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[资源] Measurements and Their Uncertainties A Practical Guide to Modern Error Analysis

Contents
1 Errors in the physical sciences 1
1.1 The importance of error analysis 1
1.2 Uncertainties in measurement 2
1.2.1 Terminology 3
1.2.2 Random errors 3
1.2.3 Systematic errors 4
1.2.4 Mistakes 4
1.3 Precision of measurements 5
1.3.1 Precision of an analogue device 5
1.3.2 Precision of a digital device 6
1.4 Accuracy of measurements 6
Chapter summary 7
2 Random errors in measurements 9
2.1 Analysing distributions: some statistical ideas 9
2.2 The mean 9
2.3 The width of the distribution: estimating the precision 10
2.3.1 Rough-and-ready approach to estimating the width 10
2.3.2 Statistical approach to estimating the width 11
2.4 Continuous distributions 12
2.5 The normal distribution 13
2.6 Sample and parent distribution 13
2.7 The standard error 14
2.7.1 The error in the error 16
2.8 Reporting results 17
2.8.1 Rounding and significant figures 18
2.9 The five golden rules 19
Chapter summary 19
Exercises 20
3 Uncertainties as probabilities 23
3.1 Distributions and probability 23
3.2 The Gaussian probability distribution function 24
3.2.1 Probability calculations 24
3.2.2 Worked example—the error function 25
3.3 Confidence limits and error bars 25
3.3.1 Extended ranges 26
3.3.2 Rejecting outliers 26
x Contents
3.3.3 Experimental example of a Gaussian distribution 27
3.3.4 Comparing experimental results with an accepted value 28
3.4 Poisson probability function for discrete events 28
3.4.1 Worked example—Poisson counts 29
3.4.2 Error bars and confidence limits for Poisson statistics 30
3.4.3 Approximations for high means 30
3.5 The central limit theorem 31
3.5.1 Examples of the central limit theorem 33
Chapter summary 34
Exercises 35
4 Error propagation 37
4.1 Propagating the error in a single-variable function 37
4.1.1 The functional approach for single-variable
functions 38
4.1.2 A calculus-based approximation for single-variable
functions 38
4.1.3 Look-up table for common single-variable functions 39
4.1.4 Worked example—single variable function 39
4.2 Propagating the error through a multi-variable function 40
4.2.1 The functional approach for multi-variable functions 41
4.2.2 Worked example—functional approach for
multi-variable functions 42
4.2.3 A calculus approximation for multi-variable functions 43
4.2.4 A look-up table for multi-variable functions 43
4.2.5 Comparison of methods 44
4.2.6 Percentage errors—dominant error 45
4.2.7 Using the look-up tables 45
4.2.8 Using the look-up tables—health warning 46
4.3 Propagating errors in functions—a summary 47
4.4 Experimental strategy based on error analysis 47
4.4.1 Experimental strategy for reducing the dominant error 49
4.5 Combined experiments—the weighted mean 49
4.5.1 The error in the mean—a special case of the
weighted mean 50
Chapter summary 51
Exercises 51
5 Data visualisation and reduction 53
5.1 Producing a good graph 53
5.1.1 The independent and dependent variables 54
5.1.2 Linearising the data 54
5.1.3 Appropriate scales for the axes 54
5.1.4 Labelling the axes 54
5.1.5 Adding data points and error bars to graphs 55
5.1.6 Adding a fit or trend line 56
5.1.7 Adding a title or caption 57
Contents xi
5.2 Using a graph to see trends in the data 57
5.2.1 Adding a linear trend line 57
5.2.2 Interpolating, extrapolating and aliasing 59
5.3 Introduction to the method of least squares and maximum
likelihood 59
5.3.1 Example using the method of least squares 61
5.4 Performing a least-squares fit to a straight line 61
5.5 Using graphs to estimate random and systematic errors 62
5.6 Residuals 63
Chapter summary 64
Exercises 64
6 Least-squares fitting of complex functions 67
6.1 The importance of χ2 in least-squares fitting 67
6.1.1 χ2 for data with Poisson errors 68
6.2 Non-uniform error bars 68
6.3 A least-squares fit to a straight line with non-uniform
error bars 69
6.3.1 Strategies for a straight-line fit 71
6.3.2 Analysis of residuals with non-uniform error bars 72
6.4 Performing a weighted least-squares fit—beyond straight
lines 72
6.4.1 Least-squares fit to an nth-order polynomial 72
6.4.2 Least-squares fit to an arbitrary nonlinear function 72
6.5 Calculating the errors in a least-squares fit 74
6.5.1 The error surface 74
6.5.2 Confidence limits on parameters from weighted
least-squares fit 75
6.5.3 Worked example 1—a two-parameter fit 77
6.5.4 Worked example 2—a multi-parameter fit 79
6.6 Fitting with constraints 79
6.7 Testing the fit using the residuals 81
Chapter summary 82
Exercises 83
7 Computer minimisation and the error matrix 85
7.1 How do fitting programs minimise? 85
7.1.1 Iterative approaches 86
7.1.2 Grid search 86
7.1.3 The gradient-descent method 87
7.1.4 Second-order expansion: the Newton method 88
7.1.5 Second-order expansion: the Gauss–Newton method 89
7.1.6 The Marquardt–Levenberg method 90
7.2 The covariance matrix and uncertainties in fit parameters 92
7.2.1 Extracting uncertainties in fit parameters 92
7.2.2 Curvature matrix for a straight-line fit 93
7.2.3 Scaling the uncertainties 93
xii Contents
7.3 Correlations among uncertainties of fit parameters 93
7.3.1 Correlation coefficients—off-diagonal elements of the
covariance matrix 93
7.4 Covariance in error propagation 95
7.4.1 Worked example 1—a straight-line fit 95
7.4.2 Worked example 2—a four-parameter fit 96
Chapter summary 97
Exercises 97
8 Hypothesis testing—how good are our models? 101
8.1 Hypothesis testing 101
8.2 Degrees of freedom 102
8.2.1 Data reduction and the number of degrees of
freedom 103
8.3 The χ2 probability distribution function 104
8.3.1 χ2 for one degree of freedom 105
8.4 Using χ2 as a hypothesis test 105
8.4.1 The reduced χ2 statistic 107
8.4.2 Testing the null hypothesis—a summary 108
8.5 Testing the quality of a fit using χ2 108
8.5.1 Worked example 1—testing the quality of a fit 109
8.5.2 Worked example 2—testing different models
to a data set 109
8.5.3 What constitutes a good fit? 110
8.6 Testing distributions using χ2 111
8.6.1 Worked example 3—testing a discrete distribution 112
8.6.2 Worked example 4—testing a continuous
distribution 113
8.7 Occam’s razor 114
8.8 Student’s t-distribution 115
8.9 Scaling uncertainties 115
8.10 Summary of fitting experimental data to a theoretical
model 116
Chapter summary 117
Exercises 118
9 Topics for further study 121
9.1 Least-squares fitting with uncertainties in both variables 121
9.1.1 Fitting to a straight line 121
9.1.2 Fitting to a more general function 122
9.1.3 Orthogonal distance regression 122
9.2 More complex error surfaces 123
9.2.1 Simulated annealing 123
9.2.2 Genetic algorithms 124
9.3 Monte Carlo methods 125
9.3.1 Introduction to Monte Carlo methods 125
9.3.2 Testing distributions with Monte Carlo methods 125
9.4 Bootstrap methods 126
Contents xiii
9.5 Bayesian inference 127
9.6 GUM—Guide to the Expression of Uncertainty
in Measurement 129
Bibliography 131
Index 133

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