GEODETIC INSTITUTE LEIBNIZ UNIVERSITY OF HANNOVER GERMANY Ingo Neumann and Hansjörg Kutterer The...
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GEODETIC INSTITUTE LEIBNIZ UNIVERSITY OF HANNOVER GERMANY Ingo Neumann and Hansjörg Kutterer The probability of type I and type II errors in imprecise hypothesis testing REC 2008 Reliable Engineering Computing February 20, 2008
GEODETIC INSTITUTE LEIBNIZ UNIVERSITY OF HANNOVER GERMANY Ingo Neumann and Hansjörg Kutterer The probability of type I and type II errors in imprecise
GEODETIC INSTITUTE LEIBNIZ UNIVERSITY OF HANNOVER GERMANY Ingo
Neumann and Hansjrg Kutterer The probability of type I and type II
errors in imprecise hypothesis testing REC 2008 Reliable
Engineering Computing February 20, 2008
Slide 2
Motivation 2 February 20, 2008 The probability of type I and
type II errors 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!?
Slide 3
Motivation 3 February 20, 2008 The probability of type I and
type II errors
Slide 4
Motivation 4 February 20, 2008 The probability of type I and
type II 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. Why non-stochastic
errors?
Slide 5
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 5
February 20, 2008 The probability of type I and type II errors
Agenda
Slide 6
Uncertainty Modeling 6 February 20, 2008 The probability of
type I and type II errors Systematic effects Measurement process:
-Stochasticity -Observation imprecision -(Outliers) Object
fuzziness, etc... Stochastics (Bayesian approach) Interval
mathematics Fuzzy theory Occurring uncertainties in this
presentation :
Slide 7
Requirements: Adequate description of Stochastics Adequate
description of Imprecision Solution: Describing the influence
factors for the preprocessing step of the originary observation
with fuzzy sets e. g., LR-fuzzy-number 7 February 20, 2008 The
probability of type I and type II errors Uncertainty Modeling
Slide 8
Sensitivity analysis for the calculation of the imprecision of
some parameters x of interest: - Instrumental error sources -
Uncertainties in reduction and corrections - Rounding errors
Influence factors (p) Linearization Partial derivatives for all
influence factors Imprecision of the influence factors 8 February
20, 2008 The probability of type I and type II errors Uncertainty
Modeling
Slide 9
9 February 20, 2008 The probability of type I and type II
errors Uncertainty Modeling -discitization Imprecise analysis ( ):
Observation Imprecision Stochastic (Bayesian approach)
Slide 10
Parameter Estimation (Linear case) Global test Outlier
detection Congruence tests Model selection Sensitivity analysis
These tests are based on a linear hypothesis Statistical hypothesis
tests Tasks in (linear) parameter estimation: 10 February 20, 2008
The probability of type I and type II errors
Slide 11
General form of a linear hypothesis: Statistical hypothesis
tests n:= number of observations u:= number of parameters precise
case Introduction of a linear hypothesis: 11 February 20, 2008 The
probability of type I and type II errors
Slide 12
Introduction of a quadratic form / test value: Test decision:
And with imprecision? 12 February 20, 2008 The probability of type
I and type II errors General form of a linear hypothesis:
Statistical hypothesis tests
Slide 13
13 February 20, 2008 The probability of type I and type II
errors General form of a linear hypothesis: Statistical hypothesis
tests With imprecise influence factors p Introduction of a
quadratic form / test value:
Slide 14
14 February 20, 2008 The probability of type I and type II
errors General form of a linear hypothesis: Statistical hypothesis
tests Resulting test scenario 1D comparison Final decision based on
the comparison of the tests value with the regions of acceptance
and rejection (card criterion)
Slide 15
Degree of agreement Degree of disagreement Basic idea 15
February 20, 2008 The probability of type I and type II errors
Statistical hypothesis tests With:
Slide 16
Degree of rejectability Test decision: Degree of agreement
Degree of disagreement 16 February 20, 2008 The probability of type
I and type II errors Statistical hypothesis tests
Slide 17
The card criterion: with: ~~~ Overlap region 17 February 20,
2008 The probability of type I and type II errors with: Statistical
hypothesis tests
Slide 18
With denoting the inverse function Probability of a type I
error in the imprecise case: Statistical hypothesis tests.. 18
February 20, 2008 The probability of type I and type II errors
Slide 19
(3) Find in such a way that the following Equation is fulfilled
within a negligible threshold: Probability of a type I error in the
imprecise case: Statistical hypothesis tests.. (1) Choose an
adequate value for : (2) Compute. 0.9 19 February 20, 2008 The
probability of type I and type II errors
Slide 20
Probability of a type II error in the imprecise case:
Statistical hypothesis tests.. 20 February 20, 2008 The probability
of type I and type II errors Choose the non-centrality parameter or
the probability of a type II error in the imprecise case
Slide 21
(3) Find in such a way that the following Equation is fulfilled
within a negligible threshold: Probability of a type II error in
the imprecise case: Statistical hypothesis tests.. (2) Choose an
adequate value for : 21 February 20, 2008 The probability of type I
and type II errors (1) Compute the probability of a type I error in
the im- precise case
Slide 22
(3) Find in such a way that the following Equation is fulfilled
within a negligible threshold: Non-centrality parameter in the
imprecise case: Statistical hypothesis tests.. (2) Choose an
adequate value for : 22 February 20, 2008 The probability of type I
and type II errors (1) Compute the probability of a type I error in
the im- precise case
Slide 23
Geodetic applications.. 23 February 20, 2008 The probability of
type I and type II errors A geodetic monitoring network of a lock:
The lock Uelzen I Monitoring network monitoring the actual
movements of the lock:
Slide 24
Geodetic applications.. 24 February 20, 2008 The probability of
type I and type II errors A geodetic monitoring network of a lock:
Measurements: - horizontal directions (a) - zenith angles (b) -
distances (c)
Slide 25
Geodetic applications.. 25 February 20, 2008 The probability of
type I and type II errors Probability of a type I error in the
imprecise case: A single outlier test (weak imprecision):
Slide 26
Geodetic applications.. 26 February 20, 2008 The probability of
type I and type II errors A single outlier test (weak imprecision):
Probability of a type I error in the imprecise case:
Slide 27
Geodetic applications.. 27 February 20, 2008 The probability of
type I and type II errors Probability of a type I error in the
imprecise case for a multiple outlier test: Non-centrality
parameter for a multiple outlier test:
Slide 28
Geodetic applications.. 28 February 20, 2008 The probability of
type I and type II errors epoch 1999epoch 2004 observations317144
parameters6039 6 identical points Congruence Test (strong
imprecision):
Slide 29
Geodetic applications.. 29 February 20, 2008 The probability of
type I and type II errors Congruence test (strong imprecision):
Test situation: Probability of a type I error in the imprecise case
for the congruence test:
Slide 30
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 -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 30 February 20, 2008 The
probability of type I and type II errors Conclusions and future
work In progress: Assessment and validation using real data Reduce
the computational complexity Take object fuzziness into
account
Slide 31
The presented results are developed within the research project
KU 1250/4 Geodtische Deformationsanalysen unter Verwendung von
Beobachtungsimprzision und Objektun- schrfe, which is funded by the
German Research Foundation (DFG). This is gratefully acknowledged
Thank you for your attention! 31 February 20, 2008 The probability
of type I and type II errors Acknowledgements
Slide 32
Contact information Ingo Neumann and Hansjrg Kutterer Geodetic
Institute Leibniz University of Hannover Nienburger Strae 1,
D-30167 Hannover, Germany Tel.: +49/+511/762-4394 E-Mail: {neumann,
kutterer}@gih.uni-hannover.deneumann, [email protected]
www.gih.uni-hannover.de The probability of type I and type II
errors in imprecise hypothesis testing 32 February 20, 2008 The
probability of type I and type II errors
Slide 33
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Slide 34
Tasks and methods Determination of relevant quantities /
parameters Calculation of observation imprecision Propagation of
observation imprecision 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 34
February 20, 2008 The probability of type I and type II errors
Uncertainty Modeling
Slide 35
Schnitt Optimierung Methoden zur Analyse der Imprzision
Slide 36
cut optimization
Slide 37
1 x Precise case (1D) Example: Two-sided comparison of a mean
value with a given value Null hypothesis H 0, alternative
hypothesis H A, error probability Definition of regions of
acceptance A and rejection R Clear and unique decisions ! 37
February 20, 2008 The probability of type I and type II errors
Statistical hypothesis tests
Slide 38
38 February 20, 2008 The probability of type I and type II
errors Statistical hypothesis tests 1 x Imprecise case
Consideration of imprecision Precise case x 1 Imprecision of test
statistics due to the imprecision of the observations
Slide 39
1 xx 1 Fuzzy-interval Imprecise casePrecise case Imprecision of
the region of acceptance due to the linguistic fuzziness or modeled
regions of transition 39 February 20, 2008 The probability of type
I and type II errors Statistical hypothesis tests Consideration of
imprecision
Slide 40
40 February 20, 2008 The probability of type I and type II
errors Statistical hypothesis tests 11 xx Imprecise casePrecise
case Imprecision of the region of rejection as complement of the
region of acceptance Consideration of imprecision
Slide 41
11 xx Imprecise casePrecise case Conclusion: Transition regions
prevent a clear and unique test decision ! 41 February 20, 2008 The
probability of type I and type II errors Statistical hypothesis
tests Consideration of imprecision
Slide 42
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 Statistical hypothesis
tests