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 MEASUREMENT UNCERTAINTIES IN SCIENCE AND TECHNOLOGY

 

Michael Grabe

 

Springer  Textbook, Physik & Astronomie, 10. März 2005, 269 p. 56 Illus. ISBN: 3-540-20944-1



About this book ...

At the turn of the 19th century, Carl Friedrich Gauß founded error calculus by predicting the then unknown position of the planet Ceres. Ever since, error calculus has occupied a place at the heart of science.

In this book, Grabe illustrates the breakdown of traditional error calculus in the face of modern measurement techniques.  Revising Gauß' error calculus ab initio, he treats random and unknown systematic errors on an equal footing from the outset. Furthermore, Grabe also proposes what may be called well defined measurement conditions, a prerequisite for defining confidence intervals that are consistent with basic statistical concepts. The resulting measurement uncertainties  are as robust and reliable as required by modern-day science, engineering and technology.





CONTENTS

 

CHARACTERIZATION, COMBINATION AND PROPAGATION OF ERRORS

1. PROPERTIES OF RESULTS OF MEASUREMENT
    1.1 Basic ideas of measurement
    1.2 Quotation of numerical results and rounding 

2. FORMALISATION OF PROCESSES OF MEASUREMENT
   2.1 Fundamental equation of measurement
   2.2 Random variables, probability densities and parent distributions
   2.3 Elements of evaluation

3. DENSITIES DERIVED FROM NORMAL PARENT DISTRIBUTIONS
   3.1 One dimensional normal density
   3.2 Multidimensional density
   3.3 Chi-Square density 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 Statistical ensemble
   4.2 Variances and Covariances
   4.3 Elementary model of analysis of variance

5. COMBINATION OF MEASUREMENT ERRORS
   5.1 Expectation of the arithmetic mean
   5.2 Uncertainty of the arithmetic mean
   5.3 Uncertainty of a function of one variable
   5.4 Systematic errors of measurands 

6. PROPAGATION OF MEASUREMENT ERRORS
   6.1 Taylor's series expansion
   6.2 Expectation of the arithmetic mean
   6.3 Uncertainty of the arithmetic mean
   6.4 Uncertainties of functional relationships
   6.5 Error propagation covering several stages

 

LEAST SQUARES ADJUSTMENT

7. LEAST SQUARES FORMALISM
   7.1 Geometry of adjustment
   7.2 Unconstrained adjustment
   7.3 Constrained adjustment

8. CONSEQUENCES OF SYSTEMATIC ERRORS
   8.1 Structure of solution vector
   8.2 Vector of input data
   8.3 Gauß-Markoff Theorem
   8.4 Choice of the weights
   8.5 Consistency of the input data

9. UNCERTAINTIES OF LEAST SQUARES ESTIMATORS
   9.1 Empirical variance-covariance matrix
   9.2 Propagation of systematic errors
   9.3 Overall uncertainties
   9.4 Uncertainty iof a function
   9.5 Uncertainty spaces

                                 

LINEAR AND LINEARIZED SYSTEMS

10. SYSTEMS WITH TWO PARAMETERS
    10.1 Straight lines
    10.2 Linearisation
    10.3 Transformation

11. SYSTEMS WITH THREE PARAMETERS
    11.1 Planes
    11.2 Parabolae
    11.3 Approximation by piecewise smooth functions
    11.4 Circles

12. SPECIAL METROLOGICAL SYSTEMS
    12.1 Dissemination of units
    12.2 Mass decades
    12.3 Pairwise comparisons
    12.4 Fundamental constants of physics
    12.5 Empirical function 

 

APPENDIX

A. On the rank of some matrices
B. Variance-Covariance matrices
C. Linear funktions of normally distributed variables
D. Orthogonal projections
E. Least squares adjustments
F. Expansion of solution vectors
G. Student's density
H. Quantiles of  Hotelling's density
 

REFERENCES

INDEX
  

 

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