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GENERALIZED GAUSSIAN ERROR CALCULUS

 

 

Michael Grabe

 

Springer  Textbook, Physik, February  2010, 301 p. 100 illus., 50 in color, ISBN: 978-3-642-03304-9

 

 About this book ...

The proposed book Generalized Gaussian Error Calculus addresses for the first time since 200 years a rigorous, complete and self-consistent revision of the Gaussian error calculus. Since experimentalists realized that measurements in general are burdened by unknown systematic errors, the classical, widespread used evaluation procedures scrutinizing the consequences of random errors alone turned out to be obsolete. As a matter of course, the error calculus to-be, treating random and unknown systematic errors side by side, should ensure the consistency and traceability of physical units, physical constants and physical quantities at large.

The generalized Gaussian error calculus considers unknown systematic errors to spawn biased estimators. Beyond, random errors are asked to conform to the idea of what the author calls well-defined measuring conditions.

The approach features the properties of a building kit: any overall uncertainty turns out to be the sum of a contribution due to random errors, to be taken from a confidence interval as put down by Student, and a contribution due to unknown systematic errors, as expressed by an appropriate worst case estimation.

 

CONTENTS

 

BASICS OF METROLOGY

 

1. TRUE VALUES AND TRACEABILITY

1.1 Metrology

1.2 Traceability

1.3 Measurement Errors

1.4 Precision and Accuracy

1.5 Measurement Uncertainty

1.6 Measuring Result

1.7 Rivaling Physical Approaches

 

2.  MODELS AND APPROACHES

2.1 Gaussian Error Model

2.2 Generalized Gaussian Approach

2.3 Robust Testing Conditions

2.4 Linearizations

2.5 Quiddity of Least Squares

2.6 Analysis of Variance

2.7 Road Map

 

 

GENERALIZED GAUSSIAN ERROR CALCULUS

 

3.  THE NEW UNCERTAINTIES

3.1 Gaussian Versus Generalized Gaussian Approach

3.2 Uncertainty and True Value

3.3 Designing Uncertainties

3.4 Quasi Safeness

 

4.  TREATMENT OF RANDOM ERRORS

4.1 Well-Defined Measuring Conditions

4.2 Multidimensional Normal Model

4.3 Permutation of Repeated Measurements

 

5.  TREATMENT OF SYSTEMATIC ERRORS

5.1 Repercussion of Biases

5.2 Uniqueness of Worst-Case Assessments

 

 

ERROR PROPAGATION

 

6. MEANS AND MEANS OF MEANS 

6.1 Arithmetic Mean

6.2 Extravagated Averages

6.3 Mean of Means

6.4 Individual Mean Versus Grand Mean

 

7.  FUNCTIONS OF ERRONEOUS VARIABLES

7.1 One Variable

7.2 Two Variables

7.3 More Than Two Variables

7.4 Concatenated Functions

7.5 Elementary Examples

7.6 Test of Hypothesis

 

8.  METHOD OF LEAST SQUARES

8.1 Empirical Variance–Covariance Matrix

8.2 Propagation of Systematic Errors

8.3 Uncertainties of the Estimators

8.4 Weighting Factors

8.5 Example

 

 

ESSENCE OF METROLOGY

 

9.  DISSEMINATION OF UNITS  

9.1 Working Standards

9.2 Key Comparisons

 

10.  MULTIPLES AND SUBMULTIPLES

10.1 Calibration Chains

10.2 Pairwise Comparisons

 

11.  FOUNDING PILLARS

11.1 Consistency

11.2 Traceability

 

 

FITTING OF STRAIGHT LINES

 

12. PRELIMINARIES

12.1 Distinction of Cases

12.2 True Straight Line

 

13.  STRAIGHT LINES: CASE (i)

13.1 Fitting Conditions

13.2 Orthogonal Projection

13.3 Uncertainties of the Input Data

13.4 Uncertainties of the Components of the Solution Vector

13.5 Uncertainty Band

13.6 EP-Region

 

14.  STRAIGHT LINES: CASE (ii)

14.1 Fitting Conditions

14.2 Orthogonal Projection

14.3 Uncertainties of the Components of the Solution Vector . . . . . 132

14.4 Uncertainty Band

14.5 EP-Region

 

15.  STRAIGHT LINES: CASE (iii)

15.1 Fitting Conditions

15.2 Orthogonal Projection

15.3 Series Expansion of the Solution Vector

15.4 Uncertainties of the Components of the Solution Vector

15.5 Uncertainty Band

15.6 EP-Region

 

 

FITTING OF PLANES

 

16.  PRELIMINARIES

16.1 Distinction of Cases

16.2 True Plane

 

17.  PLANES: CASE (i)

17.1 Fitting Conditions

17.2 Orthogonal Projection

17.3 Uncertainties of the Input Data

17.4 Uncertainties of the Components of the Solution Vector

17.5 EPC-Region

 

18. PLANES: CASE (ii)

18.1 Fitting Conditions

18.2 Orthogonal Projection

18.3 Uncertainties of the Components of the Solution Vector

18.4 Confidence Intervals and Overall Uncertainties

18.5 Uncertainty

18.6 EPC-Region

 

19 PLANES: CASE (iii)

19.1 Fitting Conditions

19.2 Orthogonal Projection

19.3 Series Expansion of the Solution Vector

19.4 Uncertainties of the Components of the Solution Vector

19.5 Uncertainty Bowls

19.6 EPC-Region

 

 

FITTING OF PARABOLAS

 

20.  PRELIMINARIES

20.1 Distinction of Cases

20.2 True Parabola

 

21.  PARABOLAS: CASE (i)

21.1 Fitting Conditions

21.2 Orthogonal Projection

21.3 Uncertainties of the Input Data

21.4 Uncertainties of the Components of the Solution Vector

21.5 Uncertainty Band

21.6 EPC-Region

 

22.  PARABOLAS: CASE (ii)

22.1 Fitting Conditions

22.2 Orthogonal Projection

22.3 Uncertainties of the Components of the Solution Vector

22.4 Uncertainty Band

22.5 EPC-Region

 

23 PARABOLAS: CASE  (iii)

23.1 Fitting Conditions

23.2 Orthogonal Projection

23.3 Series Expansion of the Solution Vector

23.4 Uncertainties of the Components of the Solution Vector

23.5 Uncertainty Band

23.6 EPC-Region

 

 

NON-LINEAR FITTING

 

24.  SERIES TRUNCATION

24.1 Homologous True Function

24.2 Fitting Conditions

24.3 Orthogonal Projection

24.4 Iteration

24.5 Uncertainties of the Components of the Solution Vector

 

25.  TRANSFORMATION  

25.1 Homologous True Function

25.2 Fitting Conditions

25.3 Orthogonal Projection

25.4 Uncertainties of the Components of the Solution Vector

 

 

APPENDICES

 

A Graphical Scale Transformations

 

B Expansion of Solution Vectors

 

C Special Confidence Ellipses and Ellipsoids

 

D Extreme Points of Ellipses and Ellipsoids

 

E Drawing Ellipses and Ellipsoids

 

F Security Polygons and Polyhedra

 

G EP Boundaries and EPC Hulls

 

H Student’s Density

 

I Uncertainty Band Versus EP-Region

 

J Quantiles of Hotelling’s Density

 

References

 

Index

 

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