Measurement uncertainties in science and technology Michael Grabe

Format:

Book

Author/Creator:

Grabe, Michael, author.

Language:

English

Subjects (All):

Measurement.

Error analysis (Mathematics).

Instrumental variables (Statistics).

Physics.

Measurement Science and Instrumentation.

Appl. Mathematics/Computational Methods of Engineering.

Mathematical Methods in Physics.

measuring.

Local Subjects:

Physics.

Measurement Science and Instrumentation.

Appl. Mathematics/Computational Methods of Engineering.

Mathematical Methods in Physics.

Physical Description:

1 online resource

Edition:

Second edition

Place of Publication:

Cham Springer 2014

Summary:

This book recasts the classical Gaussian error calculus from scratch, the inducements concerning bothrandom and unknown systematic errors. The idea of this bookis to create a formalism being fit to localize the true values of physical quantities considered - truewith respectto the set of predefined physical units. Remarkably enough, the prevailingly practiced forms of error calculus do not feature this property which however proves in every respect, to be physically indispensable. The amended formalism, termed Generalized Gaussian Error Calculus by the author, treats unknown systematic errors as biases and brings random errors to bear via enhanced confidence intervals as laid down by students. The significantly extended second edition thoroughly restructures and systematizes the text as a whole and illustrates the formalism by numerous numerical examples. They demonstrate the basic principles of how tounderstand uncertainties to localize the true values of measured values - a perspective decisive in view of the contested physical explorations

Contents:

Characterization, Combination of Propagation of Errors

Least Squares Adjustment

Linear and Linearized Systems

Generic Metrological Issues

Machine generated contents note: pt. I Characterization, Combination and Propagation of Errors

1. Basic Ideas of Measurement

1.1. Prologue

1.2. True Values

1.3. Measurement Uncertainties

1.4. Breakdown of the Gaussian Approach

1.5. Non-Gaussian Prospect

1.6. Appraising Measurement Uncertainties

1.7. Quotation of Numerical Values

2. Formalization of Measuring Processes

2.1. Random Variables

2.2. Probability Densities

2.3. Generalized Gaussian Error Calculus

2.4. Elements of Evaluation

2.5. Consistency

2.6. Traceability

3. Normal Parent Distributions

3.1. One-Dimensional Normal Density

3.2. Multidimensional Normal Density

3.3. Chi-Square and F Density

3.4. Student's (Gosset's) Density

3.5. Fisher's Density

3.6. Hotelling's Density

4. Estimators and Expectations

4.1. Expectations

4.2. Statistical Ensemble

4.3. Estimators

4.4. Behrens

Fisher Problem

4.5. Downfall of the Analysis of Variance

5. Bias and Randomness

5.1. Localization of the True Value x0

5.2. Localization of the Parameter μx

5.3. Uncertainty of the Arithmetic Mean

5.4. Second Thoughts About Systematic Errors

5.5. Function of a Measurand

5.6. Systematic Error of the Measuring Object

6. Error Propagation, Two Variables

6.1. Series Expansions

6.2. Bias

6.3. Random Errors

6.4. Overall Uncertainty

7. Error Propagation, m Variables

7.1. Series Expansions

7.2. Bias

7.3. Random Errors

7.4. Overall Uncertainties

7.5. Traceability

8. Concatenated Functions

8.1. Series Expansions

8.2. Biases

8.3. Random Errors

8.4. Overall Uncertainty

8.5. Composed Covariance

pt. II Least Squares Adjustment

9. Least Squares Formalism

9.1. Geometry of Adjustment

9.2. Unconstrained Adjustment

9.3. Constrained Adjustments

9.4. Quintessence of the Method of Least Squares

10. Consequences of Systematic Errors

10.1. Structure of the Solution Vector

10.2. Minimized Sum of Squared Residuals

10.3. Gauss

Markoff Theorem

10.4. Choice of Weights

10.5. Consistency of the Input Data

11. Uncertainties of Least Squares Estimators

11.1. Non-weighted Adjustment

11.2. Weighted Adjustment

11.3. Weighted and Non-weighted Mean of Means

11.4. Function of Least Squares Estimators

12. Uncertainty Spaces

12.1. New Solids

12.2. Confidence Ellipsoids

12.3. Security Polytopes

12.4. Composite Bounding Surfaces

pt. III Linear and Linearized Systems

13. Straight Lines

13.1. Fitting Conditions

13.2. Straight Line (i)

13.3. Straight Line (ii)

13.4. Straight Line (iii)

14. Exponentials

14.1. Fitting Conditions

14.2. Series Expansion

14.3. Transformation

15. Planes

15.1. Fitting Conditions

15.2. Plane (i)

15.3. Plane (ii)

15.4. Plane (iii)

16. Circles

16.1. Fitting Conditions

16.2. Series Expansion

16.3. Stereographic Projection

17. Parabolas

17.1. Fitting Conditions

17.2. Parabola (i)

17.3. Parabola (ii)

17.4. Parabola (iii)

18. Least Squares Trigonometric Polynomials

18.1. Representation

18.2. Fitting Conditions

18.3. Uncertainty Band

pt. IV Generic Metrological Issues

19. Dissemination of Units

19.1. Transfer of True Values

19.2. Working Standards

19.3. Key Comparisons

20. Calibration Chains

20.1. Mass Decades

20.2. Weighing Scheme

20.3. Least Squares Approach

20.4. Uncertainties

21. Pairwise Comparisons

21.1. Singular Design Matrix

21.2. On Constraints

21.3. Measuring Conditions

21.4. Least-Squares Approach

21.5. Uncertainties

22. Fundamental Constants of Physics

22.1. Constants at Present

22.2. Concatenations

22.3. Input Data

22.4. Linearization

22.5. Least Squares Estimators

22.6. Uncertainties

23. Essence of Metrology

23.1. Consistency

23.2. Traceability

Notes:

Online resource; title from PDF title page (SpringerLink, viewed May 21, 2014)

Includes bibliographical references and index

Other Format:

Printed edition:

ISBN:

9783319048888

3319048880

3319048872

9783319048871

OCLC:

880351344

Access Restriction:

Restricted for use by site license