REPORT OF THE IAG SSG 4.190:

 'NON-PROBABILISTIC ASSESSMENT IN GEODETIC DATA ANALYSIS'

IAG SCIENTIFIC ASSEMBLY 2001, BUDAPEST

Hansjörg Kutterer

DGFI Munich

 Marstallplatz 8

 D-80539 Munich

E-Mail: kutterer@dgfi.badw.de

1. Introduction

Geometrical and physical models can only be approximations of the reality. Hence the difference between the chosen model and the data remains uncertain. In Geodesy, these differences are - after some pre-processing - exclusively considered as random. Mathematically they are treated by means of stochastics. As a consequence, this proceeding is normative since the use of stochastic methods restricts in turn the considered type of uncertainty to random variability of the data. Contrary to the classical approach there are cases when stochastics is not the adequate theoretical basis to handle all 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, the common empirics-based formulation of the stochastic model in adjustment calculus implies a source of non-random uncertainty. Thus, it is not recommended to consider only random-type uncertainties.

To establish a general methodology for the comprehensive assessment of uncertainty in geodetic data analysis it is necessary to identify and to classify the occuring uncertainties in typical geodetic applications (qualification of uncertainty in observation, modelling, and inference). In addition, the elaboration of a proper terminology and the compilation of a bibliography are required. Within the work of the IAG SSG 4.190 (SSG) at least three fields of application are considered: GPS data processing, deformation analysis, and GIS. The relevant uncertainties have to be quantified regarding the respective application. The main points of interest are the data handling in the acquisition and preprocessing steps and the corresponding setup of models. As an example the uncertainty of GPS results introduced by different operators and different software packages is mentioned.

Furtheron it is necessary to collect and to characterize different non-standard approaches to deal with uncertainty and to infer under uncertainty like robust statistics, fuzzy theory, possibility theory, evidential reasoning, etc, in addition to the well-known concepts of approximation theory and stochastics. The applicability of the different approaches to the data analysis in the mentioned fields of geodetic interest needs to be discussed. Looking at the possible scientific interpretations of the quantities resulting from the data analysis it is essential to assess the corresponding (types of) uncertainty qualitatively and numerically.

Undoubtedly, there is in several cases a competition between the different approaches. In other cases with a clear distinction between the immanent uncertainties it is worthwhile to study the combination of the mathematical approaches for a more adequate use in geodetic practice. Statistics with data which are both random and imprecise can be mentioned as an example.

2. Organizational notes

Up to now (April 2001) two working meetings of the SSG have been held. The first meeting took place on April 7, 2000 in Karlsruhe, Germany. Eleven SSG members participated with oral presentations of their SSG-related work and discussions. The participation of E. A. Shyllon was funded by the IAG. This is gratefully acknowledged. On this occasion it was decided to organize an international symposium on the main topics of the SSGs work, i.e. robust estimation and fuzzy techniques. This symposium took place in Zurich, Switzerland, from March 12 to March 16, 2001. A proceedings volume is edited by Carosio and Kutterer (2001). A second SSG working meeting was held during this symposium. Further working meetings will take place on a half-annual or annual basis.

3. SSG website and mailing list

The SSG maintains the website www.dgfi.badw.de/ssg4.190 which is updated regularly. The site contains formal details (terms of reference, objectives, list of members), information on the work of the SSG (notes, papers, minutes of the working meetings, Zurich symposium report, bibliography) and a SSG mailing list. Feedback and criticism concerning the web presentation of the SSG and the contents of the website are highly appreciated.

4. Membership structure

5. Classification of uncertainty

It is well-known that the complete procedure of (geodetic) data management consists of data aquisition, data pre-processing (reduction of the 'raw' data to fit the geodetic observables which serve as an interface to the scientific models), inference (estimation and prediction of model parameters and derived quantities). Finally, regarding the general objectives of geodetic work the obtained results are interpreted in a scientific framework. For a general starting point of uncertainty assessment and management in the complete procedure, several types of uncertainty have to be distinguished. In the following, uncertainty is used as a generic expression. For more details see Kutterer (2001).

The modelling part of data analysis has to be separated into the set-up of the measurement or observation model (e.g., application of atmospheric corrections) and into the set-up of the model of main scientific interest (e.g., plate-kinematic model). A global distinction is between uncertainties of the model (or of the concept), uncertainties of the data (measurements, observations) and uncertainties introduced by the estimation or inference procedures.

The classical uncertainty concept in Geodesy is based on three classes of errors: gross errors, systematic errors, and random errors. Gross errors have to be avoided or detected by control methods, whereas systematic errors have to be eliminated by the observation set-up and correction methods. The remaining errors are considered as random. Thus, the distinction between randomness and systematics is based on the observation frame: Only those systematic errors are eliminated that can be modelled mathematically, whereas the others are neglected.

The decision about an observation value being biased by a gross error is usually based on human experiences, machine threshold values, or critical values of statistical tests. Therefore, there is some imprecision or fuzziness in the concept of gross errors. It should be noticed that 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 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).

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.

As a pragmatic matter of fact, the uncertainty of the uncertainties (uncertainty modelling) can additionally be taken into account. This comprises the uncertainty model for the observed values, the uncertainty model for the introduced prior information and the uncertainty model for the scientific (geodetic) model.

6. Mathematical theories for the assessment of uncertainty

Mathematical theories which are adequate for (at least) some parts of uncertainty modelling and handling can be separated 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) or theory of hints (Kohlas and Monney, 1995), repectively. These last two theories are more or less identical. They 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 or artificial neural networks. 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 generalization of interval mathematics. As a perspective, there are approaches to combine probabilistic and non -probabilistic approaches like, e.g., by Viertl (1996) who develops a statistics for imprecise data with extensions to Bayesian statistics.

The above-mentioned mathematical theories are (partly) different in the way of modelling and assessing the specific uncertainty. For example, there is no difference between approximation theory and stochastics or robust statistics, respectively, if only a best-fit is needed. But there is a big difference if an inference-based decision (like e.g., outlier rejection) is required because a criterion has to be specified. Thus, 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 is 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.

7. Registration of uncertainty

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 components and steps. 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.

The main steps of each geodetic data analysis are data acquisition, data pre-processing, and inference. Besides the analysis, a general description is needed as a frame for the questionnaire to identify the specific application and to make the results comparable with others. Finally, conclusions have to be drawn on the consistency of the data processing and analysis, on the adequate treatment of the existing types of uncertainty, and on the assessment of the data acquisition and analysis procedure in use. Usually, the results of a data analysis are interpreted scientifically. Thus, their genesis has to be understood thoroughly. This means particularly the sources for and the propagation of the immanent and the introduced types of uncertainty.

A proposed questionnaire can be found on the SSG webpages. Such a questionnaire is 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.

8. Status quo and future work

Information concerning the first two items of the SSG objectives is now available: The relevant types of uncertainty are characterized; a variety of mathematical methods exists which are more or less elaborated for use in geodetic data analysis. The 'First Symposium on Robust Statistics and Fuzzy Techniques' in March 2001 in Zurich which was organized by the SSG showed improvements of robust estimation techniques mainly for geodetic networks but also for the analysis of real time GPS phase data. Applications of fuzzy theory to deformation analysis, to GPS ambiguity resolution and to modelling in GIS were presented; see the proceedings for details. Within the SSG there will be further application directed studies on robust statistics, Bayes theory, interval mathematics, fuzzy theory, and artificial neural networks. A prominent task of the SSG for the period from 2001 to 2003 is the comparison of the applicability of different mathematical theories for uncertainty assessment to particular data analytical problems like, e.g., temporal or spatial prediction.

References

Alefeld G.; Herzberger J. (1983): Introduction to Interval Computations. Academic Press, New York.

Carosio A.; Kutterer H. (Ed.) (2001): Proceedings of the First International Symposium on Robust   Statistics and Fuzzy Techniques in Geodesy and GIS. Swiss Federal Institute of Technology Zurich, Institute of Geodesy and Photogrammetry - Report No. 295.

Dubois D.; Prade H. (1980): Fuzzy Sets and Systems. Academic Press, New York.

Dubois D.; Prade H. (1988): Possibility Theory, Plenum Press, New York.

Koch K. R. (1990): Bayesian Inference with Geodetic Applications. Springer, Berlin Heidelberg New York.

Kohlas J.; Monney P.-A. (1995): A Mathematical Theory of Hints. Springer, Berlin Heidelberg.

Kutterer H. (2001): Uncertainty assessment in geodetic data analysis. In: Carosio, A. und H. Kutterer  (Ed.) (2001)

Shafer G. (1976): A Mathematical Theory of Evidence. Princeton University Press, Princeton.

Viertl R. (1996): Statistical Methods for Non-Precise Data. CRC Press, Boca Raton New York London Tokyo.

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