Data Modeling for Metrology and Testing in Measurement Science

498 Pages·2009·11.851 MB·English

by  Franco Pavese (auth.), Franco Pavese, Alistair B. Forbes (eds.)

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Data Modeling
for Metrology and Testing
in Measurement Science
Franco Pavese
Alistair B. Forbes
Editors
Birkhäuser
Boston • Basel • Berlin
FrancoPavese AlistairB.Forbes
IstitutoNazionalediRicerca NationalPhysics
Metrologica,Torino,Italy Laboratory,Middlesex,UK
[email protected] [email protected]
[email protected]
ISBN978-0-8176-4592-2 e-ISBN978-0-8176-4804-6
DOI10.1007/978-0-8176-4804-6
LibraryofCongressControlNumber:2008938171
MathematicsSubjectClassification(2000):60–80,60K10,60K40,60K99,62-XX,65C50.
©BirkhäuserBoston,apartofSpringerScience+BusinessMedia,LLC2009
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Preface
The aim of this book is to provide, firstly, an introduction to probability and
statistics especially directed to the metrology and testing fields and secondly,
a comprehensive, newer set of modelling methods for data and uncertainty
analysis that are generally not considered yet within mainstream methods.
The book brings, for the first time, a coherent account of these newer meth-
ods and their computational implementation. They are potentially important
because they address problems in application fields where the usual hypothe-
sesthatareatthebasisofmostofthetraditionalstatisticalandprobabilistic
methods, for example, relating to normality of the probability distributions,
arefrequentlynotfulfilledtosuchanextentthatanaccuratetreatmentofthe
calibration or test data using standard approaches is not possible. Addition-
ally, the methods can represent alternative ways of data analysis, allowing a
deeper understanding of complex situations in measurement. The book lends
itself as a possible textbook for undergraduate or postgraduate study in an
area where existing texts focus mainly on the most common and well-known
methodsthatdonotencompassmodernapproachestocalibrationandtesting
problems.
The book is structured in such a way to guide readers with only a gen-
eralinterestinmeasurementissuesthroughaseriesofreviewpapers,froman
initial introduction to modelling principles in metrology and testing, to the
basic principles of probability in metrology and statistical approaches to un-
certainty assessment. Later chapters provide a survey of the newer methods,
from an introduction to the alternative approach of interval mathematics to
the latest developments in data analysis using least squares, FFT, wavelets,
andfuzzymethods;fromdatafusion(includingdecisiontakingandriskanal-
ysis), to tools for combining data of complex statistical structure; and from
uncertaintyissuesrelatedtomodelimperfection,tothoserelatedtocombining
testingdata.Thebookalsoincludeschaptersonmoderncomputationalissues
relatedtomeasurement:acomputer-assistedsimplifiedrigourousapproachto
data evaluation, an analysis of the strategies to adopt for measurement soft-
ware validation, an introduction to the virtual instrument approach, and an
V
VI Preface
overview of the main IT applications in metrology. The book does not con-
centrateonanyparticularfieldofapplication,becausetheapplicationsinthe
frames of metrology and testing cover so broad a range that it would be diffi-
culttomakearankingoftheirimportanceoreventoattemptagroupinginto
categories with homogeneous needs. On the other hand, most of the various
techniquesillustratedinthechaptersofthebookcanfindapplicationtomany
different issues related to these application fields.
A DVD is attached to the book, containing software for free use (under
thespecifiedconditions),rangingfromtutorialstosamplecodesoftheimple-
mentationofmethodsdescribedinthebook,tosoftwarepackageswithdemos
of methods and tools, allowing the reader to try to see especially the newer
tools at work with the minimum effort, without the need of implementing his
or her own code.
Theauthorsaremainlyselectedfromaninternationalcollaborativeframe-
work (http://www.imeko-tc21.org, http://www.imeko.org), established
in the early 1990s as ‘AMCTM’ (http://www.amctm.org), that has allowed
a community of metrologists, mathematicians, statisticians, and software/IT
engineerstoworktogether,socreatingacommonunderstandingoftheissues
discussed in this book.
F. Pavese, Istituto Nazionale di Ricerca Metrologica
A B Forbes, National Physical Laboratory
Torino
June 2008
Contents
Preface ........................................................ V
List of Contributors ...........................................XV
An Introduction to Data Modelling Principles in Metrology
and Testing
Franco Pavese ................................................... 1
1 Introduction ................................................. 1
2 Uncertaintycomponentsofthemeasurementprocess:Repeatability,
reproducibility, accuracy ...................................... 2
2.1 Basic nomenclature ...................................... 3
3 The GUM approach to the measurement process: Type A and
Type B components of uncertainty ............................. 8
3.1 Basic nomenclature ...................................... 9
4 Other approaches to errors in the measurement process ........... 11
4.1 The total error and its shared and specific components ....... 12
4.2 A distinction between measurement and measurand .......... 12
5 Data modelling in metrology and in testing for intralaboratory
measurements................................................ 13
5.1 Repeated measurements .................................. 13
5.2 Nonrepeated measurements on the same standard
within each laboratory ................................... 14
6 Data modelling in metrology and in testing for interlaboratory
measurements (intercomparisons); comparison specific problems.... 15
6.1 Data modelling.......................................... 16
6.2 Specific problems and outcomes of the intercomparisons,
namely the MRA key comparisons ......................... 19
References ...................................................... 24
VII
VIII Contents
Probability in Metrology
Giovanni B. Rossi ............................................... 31
1 Probability, statistics, and measurement – An historical
perspective .................................................. 31
1.1 The origins: Gauss, Laplace, and the theory of errors......... 31
1.2 Orthodox statistics....................................... 36
1.3 The Guide to the Expression of Uncertainty
in Measurement ......................................... 42
1.4 Issues in the contemporary debate on measurement
uncertainty ............................................. 44
2 Towards a probabilistic theory of measurement................... 55
2.1 Origin and early development of the formal theory
of measurement.......................................... 55
2.2 The representational theory of measurement ................ 59
2.3 A probabilistic theory of measurement...................... 63
3 Final remarks................................................ 67
References ...................................................... 69
Three Statistical Paradigms for the Assessment and
Interpretation of Measurement Uncertainty
William F. Guthrie, Hung-kung Liu, Andrew L. Rukhin, Blaza Toman,
Jack C. M. Wang, Nien-fan Zhang ................................. 71
1 Introduction ................................................. 71
1.1 Notation................................................ 74
1.2 Statistical paradigms..................................... 74
1.3 Examples............................................... 76
2 Frequentist approach to uncertainty assessment .................. 77
2.1 Basic method ........................................... 77
2.2 Example 1 .............................................. 84
3 Bayesian paradigm for uncertainty assessment.................... 87
3.1 Basic method ........................................... 87
3.2 Example 1 .............................................. 89
4 Fiducial inference for uncertainty assessment..................... 93
4.1 Basic method ........................................... 93
4.2 Example 1 .............................................. 96
5 Example 2...................................................100
5.1 Frequentist approach.....................................101
5.2 Bayesian approach .......................................104
5.3 Fiducial approach........................................107
6 Discussion...................................................109
7 Chapter summary ............................................112
References ......................................................114
Contents IX
Interval Computations and Interval-Related Statistical
Techniques: Tools for Estimating Uncertainty of the Results
of Data Processing and Indirect Measurements
Vladik Kreinovich................................................117
1 Importance of data processing and indirect measurements .........117
2 Estimating uncertainty for the results of data processing and
indirect measurements: An important metrological problem........119
3 Uncertainty of direct measurements: Brief description, limitations,
need for overall error bounds (i.e., interval uncertainty) ...........119
4 Data processing and indirect measurements under interval
uncertainty: The main problem of interval computations...........120
5 Uniform distributions: Traditional engineering approach
to interval uncertainty ........................................121
6 Techniques for estimating the uncertainty of the results of indirect
measurements in situations when the measurement errors of direct
measurements are relatively small ..............................123
7 Techniques for error estimation in the general case
of interval uncertainty ........................................127
8 Situations when, in addition to the upper bounds on the
measurement error, we also have partial information about the
probabilities of different error values ............................136
9 Final Remarks ...............................................143
References ......................................................144
Parameter Estimation Based on Least Squares
Methods
Alistair B. Forbes ................................................147
1 Introduction .................................................147
2 Model fitting in metrology.....................................148
3 Linear least squares problems (LS) .............................148
3.1 Orthogonal factorisation method to determine parameter
estimates ...............................................149
3.2 Minimum variance property of the least squares estimate .....150
3.3 Linear Gauss–Markov problem (GM).......................151
3.4 Generalised QR factorisation approach to the Gauss–
Markov problem .........................................152
3.5 Linear least squares, maximum likelihood estimation,
and the posterior distribution p(a|y) ......................152
4 Nonlinear least squares (NLS)..................................154
4.1 The Gauss–Newton algorithm for nonlinear least squares .....154
4.2 Approximate uncertainty matrix associated with a ........155
NLS
4.3 Nonlinear Gauss–Markov problem (NGM) ..................156
4.4 Approximate uncertainty matrix associated with a .......156
NGM
4.5 Nonlinear least squares, maximum likelihood estimation,
and the posterior distribution p(a|y) ......................156
X Contents
5 Exploiting structure in the uncertainty matrix ...................158
5.1 Structure due to common random effects, linear case .........158
5.2 Structure due to common random effects, nonlinear case......160
6 Generalised distance regression (GDR) ..........................161
6.1 Algorithms for generalised distance regression ...............162
6.2 Sequential quadratic programming for the footpoint
parameters..............................................164
6.3 Example application: Response and evaluation calibration
curves..................................................166
6.4 Example: GDR line ......................................167
7 Generalised Gauss–Markov regression (GGM)....................167
7.1 Structured generalised Gauss–Markov problems..............168
8 Robust least squares (RLS)....................................169
8.1 Empirical implementation of RLS..........................171
8.2 One-sided RLS ..........................................171
8.3 RLS and the Huber M-estimator...........................172
8.4 Algorithms for robust least squares ........................173
9 Summary and concluding remarks ..............................174
References ......................................................174
Frequency and Time–Frequency Domain Analysis Tools
in Measurement
Pedro M. Ramos, Raul C. Martins, Sergio Rapuano,
Pasquale Daponte ................................................177
1 Introduction .................................................177
2 Fourier Analysis..............................................179
3 How to use the FFT ..........................................181
3.1 Aliasing ................................................182
3.2 Spectral leakage .........................................182
3.3 Windowing .............................................183
4 Example of spectral analysis application using FFT...............184
5 Wavelet transform............................................187
6 The wavelet transform: Theoretical background
and implementation ..........................................188
6.1 Continuous wavelet transform .............................189
6.2 Discrete time wavelet transform ...........................193
7 Chirplet transform ...........................................199
8 Wavelet networks.............................................201
References ......................................................202
Data Fusion, Decision-Making, and Risk Analysis:
Mathematical Tools and Techniques
Pedro S. Gira˜o, Octavian Postolache, Jos´e M. D. Pereira .............205
1 Data fusion..................................................205