Evaluating Measurement Accuracy: A Practical Approach

323 Pages·2013·2.06 MB·English

by  Semyon G. Rabinovich (auth.)

Semyon G. Rabinovich
Evaluating
Measurement
Accuracy
A Practical Approach
Second Edition
Evaluating Measurement Accuracy

Semyon G. Rabinovich
Evaluating Measurement
Accuracy
A Practical Approach
Second Edition
SemyonG.Rabinovich
ClevelandHts,OH,USA
ISBN978-1-4614-6716-8 ISBN978-1-4614-6717-5(eBook)
DOI10.1007/978-1-4614-6717-5
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013935124
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Preface
Preface to the 2nd Edition
Theimpetusforthe2ndEditionofthismonographwasgivenbytheemergenceof
the new method of experimental data processing in indirect measurements – the
method of enumeration. This method applies to multiple indirect measurements
withindependentarguments.The1stEditionsuggestedthemethodoftransforma-
tionforthisclassofmeasurements.Themethodoftransformationisbetterthanthe
traditional method because it removes the need for Taylor series for linearizing
nonlinearmeasurementequationsandcalculatingtherandomerror.However,when
the number of arguments is greater than two, the method of transformation still
resortstoTaylorseries incomputingthe systematic error.Thus, while better than
the traditional method, the method of transformation still has not removed
completely the usage of the Taylor series, leading to certain inaccuracy in error
estimation.
The method of enumeration completely removes the need for Taylor series in
calculatingboththerandomandsystematicerrors.Further,similartothemethodof
reduction for dependent indirect measurements, the method of enumeration for
independent indirect measurements does not require any assumptions regarding
distributionfunctionsoftheexperimentaldataandutilizesfullinformationcontained
intheexperimentaldatainestimatingthemeasurandandmeasurementinaccuracy.
Therefore, with the inclusion of the method of enumeration, the 2nd Edition
presentsacompletetheoryofindirectmeasurements.Incontrasttothetraditional
theory, the new theory gives a theoretically grounded and maximally accurate
estimateoftheaccuracyofanindirectmeasurement.Thisresultisanotheroutcome
ofthephysicaltheoryofmeasurementslaidoutinthisbook.
There are anumber ofothernoteworthyadditionstothe 2ndEdition.The new
editionincludestheanalysisofthereliabilityoftraditionalmethodsofestimationof
varianceandstandarddeviationofthemeasurementresultandadeepenedanalysis
ofthefoundationaldocumentsofmetrology:InternationalVocabularyofMetrology
v
vi Preface
(VIM) [1] and Guide to the Expression of Uncertainty in Measurement [2]. Fully
re-written are the sections devoted tothe Monte Carlo method and toaccuracy of
measurementsperformedwithachainofmeasuringinstruments.Theannotationof
book chapters below describes the additions and changes to this edition in more
detail.Besidesthesechanges,thiseditionalsocorrectsanumberoftypographical
andothererrorsthroughoutthebook.
The book contains nine chapters. Chapter 1 gives the general introduction
to measurements and metrology and outlines major changes that occurred in
metrology during the last two decades. Although this chapter is of introductory
nature,itpresentssomeimportantgeneralperspectivesonthesubject.Inparticular,
it includes a classification of measurements and measurement inaccuracy and
formulatespostulatesofthetheoryofmeasurements.
Chapter 2 is devoted to measuring instruments. It describes conventional
methods of representing their metrological characteristics as well as the methods
ofcontrollingthesecharacteristicsthroughcalibrationorverification.Thechapter
alsoanalyzeserrorsoflargenumbersofinstrumentsofseveraltypesandshowsthat
thedistributionfunctionsoftheseerrorsareusuallynon-stationary.
Chapter 3 contains basic statistical methods of experimental data processing.
These methods are directly applicable to the idealized multiple measurements.
Theyarealsonecessarywhenusingstatisticalmodelsofelementarymeasurement
errors and for obtaining confidenceintervals in the courseof measurement uncer-
tainty calculations. New to this edition are Sects. 3.6 and 3.7, which contain the
analysisofthereliabilityofestimatesofthevarianceandstandarddeviationofthe
measurementresult.
Chapter 4 is devoted to direct measurements. It presents a step-by-step proce-
dureforthecalculationsoftheinaccuracyofsinglemeasurements.Thecalculation
of uncertainty of a multiple measurement is then derived as a summation of the
inaccuracy of the underlying single measurement with the random error of the
multiplemeasurement,whichisestimatedfromtherepeatedsinglemeasurements.
The chapter describes a general summation method and the advantages of this
methodoverthecommonempiricalmethodsofsummatingsystematicandrandom
errors.Finally,thechapterbrieflydescribesnonparametricandrobustmethodsfor
processing direct measurement data. In this edition, the old Sect. 4.8 is split into
threesectionstoimprovetheclarityofpresentation.Thefirstofthesenewsections
describes the estimation of accuracy of multiple measurements, the second
considerstheuniversalmethodforthesummationofrandomandsystematicerrors,
andthethirdcontainsanalysisoftheaccuracyofthatmethod.
Chapter 5 presents the theory of indirect measurements. In particular, it
describes three new methods. The first one is the method of reduction, which
handles indirect measurements with dependent arguments. This method, which
weproposedpreviouslybutwhichisnotyetwidelyknown,isthefirsttoproduce
reliableestimatesofuncertaintyofthesetypesofmeasurements.Atthesametime,
iteliminatestheneedtocalculatethe correlation coefficients–amajor stumbling
blockinthesemeasurements.Thesecondmethodisthemethodoftransformation
for indirect measurements with independent arguments, which complements the
Preface vii
traditional method. The third method – new to this edition – is the method of
enumeration.Alongwiththemethodoftransformation,themethodofenumeration
results in a more accurate and grounded theory of indirect measurements. This
chapter also applies the new general method for the summationofsystematic and
random errors from Chap. 4 to indirect measurements, thus removing the need to
use the Monte Carlo method with itsknown limitations (the reliance on unknown
distribution functions and the complexity of implementation) for this purpose.
The current edition adds two new sections, Sects. 5.3 and 5.8, to describe and
justifythemethodofenumeration.Section5.12,devotedtotheapplicationsofthe
MonteCarlomethod,iscompletelyre-written.
Chapter 6 treats simultaneous and combined measurements, using the well-
known least-squares method, which is commonly applied for these measurement
types.
Chapter 7 contains methods for combining measurement data or measurement
results. This problem arises when the same measurand is measured in different
laboratories,andthefinalresultshouldreflectallthesemeasurements.Alongwitha
traditional solution, which takes into consideration only random errors, Chap. 7
includesamethodaccountingforthesystematicerrorsaswell.
Chapter 8 includes a number of concrete examples of measurement data
processingandevaluatingmeasurementaccuracy.Thebookistargetedforpractical
use, and these examples can serve as specific blueprints for addressing typical
measurement data processing needs faced by experimenters. In the 2nd Edition,
Sect. 8.1.3 is expanded to show how one can account the interaction of the
measuring instrument with the object under measurement. A new Sect. 8.6.3,
“ApplicationofMethodofEnumeration,”isadded.
ThecurrenteditionaddsanewChap.9,whichisdevotedtothenewlypublished
International Vocabulary of Metrology (VIM) [1] and Guide to the Expression of
Uncertainty in Measurement [2]. This chapter highlights the need for these
documentsbutatthesametimeshowsthattheyinthecurrentformareunsuitable
to practice and scientifically ungrounded. The chapter further formulates
recommendationsforarevisionofthesedocumentsandsketchesdirectionstoward
reorganizationoftheframeworkforpreparationofsimilarlyimportantdocuments
inthefuture.
TheConclusionpresentssomeofmythoughtsonthepast,present,andfutureof
thescienceofmeasurements.Ithasbeenexpandedinthecurrentedition,whilethe
sectionsdiscussingVIMandGUMweremovedfromittothenewChap.9.
Thiseditionwasagaintranslatedandeditedbymyson,Dr.MichaelRabinovich
ofCaseWesternReserveUniversity.Thankyou,Misha!
ClevelandHts,OH,USA SemyonG.Rabinovich
viii Preface
From the Preface to the 1st Edition
The goal of this book is to present methods for estimating the accuracy of real
measurements, that is, measurements performed in industry, trade, scientific
research – wherever the production process, quality control decision, or the inter-
pretation of an experiment depends on measurement accuracy. The necessity for
this book arises from the fact that the existing theory of measurement errors
contains significant gaps. In particular, the current theory focuses exclusively on
multiple measurements and overlooks single measurements. Meanwhile, single
measurementsaretheonesmostcommonlyusedinpractice.Moreover,thecurrent
theoryisincompleteevenwithinthescopeofmultiplemeasurements.Forexample,
it does notprovide answerstosuch fundamentalquestions ashow totranslate the
inaccuracy of a measuring instrument into the inaccuracy of a measurement
utilizing this instrument, or how to find the full uncertainty of a measurement
result,i.e.,theuncertaintythatreflectsbothsystematicandrandomerrors.
Thescienceofmeasurements–metrology–enteredaperiodofrapiddevelop-
ment several decades ago, prompted by the growth in international trade, globali-
zation of industrial production, demands of medicine and pharmacology, the
increasedattentiontofoodqualityandenvironment,andotherneedsofthemodern
society.However,metrologywillnotfulfilltheseneedswithoutremovingtheblind
spots mentioned above. I devoted many years of research filling these gaps. This
bookgeneralizesandputsintoacoherentwholetheresultsofthiseffort.
Thebookdevelopsthegeneraltheoryofprocessingexperimentalmeasurement
data,whichaddressestheneedtoobtainthevalueofaquantitybeingmeasuredand
theaccuracyofthisestimate.Forthefirsttime,thisbookpresentsthepostulatesof
the theory of measurements. It introduces the term measurement inaccuracy as a
generaltermthatreflectsmeasurementuncertaintyinsomesituationsandlimitsof
error (or even errors themselves) in others. The book shows the relationship
between the accuracy of measuring instruments and measurements utilizing these
instruments. It presents methods of estimating the accuracy of both single and
multiple measurements. Moreover, it formulates these methods in a systematic
and unified way by formulating and utilizing a new perspective that single
measurements are the basic type of measurements and multiple measurements
represent a series of repeated single measurements. This approach, besides being
logical and intuitive, makes accounting for the measuring instruments inaccuracy
aninherentpartofthecalculationsoftheinaccuracyofthemeasurement.Thebook
offers well-grounded and practical methods for combining the components of
measurement inaccuracy. In particular, it describes how to combine the limits of
elementarysystematicerrors andhow toestimatetheoverallmeasurement uncer-
taintyaccountingforboththesystematicandrandomerrors.
Aspartofthegeneraltheoryofmeasurements,thebookdevelopsthetheoryof
indirect measurements. For dependent indirect measurements, the book proposes
the method of reduction in place of the traditional method based on the Taylor
series. This method is more accurate, simpler, and most importantly allows to
Preface ix
calculatetheconfidencelimitsoftheinaccuracyofthesemeasurements,ratherthan
juststandarddeviationofthemeasurementresultasinthetraditionalmethods.At
thesametimeitremovestheneedtoaccountforthecorrelationcoefficient,which
had been a thorny issue in this area. The book also proposes a new method of
transformationforindependentindirectmeasurements.Thebookfurtherincludesa
discussionoftheapplicabilityoftheBayes’TheoremandMonteCarlomethodsin
measurement data processing, the topics that have been actively discussed in the
metrologicalresearchpapers.
Asaresult,thisbookcanserveasacomprehensivereferencefordataprocessing
ofalltypesofmeasurements,includingsingleandmultiplemeasurements,depen-
dent and independent indirect measurements, and combined and simultaneous
measurements.Itincludesmanyconcreteexamplesthatillustratetypicalproblems
encounteredinmeasurementpractice.Thus,thebookencompassestheentirearea
ofmeasurementdataprocessing,fromgeneraltheorytopracticalapplications.
<...>
This book is intended for anyone who is concerned with measurements in any
field of science or technology, who design technological processes and chooses
instrumentswithappropriateaccuracyaspartoftheirdesign,andwhodesignand
testnewmeasuringdevices.Thisbookshouldalsobeusefultouniversitystudents
pursuing science and engineering degrees. Indeed, measurements are of such
fundamental importance for modern science and engineering that everyone in
these fields must know the basics of the theory of measurements and especially
howtoevaluatetheiraccuracy.
<...>
Inconclusion,IwouldliketothankDr.AbramKagan,ProfessorattheUniver-
sityofMaryland,CollegePark,formanyyearsofcollaborationandfriendship.This
bookbenefitedfromourdiscussionsonvariousmathematicalproblemsinmetrology.
IwouldalsoliketothankDr.IlyaGertsbach,ProfessorattheBenGurionUniversity
of Beersheva (Israel), for our discussions over the theory of independent indirect
measurements.Iwouldliketoexpressmyspecialgratitudetomyson,Dr.Michael
Rabinovich,ProfessoratCaseWesternReserveUniversity.Heprovidedsupportand
assistancethroughoutmyworkonthisbookfromeditingtheproposalforpublication
todiscussingnewresultsandthepresentationtoeditingthewholebook.Thisbook
wouldnotbepossiblewithouthishelp.
BaskingRidge,NJ,USA SemyonG.Rabinovich