Measurements and their Uncertainties

Name: Measurements and their Uncertainties | A practical guide to modern error analysis
Brand: OUP eBook
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A practical guide to modern error analysis

Ifan Hughes Thomas Hase(Autor*in)

OUP eBook (Verlag)

1. Auflage

Erschienen am 2. Juli 2010

153 Seiten

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Inhalt

Intro
Contents
1 Errors in the physical sciences
1.1 The importance of error analysis
1.2 Uncertainties in measurement
1.2.1 Terminology
1.2.2 Random errors
1.2.3 Systematic errors
1.2.4 Mistakes
1.3 Precision of measurements
1.3.1 Precision of an analogue device
1.3.2 Precision of a digital device
1.4 Accuracy of measurements
Chapter summary
2 Random errors in measurements
2.1 Analysing distributions: some statistical ideas
2.2 The mean
2.3 The width of the distribution: estimating the precision
2.3.1 Rough-and-ready approach to estimating the width
2.3.2 Statistical approach to estimating the width
2.4 Continuous distributions
2.5 The normal distribution
2.6 Sample and parent distribution
2.7 The standard error
2.7.1 The error in the error
2.8 Reporting results
2.8.1 Rounding and significant figures
2.9 The five golden rules
Chapter summary
Exercises
3 Uncertainties as probabilities
3.1 Distributions and probability
3.2 The Gaussian probability distribution function
3.2.1 Probability calculations
3.2.2 Worked example-the error function
3.3 Confidence limits and error bars
3.3.1 Extended ranges
3.3.2 Rejecting outliers
3.3.3 Experimental example of a Gaussian distribution
3.3.4 Comparing experimental results with an accepted value
3.4 Poisson probability function for discrete events
3.4.1 Worked example-Poisson counts
3.4.2 Error bars and confidence limits for Poisson statistics
3.4.3 Approximations for high means
3.5 The central limit theorem
3.5.1 Examples of the central limit theorem
Chapter summary
Exercises
4 Error propagation
4.1 Propagating the error in a single-variable function
4.1.1 The functional approach for single-variable functions
4.1.2 A calculus-based approximation for single-variable functions
4.1.3 Look-up table for common single-variable functions
4.1.4 Worked example-single variable function
4.2 Propagating the error through a multi-variable function
4.2.1 The functional approach for multi-variable functions
4.2.2 Worked example-functional approach for multi-variable functions
4.2.3 A calculus approximation for multi-variable functions
4.2.4 A look-up table for multi-variable functions
4.2.5 Comparison of methods
4.2.6 Percentage errors-dominant error
4.2.7 Using the look-up tables
4.2.8 Using the look-up tables-health warning
4.3 Propagating errors in functions-a summary
4.4 Experimental strategy based on error analysis
4.4.1 Experimental strategy for reducing the dominant error
4.5 Combined experiments-the weighted mean
4.5.1 The error in the mean-a special case of the weighted mean
Chapter summary
Exercises
5 Data visualisation and reduction
5.1 Producing a good graph
5.1.1 The independent and dependent variables
5.1.2 Linearising the data
5.1.3 Appropriate scales for the axes
5.1.4 Labelling the axes
5.1.5 Adding data points and error bars to graphs
5.1.6 Adding a fit or trend line
5.1.7 Adding a title or caption
5.2 Using a graph to see trends in the data
5.2.1 Adding a linear trend line
5.2.2 Interpolating, extrapolating and aliasing
5.3 Introduction to the method of least squares and maximum likelihood
5.3.1 Example using the method of least squares
5.4 Performing a least-squares fit to a straight line
5.5 Using graphs to estimate random and systematic errors
5.6 Residuals
Chapter summary
Exercises
6 Least-squares fitting of complex functions
6.1 The importance of ?[sup(2)] in least-squares fitting
6.1.1 ?[sup(2)] for data with Poisson errors
6.2 Non-uniform error bars
6.3 A least-squares fit to a straight line with non-uniform error bars
6.3.1 Strategies for a straight-line fit
6.3.2 Analysis of residuals with non-uniform error bars
6.4 Performing a weighted least-squares fit-beyond straight lines
6.4.1 Least-squares fit to an n[sup(th)]-order polynomial
6.4.2 Least-squares fit to an arbitrary nonlinear function
6.5 Calculating the errors in a least-squares fit
6.5.1 The error surface
6.5.2 Confidence limits on parameters from weighted least-squares fit
6.5.3 Worked example 1-a two-parameter fit
6.5.4 Worked example 2-a multi-parameter fit
6.6 Fitting with constraints
6.7 Testing the fit using the residuals
Chapter summary
Exercises
7 Computer minimisation and the error matrix
7.1 How do fitting programs minimise?
7.1.1 Iterative approaches
7.1.2 Grid search
7.1.3 The gradient-descent method
7.1.4 Second-order expansion: the Newton method
7.1.5 Second-order expansion: the Gauss-Newton method
7.1.6 The Marquardt-Levenberg method
7.2 The covariance matrix and uncertainties in fit parameters
7.2.1 Extracting uncertainties in fit parameters
7.2.2 Curvature matrix for a straight-line fit
7.2.3 Scaling the uncertainties
7.3 Correlations among uncertainties of fit parameters
7.3.1 Correlation coefficients-off-diagonal elements of the covariance matrix
7.4 Covariance in error propagation
7.4.1 Worked example 1-a straight-line fit
7.4.2 Worked example 2-a four-parameter fit
Chapter summary
Exercises
8 Hypothesis testing-how good are our models?
8.1 Hypothesis testing
8.2 Degrees of freedom
8.2.1 Data reduction and the number of degrees of freedom
8.3 The ?[sup(2)] probability distribution function
8.3.1 ?[sup(2)] for one degree of freedom
8.4 Using ?[sup(2)] as a hypothesis test
8.4.1 The reduced ?[sup(2)]statistic
8.4.2 Testing the null hypothesis-a summary
8.5 Testing the quality of a fit using ?[sup(2)]
8.5.1 Worked example 1-testing the quality of a fit
8.5.2 Worked example 2-testing different models to a data set
8.5.3 What constitutes a good fit?
8.6 Testing distributions using ?[sup(2)]
8.6.1 Worked example 3-testing a discrete distribution
8.6.2 Worked example 4-testing a continuous distribution
8.7 Occam's razor
8.8 Student's t-distribution
8.9 Scaling uncertainties
8.10 Summary of fitting experimental data to a theoretical model
Chapter summary
Exercises
9 Topics for further study
9.1 Least-squares fitting with uncertainties in both variables
9.1.1 Fitting to a straight line
9.1.2 Fitting to a more general function
9.1.3 Orthogonal distance regression
9.2 More complex error surfaces
9.2.1 Simulated annealing
9.2.2 Genetic algorithms
9.3 Monte Carlo methods
9.3.1 Introduction to Monte Carlo methods
9.3.2 Testing distributions with Monte Carlo methods
9.4 Bootstrap methods
9.5 Bayesian inference
9.6 GUM-Guide to the Expression of Uncertainty in Measurement
Bibliography
Index
A
B
C
D
E
F
G
H
I
J
L
M
N
O
P
Q
R
S
T
U
V
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Z

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