Drosg Dealing with Uncertainties

A Guide to Error Analysis
2. Auflage 2009
ISBN: 978-3-642-01384-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 243 Seiten

Dealing with Uncertainties is an innovative monograph that lays special emphasis on the deductive approach to uncertainties and on the shape of uncertainty distributions. This perspective has the potential for dealing with the uncertainty of a single data point and with sets of data that have different weights. It is shown that the inductive approach that is commonly used to estimate uncertainties is in fact not suitable for these two cases. The approach that is used to understand the nature of uncertainties is novel in that it is completely decoupled from measurements. Uncertainties which are the consequence of modern science provide a measure of confidence both in scientific data and in information in everyday life. Uncorrelated uncertainties and correlated uncertainties are fully covered and the weakness of using statistical weights in regression analysis is discussed. The text is abundantly illustrated with examples and includes more than 150 problems to help the reader master the subject.

Autoren/Hrsg.

Drosg, Manfred

Inhaltsverzeichnis

1;Preface;6
2;Preface to the First Edition;7
3;Prolog. Seven Myths in Error Analysis;9
4;Contents;11
5;Introduction;13
5.1;The Exactness of Science;14
5.1.1;Falsification vs. Verification;16
5.2;Data without Uncertainty;16
5.3;Uncertainties in Every-Day Life;17
6;Basics on Data;19
6.1;Measurement Data;19
6.1.1;The Best Estimate;20
6.1.2;Directly Measured Data;21
6.1.3;Indirectly Measured Data;21
6.1.4;Counting vs. Measuring;22
6.1.5;The Basic Counting Model;23
6.1.6;Indirect Counting;24
6.2;Analog vs. Digital;24
6.2.1;``Analog'' Measurements;25
6.3;Dealing with Data (Numerals);26
6.3.1;Valid Digits;26
6.3.2;Truncation of Numbers;26
6.3.3;Rounding;26
7;Basics on Uncertainties;29
7.1;General Characteristics of Uncertainties;31
7.1.1;Shape of Uncertainty Distributions;32
7.2;Definitions;34
7.2.1;Terminology;34
7.2.2;Necessary Requirements;34
7.2.3;Deviations;35
7.2.4;Random Uncertainties;37
7.2.5;Maximum Uncertainties (Tolerances);41
7.2.6;Limits;42
7.2.7;Outliers (Flyers);43
7.3;Uncertainty of Data Depending on One Variable;45
7.3.1;Length;45
7.3.2;Circular Area;45
7.4;Multiple Uncertainty Components (Quadratic Sum);47
7.4.1;Properties of the ``Quadratic'' Sum;49
7.4.2;Subtraction in Quadrature;51
7.4.3;Combined Standard Uncertainty;51
7.5;Uncertainty Evaluations (Error Analysis);52
7.5.1;Inductive Method;52
7.5.2;Deductive Method;53
7.6;Experimental Uncertainty;53
7.6.1;Counting and Measurement Uncertainties;54
7.6.2;Parameter Uncertainties;56
7.6.3;Model Uncertainties;56
7.6.4;Additional Experimental Uncertainties;60
8;Radioactive Decay, a Model for Random Events;61
8.1;Time Interval Distribution of Radioactive Events;61
8.1.1;Prescaling;63
8.1.2;Counting Loss (Dead Time);64
8.1.3;Direct Correction of Dead Time;65
8.1.4;Direct Correction for Lost Counts;67
8.1.5;Loss Correction Using a Pulse Generator;67
8.2;Inductive Approach to Uncertainty (Example);69
8.2.1;Properties of Data Sets and Arrays;69
8.2.2;Reproducibility within Data Sets;76
8.2.3;Linear Regression (Least-Squares Method);77
9;Frequency and Probability Distributions;80
9.1;Frequency Distribution (Spectrum);80
9.1.1;Characteristics of Distributions;83
9.1.2;Effect of Data Uncertainty on the Distribution;85
9.2;Probability Distributions;88
9.2.1;Binomial Distribution;88
9.2.2;Poisson Distribution;89
9.2.3;Normal (or Gaussian) Distribution;91
9.2.4;Finite Distributions;95
9.2.5;Convolution of Uncertainty Distributions;97
9.3;Statistical Confidence;98
9.4;Dealing with Probabilities;99
10;Deductive Approach to Uncertainty;104
10.1;Theoretical Situation;104
10.2;Practical Situation;104
10.2.1;Best Estimates Using Internal Uncertainties;106
10.2.2;Deductive vs. Inductive Uncertainties;109
10.2.3;The Sign of an Uncertainty;110
10.2.4;Benefits of Repeated Measurements;111
10.3;Regression Analysis (Least-Squares Method);116
10.3.1;Weighted Mean;118
10.3.2;Weighted Linear Regression;119
10.3.3;General Regression Analysis;121
10.3.4;Benefits of Regression Analysis;122
10.4;Data Consistency within Data Sets;123
10.4.1;Criterion of Chauvenet;123
10.4.2;Discarding Data with Internal Uncertainties;125
11;Correlation;126
11.1;Introduction;126
11.1.1;Measure of Relation;127
11.2;Correlated (Systematic) Uncertainties;128
11.2.1;Sign of a ``Systematic'' Uncertainty;129
11.2.2;Differentiation from Uncorrelated Uncertainties;130
11.2.3;More Examples of Correlated Uncertainties;137
11.2.4;External Scale Uncertainties?;141
11.3;Differentiation from ``Systematic Errors'';141
11.3.1;Gross Mistakes;142
11.3.2;Corrections;142
11.4;Correlation in Cases of Linear Regression;148
11.4.1;Weighted Linear Regression (Example);148
11.4.2;Linear Regression without Weighting (Example);149
11.5;Consistency among Data Sets;152
11.5.1;Contradictory Data Sets;154
11.5.2;Dependent (Correlated) Data Sets;156
11.6;Target Shooting as a Model for Uncertainties;158
11.6.1;Accuracy vs. Precision;160
12;Dealing with Internal Uncertainties;161
12.1;Calculations with Both Types of Uncertainties;164
12.1.1;Uncertainty of a Sum;165
12.1.2;Uncertainty of a Difference;166
12.1.3;Uncertainty of a Product;169
12.1.4;Uncertainty of a Ratio;170
12.1.5;Uncertainty of a Power (Root);171
12.1.6;Uncertainty of More Exotic Functions;172
12.2;Total Uncertainty;173
12.2.1;Adding Correlated to Uncorrelated Uncertainties;173
12.2.2;Combined Uncertainty of a Single Best Estimate;174
12.2.3;Combined Uncertainty of Data Sets;174
12.3;Using Internal Uncertainties for Diagnosis;176
12.3.1;Quality Assurance;176
12.3.2;Analogy to Bayes' Principle;178
12.3.3;Chi-Squared Test;179
13;Presentation and Estimation of Uncertainties;183
13.1;Graphic Presentation, Also of Uncertainties;183
13.1.1;Uncertainty Bars (Error Bars);184
13.1.2;Charts;185
13.1.3;Transformations;185
13.2;Correct Presentation of Uncertainties;190
13.3;Finding the Size of Internal Uncertainties;192
13.3.1;Ideal Situation in Measurements;193
13.3.2;Pragmatic Solution for Measurements;193
13.4;Estimating the Size of Uncertainties;194
13.4.1;Finding Upper Limits;195
13.4.2;Finding Lower Limits;197
14;Feedback of Uncertainties on Experiment Design;198
14.1;Optimizing Experiments;198
14.1.1;Reasons For and Against Optimization;199
14.1.2;Prevalent Design Criteria;200
14.2;Optimizing Background Measurements;201
14.2.1;Optimized Simple Background Measurement;201
14.2.2;Optimized Complex Background Measurement;202
14.3;Optimizing with Respect to Dead Time;204
14.4;Optimizing in View of the Mathematical Presentation;206
14.4.1;Optimizing Flat Dependences;207
14.4.2;Optimizing Linear Dependences;207
14.4.3;Optimum Angles for Cross Section Measurements;208
14.5;Achieving the Smallest Overall Uncertainty;209
14.5.1;The Ratio Method;209
15;Solutions;218
16;Index;231