Uncertainty analysis of experimental data with R

207 Pages·2017·4.727 MB·English

by  Benjamin David Shaw

Uncertainty Analysis
of Experimental Data
with R

Uncertainty Analysis
of Experimental Data
with R
Benjamin D. Shaw
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
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Library of Congress Cataloging‑in‑Publication Data
Names: Shaw, Ben D.
Title: Uncertainty analysis of experimental data with R / Benjamin
David Shaw.
Description: Boca Raton : CRC Press, 2017. | Includes bibliographical
references.
Identifiers: LCCN 2016048268 | ISBN 9781498797320
Subjects: LCSH: Uncertainty (Information theory)--Textbooks. |
Probabilities--Textbooks. | R (Computer program language)
Classification: LCC Q375 .S53 2017 | DDC 001.4/34028553--dc23
LC record available at https://lccn.loc.gov/2016048268
Visit the Taylor & Francis Web site at
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I dedicate this book, with love, to my family.

Contents
1. Introduction .....................................................................................................1
1.1 What Is This Book About? ...................................................................1
1.2 Units ........................................................................................................3
1.3 Physical Constants and Their Uncertainties .....................................5
1.4 Dimensionless Quantities ....................................................................5
1.5 Software ..................................................................................................6
1.6 Topics Covered ......................................................................................7
Problems ............................................................................................................7
References .........................................................................................................8
2. Aspects of R ...................................................................................................11
2.1 Getting R ...............................................................................................11
2.2 Using R .................................................................................................11
2.3 Getting Help ........................................................................................12
2.4 Libraries and Packages .......................................................................13
2.5 Variables ...............................................................................................13
2.6 Vectors ...................................................................................................14
2.7 Arithmetic ............................................................................................15
2.8 Data Frames .........................................................................................16
2.9 Exporting Data ....................................................................................17
2.10 Importing Data ....................................................................................18
2.11 Internal Mathematical Functions ......................................................18
2.12 Writing Your Own Functions ............................................................19
2.13 Plotting Mathematical Functions ......................................................19
2.14 Loops .....................................................................................................20
2.15 Making Decisions ................................................................................21
2.16 Scripts....................................................................................................23
2.17 Reading Data from Websites .............................................................25
2.18 Matrices and Linear Algebra .............................................................25
2.19 Some Useful Functions and Operations ..........................................28
2.19.1 Data Frames ...........................................................................28
2.19.2 Vectors .....................................................................................29
2.19.3 Probability and Statistics ......................................................29
2.19.4 Plotting ...................................................................................29
2.19.5 Matrices and Linear Algebra ...............................................30
2.19.6 Data, Functions, Libraries, and Packages ..........................30
2.19.7 Various Other Functions and Operations ..........................30
Problems ..........................................................................................................30
References .......................................................................................................32
vii
viii Contents
3. Statistics ..........................................................................................................33
3.1 Populations and Samples ...................................................................33
3.2 Mean, Median, Standard Deviation, and Variance of a Sample ......34
3.3 Covariance and Correlation ...............................................................35
3.4 Visualizing Data ..................................................................................36
3.4.1 Histograms ..............................................................................36
3.4.2 Box Plots ..................................................................................37
3.4.3 Plotting Data Sets ...................................................................39
3.4.4 Some Plotting Parameters and Commands .......................42
3.5 Estimating Population Statistics .......................................................43
3.5.1 Confidence Interval for the Population Mean Using
Student’s t Variables ..............................................................43
3.5.2 Confidence Interval for the Population Variance
Using Chi-Square Variables ..................................................44
3.5.3 Confidence Interval Interpretation ......................................45
3.6 Comparing the Means of Two Samples ...........................................46
3.7 Testing Data for Normality ................................................................47
3.8 Outlier Identification ..........................................................................49
3.8.1 Modified Thompson τ Technique ........................................50
3.8.2 Chauvenet’s Criterion ...........................................................51
Problems ..........................................................................................................54
References .......................................................................................................56
4. Curve Fits ........................................................................................................57
4.1 Linear Regression ................................................................................57
4.2 Nonlinear Regression .........................................................................64
4.3 Kernel Smoothing ...............................................................................67
Problems ..........................................................................................................71
References .......................................................................................................73
5. Uncertainty of a Measured Quantity ........................................................75
5.1 What Is Uncertainty? ..........................................................................75
5.2 Random Variables ...............................................................................75
5.3 Measurement Uncertainties ...............................................................78
5.4 Elemental Systematic Errors ..............................................................82
5.4.1 Normal Distributions ............................................................83
5.4.2 Uniform Distributions ...........................................................83
5.4.3 Triangular Distributions .......................................................84
5.5 Coverage Factors .................................................................................85
Problems ..........................................................................................................89
References .......................................................................................................90
6. Uncertainty of a Result Calculated Using Experimental Data .............91
6.1 Taylor Series Approach ......................................................................92
6.2 Coverage Factors .................................................................................99
Contents ix
6.3 The Kline–McClintock Equation .....................................................102
6.4 Balance Checks ..................................................................................103
Problems ........................................................................................................103
References .....................................................................................................105
7. Taylor Series Uncertainty of a Linear Regression Curve Fit ..............107
7.1 Curve-Fit Expressions ......................................................................107
7.2 Cases to Consider ..............................................................................110
7.2.1 Case 1: No Errors in x and No Correlations ....................110
i
7.2.2 Case 2: Random Errors Only ..............................................112
7.2.3 Case 3: Random and Systematic Errors ............................114
7.3 General Linear Regression Theory .................................................117
7.4 Uncertainties in Regression Coefficients .......................................126
7.5 Evaluating Uncertainties with Built-in R Functions ....................127
Problems ........................................................................................................128
References .....................................................................................................129
8. Monte Carlo Methods ................................................................................131
8.1 Overall Monte Carlo Approach ......................................................131
8.2 Random Number Generation ..........................................................132
8.2.1 Accept/Reject Method ........................................................133
8.2.2 Inverse-cdf Method .............................................................136
8.3 Random Sampling ............................................................................137
8.4 Uncertainty of a Measured Variable ...............................................138
8.5 Bootstrapping with Internal Functions in R ..................................141
8.6 Monte Carlo Convergence Criteria.................................................143
8.7 Uncertainty of a Result Calculated Using Experimental Data ...144
8.8 Uncertainty Bands for Linear Regression Curve Fits ..................149
8.9 Uncertainty Bands for a Curve Fit with Kernel Smoothing........152
Problems ........................................................................................................155
References .....................................................................................................157
9. The Bayesian Approach .............................................................................159
9.1 Bayes’ Theorem for Probability Density Functions......................164
9.2 Bayesian Estimation of the Mean and Standard Deviation
of a Normal Population ....................................................................167
Problems ........................................................................................................172
References .....................................................................................................174
Appendix: Probability Density Functions ....................................................175
Index .....................................................................................................................191