H. Kutterer, kutterer@dgfi.badw.de
Motivation
Geometrical and physical models can only be
approximations of the reality. Hence, the difference between the selected model
and the data is uncertain. In Geodesy, these differences either remain
unclassified or are considered as exclusively random. The first case leads to
estimation theory including robust techniques. In the second case stochastics
based on probability theory supplies a variety of methods for modelling as well
as for data analysis and assessment. Contrary to this, stochastics is not
always the adequate theoretical basis to handle problem‑immanent
uncertainties. Two examples may give an idea. In applications like, e.g.,
Real-Time Kinematic Differential GPS, imprecision due to unknown systematic
effects is the most relevant type of uncertainty. Besides, in GIS modelling
there are often fuzzy or vague transitions between spatial objects regarding
the respective semantics.
The probabilistic point of view is normative. It does
not allow to handle different types of uncertainty in a distinct way. In order
to establish a general methodology for the comprehensive assessment of
uncertainty in data analysis it is necessary to identify and to classify the
occuring uncertainties in typical geodetic applications. Within the work of the
IAG SSG 4.190 (SSG) several mathematical theories were considered which are not
based on probability theory such as interval mathematics and fuzzy theory. One
prominent topic was the joint treatment of random-type uncertainty
(stochasticity) and imprecision of the data. Three fields of application were
studied: GPS data processing, deformation analysis, GIS.
Chairman: H.
Kutterer (Germany)
Members:
O. Akyilmaz (Turkey) M.
Brovelli (Italy)
A. Brunn (Germany) A.
Carosio (Switzerland)
B. Crippa (Italy) G.
Joos (Germany)
K. Heine (Germany) S. Leinen (Germany)
B. Merminod (Switzerland) F. Neitzel (Germany)
W. Niemeier (Germany) J. Ou (China)
D. Rossikopoulos (Greece) B. Schaffrin (U.S.A.)
E. A. Shyllon (Nigeria) A. Stein (The Netherlands)
J. Wang (Australia) Y.
Yang (China)
J. Zavoti (Hungary)
Corresponding Members:
R. Fletling (Germany) J.
B. Miima (Germany)
M. Molenaar (The Netherlands) S. Schön (Germany)
W.-D. Schuh (Germany) R.
Viertl (Austria)
A. Wieser (Austria)
During the period from 1999 to 2003 one scientific
symposium and three working meetings were held within the frame of the SSG. The
`First Symposium on Robust Statistics and Fuzzy Techniques in Geodesy and GISA
took place in Zurich, Switzerland, from March 12 to March 16, 2001. It was
organized by A. Carosio and H. Kutterer who also edited the proceedings volume
(Carosio and Kutterer, 2001). A first working meeting took place on April 7,
2000 in Karlsruhe, Germany. The participation of E. A. Shyllon was funded by
the IAG what is gratefully acknowledged. A second SSG working meeting was held
during the Zurich symposium. The third and last meeting was organized on the
occasion of the IAG Scientific Assembly 2001 in Budapest, Hungary.
The SSG maintains the website www.dgfi.badw.de/ssg4.190.
The site contains formal details (terms of reference, objectives, list of
members), information on the work of the SSG (notes, minutes of the working
meetings, Zurich symposium report, bibliography) and a mailing list. It is
planned to keep the site beyond the IUGG General Assembly 2003 in Sapporo,
Japan, and to update it when required.
Qualification and quantification of uncertainty
Uncertainties of the data are due to the random
selection of the data, the random variability of the data (central limit
theorems), imprecision of the observation procedure and instruments (round‑off
errors, recording of correction data), lacking reliability of the data, reduced
credibility of the data (data are recorded reliably, but their adequacy for the
modelled situation is questionable), data gaps, or lacking consistency of data
coming from different sources. Uncertainties of the estimation or inference
procedures result from simplifications for (convenient) mathematical treatment
(e.g., linearized models), (ambiguous) choice of the optimum principle of
parameter estimation, or decisions based on discrete alternatives and on
threshold values.
The uncertainty of models or concepts is not
considered in classical Geodesy. Nevertheless, there are uncertainties of the
model because of the incomplete (human) knowledge (modelling of the 'state of
the art'), necessary simplifications due to the complexity of the real world
(naming of and restriction to the relevant characteristics), modelling of a
substitute ('proxy') situation (discretization of continuous objects and
processes), fuzziness or imprecision of linguistic expressions or descriptions
('gross error', 'high temperature'), imprecision or inaccuracy of some 'known'
model parameters, ambiguity (non‑uniqueness in a crisp sense), or
vagueness (non‑uniqueness in a fuzzy sense, non‑specificity).
Aiming at the assessment and management of
uncertainty in typical geodetic data analysis it is indispensable to register,
to characterize, and to categorize the essential contributors and effects. The
set‑up of a corresponding questionnaire is the key to the assessment of
uncertainty. It can serve as a basis for the improvement of particular
procedures in use and for the comparison of procedures. A proposed
questionnaire can be found on the SSG webpages. Such a questionnaire is
particularly recommended as a basis for the assessment of routine data analysis
like in the IAG data services. This could help to get a deeper understanding of
the data products to be used or interpreted.
Mathematical theories for the assessment of uncertainty
Mathematical theories which are adequate for (at
least) some parts of uncertainty modelling and handling can be divided into
theories which are more or less based on the theory of probability and into
theories which are not. The approximation theory is the most fundamental
approach since uncertainty is considered in terms of approximation errors which
are minimized by minimizing a suitable measure for the distance between model
and data. Probabilistic theories are the theory of stochastics with uncertainty
modelled by means of random variables, the Bayes theory allowing the use of
stochastic (sometimes subjective) prior knowledge (Koch, 1990) and the evidence
theory (Shafer, 1976). The last theory can be understood as a generalization of
the Bayes theory; uncertain prior knowledge is modelled and assessed using
credibility and plausibility measures. Finally, robust statistics has to be
settled between pure approximation theory and stochastics.
Non-probabilistic theories are interval mathematics
(Alefeld and Herzberger, 1983), fuzzy theory (Dubois and Prade, 1980),
possibility theory (Dubois and Prade, 1988), the theory of rough sets and the
artificial neural networks (ANN). Interval mathematics allows to consider
imprecise data whereas fuzzy theory comprises both fuzziness (or imprecision)
of the model and of the data. The main branches of fuzzy theory are fuzzy logic
and fuzzy data analysis. The latter can be understood as a generalization of
interval mathematics. There are approaches to combine probabilistic and
non-probabilistic theories like, e.g., by Viertl (1996) who develops a
statistics for imprecise data with extensions to Bayesian statistics.
There is a need in geodetic data analysis for the
selection of the adequate kind of mathematics, for the definition of particular
measures of uncertainty, and for the combination of the most suitable
mathematical theories if several types of uncertainty occur in the
applications. For further information and for a extended list of references see
the SSG website. Within the SSG the main focus was on robust statistics and on
geodetic applications of both fuzzy logic and fuzzy data analysis to handle
classical model-data deviations in general and to consider (non-random) data
and model imprecision.
Interval mathematics and fuzzy data analysis
Imprecision of the data can be taken into account
either by intervals or fuzzy numbers. An interval is defined by its lower and
its upper bound or, equivalently, by its meanpoint and its radius. The radius
of the interval is a proper measure of imprecision. A fuzzy number is typically
defined by its membership function which is controlled by a meanpoint parameter
and a spread parameter. Fuzzy numbers can be understood as generalized
intervals. Imprecision measures based on fuzzy numbers are proportional to the
spreads (Kutterer, 2002a). The definition of vectors is unique in case of
intervals. In fuzzy theory there are several ways like, e.g., the use of the
minimum rule or of quadratic forms (Kutterer, 2001c, 2002a).
Least-squares (LS) adjustment is a typical technique
in geodetic data analysis. The case of observation intervals and their impact
on the LS estimator (LSE) was discussed by Schön and Kutterer (2001a, b, 2002)
in order to solve two problems. First, in geodetic practice imprecision needs
to be quantified. Second, the impact of observation imprecision on the
estimated parameters can be reduced using mathematical optimization techniques.
Both aspects are studied in detail by Schön (2003). Kutterer (2001c) discusses
two fuzzy-theoretical ways to introduce imprecision to the LSE in a
Gauss-Markov model. The first one is the fuzzy-extended LSE where the imprecise
observations vector is inserted into a consistent extension of the LSE. The
meanpoint of the fuzzy-extended LSE is equivalent to the classical LSE. Its
spreads quantify the imprecision in addition to the variance-covariance matrix
which represents stochastic dispersion. The second one is based on the maximum
similarity principle and leads to the fuzzy LSE in the strict sense and to the
hybrid fuzzy LS approximation.
As in geodetic practice neither the stochastic nor
the interval or fuzzy approach are adequate to jointly handle stochasticity and
imprecision, fuzzy extensions of confidence regions and hypothesis tests for
stochastic and imprecise data have been developed. The main idea is to apply
the extension principle of fuzzy theory to the respective mathematical
relations. The derivation of fuzzy-extended confidence regions to the resolution
of GPS phase ambiguity parameters was studied in Kutterer (2001b, 2002b).
Statistical hypothesis tests for imprecise data are derived in Kutterer (2003).
Fuzzy systems and artificial neural networks
In many applications of control theory and of decision
theory it is either expensive or just impossible to acquire complete and
precise information on all relevant parameters and relations. Hence, the need
for moderate complex but adequate models leads to the development of fuzzy
systems which are based on fuzzy logic. Fuzzy systems consist of four
components: an input component, an inference component, an output component and
a feedback connection from the output to the input. The input component
comprises an interface which allows to fuzzify real input data by means of
linguistic variables such as @length@ with the fuzzy states @short@, @medium@,
@long@. The inference component consists of a fuzzy rule base and a method for
the aggregation of the resulting fuzzy set. The output component yields real parameters
which are derived from the fuzzy result by a defuzzification method.
Fuzzy models were used successfully by Heine (1999,
2001) for the modelling of deformation processes. Leinen (2001) studied the
On-The-Fly resolution of GPS phase ambiguities as a multi-attribute decision
making process in order to assess the importance of some evidence pro or contra
a particular candidate solution. Aliosmanoglu and Akyilmaz (2002) considered
outlier detection in geodetic networks based on fuzzy logic. Joos (2001)
presented the @egg-yolk@ approach which allows to model fuzzy transition zones
between spatial objects in GIS such as meadows and forests. Graeff (2002)
compared probabilistic and fuzzy approaches in case of template matching for
GIS data acquisition. An adaptive network fuzzy inference system (ANFIS) was
studied by Akyilmaz and Kutterer (2003) for the short-term prediction of Earth
orientation parameters. Wieser (2001a, b) and Wieser and Brunner (2002a, b)
show the benefit of fuzzy logic for the determination of a realistic variance
model for GPS data and correspondingly for improved robust estimates for
kinematic processing of short baselines.
The artificial neural networks (ANN) represent an
independent methodology to handle various types of uncertainty. They have been
developed in order to imitate human thinking. They are composed of a large
number of simple processors (neurons) that are massively connected and operate
simultaneously. ANN are trained based on examples what is called machine
learning. During the training process individual weights are assigned to the
neurons. Some work has been done on the application of ANN to geodetic
problems. Heine (1999) and Niemeier and Miima (2001) considered the modelling
of deformation processes. Note that both fuzzy systems and ANN are solely
mathematical representations of the underlying physical processes. They offer
easy-to-handle best-fit solutions but they are - at least at first glance -
inadequate for physical interpretation.
Estimation theory including robust techniques
The use of robust techniques for the reliable
estimation of parameters and derived quantities is important in case of
insufficient knowledge of the statistical properties of the data. Within the
frame of the SSG several progress has been achieved. Wicki (2001) describes the
BIBER estimator which is used by the Swiss Federal Office of Topography for
geodetic networks. Kanani and Carosio (2001) use the BIBER estimator for the
automatic vectorization of areal objects from digital topographic maps. Yang
studies - together with varying co-authors - robust estimators in applications
such as satellite laser ranging in order to handle systematic errors (Yang et
al., 1999a), kinematic GPS positioning (Yang et al., 2001b), and dynamical sea
surface models (Yang et al, 1999a).
Maximum correlation adjustment (MCA) is based on the
quantitative comparison of data and model by means of their correlation
coefficient (Neitzel, 2001). A set of solutions is obtained by maximizing the
correlation coefficient. In geometric deformation analysis the Helmert
transformation is one of the MCA solutions. Finally, some extensions of
classical estimation theory are mentioned which are relevant in the scope of
the SSG. Schaffrin and Iz (2001) study an extended estimator which is adequate
to handle partial inconsistencies when integrating heterogeneous data sets. The
Best Linear Minimum Partial Unbiased Estimator (BLIMBPE) is described in
Schaffrin and Iz (2002).
Perspectives
Two main outcomes of the SSG>s work should be
emphasized. First, there is a clear need for a distinct modelling and handling
of the different types of problem-immanent uncertainties. Second, there are
different mathematical theories which allow a dedicated treatment. There is
certainly a competition of methodologies in some fields but in most parts they
are complementing each other. One example is represented by the soft computing
techniques which comprise fuzzy techniques, ANN and genetic algorithms. A
second example in data analysis is a joint modelling and inference strategy in
a combined stochastic, fuzzy and Bayesian framework. Once established, this
allows to integrate stochasticity and imprecision of the data as well as model
and prior information uncertainty. Both given examples reflect essential and de
manding challenges in geodetic data analysis for the near future.
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