influence of systematic effects in stochastic and functional models

INFLUENCE OF SYSTEMATIC EFFECTS
IN STOCHASTIC AND FUNCTIONAL MODELS
Hans PELZER
In: BORRE, Kai / WELSCH, Walter (Eds.) [1982]:
International Federation of Surveyors - FIG Proceedings Survey Control Networks
Meeting of Study Group 5B, 7th - 9th July, 1982, Aalborg University Centre, Denmark
Schriftenreihe des Wissenschaftlichen Studiengangs Vermessungswesen der Hochschule der Bundeswehr München, Heft 7, S. 309-320
ISSN: 0173-1009
INFLUENCE OF SYSTEMATIC EFFECTS
IN STOCHASTIC AND FUNCTIONAL MODELS
Hans PELZER
Hannover, Federal Republic of Germany
1. Insufficient adjustment models
In the adjustment of geodetic networks as well as in other
adjustment problems we describe the relationships between
observed quantities and unknown parameters by a functional
model, e. g. in the linearized form
� = A x
l
� ,
(1.1)
� : true values of observed quantities
l
(n x 1 vector),
x
� : true values of unknown parameters
(u x 1 vector),
A : n x u matrix of coefficients.
� in (1.1) by its
Replacing the true observation vector l
observed value l we get the observation equation
l+v = A x
�
(1.2)
v : vector of residuals,
x
� : parameter vector to be estimated.
The stochastic properties of l are described by
�
E�l� = μ = l
(1.3)
l
ε
= l-μ
Σll
= E�εεT � = σ2o Q
μ
l
,
(1.4)
l
ll
= σ2o P-1
: expected value of l,
Σll
: covariance matrix of l,
Q
: cofactor matrix,
P
: weight matrix,
σ2o
: variance of unit weight.
ll
309
(1.5)
The covariance matrix Σll together with equation (1.3) is
called the stochastic model, and in combination with the
functional model (1.1) the least square solution of (1.2)
is given by the well-known formulae
Nxx x
� - nx = O ,
(1.6)
= N-1
,
xx
(1.8)
= Q
(1.9)
Nxx = AT P A ; nx = AT P l ,
Q
x
�
xx
xx
nx ,
Σxx = σ2o Q
xx
,
(1.7)
(1.10)
Σxx : covariance matrix of x
�.
Of course, the adjustment model cannot be more than an approximation to the physical reality, which may be sufficient or
not. If the model is not sufficient, the residual vector v
does not agree with its covariance matrix
Σvv = σ2o �Q - AT Q
ll
xx
A� ,
(1.11)
detectable by statistical tests.
In the case of insufficient adjustment model the functional
model (1.1) has to be extended. Generally this can be done
by additional parameters, forming a vector ξ,
x
�
⋯
�
�
A
⋮
B
] � �
l = A x+B ξ = [
�
ξ
� : true value of additional parameters
ξ
(m x 1 vector),
B : n x m matrix of coefficients.
310
(1.12)
� is known, its influence to the observaIf the true value ξ
tion vector l may be considered as a systematic model error
∆,
�-∆
� = A x
l
� ,
�
∆
�
∆
(1.13)
� ,
= B ξ
(1.14)
: systematic model error in
(n x 1 vector).
In this case in (1.2) and (1.7) the observation vector l
has to be replaced by
�
� = l-B ξ
l = l-∆
(1.15)
and the ordinary solution (1.6) – (1.10) can used now as
before.
� is not known, i.e. the
Normally, however, the true value ξ
model (1.12) leads to another solution.
2. Typical examples
2.1 Calibration parameters of EDM-instruments
Normally the calibration of EDM-instruments, i.e. in simple
cases the determination of the scale factors and additive
constants, will be carried out in special calibration networks
or by laboratory tests. But, as an alternative procedure, these
calibration parameters or some of them can be determined directly
from the network observations, this method is often called "onthe-job-calibration". In these cases the calibration parameters
play the role of the additional parameters ξ in equation
(1.12).
311
On the other hand, even if a special calibration measurement
has been carried out, the resulting parameters are of limited accuracy and may be improved by the network adjustment.
2.2 Centering errors
In the functional model (1.1) all observations from or to a
certain network station are considered as centered to one and
the same point. Of course, this model is disturbed if one or
more centering errors occur, such centering errors may be
considered as additional parameters ξ in the extended model
(1.12).
In practice we have to distinguish two cases. In the first one
we have only a limited number of possibly gross centering errors
in the network; these errors can be estimated in the extended
model (1.12). In the second case, however, all observations
contain small centering errors which can be described by their
statistical distribution, this case is considered in Ch. 4 and 5.
3. Estimation of additional parameters without stochastic
information
In this model the additional parameters ξ in (1.12) are considered as nonstochastical quantities like the main parameters x;
i.e. they have to be estimated in the same manner. The model
may be used, for example, in the case of on-the-job-calibration
(Ch. 2.1), or for the purpose of determination of gross centering errors (Ch. 2.2).
The extended model (1.12) leads to the extended observation
equation
x
�
⋯
A
⋮
B
[
]
l+v =
� �
(3.1)
�
ξ
� : estimated value of ξ
�.
ξ
312
and together with the unchanged stochastic model (1.5) we
get the normal equation
x
�
A T P A | AT P B
AT P l
� ------+------ � �--� - � ------ � = O
�
ξ
B T P A | BT P B
BT P l
(3.2)
or, with other symbols for abbreviation,
nx
x
�
� ------+------ � �--� - � ------ � = 0 .
nξ
�
N
|
N
ξ
Nxx
|
Nxξ
ξx
(3.3)
ξξ
With
Q
xx
|
Q
Nxx
xξ
|
Nxξ
�------+------� = �------+------�
Q
| Q
Nξx | Nξξ
ξx
-1
,
(3.4)
ξξ
the solution of (3.2) resp. (3.3) is
Q
| Q
nx
x
�
xx
xξ
�--� = �------+------� �--� .
nξ
�
Q
| Q
ξ
ξx
(3.5)
ξξ
This solution is advantageous if one is interested in the
�, e.g. if the statistical significance
numerical value of ξ
� has to be tested.
of ξ
� is not of special
If, however, the numerical value of ξ
� may be eliminated beforehand by
interest ξ
� = N - N N-1 N
N
,
xx
xx
xξ ξξ ξx
n
�x = nx - Nxξ N-1
n ,
ξξ ξ
� x
N
�-n
�x = O .
xx
313
(3.6)
(3.7)
(3.8)
The solution of (3.8) is simply
�-1 n
x
� = N
� = Q
xx x
xx
Q
xx
�-1 ,
= N
xx
n
�x ,
(3.9)
(3.10)
x
� normally distributed with expected value
E�x
�� = x
�
(3.11)
and covariance matrix
Σxx = σ2o Q
.
xx
(3.12)
�
From (3.6) follows that the matrix N
may be singular
xx
though Nxx is not, i.e. the additional parameters ξ may be
not estimable from the observation vector l. But even in
�
�-1 exists, the
is regular and Q = N
the case where N
xx
xx
xx
inequality.
tr �Q � ≥ tr �N-1
�
xx
xx
holds, i.e. the determination of ξ in the extended model is
connected with a loss in precision of the main parameters x.
That is the price we have to pay for an unbiased estimation
of x.
In order to proof (3.12) we take from (3.4)
Q
xx
= N-1
+ N-1
N Q
xx
xx xξ
ξξ
Nξx N-1
= N-1
+ F FT ,
xx
xx
(3.14)
with an auxiliary matrix
1
F = N-1
N Q2
xx xξ
ξξ
.
(3.15)
Therefore
tr �Q � = tr �N-1
� + tr �F FT �
xx
(3.16)
tr �F FT � ≥ O .
(3.17)
xx
and, for any real matrix F,
314
4. Estimation of additional parameters as pseudo observations
In many cases it is impossible to follow the solution of Ch. 3
�
in (3.10) is singular and no simple inverse
because the matrix N
xx
Q
exists. In such cases, however, we may often regard the additioxx
nal parameters ξ as stochastic quantities with expected value
and given matrix
E �ξ� = O
(4.1)
Σξξ = σ2o Q
ξξ
.
(4.2)
�
But, independent of the question whether or not the matrix N
is
xx
singular, the parameter vector ξ often is of stochastic nature
and should be treated in the way described below.
For example in the case of determination of calibration parameters
(Ch. 2.1) preliminary values of these parameters may be known from
calibration procedures, together with the corresponding covariance
matrix Σξξ . Then we may interpretate the remaining errors in the
parameters as stochastic values with properties (4.1) and (4.2)
and the network adjustment only as a possibility to improve these
calibration parameters.
As another example we may consider the centering errors (Ch. 2.2)
as random errors to be described by (4.1) and (4.2), where the
covariance matrix Σξξ has to be estimated based on practical
experience.
On the basis of (4.1) and (4.2) we may extend the observation
equation (3.1) by addition of pseudo observations
v
x
�
A | B
l
�--� + �--� = �----+----� �--� ,
�
ξ
�
O
O | I
ξ
(4.3)
corresponding to the stochastical model
Σll | O
Q
ll
|O
P
|O
Σ = �---+---� = σ2o �---+---� = σ2o �---+---�
O |Q
O | Σξξ
O | Pξξ
ξξ
315
-1
.
(4.4)
The resulting normal equation is, with symbols defined in
(3.3)
nx
x
�
�----+-------� � --- � - � --- � = O
nξ
�
Nξx | Nξξ +Pξξ
ξ
Nxx |
Nxξ
(4.5)
with the solution
� | Q
�
Q
nx
x
xx
xξ
�---� = �----+----� �---� ,
ξ
nξ
� | Q
�
Q
ξx
ξξ
�,
where the cofactor matrix Q
�
Q
xx
�
|Q
(4.6)
Nxx |
xξ
Nxξ
� = �----+----� = �---- + -------�
Q
Nξx | Nxx +Pξξ
� |Q
�
Q
ξx
-1
,
(4.7)
ξξ
is not identical with the corresponding matrix in (3.4).
Similar to Ch. 3, the vector 𝜉̅ can be eliminated beforehand by
� = N - N �N +P �
N
xx
xx
xξ
ξξ
ξξ
n
�x = nx - Nxξ �Nξξ +Pξξ �
-1
Nξx ,
(4.8)
-1
nξ ,
(4.9)
leading to the reduced normal equation
� x-n
N
�x = O
xx
(4.10)
�
Q
(4.11)
with the solution
x
�
xx
�-1 ,
= N
xx
�
= Q
xx
n
�x ,
(4.12)
316
where, of course, the parameter vector x
� is different from
the corresponding vector x
� in (3.9).
Evidently, the model considered in this chapter is a general
one and contains both, the model of Ch. 1 without additional
parameters as well as that of Ch. 3. The first model is defined by
Σξξ = O , Pξξ → ∞ I , �Nξξ +Pξξ �
-1
→ O ,
(4.13)
and the second one follows simply from
Σξξ → ∞ I , Pξξ = O ,
(4.14)
see (4.8) and (4.9).
5. Correlation model
On the same conditions as in Ch. 4, i.e. if the additional
parameters ξ can be considered as random variates with properties (4.1) and (4.2), we may introduce a quite different
adjustment model. The model equation (1.12) we can write in
the form
�-B ξ
� = l
�-∆
� = A x
l
� ,
(5.1)
cf. (1.13), leading to the observation equation
l-∆ = A x
� ,
(5.2)
E�∆� = E �B ξ� = B E �ξ� = O
(5.3)
where ∆ is a random vector of systematic effects with expected
value (cf. (4.1) and (1.14))
317
and variance matrix Σ∆∆ ,
Σ∆∆ = σ2o Q
where Q
∆∆
,
∆∆
(5.4)
follows with (1.14) from the law of error
propagation,
Q
∆∆
= B Q BT .
(5.5)
ξξ
In accordance with (4.3) the observed value ∆ of systematic
model errors in l may be zero,
∆ = Bξ = O .
(5.6)
In this case the functional model (1.1) resp. the observation
equation (1.2) remains unchanged
l-Δ+v = l+v = Ax
� ,
(5.7)
but the stochastic model, i.e. the covariance matrix Σll
of l changes to
�
� = σ2o Q
Σ
ll
= σ2o �Q +Q � = σ2o �Q +B Q BT � .
ll
ll
ΔΔ
ξξ
ll
(5.8)
The resulting normal equation is
AT �Q +B Q BT �
ξξ
ll
-1
Ax
� - AT �Q +B Q BT �
ξξ
ll
-1
l = O
(5.9)
its solution is identical with the solution of (4.10), because
in both cases the same functional and stochastical information
is used; for a numerical verification see EBNER (1973). For the
�
same reason Q
from (4.11) is equal to
xx
�
Q
= �A �Q +B Q B �
T
xx
T
ll
ξξ
318
-1
-1
A�
.
(5.10)
6. Conclusion
Systematic errors in the observations may be interpreted
as errors in the mathematical adjustment model, which can
be eliminated by insertion of additional parameters ξ in the
model. A general solution of this problem can be found if
these additional parameters are considered as random variables.
In this solution, all information about these parameters is
concentrated in the covariance matrix Σξξ of ξ.
With any matrix norm ‖∙∙‖ we get
� O � ≤ � Σξξ � < � ∞ I � .
(6.1)
In the first (left hand) extreme case �� O � = O� we have no
further information about the systematic effect and have to
detect them from the network adjustment alone (s. Ch. 3). In
the right hand extreme case, however, we know exactly the true
values ξ of the systematic errors and can reduce our observations l before carrying out the network adjustment.
In most practical cases the covariance matrix Σξξ is neither
the zero nor an infinite matrix and therefore one of the solutions given in Ch. 4 and 5 have to be used. Because the results of both solutions are identical, we can use one of these
solutions with respect to the practical computation. From this
point of view, normally the solution given in Ch. 4 may be more
suitable.
319
REFERENCES
EBNER, H., 1973: Zusätzliche Parameter in Ausgleichungen.
Zeitschrift für Vermessungswesen 98, p. 385-391.
MIKHAIL, E.M., 1976:
New York.
Observations and least squares.
PELZER, H., 1981: Leistungsfähigkeit und Grenzen des Funktionalmodells. Meeting of the Working Group III/1 of the
International Society of Photogrammetry and Remote Sensing,
Stuttgart, November 26-27, 1981.
SCHMID, H.H. and SCHMID, E., 1965: A Generalized Least
Squares Solution for Hybrid Measuring Systems. The Canadian
Surveyor XIX, p. 27-41.
320

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