Generalized Gaussian Error Calculus (eBook)

1967 Graduation in Physics at the Technical University of Stuttgart1970 Doctorate at the Technical University of Braunschweig1970 – 1975 Scientific assistant and lecturer at the Technical University of Braunschweig1975 – 2004 Member of Staff at the Physikalische Technischer Bundesanstalt Braunschweig, commissioned to legal metrology, computerized interferometric measurment of length, measurement uncertainties and the adjustment of physical constants

Preface 8
Contents 10
Part I Basics of Metrology 15
True Values and Traceability 16
Metrology 16
Traceability 16
Measurement Errors 17
Precision and Accuracy 19
Measurement Uncertainty 19
Measuring Result 19
Rivaling Physical Approaches 21
Models and Approaches 22
Gaussian Error Model 22
Generalized Gaussian Approach 23
Robust Testing Conditions 27
Linearizations 28
Quiddity of Least Squares 29
Analysis of Variance 33
Road Map 34
Part II Generalized Gaussian Error Calculus 35
The New Uncertainties 36
Gaussian Versus Generalized Gaussian Approach 36
Uncertainty and True Value 36
Designing Uncertainties 37
Quasi Safeness 40
Treatment of Random Errors 41
Well-Defined Measuring Conditions 41
Multidimensional Normal Model 42
Permutation of Repeated Measurements 43
Treatment of Systematic Errors 44
Repercussion of Biases 44
Uniqueness of Worst-Case Assessments 45
Part III Error Propagation 46
Means and Means of Means 47
Arithmetic Mean 47
Extravagated Averages 49
Mean of Means 49
Individual Mean Versus Grand Mean 55
Functions of Erroneous Variables 61
One Variable 61
Two Variables 64
More Than Two Variables 69
Concatenated Functions 74
Elementary Examples 76
Test of Hypothesis 82
Method of Least Squares 87
Empirical Variance--Covariance Matrix 87
Propagation of Systematic Errors 90
Uncertainties of the Estimators 91
Weighting Factors 92
Example 95
Part IV Essence of Metrology 97
Dissemination of Units 98
Working Standards 98
Key Comparisons 102
Multiples and Sub-multiples 108
Calibration Chains 108
Pairwise Comparisons 117
Founding Pillars 120
Consistency 120
Traceability 121
Part V Fitting of Straight Lines 122
Preliminaries 123
Distinction of Cases 123
True Straight Line 124
Straight Lines: Case (i) 126
Fitting Conditions 126
Orthogonal Projection 126
Uncertainties of the Input Data 128
Uncertainties of the Components of the Solution Vector 129
Uncertainty Band 131
EP-Region 132
Straight Lines: Case (ii) 135
Fitting Conditions 135
Orthogonal Projection 136
Uncertainties of the Components of the Solution Vector 136
Uncertainty Band 139
EP-Region 141
Straight Lines: Case (iii) 145
Fitting Conditions 145
Orthogonal Projection 146
Series Expansion of the Solution Vector 147
Uncertainties of the Components of the Solution Vector 149
Uncertainty Band 151
EP-Region 152
Part VI Fitting of Planes 157
Preliminaries 158
Distinction of Cases 158
True Plane 158
Planes: Case (i) 160
Fitting Conditions 160
Orthogonal Projection 160
Uncertainties of the Input Data 161
Uncertainties of the Components of the Solution Vector 162
EPC-Region 164
Planes: Case (ii) 167
Fitting Conditions 167
Orthogonal Projection 168
Uncertainties of the Components of the Solution Vector 168
Confidence Intervals and Overall Uncertainties 170
Uncertainty Bowls 171
EPC-Region 173
Planes: Case (iii) 180
Fitting Conditions 180
Orthogonal Projection 181
Series Expansion of the Solution Vector 182
Uncertainties of the Components of the Solution Vector 184
Uncertainty Bowls 186
EPC-Region 188
Part VII Fitting of Parabolas 192
Preliminaries 193
Distinction of Cases 193
True Parabola 193
Parabolas: Case (i) 195
Fitting Conditions 195
Orthogonal Projection 195
Uncertainties of the Input Data 196
Uncertainties of the Components of the Solution Vector 197
Uncertainty Band 199
EPC-Region 199
Parabolas: Case (ii) 202
Fitting Conditions 202
Orthogonal Projection 203
Uncertainties of the Components of the Solution Vector 203
Uncertainty Band 206
EPC-Region 208
Parabolas: Case (iii) 212
Fitting Conditions 212
Orthogonal Projection 213
Series Expansion of the Solution Vector 214
Uncertainties of the Components of the Solution Vector 216
Uncertainty Band 218
EPC-Region 219
Part VIII Non-linear Fitting 224
Series Truncation 225
Homologous True Function 225
Fitting Conditions 226
Orthogonal Projection 226
Iteration 228
Uncertainties of the Components of the Solution Vector 229
Transformation 234
Homologous True Function 234
Fitting Conditions 234
Orthogonal Projection 235
Uncertainties of the Components of the Solution Vector 236
Part IX Appendices 240
Graphical Scale Transformations 242
Expansion of Solution Vectors 248
Special Confidence Ellipses and Ellipsoids 254
Extreme Points of Ellipses and Ellipsoids 258
Drawing Ellipses and Ellipsoids 262
Security Polygons and Polyhedra 264
EP Boundaries and EPC Hulls 274
Student's Density 280
Uncertainty Band Versus EP-Region 284
Quantiles of Hotelling's Density 292
References 294
Index 296