Application of Robust Estimation Methods for the Analysis of Outlier Measurements / Aplikácia Robustných Odhadovacích Metód Pri Analýze Odľahlých Meraní

Abstract

The basis of mathematical analysis of geodetic measurements is the method of least squares (LSM), whose bicentenary we celebrated in 2006. In geodetic practice, we quite often encounter the phenomenon when outlier measurements penetrate into the set of measured data as a result of e.g. the impact of physical environment. That fact led to modifications of LSM that have been increasingly published mainly in foreign literature in recent years. The mentioned alternative estimation methods are e.g. robust estimation methods and methods in linear programming. The aim of the present paper is to compare LSM with the robust estimation methods on an example of a regression line.

Full text

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GeoScience Engineering Volume LVII (2011), No.3
http://gse.vsb.cz p. 14-29, ISSN 1802-5420
APPLICATION OF ROBUST ESTIMATION METHODS FOR THE
ANALYSIS OF OUTLIER MEASUREMENTS
APLIKÁCIA ROBUSTNÝCH ODHADOVACÍCH METÓD PRI ANALÝZE
ODĽAHLÝCH MERANÍ
Silvia GAŠINCOVÁ1, Juraj GAŠINEC2, Gabriel WEISS3, Slavomír LABANT4
1 Ing., PhD., Institute of Geodesy, Cartography and Geographic Information Systems, Faculty of
Mining, Ecology, Process Control and Geotechnologies, Technical University of Košice
Park Komenského 19, 043 84 Košice, Slovak Republic, +421 55 602 2846
e-mail: silvia.gasincova@tuke.sk
2 doc., Ing., PhD., Institute of Geodesy, Cartography and Geographic Information Systems, Faculty of
Mining, Ecology, Process Control and Geotechnologies, Technical University of
Košice, Park Komenského 19, 043 84 Košice, Slovak Republic, +421 55 602 2846
e-mail: juraj.gasinec@tuke.sk
3 prof.,Ing., PhD., Institute of Geodesy, Cartography and Geographic Information Systems, Faculty of
Mining, Ecology, Process Control and Geotechnologies, Technical University of
Košice, Park Komenského 19, 043 84 Košice, Slovak Republic, +421 55 602 2896
e-mail: gabriel.weiss@tuke.sk
4 Ing., PhD., Institute of Geodesy, Cartography and Geographic Information Systems, Faculty of
Mining, Ecology, Process Control and Geotechnologies, Technical University of Košice,
Park Komenského 19, 043 84 Košice, Slovak Republic, +421 55 602 2989
e-mail: slavomir.labant@tuke.sk
Abstract
The basis of mathematical analysis of geodetic measurements is the method of least squares (LSM),
whose bicentenary we celebrated in 2006. In geodetic practice, we quite often encounter the phenomenon when
outlier measurements penetrate into the set of measured data as a result of e.g. the impact of physical
environment. That fact led to modifications of LSM that have been increasingly published mainly in foreign
literature in recent years. The mentioned alternative estimation methods are e.g. robust estimation methods and
methods in linear programming. The aim of the present paper is to compare LSM with the robust estimation
methods on an example of a regression line.
Abstrakt
Základom matematickej analýzy dát je metóda najmenších štvorcov (MNŠ), ktorej dvesté výročie sme si
pripomenuli v roku 2006. V geodetickej praxi sa v dôsledku napr. vplyvu fyzikálneho prostredia pomerne
často stretávame s javom, že do súboru meraných dát prenikajú odľahlé merania. Práve táto skutočnosť viedla
k modifikáciám MNŠ ktoré sa v posledných rokoch čoraz častejšie publikujú predovšetkým v zahraničnej
literatúre. Spomínanými alternatívnymi odhadovacími metódami, sú napr. robustné odhadovacie metódy
a metódy lineárneho programovania. Náplňou predloženého príspevku je porovnanie MNŠ s robustnými
odhadovacími metódami na príklade regresnej priamky.
Key words: robust estimations, method of least squares, outlier detection
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1 INTRODUCTION
As declared above the subject of the present contribution is to refer also to other ways of processing the
results of geodetic measurements as compared to the standard method being used, which is the least squares
method (LSM). Although this method is completely remade, when using it a "smudge" of undetected blunders
penetrated into the set of measured values may occur, mostly as a result of the undue impact of physical
environment on measurements. For this reason, the above mentioned alternative methods of statistical processing
appeared in geodetic practice. Given the fact that the alternative estimation methods are used primarily for the
detection of troublesome effects on measurements, the properties of these processing methods are applied to an
illustrative example of a regression line weighted first by a normal distribution and then to an example of a line
weighted by an experimental outlier. In this paper the following methods are compared with LSM: the robust M-
estimator by Huber, the robust M-estimator by Hampel and the Danish method.
2 THE METHOD OF LEAST SQUARES
The need of an adjustment calculus and the discovery of LSM itself resulted from advances in technology
for measuring and accumulation of surveying material in the field of astronomy and geodesy at the end of the
17th century. The discoverer of the least squares method, having become a classic tool for the theory of errors, is
Carl F. GAUSS (1777-1855) [7]. The essence of this method consists in minimizing the sum of squared
deviations occurred during the measurement of the behaviour of a quantity or physical phenomenon (3). The
least squares method results from the condition of a so-called L2-norm (2), whereas the norm is the number
assigned to each n-dimensional vector
),( ,,21 n
vvvv 

characterizing its size in some sense [1] , [2]. In geodesy,
the objective functions of the following type are used most frequently:
n1,i .min)(
1
1










p
p
n
ii
vv

(1)
where:
p - parameter defining the special type of objective function,
in i - vector of corrections.
Assuming
2p
(L2 norm) the objective function is as follows:
2
1
1
2
)( 







n
ii
vv

(2)
leading to the least squares method that, under certain conditions, leads to the most reliable estimators of
unknown quantities, and hence this is the method most commonly used in geodetic practice for processing
measured data. The mathematical formulation of this method is as follows:
 .min
2PvvT
iivp















n
v
v
v

2
1
v















n
p
p
p
000
....
0.0
0.0
2
1
P
k const., for i=1,2, ... n (3)
where:
v - vector of corrections
P - matrix of weighting coefficients (weights of p measurements) being ordered along the diagonal of
the weighting matrix.
The measurement weights are proportional numbers, qualitatively evaluating the achieved
measurement result. Introducing the weights we prefer a more accurate measurement that takes a share
in the measurand adjustment.
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The least squares method will also be explained on a one-dimensional linear model, while all
the estimation methods will be demonstrated on an example of a regression line. Let the following
linear relationship exists between the variable y, variables X (Fig. 1) and the random component ui:
equation modellinear y
n1,2i kde
0
22110
uX
uXXXy iikkiii





(4)
where:
y i - dependent variables
X1 , X2, ... X k - independent variables

0 - model parameter expressing the value attained by y when X
equals zero, called intercept

- regression coefficient indicating the slope of the regression line, i.e. the rate at which y i increases
per unit increase in X 1, X 2, ... Xk
ui – random component.
Fig. 1 One-dimensional linear model
We may rewrite the model using the following matrix notation:
uXβy
, (5)
where:
y - n-dimensional vector of the dependent variables,
X - matrix (k +1) independent variables
u - n-dimensional vector of the random components,

- unknown (unobservable) parameters we need to determine; in geodesy
we use the term estimators.
Thus, when deriving an estimator we result from the matrix notation (5). The problem lies in finding
the estimator

ˆ
so that the estimated regression line
uXy 

ˆ
ˆ
approximates best the regression line
uXy 

.
The following applies to the difference
yy ˆ

, the so-called vector of residuals (the vector of
corrections in geodesy)
n
vvvv ,,, 21 
:
0200 400 600 800 1000 1200 1400 1600 1800 2000
0
1
2
3
4
5
6
7
8
9
10
x - nezávislé premenné
y - závislé premenné
y=   X

1 jednotka


jednotiek
x- independent variables
y-dependent variables
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vβXy
βXyyyv


ˆ
čoho z
ˆ
ˆ
(6)
whereof
Using LSM, we search for the estimator
β
ˆ
of the vector of parameters

so that the sum of the
squared residuals (3) is minimum.
)
ˆ
()
ˆ
(
1
2
iβXyβXyvv 


TT
n
i
v
(7)
Multiplying (7) we get:
.
ˆˆˆˆ
ˆˆˆˆ
2
ˆˆ
2
βXβXβXyyβXyy
βXβXβXβXβXyyβXyβXyyv T
TTTTTT
TTTTTTTTTT v


(8)
Let us differentiate the equation (8) with respect to
T
β
ˆ
:
XβXyX
β
vv TT
T
T



ˆ
(9)
If we set the derivative (9) equal to zero, then we obtain the relationship
yXβXX TT 
ˆ
(10)
that is the matrix notation of normal equations. The matrix
XXT
is nonsingular, i.e. it is invertible.
Therefore, the following applies to the vector of unknown parameters from the relation (10):
yXXXβTT 1
)(
ˆ

(11)
If the line equation (Fig. 2) has the generally known form
baXy
(12)
where a and b are the searched estimators to be determined, the line equation has the following form
in a matrix notation:
vAy  X
, (13)
where:
v: vector of corrections,
A: configuration matrix, the matrix of partial derivatives,
X: searched estimators (in geodetic calculations, X is replaced by the parameter

,
these are coordinates as a rule).
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Fig. 2 Regression line in general
Whereas the relationships between the measured and the unknown variables are expressed by an
intermediary function of the searched unknown parameters (estimators), the given model may be
rewritten into the following form:
ΘAvl
vly
ˆ


(14)
b
y
a
y




,A
(15)




















































b
a
x
x
x
nnn
ˆ
ˆ
.
1
1
1
ˆ
2
1
2
1
2
1


v
v
v
y
y
y
Avl
yl
(16)
The following applies to the vector of residuals:
lAv  ˆ
, (17)
which leads to the Guss-Markov model:
l
lQ
lAv
2
0
ˆ



(18)
When using LSM the following applies to the sought estimators after adjusting the parameters
(indices):
0200 400 600 800 1000 1200 1400 1600 1800 2000
0
1
2
3
4
5
6
7
8
9
10
x - nezávislé premenné
y - závislé premenné
a
y=aX + b
configuration matrix
x- independent variables
y-dependent variables
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 
 
(11)y analogousl)(
ˆ
)(unit matrix
,)(
ˆ
1
111
lAAAΘ
IQ
lQAAQAΘ
TT
l
l
T
l
T





. (19)
3 ROBUST ESTIMATION METHODS
In the second half of the last century unconventional estimation methods were developed in the
theory of linear programming in addition to the standard estimation methods. In those methods, other
variable, not the arithmetic mean, is selected as the parameter of centrality. From these methods, the
so-called robust estimation methods are preferred in geodetic practice [10] , [9] , [8] , [6] , [5] , [4].
Two types of robust estimations are known: robust estimations applied on the basis of LSM,
when the sum of squared corrections is replaced by more appropriate correction functions: e.g.
maximum credible estimators (robust M - estimators) [10] , [9] , [8] , [6] , [5] , [4] and pure robust
methods that include linear programming methods, such as a simplex method.
The M robust method of adjustment on the principle of LSM occurs when there is no minimized
function
vvT
in the estimation process, but another suitably chosen correction function
)( i
v

called the
function of losses (estimators).
min)( 
i
v

, (20)
that generates the so-called influence function
)( i
v
for the estimation process, characterizing
the impact of errors on the adjusted values, which the following applies to:
 0)( i
v
, (21)
where:
i
i
iv
v
v

 )(
)(

. (22)
In order to be robust, the adjustment should be performed by an iterative method with variable
weights, i.e., so that the weights for observables are determined as functions of corrections in each
iteration step
v
v
vp i
)(
)( 

, (23)
where
)( i
vp
is the weighting function in the adjustment process solution. The iterative robust
estimation algorithm proceeds as follows:
1. In the first iteration step the standard LSM adjustment with weights
1
)1(
1p
is carried out ((1)-
the iterative step for the i-th observable), in case of heterogeneous measurements it is
necessary to perform their homogenization by means of the matrix of the square roots of the
weights
P
(this feature is available in a MATLAB program - rootm (P)) [4] .
2. From the corrections obtained in the first iteration step using the weighting function
)( i
vp
the
new weights are determined to be used in the next step and by analogy in further ones.
New weights are created according to a relevant regulation for the weighting function up to the
end of the iteration process being selected with a reasonable number of steps so that the acceptable
(
6)
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convergence of weights occurs in the last steps. The weighting functions are determined based on
previously theoretically and empirically investigated and verified different assumptions and for
different kinds of measurements.
Using the robust estimation techniques in the processing of horizontal geodetic control networks
a great deal of attention was paid to v [4]; for that reason the present contribution demonstrates the
following robust estimation methods with their experimented and verified constants on an example of
a regression line [3] , [4] , [5] , [6] , [8] , [9] , [10] : the robust M - estimator by Huber, the robust M -
estimator by Hampel and the Danish method.
2.1 Robust M-estimator by Huber
This estimator uses the following functions and relevant tuning- damping constants
estimation function
1,5usually isconstant damping theof valueThe
v
2
1
.
v
2
1
)(
2
2

















kkvk
kv
v

(24)
influence function
v )(.
v
)(
)(
)( 














 kvsignk
kv
v
v
v

(25)
weighting function














k
v
k
k
v
v
vp v
v 1
)(
)(
(26)
The behaviours of these individual functions may be illustrated graphically as well [4] , [10]. Due to the
scope of the contribution and the fact that the principle of robust M-estimation methods consists in the repetition
of LSM with gradual changing the weights according to the relevant regulation (23) at the used processing
methods, we present only the graph of the weighting function.
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Fig. 3 Graphic behaviour of the weighting function for the robust M-estimator by Huber
2.2 Robust M-estimator by Hampel
This estimator uses the following functions and relevant damping constants:
estimation function
 































































vcabcaab
cvb
bc
vc
bca
aab
bvaava
avv
v
)(
2
1
2
1
1
2
1
2
1
2
1
.
2
1
)(
2
2
2
2
2

(27)
influence function

























vc
cvbvsign
bc
vc
a
bvavsigna
av
v
0
)(..
)(.
v
)(
(28)
-6 -4 -2 0 2 4 6
0
0.2
0.4
0.6
0.8
1
1.2
Grafický priebeh váhovej funkcie
v - opravy
p(v) - váhy jednotlivých opráv
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weighting function
 





























vc
cvb
vbc
vc
a
bva
v
a
a
vp
0
.
v 1
)(
(29)
Fig. 4 Graphic behaviour of the weighting function for the robust M-estimator by Hampel
2.3 Danish method
The strategy of the Danish method is that it reduces (shrinks) the impact of remote
measurements (outliers) on the estimates of quantities. The principle of this method is based on the
indication of the outliers by corresponding major corrections. After the standard adjustment of the
estimations of the first-order parameters by the method of least squares through the Gauss - Markov
model, the a priori weights of measurements are replaced by correction functions. The next iteration
step is the adjustment of the initial weights of measurements according to the following relation
,2,1 )(
1
ivppp ii
(30)
which results in the increase of the absolute values of outlier measurements, while reducing their
deformation impact on the network geometry. The iterative cycle is repeated until expected results are
achieved. In the current horizontal geodetic control networks the solution requires no more than (10-
15) iterations [4] , [3]. For this method the following functions and standards have been derived:
influence function:












 cev
cv
vb
va v .
v
)(
(31)
At present different types of exponential functions are used for the weighting function [3]. In
order to demonstrate this method, the following weighting function was used in the practical solution:
-10 -8 -6 -4 -2 0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1.2
Grafický priebeh váhovej funkcie
v - opravy
p(v) - váhy meraní
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weighting function

























s
vp
pre exp
s
vp
pre 1
)(
0
ii
0
0
ii
c
cs
pv
c
vp
ii
i
(32)
with these standards:
for the 1st iteration step
0a
,
for the 2nd and 3rd iteration steps
05.0a
and throughout the iterative process
3,3  cb
.
Thus, the equation (32) is defined by the following interval:
cspc  01i /v
(33)
which the weights of measurements are started to be determined from. The constant c is chosen
usually between 2 to 3 and depends on the redundancy (excess measurements) determined by the
Gauss - Markov model and the quality of measurands. If the constant c <2 the method being used is
robust, in case c> 3 the processing method is changed to LSM.
4 EMPIRICAL DEMONSTRATION
This part of the paper is devoted to examining the properties of individual estimation methods
using an example of a regression line that may be represented from the geodetic point of view, e.g. by
the electronic rangefinder equation.
4.1 Adjustment of the regression line through LSM
The properties of estimation methods were studied on an illustrative example of a regression
line
baxy
whose modified form
ppmdab
d..

is normally used in geodesy to characterize the
accuracy of electronic rangefinders, where the parameter
a
characterizes the effect of independent
errors and the parameter
b
characterizes the effect of errors dependent on measured distance. The
constant
ppm
(parts per million - means "out of million") is equal to 10-6. The adjusted regression line
in the form
][32 mmppmdy 
was weighted with a normal distribution by means of the least squares
method (Fig. 5).
Fig. 5 Deterministic model of the regression line weighted with a normal distribution
Note:
d
STD


,
0200 400 600 800 1000 1200 1400 1600 1800 2000
2
3
4
5
6
7
8
d[m]
STD[mm]
STD=3[mm]* 2ppm*d
experimental value
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10.2478/v10205-011-0007-1 24
GeoScience Engineering Volume LVII (2011), No.3
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When using this method the basis is the model
ΘAvl ˆ

leading to the famous Gauss-Markov model
[1]. The results of adjustment of the regression line by means of LSM are demonstrated in Table 1, the
graphical interpretation of the results obtained is shown in Fig. 6.
Fig. 6 Graph of theoretical values of the regression line adjusted by means of LSM
Tab. 1 Adjustment results of the regression line weighted with a normal distribution by means of LSM
no. of
m.
Theoretical line
LSM
d
d
erm

.det
d

..exp

v

[m]
[mm]
[mm]
[mm]
[mm]
[mm]
1.
200.00
3.20
3.40
0.20
0.2473
-0.05
2.
400.00
3.40
3.80
0.40
0.4145
-0.01
3.
600.00
4.50
4.20
-0.30
-0.3182
0.02
4.
800.00
5.30
4.60
-0.70
-0.7509
0.05
5.
1000.00
4.70
5.00
0.30
0.2164
0.08
6.
1200.00
5.00
5.40
0.40
0.2836
0.12
7.
1400.00
5.70
5.80
0.10
-0.0491
0.15
8.
1600.00
6.50
6.20
-0.30
-0.4818
0.18
9.
1800.00
6.30
6.60
0.30
0.0855
0.21
10.
2000.00
6.40
7.00
0.60
0.3527
0.25
Regression line parameters:
Deterministic shape of the line y = 3,0 [mm] + 2,0*ppm*d
Parameters of the line estimated by LSM y = 3,1 [mm] + 1,8*ppm*d
Note:
3.2..  ppmdppmdabSTD d

v

,
dd
erm


..exp.det

0200 400 600 800 1000 1200 1400 1600 1800 2000
2
3
4
5
6
7
8
d[m]
STD[mm]
STD=3[mm]* 2ppm*d
experimental value
regression line by LSM
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GeoScience Engineering Volume LVII (2011), No.3
http://gse.vsb.cz p. 14-29, ISSN 1802-5420
0200 400 600 800 1000 1200 1400 1600 1800 2000
2
3
4
5
6
7
8
9
10
d[m]
STD[mm]
STD=3[mm]* 2ppm*d
experimental values
Given the fact that the present contribution is devoted to the issues of estimation methods, which allow
to reveal the so-called outlier measurements in the set of measured data and for which we do not select
the arithmetic mean as the centrality parameter, the regression line is deliberately weighted with just
one outlier, in order to track the performance of such methods, Fig. 6. Such modified regression line
was adjusted first by the method of least squares and then by the iteration-based robust M-estimation
methods.
Fig. 7 Graph of theoretical values of the proposed regression line with an experimental outlier
The graphic interpretation of the results obtained by the adjustment of the modified regression line
processed by LSM is shown in the following figure:
Fig. 8 Graphical interpretation of the adjustment results of the regression line with an experimental outlier
by LSM
The numerical results of the line adjustment by the least squares method is presented in Tab. 2.
0200 400 600 800 1000 1200 1400 1600 1800 2000
2
3
4
5
6
7
8
9
10
d[m]
STD[mm]
STD=3[mm]* 2ppm*d
experimental values
refgression line by LSM
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10.2478/v10205-011-0007-1 26
GeoScience Engineering Volume LVII (2011), No.3
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Tab. 2 Adjustment results of the regression line with an experimental outlier by LSM
no. of m.
Theoretical line
LSM
d
d
erm

.det
d

..exp

v

[m]
[mm]
[mm]
[mm]
[mm]
[mm]
1.
200.00
3.20
3.40
0.20
-0.0436
0.24
2.
400.00
3.40
3.80
0.40
0.2594
0.14
3.
600.00
4.50
4.20
-0.30
-0.3376
0.04
4.
800.00
5.30
4.60
-0.70
-0.6345
-0.07
5.
1000.00
4.70
5.00
0.30
0.4685
-0.17
6.
1200.00
5.00
5.40
0.40
0.6715
-0.27
7.
1400.00
5.70
5.80
0.10
0.4745
-0.37
8.
1600.00
6.50
6.20
-0.30
0.1776
-0.48
9.
1800.00
9.50
6.60
-2.90
-2.3194
-0.58
10.
2000.00
6.40
7.00
0.60
1.2836
-0.68
Regression line parameters:
Deterministic shape of the line y = 3.0 [mm] + 2.0*ppm*d
Parameters of the line estimated by LSM y = 2.7 [mm] + 2.5*ppm*d
4.2 Regression line adjustment by means of robust M-estimation methods
The robust M-estimator methods along with the Denmark method are based on the principle of
the least squares method. These are iterative methods (in which the adjustment process is repeated
several times); their principle consists in so-called “reweighting”, which means that the weights are
being changed intentionally. In the first iteration step, the standard method of least squares is
performed, where the weight of each measurement is the same and equal to one. In the next step, the
weights are correction functions. First the regression line weighted with a normal distribution and then
the regression line weighted with an outlier was adjusted through the above robust M-estimation
methods and the Danish method.
The adjustment results of both lines are shown in Tables 4 and 5, together with the graphical
interpretation of processing results.
Tab. 3 Comparison of the adjustment results of the regression line weighted with a normal distribution by means
of LSM and robust M-estimation methods
LSM
DANISH
HUBER
HAMPEL
d
d exper.
ddeter

v

v
p

v
p

v
p

[m]
[mm]
[mm]

[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
200.00
3.20
3.40
0 .20
0.2473
-0.05
0.2473
1
-0.05
0.2473
1
-0.05
0.2473
1
-0.05
400.00
3.40
3.80
0.40
0.4145
-0.01
0.4145
1
-0.01
0.4145
1
-0.01
0.4145
1
-0.01
600.00
4.50
4.20
-0.30
-0.3182
0.02
-0.3182
1
0.02
-0.3182
1
0.02
-0.3182
1
0.02
800.00
5.30
4.60
-0.70
-0.7509
0.05
-0.7509
1
0.05
-0.7509
1
0.05
-0.7509
1
0.05
1000.00
4.70
5.00
0.30
0.2164
0.08
0.2164
1
0.08
0.2164
1
0.08
0.2164
1
0.08
1200.00
5.00
5.40
0.40
0.2836
0.12
0.2836
1
0.12
0.2836
1
0.12
0.2836
1
0.12
1400.00
5.70
5.80
0.10
-0.0491
0.15
-0.0491
1
0.15
-0.0491
1
0.15
-0.0491
1
0.15
1600.00
6.50
6.20
-0.30
-0.4818
0.18
-0.4818
1
0.18
-0.4818
1
0.18
-0.4818
1
0.18
1800.00
6.30
6.60
0.30
0.0855
0.21
0.0855
1
0.21
0.0855
1
0.21
0.0855
1
0.21
2000.00
6.40
7.00
0.60
0.3527
0.25
0.3527
1
0.25
0.3527
1
0.25
0.3527
1
0.25
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10.2478/v10205-011-0007-1 27
GeoScience Engineering Volume LVII (2011), No.3
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Regression line parameters:
Deterministic shape of the line y = 3.0 [mm]+2,0*ppm*d
Parameters of the line estimated by LSM y = 3.1 [mm]+1,8*ppm*d
Parameters of the line estimated by the Danish method y = 3.1 [mm]+1,8*ppm*d
Parameters of the line estimated by the Huber method y = 3.1 [mm]+1,8*ppm*d
Parameters of the line estimated by the Hampel method y = 3.1 [mm]+1,8*ppm*d
Fig. 9 Graphical interpretation of the adjustment results of the regression line weighted with a normal
distribution by means of LSM and robust M-estimation methods
In case of the regression line non-weighted with any outlier the results of the adjustment by LSM and
the robust M-estimation methods are identical as demonstrated in Tab. 4 and in the graphical
interpretation of the processing results, Fig. 9. The comparison of the adjustment results of the
modified regression line (weighted with an outlier) is presented in Tab. 4 and Fig. 10.
Tab. 4 Comparison of the adjustment results of the regression line weighted with a normal distribution by means
of LSM and robust M - estimation methods
LSM
DANISH
HUBER
HAMPEL
d
d
exper.
ddeter

v

v
p

v
p

v
p

[m]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
200.00
3.20
3.40
0.20
-0.0436
0.24
0.2369
1.0000
-0.04
0.2369
1.0000
-0.04
0.2365
1.0000
0.0000
400.00
3.40
3.80
0.40
0.2594
0.14
0.4090
1.0000
-0.01
0.4090
1.0000
-0.01
0.4088
1.0000
0.2167
600.00
4.50
4.20
-0.30
-0.3376
0.04
-0.3189
1.0000
0.02
-0.3189
1.0000
0.02
-0.3189
1.0000
-0.4667
800.00
5.30
4.60
-0.70
-0.6345
-0.07
-0.7468
1.0000
0.05
-0.7468
1.0000
0.05
-0.7466
1.0000
-0.8500
1000.00
4.70
5.00
0.30
0.4685
-0.17
0.2253
1.0000
0.07
0.2253
1.0000
0.07
0.2257
1.0000
0.1667
1200.00
5.00
5.40
0.40
0.6715
-0.27
0.2974
1.0000
0.10
0.2974
1.0000
0.10
0.2980
1.0000
0.2833
1400.00
5.70
5.80
0.10
0.4745
-0.37
-0.0305
1.0000
0.13
-0.0305
1.0000
0.13
-0.0298
1.0000
0.0000
1600.00
6.50
6.20
-0.30
0.1776
-0.48
-0.4584
1.0000
0.16
-0.4584
1.0000
0.16
-0.4575
1.0000
-0.3833
1800.00
9.50
6.60
-2.90
-2.3194
-0.58
-3.0863
0.1059
0.19
-3.0863
0.4860
0.19
-3.0852
0.6483
-2.9667
2000.00
6.40
7.00
0.60
1.2836
-0.68
0.3858
1.0000
0.21
0.3858
1.0000
0.21
0.3871
1.0000
0.5500
Regression line parameters:
Deterministic shape of the line y = 3.0[mm]+2,0*ppm*d
Parameters of the line estimated by LSM y = 2.7[mm]+2,5*ppm*d
0200 400 600 800 1000 1200 1400 1600 1800 2000
2
3
4
5
6
7
8
d[m]
STD[mm]
STD=3[mm]* 2ppm*d
weighted points
regression line by LSM
regression line determined by the Danish method
regression line determined by the method by HUBER
regression line determined by the method by HAMPEL
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10.2478/v10205-011-0007-1 28
GeoScience Engineering Volume LVII (2011), No.3
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Parameters of the line estimated by the Danish method y = 3.1[mm]+1,9*ppm*d
Parameters of the line estimated by the Huber method y = 3.1[mm]+1,9*ppm*d
Parameters of the line estimated by the Hampel method y = 3.1[mm]+1,9*ppm*d
Fig. 10 Graphical interpretation of LSM and the robust M-estimation methods on the example of a
regression line weighted with an experimental outlier
It is clear from the processing results that in this case the robust M-estimation methods came to
identical results, herewith they assigned the largest correction and the smallest weight value to the
outlier (Table 4); the graphical representations of the adjustment results of such modified line by
means of the robust M-estimation methods are identical.
REFERENCES
[1] BÖHM, J., RADOUCH, V., HAMPACHER, M.: Theory of errors and adjustment calculus,
GKP Prague.
[2] BÖHM, J., RADOUCH, V.: Adjustment calculus, Prague, Cartography, 1978.
[3] CASPARY, W.F.: Concepts of network and deformation analysis, Monograph 11, School of
Surveying, The University of New South Wales, Kensington, NSW, Australia, 1987.
[4] GAŠINCOVÁ, S.: Processing of 2D networks with robust methods. PhD thesis. Košice, 2007,
str. 90.
[5] HAMPEL, F. et al: Robust statistics, the approach based of influence functions. J.Willey&Sons,
New York, 1986.
[6] HAMPEL, F.: Contribution to the theory of robust estimation. PhD. Thesis, Univ. of California,
1968.
[7] HOBST, E., HOBSTOVÁ, M.: Carl Friedrich Gauss - the founder of modern mathematics.
Advances in Mathematics, Physics and Astronomy, Vol. 52 (2007), No. 4, 296—307, Nurnberg,
2007.
[8] HUBER, P.J.: Robust statistics. Willey&Sons, New York, 1981.
0200 400 600 800 1000 1200 1400 1600 1800 2000
2
3
4
5
6
7
8
9
10
d[m]
STD[mm]
STD=3[mm]* 2ppm*d
weighted points
regression line by LMS
regression line determined by the Danish methodu
regression line determined by the method by HUBER
regression line determined by the method by HAMPEL
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10.2478/v10205-011-0007-1 29
GeoScience Engineering Volume LVII (2011), No.3
http://gse.vsb.cz p. 14-29, ISSN 1802-5420
[9] HUBER, P.J.: Robust estimation of a location parameter. Ann Math Stat 35: 73-101, 1964.
[10] J, GER, R., MÜLLER, T. Saler, H. SCHV Ä BLE, R. Klassischle und robuste
Ausgleichungsverfahren, Herbert Wichmann Verlag, Heidelberg, 2005.
RESUMÉ
V predloženom príspevku bol zámerne zvolený jednoduchý, ale o to názornejší príklad demonštrujúci
pozitívne vlastnosti alternatívnych, v spracovaní geodetických meraní čoraz častejšie používaných a
odporúčaných robustných M - odhadovacích metód na báze iteratívneho vyrovnania metódy najmenších
štvorcov s účelovo znižovaným, deformujúcim vplyvom týchto chýb na odhadované parametre a pre ich
striktnejšiu identifikáciu v súbore meraných veličín. Matematická báza týchto metód sa pomerne nenáročne
implementuje do algoritmu metódy najmenších štvorcov a v práci uvedené M–robustné odhady podľa Hubera,
Hampela alebo dánskej metódy poukazujú na vzájomnú tesnosť ich výsledkov.
Cieľom príspevku bolo poukázať na skutočnosť, že robustné odhadovanie metódy predstavujú silný
nástroj na identifikáciu odľahlých meraní, ktoré z určitých objektívnych dôvodov prenikli do súborov dát
meraných terénnych veličín. Záleží individuálne na každom prípade či odľahlé merania budú zo súborov
meraných veličín eliminované, resp. či budú nahradené novými, nezávislými meraniami alebo analýzou ich
genézy bude problém hlbšie skúmaný pre podrobnejšie a objektívnejšie popísanie experimentálneho merania
a dynamiky stavu obklopujúceho fyzikálneho prostredia komplexnejšími matematicko–štatistickými modelmi.
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