Table of Contents
Part I Basics of Metrology
1 True Values and Traceability 3
1.1 Metrology 3
1.2 Traceability 3
1.3 Measurement Errors 4
1.4 Precision and Accuracy 6
1.5 Measurement Uncertainty 6
1.6 Measuring Result 6
1.7 Rivaling Physical Approaches 8
2 Models and Approaches 9
2.1 Gaussian Error Model 9
2.2 Generalized Gaussian Approach 10
2.3 Robust Testing Conditions 14
2.4 Linearizations 15
2.5 Quiddity of Least Squares 16
2.6 Analysis of Variance 20
2.7 Road Map 21
Part II Generalized Gaussian Error Calculus
3 The New Uncertainties 25
3.1 Gaussian Versus Generalized Gaussian Approach 25
3.2 Uncertainty and True Value 25
3.3 Designing Uncertainties 26
3.4 Quasi Safeness 29
4 Treatment of Random Errors 31
4.1 Well-Defined Measuring Conditions 31
4.2 Multidimensional Normal Model 32
4.3 Permutation of Repeated Measurements 33
5 Treatment of Systematic Errors 35
5.1 Repercussion of Biases 35
5.2 Uniqueness of Worst-Case Assessments 36
Part III Error Propagation
6 Means and Means of Means 39
6.1 Arithmetic Mean 39
6.2 Extravagated Averages 41
6.3 Mean of Means 41
6.4 Individual Mean Versus Grand Mean 47
7 Functions of Erroneous Variables 53
7.1 One Variable 53
7.2 Two Variables 56
7.3 More Than Two Variables 61
7.4 Concatenated Functions 66
7.5 Elementary Examples 68
7.6 Test of Hypothesis 74
8 Method of Least Squares 79
8.1 Empirical Variance-Covariance Matrix 79
8.2 Propagation of Systematic Errors 82
8.3 Uncertainties of the Estimators 83
8.4 Weighting Factors 84
8.5 Example 87
Part IV Essence of Metrology
9 Dissemination of Units 91
9.1 Working Standards 91
9.2 Key Comparisons 95
10 Multiples and Sub-multiples 101
10.1 Calibration Chains 101
10.2 Pairwise Comparisons 110
11 Founding Pillars 113
11.1 Consistency 113
11.2 Traceability 114
Part V Fitting of Straight Lines
12 Preliminaries 117
12.1 Distinction of Cases 117
12.2 True Straight Line 118
13 Straight Lines: Case (i) 121
13.1 Fitting Conditions 121
13.2 Orthogonal Projection 121
13.3 Uncertainties of the Input Data 123
13.4 Uncertainties of the Components of the Solution Vector 124
13.5 Uncertainty Band 126
13.6 EP-Region 127
14 Straight Lines: Case (ii) 131
14.1 Fitting Conditions 131
14.2 Orthogonal Projection 132
14.3 Uncertainties of the Components of the Solution Vector 132
14.4 Uncertainty Band 135
14.5 EP-Region 137
15 Straight Lines: Case (iii) 141
15.1 Fitting Conditions 141
15.2 Orthogonal Projection 142
15.3 Series Expansion of the Solution Vector 143
15.4 Uncertainties of the Components of the Solution Vector 145
15.5 Uncertainty Band 147
15.6 EP-Region 148
Part VI Fitting of Planes
16 Preliminaries 155
16.1 Distinction of Cases 155
16.2 True Plane 155
17 Planes: Case (i) 157
17.1 Fitting Conditions 157
17.2 Orthogonal Projection 157
17.3 Uncertainties of the Input Data 158
17.4 Uncertainties of the Components of the Solution Vector 159
17.5 EPC-Region 161
18 Planes: Case (ii) 165
18.1 Fitting Conditions 165
18.2 Orthogonal Projection 166
18.3 Uncertainties of the Components of the Solution Vector 166
18.4 Confidence Intervals and Overall Uncertainties 168
18.5 Uncertainty Bowls 169
18.6 EPC-Region 171
19 Planes: Case (iii) 179
19.1 Fitting Conditions 179
19.2 Orthogonal Projection 180
19.3 Series Expansion of the Solution Vector 181
19.4 Uncertainties of the Components of the Solution Vector 183
19.5 Uncertainty Bowls 185
19.6 EPC-Region 187
Part VII Fitting of Parabolas
20 Preliminaries 193
20.1 Distinction of Cases 193
20.2 True Parabola 193
21 Parabolas: Case (i) 195
21.1 Fitting Conditions 195
21.2 Orthogonal Projection 195
21.3 Uncertainties of the Input Data 196
21.4 Uncertainties of the Components of the Solution Vector 197
21.5 Uncertainty Band 199
21.6 EPC-Region 199
22 Parabolas: Case (ii) 203
22.1 Fitting Conditions 203
22.2 Orthogonal Projection 294
22.3 Uncertainties of the Components of the Solution Vector 204
22.4 Uncertainty Band 207
22.5 EPC-Region 209
23 Parabolas: Case (iii) 213
23.1 Fitting Conditions 213
23.2 Orthogonal Projection 214
23.3 Series Expansion of the Solution Vector 215
23.4 Uncertainties of the Components of the Solution Vector 217
23.5 Uncertainty Band 219
23.6 EPC-Region 220
Part VIII Non-linear Fitting
24 Series Truncation 227
24.1 Homologous True Function 227
24.2 Fitting Conditions 228
24.3 Orthogonal Projection 228
24.4 Iteration 230
24.5 Uncertainties of the Components of the Solution Vector 231
25 Transformation 237
25.1 Homologous True Function 237
25.2 Fitting Conditions 237
25.3 Orthogonal Projection 238
25.4 Uncertainties of the Components of the Solution Vector 239
Part IX Appendices
A Graphical Scale Transformations 245
B Expansion of Solution Vectors 251
C Special Confidence Ellipses and Ellipsoids 257
D Extreme Points of Ellipses and Ellipsoids 261
E Drawing Ellipses and Ellipsoids 265
F Security Polygons and Polyhedra 267
G EP Boundaries and EPC Hulls 277
H Student's Density 283
I Uncertainty Band Versus EP-Region 287
J Quantiles of Hotelling's Density 295
References 297
Index 299
