Geodetic InstitutE Leibniz University of Hannover Germany

1. The probability of type I and type II errors in imprecise hypothesis testing Ingo Neumannand Hansjörg Kutterer GeodeticInstitutELeibniz University of HannoverGermany REC 2008 Reliable Engineering Computing February 20, 2008

2. Motivation “Sometimes the uncertainty budget in geodetic applications is too optimistic” • Examples: • A geometric leveling around a big lake in Germany, Switzerland and Austria • Control networks that are observed with two different techniques • (i) terrestrial measurements • (ii) satellite measurements • Why? • Ignorance of non-stochastic errors in the measurements and in the preprocessing steps of the measurements!? The probabilityof type I and type II errors

3. Motivation The probabilityof type I and type II errors

4. Motivation Why non-stochastic errors? • The model constants are only partially representative for the given situation (e. g., the model constants for the refraction index for distance measurements). • The number of additional information (measurements) may be too small to estimate reliable distributions. • Displayed measurement results are affected by rounding errors. • Other non-stochastic errors of the reduced observations occur due to neglected correction and reduction steps and for effects that cannot be modeled. The probabilityof type I and type II errors

5. Agenda • Motivation • Uncertainty modeling in geodetic data analysis • Statistical hypothesis tests in case of imprecise data • General form of a linear hypothesis • Probability of type I/II errors • Geodetic applications • One and multidimensional case (weak imprecision) • Congruence test (strong imprecision) • Conclusions and future work The probabilityof type I and type II errors

6. Systematic effects Uncertainty Modeling • Measurement process: • Stochasticity • Observation imprecision • (Outliers) • Object fuzziness, etc... Occurring uncertainties in this presentation: Stochastics (Bayesian approach) Interval mathematics Fuzzy theory The probabilityof type I and type II errors

7. Solution: Describing the influence factors for the preprocessing step of the originary observation with fuzzy sets a Uncertainty Modeling Requirements: • Adequate description of Stochastics • Adequate description of Imprecision e. g., LR-fuzzy-number The probabilityof type I and type II errors

8. - Instrumental error sources - Uncertainties in reduction and corrections Influence factors (p) - Rounding errors Linearization Partial derivatives for all influence factors Imprecision of the influence factors Uncertainty Modeling Sensitivity analysis for the calculation of the imprecision of some parameters x of interest: The probabilityof type I and type II errors

9. Stochastic (Bayesian approach) Observation Imprecision 1 a=0.6 0.6 0.8 a=0.2 a=0.4 a=0.8 a=1.0 -discitization 0.4 0.2 Uncertainty Modeling Imprecise analysis ( ): The probabilityof type I and type II errors

10. These tests are based on a linear hypothesis Statistical hypothesis tests Tasks in (linear) parameter estimation: The probabilityof type I and type II errors

11. Introduction of a linear hypothesis: Statistical hypothesis tests • General form of a linear hypothesis: precise case n:= number of observations u:= number of parameters The probabilityof type I and type II errors

12. Statistical hypothesis tests • General form of a linear hypothesis: Introduction of a quadratic form / test value: Test decision: And with imprecision? The probabilityof type I and type II errors

13. Statistical hypothesis tests • General form of a linear hypothesis: Introduction of a quadratic form / test value: With imprecise influence factors p The probabilityof type I and type II errors

14. Statistical hypothesis tests • General form of a linear hypothesis: Resulting test scenario  1D comparison Final decision based on the comparison of the tests value with the regions of acceptance and rejection (card criterion) The probabilityof type I and type II errors

15. Statistical hypothesis tests Basic idea Degree of disagreement Degree of agreement With: The probabilityof type I and type II errors

16. Degree of agreement Degree of disagreement Degree of rejectability Statistical hypothesis tests Test decision: The probabilityof type I and type II errors

17. Overlap region Statistical hypothesis tests The card criterion: ~ ~ ~ with: with: The probabilityof type I and type II errors

18. Statistical hypothesis tests • Probability of a type I error in the imprecise case: . With denoting the inverse function The probabilityof type I and type II errors .

19. Statistical hypothesis tests • Probability of a type I error in the imprecise case: (1) Choose an adequate value for : 0.9 (2) Compute . (3) Find in such a way that the following Equation is fulfilled within a negligible threshold: . The probabilityof type I and type II errors .

20. Statistical hypothesis tests • Probability of a type II error in the imprecise case: Choose the non-centrality parameter or the probability of a type II error in the imprecise case . The probabilityof type I and type II errors .

21. Statistical hypothesis tests • Probability of a type II error in the imprecise case: (1) Compute the probability of a type I error in the im-precise case (2) Choose an adequate value for : (3) Find in such a way that the following Equation is fulfilled within a negligible threshold: . The probabilityof type I and type II errors .

22. Statistical hypothesis tests • Non-centrality parameter in the imprecise case: (1) Compute the probability of a type I error in the im-precise case (2) Choose an adequate value for : (3) Find in such a way that the following Equation is fulfilled within a negligible threshold: . The probabilityof type I and type II errors .

23. monitoring the actual movements of the lock: Geodetic applications A geodetic monitoring network of a lock: Monitoring network The lock Uelzen I . The probabilityof type I and type II errors .

24. Geodetic applications A geodetic monitoring network of a lock: Measurements: - horizontal directions (a) - zenith angles (b) - distances (c) . The probabilityof type I and type II errors .

25. Geodetic applications A single outlier test (weak imprecision): Probability of a type I error in the imprecise case: . The probabilityof type I and type II errors .

26. Geodetic applications A single outlier test (weak imprecision): Probability of a type I error in the imprecise case: . The probabilityof type I and type II errors .

27. Geodetic applications Probability of a type I error in the imprecise case for a multiple outlier test: Non-centrality parameter for a multiple outlier test: . The probabilityof type I and type II errors .

28. Geodetic applications Congruence Test (strong imprecision): . 6 identical points The probabilityof type I and type II errors .

29. Geodetic applications Congruence test (strong imprecision): Test situation: Probability of a type I error in the imprecise case for the congruence test: . The probabilityof type I and type II errors .

30. Conclusions and future work • Statistical hypothesis tests in linear parameter estimation (impecise case) • Type I and Type II error probabilities • The non-centrality parameter in the imprecise case • 1D case is straightforward, mD case needs a-cut optimization • The difference between the precise and the imprecise case depends on the task of the test: • (i) outlier tests or • (ii) safety-relevant test • and on the order of magnitude of imprecision • In progress: Assessment and validation using real data • Reduce the computational complexity • Take object fuzziness into account The probabilityof type I and type II errors

31. Acknowledgements The presented results are developed within the research project KU 1250/4 ”Geodätische Deformationsanalysen unter Verwendung von Beobachtungsimpräzision und Objektun-schärfe”, which is funded by the German Research Foundation (DFG). This is gratefully acknowledged Thank you for your attention! The probabilityof type I and type II errors

32. Contact information The probability of type I and type II errors in imprecise hypothesis testing Ingo Neumann and Hansjörg Kutterer Geodetic Institute Leibniz University of Hannover Nienburger Straße 1, D-30167 Hannover, Germany Tel.: +49/+511/762-4394 E-Mail: {neumann, kutterer}@gih.uni-hannover.de www.gih.uni-hannover.de The probabilityof type I and type II errors

34. Uncertainty Modeling • Tasks and methods • Determination of relevant quantities / parameters • Calculation of observationimprecision • Propagation of observationimprecision to the est. parameters • Assessment of accuracy (imprecise case) • Regression and least squares adjustments • Statistical hypothesis tests • - General form of a linear hypothesis • - Probability of Type I/II errors • Optimization of configuration The probabilityof type I and type II errors

35. Methoden zur Analyse der Impräzision a-Schnitt Optimierung

36. a-cut optimization

37. Statistical hypothesis tests Precise case (1D) 1 Example: Two-sided comparison of a mean value with a given value Clear and unique decisions ! x Null hypothesis H0, alternative hypothesis HA, error probability g → Definition of regions of acceptance A and rejection R The probabilityof type I and type II errors

38. 1 x Imprecision of test statistics due to the imprecision of the observations Statistical hypothesis tests Consideration of imprecision Imprecise case Precise case 1 x The probabilityof type I and type II errors

39. Statistical hypothesis tests Consideration of imprecision Precise case Imprecise case 1 1 x x Imprecision of the region of acceptance due to the linguistic fuzziness or modeled regions of transition Fuzzy-interval The probabilityof type I and type II errors

40. Statistical hypothesis tests Consideration of imprecision Precise case Imprecise case 1 1 x x Imprecision of the region of rejection as complement of the region of acceptance The probabilityof type I and type II errors

41. Statistical hypothesis tests Consideration of imprecision Precise case Imprecise case 1 1 x x Conclusion: Transition regions prevent a clear and unique test decision ! The probabilityof type I and type II errors

42. Statistical hypothesis tests Conditions for an adequate test strategy • Quantitative comparison of the imprecise test statistics and the regions of acceptance and rejection • Precise criterion pro or con acceptance • Probabilistic interpretation of the results