösler 2016, 2017).
AAppendix 1: Analytical solution
for the transformation with xed
scale parameter
The least squares error functional Eq. (3.3) to be minimized
reads with Eq. (4.3)
Ω(^
X) =
N
i=1
pi[(Xi−^
X0−xi·^
ϵ+yi·^
ϵ)2
+ (Yi−^
Y0−xi·sin ^
ϵ−yi·^
ϵ)2] = min (A.1)
Two necessary conditions for a minimum read
0 = ∂Ω
∂^
X0
=−2
N
i=1
piXi−^
X0−xi·cos ^
ϵ+yi·sin ^
ϵ
0 = ∂Ω
∂^
Y0
=−2
N
i=1
piYi−^
Y0−xi·sin ^
ϵ−yi·cos ^
ϵ
Using Σpi= 1 gives estimates for the translation parame-
ters:
^
X0=
N
i=1
piXi−xi·cos ^
ϵ+yi·sin ^
ϵ
=X*−x*·cos ^
ϵ+y*·sin ^
ϵ
^
Y0=
N
i=1
piYi−xi·sin ^
ϵ−yi·cos ^
ϵ
=Y*−x*·sin ^
ϵ−y*·cos ^
ϵ
Substitution ^
X0,^
Y0into the least squares error functional
yields
min = Ω^
X=
N
i=1
pi∆Xi−∆xi·cos ^
ϵ+∆yi·sin ^
ϵ2
+∆Y i−∆xi·sin ^
ϵ−∆yi·cos ^
ϵ2
=
N
i=1
pi∆X2
i+∆Y 2
i+∆x2
i+∆y2
i
−2∆Xi∆xi·cos ^
ϵ−∆yi·sin ^
ϵ
−2∆Y i∆xi·sin ^
ϵ+∆yi·cos ^
ϵ
=h+H−2c·cos ^
ϵ+s·sin ^
ϵ
The third necessary condition for a minimum reads
0 = 1
2
∂Ω
∂^
ϵ=c·sin ^
ϵ−s·cos ^
ϵ
This gives the estimate for the rotation parameter
^
ϵ= arctan s
c= arcsin s
√c2+s2= arccos c
√c2+s2
This unique stationary point must be a minimum because
Ωis bounded from below. The minimum is obtained at
Ω^
X=h+H−2c2+s2
√c2+s2=h+H−2c2+s2
BAppendix 2: Analytical solution
for the transformation with xed
rotation parameter
Similar to appendix 1, but with xed rotation parameter ϵ0
and with estimated scale parameter ^
µ, we start from
Ω^
X=
N
i=1
piXi−^
X0−xi·^
µ·cos ϵ0+yi·^
µ·sin ϵ02
+(Yi−^
Y0−xi·^
µ·sin ϵ0−yi·^
µ·cos ϵ0)2= min
and obtain
^
X0=X*−x*·^
µ·cos ϵ0+y*·^
µ·sin ϵ0
^
Y0=Y*−x*·^
µ·sin ϵ0−y*·^
µ·cos ϵ0
Substitution ^
X0,^
Y0into the least squares error functional
yields
min = Ω^
X=
N
i=1
pi∆Xi−∆xi·^
µ·cos ϵ0+∆yi·^
µ·sin ϵ02
+∆Y i−∆xi·^
µ·sin ϵ0−∆yi·^
µ·cos ϵ02
=^
µ2·h+H−2c·^
µ·cos ϵ0+s·^
µ·sin ϵ0
The third necessary condition for a minimum reads
0 = 1
2
∂Ω
∂^
µ=^
µ·h−(c·cos ϵ0+s·sin ϵ0)
This gives the estimate for the scale parameter
^
µ=c·cos ϵ0+s·sin ϵ0
h
Hypothesis testing in non-linear models exemplied by the planar coordinate transformations |113
This unique stationary point must be a minimum because
Ωis bounded from below. The minimum is obtained at
Ω^
X=(c·cos ϵ0+s·sin ϵ0)2
h
+H−2c·cos ϵ0+s·sin ϵ0
h(c·cos ϵ0+s·sin ϵ0)
=H−(c·cos ϵ0+s·sin ϵ0)2
h
CAppendix 3: Analytical solution
for the four-parameter
transformation
Following the line of appendix 2, but replacing ϵ0by ^
ϵ
gives
^
µ=c·cos ^
ϵ+s·sin ^
ϵ
h
Ω^
X=H−c·cos ^
ϵ+s·sin ^
ϵ2
h
Now also minimizing Ω^
Xfor ^
ϵyields a fourth necessary
condition
0 = 1
2
∂Ω
∂^
ϵ=1
hc·cos ^
ϵ+s·sin ^
ϵc·sin ^
ϵ−s·cos ^
ϵ
At least one of the factors must be zero, therefore we obtain
two solutions
^
ϵ1= arctan s
c,^
ϵ2= arctan −c
s
but the second solution obviously belongs to a maximum
of Ωand is dropped. We thus arrive at
^
ϵ= arctan s
c
Inserting this for ^
µand Ω^
Xgives
^
µ=√c2+s2
h
Ω^
X=H−√c2+s2
h
DAppendix 4: Covariance matrix of
the linearized GMM of the
three-parameter transformation
The normal matrix of the linearized GMM with Ain
Eq. (6.4) is of the form
ATPA =⎛
⎜
⎝
u v w
v1 0
w0 1 ⎞
⎟
⎠
with
u:= Σpix2
i+y2
i,v:= −x*·sin ^
ϵ−y*·cos ^
ϵ,
w:= x*·cos ^
ϵ−y*·sin ^
ϵ.
The corresponding inverse can be readily written
down:
ATPA−1=1
u−v2−w2⎛
⎜
⎝
1−v−w
−v u −w2vw
−w vw u −v2⎞
⎟
⎠
=1
h⎛
⎜
⎝
1−v−w
−v h +v2vw
−w vw h +w2⎞
⎟
⎠
because u−v2−w2=−x2
*−y2
*+Σpix2
i+y2
i=h.
EAppendix 5: Covariance matrix of
the linearized GMM of the
four-parameter transformation
The normal matrix of the linearized GMM with Ain Eq.
(7.6) is of the form
ATPA =⎛
⎜
⎜
⎜
⎝
^
µ2·u0
0u
^
µ·v^
µ·w
w−v
^
µ·v w
^
µ·w−v
1 0
0 1
⎞
⎟
⎟
⎟
⎠
with u,v,was in appendix 4.
The corresponding inverse can be readily written
down:
ATPA−1=1
h⎛
⎜
⎜
⎜
⎝
^
µ−20
0 1
−^
µ−1v−^
µ−1w
−w v
−^
µ−1v−w
−^
µ−1w v
u0
0u
⎞
⎟
⎟
⎟
⎠
where u−v2−w2=hhas been used (see appendix 4).
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