Grabe Generalized Gaussian Error Calculus

1. Auflage 2010
ISBN: 978-3-642-03305-6
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 301 Seiten

This book addresses a rigorous, complete and self-consistent revision of the Gaussian error calculus. It integrates mathematics and its applications to physical measurements, and serves as a text for graduate students and a reference for researchers.

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

Autoren/Hrsg.

Grabe, Michael

Inhaltsverzeichnis

1;Preface;8
2;Contents;10
3;Part I Basics of Metrology;15
3.1;True Values and Traceability;16
3.1.1;Metrology;16
3.1.2;Traceability;16
3.1.3;Measurement Errors;17
3.1.4;Precision and Accuracy;19
3.1.5;Measurement Uncertainty;19
3.1.6;Measuring Result;19
3.1.7;Rivaling Physical Approaches;21
3.2;Models and Approaches;22
3.2.1;Gaussian Error Model;22
3.2.2;Generalized Gaussian Approach;23
3.2.3;Robust Testing Conditions;27
3.2.4;Linearizations;28
3.2.5;Quiddity of Least Squares;29
3.2.6;Analysis of Variance;33
3.2.7;Road Map;34
4;Part II Generalized Gaussian Error Calculus;35
4.1;The New Uncertainties;36
4.1.1;Gaussian Versus Generalized Gaussian Approach;36
4.1.2;Uncertainty and True Value;36
4.1.3;Designing Uncertainties;37
4.1.4;Quasi Safeness;40
4.2;Treatment of Random Errors;41
4.2.1;Well-Defined Measuring Conditions;41
4.2.2;Multidimensional Normal Model;42
4.2.3;Permutation of Repeated Measurements;43
4.3;Treatment of Systematic Errors;44
4.3.1;Repercussion of Biases;44
4.3.2;Uniqueness of Worst-Case Assessments;45
5;Part III Error Propagation;46
5.1;Means and Means of Means;47
5.1.1;Arithmetic Mean;47
5.1.2;Extravagated Averages;49
5.1.3;Mean of Means;49
5.1.4;Individual Mean Versus Grand Mean;55
5.2;Functions of Erroneous Variables;61
5.2.1;One Variable;61
5.2.2;Two Variables;64
5.2.3;More Than Two Variables;69
5.2.4;Concatenated Functions;74
5.2.5;Elementary Examples;76
5.2.6;Test of Hypothesis;82
5.3;Method of Least Squares;87
5.3.1;Empirical Variance--Covariance Matrix;87
5.3.2;Propagation of Systematic Errors;90
5.3.3;Uncertainties of the Estimators;91
5.3.4;Weighting Factors;92
5.3.5;Example;95
6;Part IV Essence of Metrology;97
6.1;Dissemination of Units;98
6.1.1;Working Standards;98
6.1.2;Key Comparisons;102
6.2;Multiples and Sub-multiples;108
6.2.1;Calibration Chains;108
6.2.2;Pairwise Comparisons;117
6.3;Founding Pillars;120
6.3.1;Consistency;120
6.3.2;Traceability;121
7;Part V Fitting of Straight Lines;122
7.1;Preliminaries;123
7.1.1;Distinction of Cases;123
7.1.2;True Straight Line;124
7.2;Straight Lines: Case (i);126
7.2.1;Fitting Conditions;126
7.2.2;Orthogonal Projection;126
7.2.3;Uncertainties of the Input Data;128
7.2.4;Uncertainties of the Components of the Solution Vector;129
7.2.5;Uncertainty Band;131
7.2.6;EP-Region;132
7.3;Straight Lines: Case (ii);135
7.3.1;Fitting Conditions;135
7.3.2;Orthogonal Projection;136
7.3.3;Uncertainties of the Components of the Solution Vector;136
7.3.4;Uncertainty Band;139
7.3.5;EP-Region;141
7.4;Straight Lines: Case (iii);145
7.4.1;Fitting Conditions;145
7.4.2;Orthogonal Projection;146
7.4.3;Series Expansion of the Solution Vector;147
7.4.4;Uncertainties of the Components of the Solution Vector;149
7.4.5;Uncertainty Band;151
7.4.6;EP-Region;152
8;Part VI Fitting of Planes;157
8.1;Preliminaries;158
8.1.1;Distinction of Cases;158
8.1.2;True Plane;158
8.2;Planes: Case (i);160
8.2.1;Fitting Conditions;160
8.2.2;Orthogonal Projection;160
8.2.3;Uncertainties of the Input Data;161
8.2.4;Uncertainties of the Components of the Solution Vector;162
8.2.5;EPC-Region;164
8.3;Planes: Case (ii);167
8.3.1;Fitting Conditions;167
8.3.2;Orthogonal Projection;168
8.3.3;Uncertainties of the Components of the Solution Vector;168
8.3.4;Confidence Intervals and Overall Uncertainties;170
8.3.5;Uncertainty Bowls;171
8.3.6;EPC-Region;173
8.4;Planes: Case (iii);180
8.4.1;Fitting Conditions;180
8.4.2;Orthogonal Projection;181
8.4.3;Series Expansion of the Solution Vector;182
8.4.4;Uncertainties of the Components of the Solution Vector;184
8.4.5;Uncertainty Bowls;186
8.4.6;EPC-Region;188
9;Part VII Fitting of Parabolas;192
9.1;Preliminaries;193
9.1.1;Distinction of Cases;193
9.1.2;True Parabola;193
9.2;Parabolas: Case (i);195
9.2.1;Fitting Conditions;195
9.2.2;Orthogonal Projection;195
9.2.3;Uncertainties of the Input Data;196
9.2.4;Uncertainties of the Components of the Solution Vector;197
9.2.5;Uncertainty Band;199
9.2.6;EPC-Region;199
9.3;Parabolas: Case (ii);202
9.3.1;Fitting Conditions;202
9.3.2;Orthogonal Projection;203
9.3.3;Uncertainties of the Components of the Solution Vector;203
9.3.4;Uncertainty Band;206
9.3.5;EPC-Region;208
9.4;Parabolas: Case (iii);212
9.4.1;Fitting Conditions;212
9.4.2;Orthogonal Projection;213
9.4.3;Series Expansion of the Solution Vector;214
9.4.4;Uncertainties of the Components of the Solution Vector;216
9.4.5;Uncertainty Band;218
9.4.6;EPC-Region;219
10;Part VIII Non-linear Fitting;224
10.1;Series Truncation;225
10.1.1;Homologous True Function;225
10.1.2;Fitting Conditions;226
10.1.3;Orthogonal Projection;226
10.1.4;Iteration;228
10.1.5;Uncertainties of the Components of the Solution Vector;229
10.2;Transformation;234
10.2.1;Homologous True Function;234
10.2.2;Fitting Conditions;234
10.2.3;Orthogonal Projection;235
10.2.4;Uncertainties of the Components of the Solution Vector;236
11;Part IX Appendices;240
11.1;Graphical Scale Transformations;242
11.2;Expansion of Solution Vectors;248
11.3;Special Confidence Ellipses and Ellipsoids;254
11.4;Extreme Points of Ellipses and Ellipsoids;258
11.5;Drawing Ellipses and Ellipsoids;262
11.6;Security Polygons and Polyhedra;264
11.7;EP Boundaries and EPC Hulls;274
11.8;Student's Density;280
11.9;Uncertainty Band Versus EP-Region;284
11.10;Quantiles of Hotelling's Density;292
11.11;References;294
12;Index;296