Series on Advances in Mathematics for Appli*
>ciences H o i . 72
ADVANC
MATH EM AT COMPUTATIONA IN METROLO
Editors
P Ciarlini E Filipe A B Forbes F Pavese C Perruchet B R L Siebert World Scientific
X
ADVANCED
MATHEMATICAL & COMPUTATIONAL TOOLS IN METROLOGY VII
Book Editors Patrizia Ciarlini (Istituto per le Applicazioni del Calcolo, CNR, Roma, Italy), Eduarda Filipe (Instituto Portugues da Qualidade, Caparica, Portugal) Alistair B Forbes (National Physical Laboratory, Teddington, UK), Franco Pavese (Istituto di Metrologia "G.Colonnetti", CNR, Torino, Italy), Christophe Perruchet (UTAC, Montlhery, France) Bernd Siebert (Physikalisch-Technische Bundesanstalt, Berlin, Germany) For the first six Volumes see this Series vol. 16 (1994), vol.40 (1996), vol.45 (1997), vol.53 (2000), vol.57 (2001) and vol.66 (2004)
X THEMATIC NETWORK "ADVANCED MATHEMATICAL AND COMPUTATIONAL TOOLS IN
METROLOGY" (SOFTOOLSMETRONET). Coordinator: F Pavese, Istituto di Metrologia "G.Colonnetti" ( I M G C ) , Torino, IT (EU Grant G6RT-CT-2001-05061 to IMGC) Caparica Chairperson: E. Filipe, Instituto Portugues da Qualidade, Caparica, Portugal INTERNATIONAL SCIENTIFIC COMMITTEE
Eric Benoit, LISTIC-ESIA, Universite de Savoie, Annecy, France Worfram Bremser, Federal Institute for Materials Research and Testing, Berlin, Germany Patrizia Ciarlini, Istituto per le Applicazioni del Calcolo "M.Picone", Roma, Italy Eduarda Corte-Real Filipe, Instituto Portugues da Qualidade (IPQ), Caparica, Portugal Alistair B Forbes, National Physical Laboratory (NPL-DTI), Teddington, UK Pedro Girao, Telecommunications Institute, DEEC, 1ST, Lisboa, Portugal Ivette Gomes, CEAUL and DEIO, Universidade de Lisboa, Lisboa, Portugal Franco Pavese, Istituto di Metrologia "G.Colonnetti" (IMGC), Torino, Italy Leslie Pendrill, Swedish National Testing & Research Institute (SP), Boris, Sweden Christophe Perruchet, UTAC, France Bernd Siebert, Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany
ORGANISED BY
Instituto Portugues da Qualidade (IPQ), Caparica, Portugal CNR, Istituto di Metrologia "G.Colonnetti", (IMGC) Torino, Italy IMEKO Technical Committee TC21 "Mathematical Tools for Measurement"
Sponsored by EU Thematic Network SofTools_MetroNet EUROMET IPQ, Portugal IMGC-CNR, Italy Societa' Italiana di Matematica Applicata ed Industriale (SIMAI), Italy LNE, France NPL-DTI, United Kingdom PTB, Germany SPMet, Portugal
Series on Advances in Mathematics for Applied Sciences - Vol. 72
ADVANCED
MATHEMATICAL & COMPUTATIONAL TOOLS IN METROLOGY VII
JH9 P Ciarlini CNR - Istituto di Applicazione del Calcolo, Roma, Italy
E Filipe Institute Portugues da Qualidade, Caparica, Portugal
I I I1 ItI1 I 1 1 0
A B Forbes
L
4
6
8
10
National Physical laboratory, Middlesex, UK
F Pavese CNR - Istituto di Metrologia, Torino, Italy National Institutefor Research in Metrology (INRiM), Torino, Italy
C Perruchet UTAC, Montlhery, France
B R L Siebert Physikalisch-Technische Bundesanstalt, Berlin, Germany
\[p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG
• TAIPEI • CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
ADVANCED MATHEMATICAL AND COMPUTAITONAL TOOLS IN METROLOGY VII Series on Advances in Mathematics for Applied Sciences — Vol. 72 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-256-674-0
Printed in Singapore by World Scientific Printers (S) Pte Ltd
Foreword This volume collects the refereed contributions based on the presentation made at the seventh workshop on the theme of advanced mathematical and computational tools in metrology, held at the IPQ Caparica, Portugal, in June 2005. The aims of the European Projects having supported the activities in this field were • • •
•
To present and promote reliable and effective mathematical and computational tools in metrology. To understand better the modelling, statistical and computational requirements in metrology. To provide a forum for metrologists, mathematicians and software engineers that will encourage a more effective synthesis of skills, capabilities and resources. To promote collaboration in the context of EU Programmes, EUROMET and EA Projects, MRA requirements.
•
To support young researchers in metrology and related fields.
•
To address industrial requirements.
The themes in this volume reflect the importance of the mathematical, statistical and numerical tools and techniques in metrology and also keeping the challenge promoted by the Meter Convention, to access a mutual recognition for the measurement standards. Caparica, November 2005
The Editors
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Contents
Foreword
v
Full Papers Modeling Measurement Processes in Complex Systems with Partial Differential Equations: From Heat Conduction to the Heart MBar, S Bauer, R Model andR W dos Santos
1
Mereotopological Approach for Measurement Software E Benoit and R Dapoigny
13
Data Evaluation of Key Comparisons Involving Several Artefacts M G Cox, P MHarris andE Woolliams
23
Box-Cox Transformations and Robust Control Charts in SPC MI Gomes andF O Figueiredo
35
Multisensor Data Fusion and Its Application to Decision Making P S Girao, J D Pereira and O Postolache
47
Generic System Design for Measurement Databases - Applied to Calibrations in Vacuum Metrology, Bio-Signals and a Template System H Gross, V Hartmann, KJousten and G Lindner
60
Evaluation of Repeated Measurements from the Viewpoints of Conventional and Bayesian Statistics / Lira and W Woger
73
Detection of Outliers in Interlaboratory Testing C Perruchet
85
On Appropriate Methods for the Validation of Metrological Software D Richter, N Greifand H Schrepf
98
Data Analysis - A Dialogue with the Data D S Sivia
108
Vlll
Contributed Papers A Virtual Instrument to Evaluate the Uncertainty of Measurement in the Calibration of Sound Calibrators G de Areas, MRuiz, J MLopez, MRecuero andRFraile
119
Intercomparison Reference Functions and Data Correlation Structure W Bremser
130
Validation of Soft Sensors in Monitoring Ambient Parameters P Ciarlini, U Maniscalco and G Regoliosi
142
Evaluation of Standard Uncertainties in Nested Structures E Filipe
151
Measurement System Analysis and Statistical Process Control A B Forbes
161
A Bayesian Analysis for the Uncertainty Evaluation of a Multivariate Non Linear Measurement Model G Iuculano, G Pellegrini and A Zanobini
171
Method Comparison Studies between Different Standardization Networks A Konnert
179
Convolution and Uncertainty Evaluation M J Korczynski, M G Cox and P Harris
188
Dimensional Metrology of Flexible Parts: Identification of Geometrical Deviations from Optical Measurements C Lartigue, F Thiebaut, P Bourdet andN Anwer Distance Splines Approach to Irregularly Distributed Physical Data from the Brazilian Northeastern Coast S de Barros Melo, E A de Oliveira Lima, MCde Araujo Filho and C Costa Dantas Decision-Making with Uncertainty in Attribute Sampling L R Pendrill andHKdllgren
196
204
212
IX
Combining Direct Calculation and the Monte Carlo Method for the Probabilistic Expression of Measurement Results G B Rossi, F Crenna, M G Cox and P M Harris
221
IMet - A Secure and Flexible Approach to Internet-Enabled Calibration at Justervesenet A Sand and H Slinde
229
Monte Carlo Study on Logical and Statistical Correlation B Siebert, P Ciarlini and D Sibold
237
The Middle Ground in Key Comparison Analysis: Revisiting the Median A G Steele, B M Wood and R J Douglas
245
System of Databases for Supporting Co-Ordination of Processes under Responsibility of Metrology Institute of Republic of Slovenia T Tasic, M Urleb and G Grgic
253
Short Communications Contribution to Surface Best Fit Enhancement by the Integration of the Real Point Distribution SAranda, J Mailhe, J M Linares andJMSprauel Computational Modeling of Seebeck Coefficients of Pt/Pd Thermocouple H S Aytekin, R Ince, A Tlnce and S Oguz
258
262
Data Evaluation and Uncertainty Analysis in an Interlaboratory Comparison of a Pycnometer Volume E Batista and E Filipe
267
Propagation of Uncertainty in Discretely Sampled Surface Roughness Profiles J KBrennan, A Crampton, X Jiang, R Leach andP M Harris
271
Computer Time (CPU) Comparison of Several Input File Formats Considering Different Versions of MCNPX in Case of Personalised Voxel-Based Dosimetry S Chiavassa, M Bardies, D Franck, J R Jourdain, J F Chatal and IA ubineau-Laniece
276
X
A New Approach to Datums Association for the Verification of Geometrical Specifications J Y Choley, A Riviere, P Bourdet and A Clement Measurements of Catalyst Concentration in the Riser of a FCC Cold Model by Gamma Ray Transmission C Costa Dantas, V A dos Santos, E A de Oliveira Lima and S de Barros Melo
280
284
Software for Data Acquisition and Analysis in Angle Standards Calibration MDobre and H Piree
289
Calculation of Uncertainties in Analogue Digital Converters - A Case Study MJ Korczynski and A Domanska
293
Asymptotic Least Squares and Student-? Sampling Distributions A B Forbes
297
A Statistical Procedure to Quantify the Conformity of New Thermocouples with Respect to a Reference Function D Ichim and MAstrua
301
Non-Parametric Methods to Evaluate Derivative Uncertainty from Small Data Sets D Ichim, P Ciarlini, E Badea and G Delia Gatta
306
Algorithms for Scanning Probe Microscopy Data Analysis P Klapetek
310
Error Correction of a Triangulation Vision Machine by Optimization A Meda and A Balsamo
316
Atomic Clock Prediction for the Generation of a Time Scale G Panfilo and P Tavella
320
Some Problems Concerning the Estimate of the Degree of Equivalence in MRA Key Comparisons and of Its Uncertainty FPavese
325
XI
Validation of Web Application for Testing of Temperature Software A Premus, TTasic, U Palmin and J Bojkovski
330
Measurement Uncertainty Evaluation Using Monte Carlo Simulation: Applications with Logarithmic Functions J A Sousa, A S Ribeiro, C O Costa and MP Castro
335
Realisation of a Process of Real-Time Quality Control of Calibrations by Means of the Statistical Virtual Standard VI Strunov
340
An Approach to Uncertainty Analysis Emphasizing a Natural Expectation of a Client R Willink
344
Special Issue Preparing for a European Research Area Network in Metrology: Where are We Now? M Kiihne, W SchmidandA Henson
350
Author Index and e-mail addresses
361
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Advanced Mathematical and Computational Tools in Metrology VII Edited by P. Ciarlini, E. Filipe, A. B. Forbes, F. Pavese, C. Perruchet & B. Siebert © 2006 World Scientific Publishing Co. (pp. 1-12)
MODELLING MEASUREMENT PROCESSES IN COMPLEX SYSTEMS WITH PARTIAL DIFFERENTIAL EQUATIONS: FROM HEAT CONDUCTION TO THE HEART MARKUS BAR, STEFFEN BAUER, REGINE MODEL, RODRIGO WEBER DOS SANTOS Department Mathematical Modeling and Data Analysis, Physikalisch-Technische Bundesanstalt (PTB), Abbestr. 2-12, 10587 Berlin, Germany. The modelling of a measurement process necessary involves a mathematical formulation of the physical laws that link the desired quantity with the results of the measurement and control parameters. In simple cases, the measurement model is a functional relationship and can be applied straightforward. In many applications, however, the measurement model is given by partial differential equations that usually cannot be solved analytically. Then, numerical simulations and related tools such as inverse methods and linear stability analysis have to be employed. Here, we illustrate the use of partial differential equations for two different examples of increasing complexity. First, we consider the forward and inverse solution of a heat conduction problem in a layered geometry. In the second part, we show results on the integrative modelling of the heart beat which links the physiology of cardiac muscle cells with structural information on their orientation and geometrical arrangement.
1. Introduction Analysing measurement processes in metrology requires often not only sophisticated tools of data analysis, but also numerical simulations of mathematical models and other computational tools. Here, we present a range of applications where the appropriate model is given by partial differential equations that can be solved only by numerical schemes like the methods of finite differences or finite elements. Mathematical models can serve different purposes; in the ideal case, the model is known exactly and simulations play the role of a ."virtual experiment". On the opposite side of the spectrum, models can also be used for developing hypotheses about observed behaviour in complicated systems and may serve as a guideline to design new experiments. Once plausible hypotheses have been established, models tend to evolve towards a more and more complete characterization of the system until the stage of the virtual experiment is reached. In the latter state, one may also employ inverse methods to obtain information on unknown physical parameters, functional dependencies or the detailed geometry of a measurement process from measurement data.
2
In this article, we illustrate the spectrum of possibilities of mathematical modelling by two examples. Section 2 describes finite element simulations of the heat equation and results of the inverse problem in a typical setup used for the measurement of heat conductivities as an example for a tractable virtual experiments. In Section 3, we will discuss basic aspects of the PTB heart modelling project. This project aims at the development of a ."virtual organ" and may later serve as a normal in medical physics. Modelling results are crucial in the interpretation of electro- and magneto-cardiographic measurements. The philosophy of heart modelling is to build a complete model that integrates the physiology of heart muscles cells with the dynamic geometry of the heart. Then, one can predict the impact of molecular changes or pathologies on the propagation of potential waves in the heart muscle and ultimately on the form of the ECG or MCG. Such computations may be used for an improvement of the diagnosis of heart diseases. Along this line, we present results on modelling the ventricles of the mouse heart in a realistic three-dimensional geometry and with information on the fiber orientation. 2. Determination of Heat Conduction Parameters in Layered Systems Within the process of development of new measurement procedures the part of mathematical modelling and simulation has gained importance in metrology. The so-called "virtual experiment" stands for the numerical simulation of an experiment based on a realistic mathematical model, virtual, because it proceeds in a computer instead in reality. Consequently, a virtual experiment design (VED) is a powerful numerical tool for: 1. the simulation, prediction, and validation of experiments; 2. optimization of measuring instruments, e.g., geometric design of sensors; 3. cause and effect analysis; 4. case studies, e.g., material dependencies; and 5. estimation of measurement errors. For indirect measurement problems where the physical properties of interest have to be determined by measurement of related quantities a subsequent data analysis solves an inverse problem. Here, the virtual experiment works as a solver for the forward problem within the optimization procedure for the inverse problem. In this field, a proper example is given by the determination of thermal transport properties of materials under test - an important task in the fields of engineering which try to reduce the energy involved, e.g., in process engineering and in the building industry. As it is infeasible to directly measure the thermal conductivity X and the thermal diffusivity a, the problem leads to an inverse heat transfer problem.
3
Non-steady-state methods such as the transient hot strip method (THS) [1-4] offer several advantages over conventional stationary methods, e.g., in shorter times of measurement and wider working temperature ranges. Here, a currentcarrying thin metal strip is clamped between two sample halves where it simultaneously acts as a resistive heater and a temperature sensor. The transient temperature rise is calculated from measured voltage drop with the aid of the thermometer equation and yields the input information for the subsequent data analysis.
Figure 1. Transient hot strip: Thermal part of set-up. a, sample halves; b, hot-strip of width D and length L; 1, electric current; U(f), THS voltage signal.
2.1 Mathematical Problem In case of homogeneous media a well-posed inverse problem of parameter identification has to be solved using an analytic approximation solution for the heat conduction equation. More complicated is the situation for layered composites where no adequate analytic solution is at hand and, furthermore, the standard measurement situation violates the unique solvability [5]. It is known from experiments and theory that the set-up may be treated mathematically as a two-dimensional system if the strip is sufficiently long, i.e., L > 100 mm. In that case, heat losses at both ends are generally negligible. Therefore, the underlying three-dimensional model discussed so far can be limited to two spatial dimensions. Hence the problem may be defined in a cross sectional area perpendicular to the strip as shown for a two-layered sample in Figure 2.
4
2.2 Forward Problem For symmetry reasons, the numerical integration domain can be reduced to a quarter of the cross sectional area. On the cut boundaries of the quadrant considered, homogeneous boundary conditions of the second kind are assumed, i.e., any heat flux vanishes. Then, for two concentric homogeneous layers the heat conduction equation is specified as
v
*
out ci
tetytae
layer
**
1
•••
C ~j
» •
integration domain
Figure 2. Schematic cross-section through the sample perpendicular to the strip. The thickness of the metal strip (0.01mm) is exaggerated.
pcp
d T { x
^
j )
= X(TXX (x, y, t) + Tyy (x, y, t)) +
q(x,y)
(1)
with the initial condition T(x,y, 0) = T0,
(x,y )e Q2 = [0,/] X [0,d],
the boundary conditions of the third kind and the symmetry conditions T x (x,y,t) = 0, T y (x,y,t) = 0,
x=0, 0< y < d 0< x^irr=
v ff
( in,v P(x,z) . »
proper_part_of definition PP(x,y)~defP(x,y)A^P(y,x)
(3)
(4)
17 asymmetry PP(x,y)-*^PP(y,x)
.
transitivity PP(x, y) A PP(y, z) -> PP(x, z) .
(6)
overlap 0(x,y) =
def3z(P(z,x)AP(z,y)).
(7)
From a mathematical point of view, the part_of relation is a partial order relation and the proper_part_of relation is the associated strict order relation. With the above set of relations, it is possible to write down some axioms specifying desirable properties for any system decomposition. Then we follow the PhysSys approach proposed by W. N. Borst et al. [7] where mereology is extended with topological relations (this reflects the fact that, in the configuration design, components are thought to be decomposed first and connected later on). The topology is based upon the is-connected-to relation C where individuals are said to be connected if they are able to exchange energy. Cc/*/; C(x, y)->C(y,x)
(8) •
(9)
Adopting an extension of the Clarke's mereotopology [8], we assume that connections only connect a pair of individuals. Additional relations define how individuals are connected. With the previous set of relations, the physical system description is brought back to the instrument as additional computational knowledge. 4.
The ontology-based model
The physical system is represented into a mereo-topological model where individuals are physical entities with a proper_part_of relation issued from the mereology, and a is-connected-to relation from the topology. The last relation represents energy links and data links between physical entities. This modelling gives rise to the structural knowledge of the physical system. In order to illustrate this modeling, a simple example of a Pitot tube for the measurement of the velocity and the level in a open water channel is presented bellow. A Pitot tube is a compound sensor that measures both static and dynamic pressure in two areas of the water with two pressure sensors. For this purpose, the water area is divided into a dynamic area, and a static one just behind the Pitot tube (see fig. 3). The front pressure sensor in contact with the static water
18
area performs the measurement of the dynamic pressure and the other one, i.e. the lateral pressure sensor, performs the measurement of the static pressure. The static pressure gives the level of the water, and the difference between both pressures gives the velocity.
velocity level
dynamic water area lateral pressure sensor
s ^z\ f r
static water area
Water area
Pitot tube^) ont pressure sensor
Figure 3. Example of a physical system to model. A Pitot tube into an open water channel.
The mereological representation of such system is made by the definition of the proper_part_of relation, for example the lateral pressure sensor is a proper part of Pitot tube. The dynamic water area is a part of water area. PP(lateralSensor, PitotTube)
(10)
PPifrontSensor, PitotTube)
(11)
P(staticWaterArea,waterArea)
(12)
PP(dynamicWaterArea,waterArea)
(13)
Note that when the water does not move, the static water area is the water area. This explains why the part_of relation is used instead of the proper_part_of relation, in the mereo-topological representation, the isconnected-to relation is also used. In the example, the pressure sensors are connected to a water area. C(lateralSensor,dynamicWaterArea)
(14)
C(frontSensor,staticWaterArea)
(15)
The figures 4 and 5 give two different graphic representations of this model.
frontSensor
staticWaterArea
Figure 4. Graph oriented representation of the mereo-topological model of the open water channel.
19 System waterArea
PitotTube lateralSensor
dynamic WaterArea
frontSensor
static WaterArea
Figure 5. Box oriented representation of the mereo-topological model of the open water channel.
The main advantage of the mereo-topology is to offer a simple modeling for complex systems. The level of description of an individual by its parts can be chosen in a consistent way with the use of the model. For example the front sensor is made of several parts, but at our level we do not decide to represent them. In order to improve the precision of the model, the description of the front sensor by its parts can be added. As this sensor is a common one, we can imagine its mereo-topological description to be included in a library of models. System / PitotTube
waterArea
PitotTube __ _ff_ __ frontSensor
staticWaterArea / \
membrane / piezo-crystal
library item )
Figure 6. Reduction of the model and expansion of the model of the front sensor. The model of the front sensor can be an item in a library of mereotopological models.
The functionality of the measurement system is also represented into the mereology theory. Individuals are now goals. A goal is a part_of another goal if and only if the achievement of the first one is required to achieve the second one. goal are defined by the triplet (action verb, physical role, physical entity) where the physical role is a physical quantity or a vector of physical quantities. For example a general goal can be (to_measure,{velocity,level],waterArea). gl = (to_acquire, {pressure}, {staticWaterArea})
(16)
g2 = {to_acquire, {pressure}, {dynamicWaterArea})
(17)
g3 = (to_compute, {velocity,level], {waterArea})
(18)
g4 = (to_send, {velocity, level], {waterArea})
(19)
20
G = {tojneasure, {velocity, level}, {waterArea})
(20)
The behavioral model is based on an event-based approach (Event Calculus) in order to include the temporal precedence. It is represented by the causal graph and can be interpreted as an ontology where the concepts are the nodes of the graph and the order relation is the causality relation. The concepts are actions, and are linked to the action verbs of the goals. The link between the physical modelling, the behavioural modelling and the functional one is performed by the application of the definition: Goals are reached by performing actions on physical quantities associated to physical entities. Practically, the concepts associated to each ontology are tuples of common entities. • Concepts in the physical ontology are defined as follows: Given R, a finite set of physical roles and 0, the finite set of physical entities, a physical context c is a tuple: c= (r; n(r), §\, 2, ...,n(r))> where r e / ? , denotes its physical role (e.g., a physical quantity), /J:R —>Nat, a function assigning to each role its arity (i.e., the number of physical entities related to a given role) and {2, ...,M(r))}, a set of entities describing the spatial locations where the role has to be taken.• Concepts in the functional ontology are goals defined such as the pair (A,Q, where A is an action symbol and C is a non-empty set of physical contexts defined as above. • Concepts in the behavioral ontology are software actions (i.e., functions) guarded by input conditions specifying the way of achievement of a given goal. In [9], Richard Dapoigny shows how the physical and the functional ontologies give enough constraints to automate the design of the behavioral causal graph. 5.
Implementation
These results are right now partially implemented into a specific software design tool named Softweaver. This tool is adapted to different kind of designer competencies including the application designer one. It includes two GUIs adapted to the definition of the physical system, a mereology checking, and a code generator for the intelligent sensors that will run the application software.
21
Machineunderstandable
Developer
Internal service implemen -tation Designer Internal seivices library Intelligent sensors
Figure 7. General behaviour of the Softweaver design tool.
* Generator
Systerr
StaticWaterArea
Oyn a mi cWate fAte a
Figure 8. Implementation of the example on both Softweaver GUIs.
Lateral Sensor
O Co
22
6.
Conclusion
After the graphical approaches, the software design tools have actually reached a new level in the evolution toward an improved interaction with the knowledge of instrument designers. They now include a high level knowledge representation and are able to handle concepts and relations organized into ontologies. In the specific field of measurement, this knowledge needs to be functional, behavioral and structural. The presented approach proposes to start a software development with the description of the physical system together with the goals of the software. Several studies had shown that this approach can be used to automate the design of measurement software. The measurement process maps the real world onto a model that is an approximation of the reality. A mereo-topological based model is less precise than any analytical model but it allows to represent complex systems and to perform easily distributed reasoning. It can be a solution to understand complex systems, to check for inconsistent behaviors, and to prevent from wrong designs. References 1. L. Finkelstein, Analysis of the concepts of measurement, information and knowledge, XVIIIMEKO World Congress, 1043-1047, (2003). 2. L. Mari, The Meaning of "quantity" in measurement, Measurement, 17/2, 127-138, (1996). 3. K. H. Ruhm, Science and technology of measurement - a unifying graphic-based approach, 10th IMEKO TC7 International Symposium, 77-83, (2004). 4. R. Dapoigny, E. Benoit and L. Foulloy, Functional ontologies for intelligent instruments, Foundations of Intelligent Systems, Lecture Notes in Computer Science (Springer), 2872, (2003). 5. E. Benoit, L. Foulloy, J. Tailland, InOMs model: a Service Based Approach to Intelligent Instruments Design, SCI 2001, XVI, 160-164 (2001). 6. T. Gruber, Toward Principles for the Design of Ontologies Used for Knowledge Sharing, Int. J. of Human and Computer Studies, 43(5/6), 907-928 (1995). 7. W.N. Borst, J.M. Akkermans, A. Pos and J.L. Top, The PhysSys ontology for physical systems, Ninth International Workshop on Qualitative Reasoning QR'95, 11-21 (1995) 8. B. Clarke, A calculus of individuals based on v connection', Notre Dame Journal of Formal Logic, 23/3, 204-218 (1981). 9. R. Dapoigny, N. Mellal, E. Benoit, L. Foulloy, Service Integration in Distributed Control Systems: an approach based on fusion of mereologies, IEEE Conf. on Cybernetics and Intelligent Systems (CIS'04), 1282-1287 (2004).
Advanced Mathematical and Computational Tools in Metrology VII Edited by P. Ciarlini, E. Filipe, A. B. Forbes, F. Pavese, C. Perruchet & B. Siebert © 2006 World Scientific Publishing Co. (pp. 23-34)
DATA EVALUATION OF K E Y COMPARISONS INVOLVING SEVERAL ARTEFACTS
M. G. C O X , P. M. H A R R I S A N D E M M A W O O L L I A M S Hampton
National Physical Laboratory, Road, Teddington, Middlesex TW11 E-mail:
[email protected] OLW,
UK
Key comparison data evaluation can be influenced by many factors including artefact instability, covariance effects, analysis model, and consistency of model and data. For a comparison involving several artefacts, further issues concern the manner of determination of a key comparison reference value and degrees of equivalence. This paper discusses such issues for key comparisons comprising a series of linked bilateral measurements, proposing a solution approach. An application of the approach to the spectral irradiance key comparison C C P R K l - a is given.
1. Introduction Analysis of the data provided by the national measurement institutes (NMIs) participating in a key comparison is influenced by factors such as (a) the requirements of the CIPM Mutual Recognition Arrangement (MRA) *, (b) the guidelines for the key comparison prepared by the relevant Working Group of the appropriate CIPM Consultative Committee, (c) the number of artefacts involved, (d) artefact stability, (e) correlations associated with the data provided by the same NMI, (f) as (e) but for different NMIs, (g) the definition of the key comparison reference value (KCRV), (h) the handling of unexplained effects, and (i) the consistency of the data. This paper concentrates on key comparisons that comprise a series of linked bilateral comparisons involving multiple artefacts having nominally identical property values. In each bilateral comparison a distinct subset of the artefacts is measured by just one of the NMIs participating in the comparison and the pilot NMI. Table 1 indicates the measurement design for six NMIs, A-F, each allocated two artefacts. P denotes the pilot and Ve the property of the £th artefact. Thus, e.g., the value of VQ is measured by NMI C and the pilot. The main objective is to provide degrees of equivalence (DoEs) for the
24 Table 1. The measurement design in the case of six NMIs, each allocated two artefacts. Vi
V2
A P
A P
v3 B P
v4 B P
V5
V6
C P
C P
v7 D P
v8 D P
V9 E P
Vio E P
Vn
V12
F P
F P
comparison. A KCRV is not an intrinsic part of the approach considered. Gross changes due to handling or transportation might cause some of the data to be unsuitable for the evaluation (section 2). Also, artefacts can change randomly on a small scale (section 5.2). A basic model of the measurements can be formed, as can a model in which systematic effects are included as parameters (section 3). Model parameter determination is posed and solved as a least squares problem (section 4). A statistical test is used to assess the consistency of the model and the data and strategies to address inconsistency are considered (section 5). The degrees of equivalence required by the MRA can be formed from the solution values of the model parameters (section 6). A key comparison of spectral irradiance 17 required consideration of all factors (a)-(i) above (section 7).
2. Artefact stability To obtain information about artefact instability, artefacts are measured in a sequence such as PNP or PNPN, where P denotes measurement by the pilot NMI and N that by the NMI to which an artefact is assigned. The data provided by an NMI for the artefacts assigned to it are taken within a measurement round, or simply round. For the sequence PNP, e.g., the pilot NMI measures within two rounds and other NMIs within one round. The data provided for each artefact can be analyzed to judge whether the artefact property value has changed grossly during transportation and, if so, consideration given to excluding the data relating to that artefact from the main data evaluation 17 . The use of such an initial screening process would not account for subtle ageing effects that might have occurred. Such considerations can be addressed in the main data evaluation (section 5).
3. Model formation Each data item provided by each participating NMI relating to each artefact estimates that artefact's property value and is generally influenced by: (1) A random effect, unrelated to other effects;
25
(2) A systematic effect, applying to all data provided by that NMI, but unrelated to other NMIs' data; (3) A systematic effect, applying to all data provided by a group of NMIs containing that NMI, where a non-participating NMI provides traceability to the NMIs in the group; (4) As (3) but where one member of the group provides traceability to the others in the group. A common underlying systematic effect is taken into account by augmenting the uncertainty matrix (covariance matrix) associated with the set of NMI's measurement data by diagonal (variance) and off-diagonal (covariance) terms n (section 3.1). Each such term is either associated with a pair of data items (a) provided by that NMI or (b) provided by that NMI and another participating NMI. Alternatively, quantities corresponding to systematic effects can be included explicitly in the model 5>13>14 (section 3.2).
3.1. Basic
model
A lower case letter, say q, denotes an estimate of the value of a quantity denoted by the corresponding upper case letter (Q), and u(q) the standard uncertainty associated with q. Uq denotes the uncertainty matrix associated with the estimate q of the value of a vector quantity Q. Denote by xgti>r the data item relating to artefact Ps property value provided by NMI i in its rth measurement round. It is an estimate of the value of X(titr, the quantity represented by NMI i's measurement scale of artefact Ps property in round r. Let Ve denote artefact's Ps property, the value of which is to be estimated. A model of the measurement is Xt,i,r = Ve.
(1)
Let X denote the vector of the X(^ 15 . Additional terms can be included in the model to account for deterministic drift 8 - u ' 1 8 . 6. Determination of the degrees of equivalence For a comparison involving one stable artefact measured by participating NMIs, the KCRV can be taken as the least squares estimate of the artefact property value 4 . A unilateral DoE would be the deviation of an NMFs measurement of the property value from the KCRV, with the uncertainty associated with that deviation at the 95 % level of confidence. A bilateral DoE would be the difference of two such deviations, with the uncertainty associated with that difference at the 95 % level of confidence. For the comparison considered here, again least squares provides best estimates of the artefact property values and the NMI systematic effects. There is not a natural choice of overall KCRV for such a comparison although, for each £, the best estimate of the value of artefact £'s property Vg can be taken as a reference value for that property. Indeed, an overall KCRV is not needed in forming unilateral and bilateral DoEs (sections 6.1 and 6.2). 6.1. Unilateral
degrees of
equivalence
From model (2), X(,i,r — Ve = Si, the left-hand side of which is the deviation of the quantity represented by NMI i's measurement scale of artefact Ts property in round r from that property. The right-hand side is NMI i's systematic effect. Taking expectations, E(X^j, r ) — E(V^) = E(Si), i.e., E(Xe,i,r) -vt
= Si,
(6)
31 where ve is the best estimate of the value of Ve and Sj that of Si provided by least squares. The left-hand side of expression (6) constitutes the unilateral DoE for NMI i, being the deviation of (an expected) measurement data item relating to artefact £'s property value from (the best estimate of) that property value. Thus, the value component of the unilateral DoE for NMI % is given by di = Sj and the associated uncertainty by u(di) = u(si). Under a normality assumption, the unilateral DoE for NMI i would be taken as (s*, 2u(si)). The u2(si) are relevant diagonal elements of Uy. However, U(SJ) so obtained would be too small as a result of the effect of processing a quantity of data on the uncertainties associated with random effects. Hence, the value obtained can be augmented in quadrature by a typical uncertainty associated with random effects as reported by NMI i. The vector s and hence the unilateral DoEs will be influenced by the choice of resolving condition (section 3.2). In particular, the vector estimate of the value of S corresponding to two different choices of resolving condition will differ by a vector of identical constants. This situation corresponds to that in a key comparison consisting of the circulation of a single, stable artefact among participating NMIs 4 , in which different choices of KCRV yield unilateral DoEs whose values differ collectively by a constant.
6.2. Bilateral
degrees of
equivalence
The bilateral DoEs can be obtained similarly. Again from the model (2), (Xt^r — Ve) — (X^^y — Ve) = Si — S^, the left-hand side of which constitutes the difference between (a) the deviation of the quantity represented by NMI i's measurement scale of artefact £'s property in round r from that artefact property and (b) the counterpart of (a) for NMI i', artefact £' and round r'. The right-hand side is the difference between NMI i's and NMI i"s systematic effects. Taking expectations as before, the value component of the bilateral DoE for NMI i and NMI i' is given by dit»/ = s$ — s^ and the associated uncertainty by u(di) = u(s»— s,/). Under a normality assumption, the bilateral DoE for NMI i would be taken as (sj — Sj/, 2U(SJ —Sj/)). The variance U 2 (SJ —s^) =u2(si)+u2(si') — 2COV(SJ,SJ/) can be formed from the diagonal elements of the uncertainty matrix Uy corresponding to s, and Sj' and the off-diagonal element lying in the corresponding row and column position. The estimate Sj — s^ and hence the corresponding bilateral DoE will not be influenced by the choice of resolving condition (section 3.2). The reason is that the only freedom in the model solution is that s can be adjusted by
32
a vector of identical constants, and v accordingly (section 3.2), and hence differences between the elements of s are invariant.
7. Application The approach described in this paper was applied to the spectral irradiance key comparison CCPR Kl-a 1 7 . This comparison was carried out separately, as stipulated in its protocol, for each of a number of wavelengths. Each participating NMI was assigned several lamps. The comparison design involved the measurement sequence PNPN or NPNP. Nominally, there were 12 participating NMIs each measuring three lamps in two measurement rounds. The problem thus involved, nominally, 144 measurements (four measurements made by an NMI and the pilot NMI of 12 x 3 lamps) and 49 model parameters (corresponding to 36 artefact properties and 13 systematic effects). The measurement data was in fact incomplete because some NMIs measured fewer than three lamps and some measurement data items were excluded as part of the initial data screening (section 2). Let zt,i,r denote the measurement data item for artefact ts property value, a lamp spectral irradiance value, made by NMI i in its round r, as an estimate of the value of ^,i, r > the quantity provided by NMI i's measurement scale of artefact £'s property in round r. Round uncertainties were incorporated to account for changes made to some NMIs' measurement scales between rounds. Each ze,i,r has an associated fractional standard uncertainty declared by NMI i and appropriate pairs of these data items, as a consequence of measurement data being provided by the same NMI and of traceability, have associated fractional covariances. Overall, the set of ztj 0,
with \x = 58 > 0 and a = 6y/8 > 0, or has a Weibull p.d.f.,
40
fw(x;6,6)
e
-(^)e~1e-^e,
=
x>0,
with fi = 8 r ( l / 0 + 1) and a = 5yjT{2/9 + 1) - T2{\/6 +1), and T denoting, as usual, the complete Gamma function. If S = 1, we shall use the notations Ga(6) and W{6), respectively. In order to have distributions with different skewness and tail weight coefficients, we consider 9 = 0.5, (0.25), 2. Denoting F*~ and $*~ the inverse functions of F and of the standard normal d.f. *~ (0.5) s $^(0.75)-*^(0.5) )
-(0.99)-*"" (0.5) j _ F~(0.5)-F~(0.01)\ -(0.75)-F~(0.5) F - (0.5)- F - (0.25)
•
and the skewness coefficient, /i3//x2 ,
7
where iir denotes the r-th central moment of F.
In Fig. 1 we present the most efficient estimator for the mean and for the standard deviation, among the estimators under study. In this figure the overall set of considered distributions is ordered by the coefficient T. Estinu Mono the mean value 1 2,260 1,305 1.218 1,105 1,062 1,062 1,042 1,030 1,023 1,018 0,972 0,933 0,916 0,911
y 9,302 7,556 2,828 2,309 2,000 8,000 1,789 1,633 1,512 1,414 8,883 9.909 10,995 12,109
F W(0,5) W(0,75) Ga(0,5) Ga(0,75) Ga(1) W(1) Ga(1,25) Ga(1,5) Ga(1,75) Ga(2) W(1,25) W(1,5) W(1,75) W(2)
Estimation of the standard deviation
3
4
5
6
7
8
TMd TMd TMd TMd TMd TMd TMd TMd TMd TMd TMd
TMd TMd TMd TMd TMd TMd TMd TMd TMd TMd TMd
TMd TMd TMd TMd TMd TMd TMd TMd TMd
TMd TMd TMd
TMd
TMd
TMd TMd
TMd TMd
TMd TMd TMd TMd
Mean Mean Mean Mean
Mean Mean Mean Mean
Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean
Figure 1.
9 TMd
10 TMd
Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean Mean
3 TR TR TR TR TR TR TR TR TR TR TR TR TR TR
4 TR TR TR TR TR TR TR TR TR TR TR TR TR TR
5 TR TR TR TR TR TR TR TR TR TR TR TR TR TR
6
7
TR TR TR TR TR TR TR TR TR TR TR TR TR TR
TR TR TR TR TR TR TR TR TR TR TR TR TR TR
8 TR TR TR TR TR TR TR TR TR TR TR TR TR TR
9 TR TR TR TR TR TR TR TR TR TR TR TR TR TR
10 TR TR TR TR TR TR TR TR TR TR TR TR TR TR
Most efficient estimator.
From this figure we can observe that the bootstrap median and range, as well as the classical S and R, are not at all competitive. The TMd statistic in Eq. (1) is the most efficient estimator for the mean value, when we consider small-to-moderate sample sizes or heavy-tailed distributions, and we advise the use of the sample mean only for large samples of not-heavy tailed distributions. The TR statistic in Eq. (2) is without doubt the most efficient estimator for the process standard deviation. In Fig. 2 we picture the "degree of robustness" of these estimators, i.e., the indicator in Eq. (6), over the same set of distributions. Now we observe that besides the efficiency of the TMd and the TR statistics, they are much
41 Estimation of the mean value
Figure 2.
Estimation of the standard deviation
Degree of robustness of the different estimators.
more robust than the usual X, R and S estimators, and this fact justifies the use, in practice, of the associated control charts. Similar simulation studies for other symmetric and asymmetric distributions have been done to compare several location and scale estimators in terms of efficiency and robustness. For details see, for instance, Chan et al. [4], Cox and Iguzquiza [7], Figueiredo [8], Figueiredo and Gomes [9], Lax [12], Tatum [19]. 3. The Use of Box-Cox Transformations in SPC The hypothesis of independent and normally distributed observations is rarely true in practice. In the literature, many data-transformations have been suggested in order to reach normality. Given the original observations (zi, £2, • • • , xn) from the process to be controlled, assumed to be positive, as usually happens with a great diversity of industrial processes' measures, we here suggest the consideration of the Box-Cox transformed sample,
>?-l)/A
-{i
In Xi
ifA^O if A = 0
Ki-*>!
f(x,v\x],...,x„)cc-^rexp
(17)
The variable v can now be "integrated out". After some manipulations and change of variables, we arrive at the marginal posterior
/(0* which is Student's t with v=n-\
n-\
(18)
degrees of freedom in the variable
T = Jn(X-x)ls.
(19)
In this derivation, x and s are mathematically equal to the sample mean and sample standard deviation of conventional statistics, but they are not interpreted as the values assumed by random variables X and S, which in this formulation are meaningless.
81 Since the expectation and variance of Student's / pdf are zero and (« - l)/(« - 3), respectively, the information contained in the final posterior can be summarized as the result x = x with standard uncertainty II = V ( « - 1 ) / ( » - 3 ) ( J / V « ) .
(20)
Finally, coverage intervals for the value of the measurand follow immediately from the meaning ascribed to the posterior. Thus, the central interval which we believe (based on the information provided solely by the data) contains the value of X with coverage probability p is x±n'l,2stv^+p)/2. This interval coincides exactly with the confidence interval derived in conventional statistics, but in the Bayesian case the interpretation is clearer. Bayesian intervals determined in this way might be called "credible" intervals to distinguish them from "confidence" intervals for the values of unknown fixed parameters of frequency distributions. 3.3. Other probability models Reconsider the problems in section 2.1 and 2.2. Suppose first that the probability model corresponds to a uniform distribution centred at X and width W, about both of which we know nothing. The likelihood for each data point is then 1
W
/«|-.
W
L(x,w,x,)oc— if x <x, <x + — and L(x,w,xi) = 0 otherwise- (/U w 2 2 Multiplying all likelihoods yields a global likelihood proportional to 1/w" foTx-w/2<xu...,x„ < x + w/2, being zero otherwise. This condition is equivalent to x - w 12 < xm and xM < x + w 12 where xm = min(xi,.. .x„) and xM = max(xi,...x„). With non informative priorsf(x) = l and f(w) = \/w (where the latter is due to W being a scale parameter) the joint posterior becomes i
1
f{x,M\xx,...,x„)v:^j-;
W
W
xM- — <x<xm+—,
w>0.
(22)
In order to quantify our state of knowledge about the width of the uniform pdf, we "integrate out" x from the joint posterior. Thus, we find f(M\Xl,...,x„)X(w-r)/wn+l;
w>r,
(23)
82 where r - xM- xmis the observed (known) range. The normalization constant can be found from the condition that the marginal posterior integrates to one; the result is K= n(n - l)r" _ 1 . The expectation and variance become: E(W\Xl,...,xn)
=^ - r ; K-2
2
V(W\xl,...,x.)=
"
r2.
(24)
(«-2) («-3)
Thus, the Bayesian analysis yields a measurement result E(fF|x,,..., xn) for the width of the distribution that is larger than the estimate r of the "true" expectation of the width obtained with conventional statistics. Note also that in the Bayesian analysis the square root of W{W\xx,...,xn) can be taken as the standard uncertainty associated with the result nrl{n-2). Finally, it is easy to check that the one sided credible interval for the width, with coverage probability p is (r, wv), where wv is the smallest root greater than r of Y7 —"--H + l = 1 - / 7 .
(25)
This interval is identical to the conventional confidence interval (see equation 3), but the Bayesian derivation is completely different. In this context, p is a probability, not a confidence level. Suppose now that (xi,...x„) are the numbers of decay events from a radioactive source observed during a fixed period of time. The quantity of interest is the emission rate X. The individual likelihoods are then of the form L(x;xt) a exp(-x)xXl . Since we know that X is positive, and that it represents a scale parameter, we use Jeffreys' prior f(x) = \lx . The posterior is then f(x\xx,..., xn) oc xN~x exp(-TTx); x > 0,
(26)
where N = Hxr We recognize this as the gamma distribution, so the normalization constant is K = nN IN\. The state of knowledge about X is summarized by the expectation and variance E(X\xu...,x„)
= x=-; n
y(X\x„...,x„)
=- , n
(27)
where, again, the average of the counts x is not the value assumed by a random variable X. The central credible interval for X with coverage probability p can now be found. Its limits, JCL and xv, are the solutions of
83
\XLxN'1 exp(-nx)dx = f xN'' »
^ . otherwise
,,M
(19)
114 1
1
1
(b)"
t .••••''" 1 /
-0.5
0
0.5
Quantity of interest
1
0.5
\n/i
i
1
H
1.5
2
Quantity of interest /J.
Figure 1. The posterior pdfs of Eqs. (10) and (21), with Peelle's measurements marked by arrows, which have been scaled vertically to have a maximum value of unity to aid comparison, (a) The posterior pdf of Eq. (21), P r ( l n ^ | l n x , / 2 ) , which is a Gaussian, (b) The same pdf transformed to linear fi, Pr(^i|lnx, I2), where it is non-Gaussian; the posterior pdf of Eq. (10), Pr(/x|x, Ii), is plotted as a dotted line.
Using Bayes' theorem, Pr(hiju|lnx,h) oc Pr(lnx|ln/u,/2) x Pr(ln/i|7 2 ) ,
(20)
where we have again omitted the denominator Pr(lnx|/2), we find that the logarithm of the posterior pdf for In /x is J02 = I n [ P r ( l n ^ | l n x , J 2 ) ] = constant
Qi
(21)
for fimm < n < /i m a x , and —00 otherwise. Thus Pr(ln fi | In x, I2) is also a Gaussian pdf which, for the case of equal relative error-bars s\ = S2, can be succinctly summarised by Infi = Iny/x\X2
± Si
(22)
The substitution of Peelle's data yields ln/x = 0.20 ± 0.21 or, through a standard (linearised) propagation of errors 4 , /x w 1.22 ±0.26. The posterior pdfs of Eqs. (10) and (21) are shown graphically in Fig. 1. 4.2. Looking at the
Evidence
The above two analyses of Peelle's data give noticeably different optimal estimates of /x, although there is a substantial degree of overlap between them. This should not be too surprising as each is predicated on a different set of assumptions, I\ and I2, corresponding to alternative interpretations of the information provided. Hanson et al.9 correctly point out that the real solution to this problem rests with the experimentalists giving more
115
details on the nature of the uncertainties in the measurements. Whatever the response, probability theory provides a formal mechanism for dealing with such ambiguities; it is based on marginalisation. If I represents the statement of Peelle's puzzle, and any other information pertinent to it, then our inference about the value of \i is encapsulated by Pr(^z|x, I). This can be related to analyses based on alternative interpretations of the data, Ii, I2, ..., IM, by M
Pr(/i|x,/) = ] T P r G u , / J - | x , / ) ,
(23)
i=i
which is a generalisation of Eq. (4). Using the product rule of probability and Bayes' theorem, each term in the summation becomes P r f o / j l x , / ) = PrQulx,/,) x ^ ' f f ^ ^
1 0
,
(24)
where the conditioning on I has been dropped, as being unnecessary, when Ij is given. Since Pr(x|J) does not depend on /J, or j , it can be treated as a normalisation constant. Without a prior indication of the 'correct' interpretation of the data, when all the Pr(7j |7) can be set equal, Eq. (23) simplifies to M
Pr(/i|x,7) ex J2 Pr(Hx,-0) * ?*(*&)
•
(25)
This is an average of the alternative analyses weighted by the evidence of the data, Pr(x|Jj). The latter, which is also known as the global or marginal likelihood, or the prior predictive, is simply the denominator term that is usually omitted in applications of Bayes' theorem as an uninteresting normalisation constant:
Using the assignments of Eqs. (7) and (9), the evidence for 7i is given by P r ( x , H / i ) dM =
—
/
/=% •
(27)
The dependence of the analysis on ^ m j n and /umax might seem surprising, but that is because their exact values tend to be irrelevant for the more familiar problem of parameter estimation: the posterior pdf Pr(/i|x,Ij) is independent of the bounds as long as they cover a sufficiently large /u-range
116
to encompass all the significant region of the likelihood function Pr(x| //, Ij). For the assignments of Eqs. (17) and (19), the corresponding evidence is best evaluated in log-space: Pr(x|7 2 ) =
Pr(lnx|7 2 ) a:iX2 InMmax
I-Q2/2
r e~Q2/2
XiX2\n[lJ,ma.x/nmin]
\
J
d l n
2irsis2 y/T-
^
(28)
where the x\ x2 in the denominator is the Jacohian for the transformation from Pr(lnx|i2) to Pr(x|/2). It should be noted that ^ m ; n and /x max do not have to have the same values in Eqs. (27) and (28): these bounds must be positive in Eq. (28), in keeping with the scale parameter view of /J, implied by I2, whereas they are free from this restriction in Eq. (27). Carrying out the evidence-weighted averaging of Eq. (25) for M = 2, with /i m ; n and /imax set somewhat arbitrarily to 0.1 and 3.0 in both Eqs. (27) and (28), we obtain the marginal posterior pdf for Peelle's problem shown in Fig. 2; it has a mean of 0.96, a standard deviation of 0.27, a maximum at 0.91 and is asymmetric with a tail towards higher /j,. Although the precise result necessarily depends on the //-bounds chosen, it does so fairly weakly. The essential conclusion is that a value of // between 1.5 and 2.0, which is on the upper-side of the larger measurement, cannot be excluded with such high certainty if the possibility of I2 is admitted (in addition to 7i).
0.5 1 1.5 2 Quantity of interest \j. Figure 2. The marginal posterior pdf of Eq. (25), Pr(/u|x, I), for M = 2. The evidenceweighted contributions from the two alternative interpretations of the data considered, P r ( / i | x , 7 i ) and Pr(fi|x,I2), are shown with a dotted and dashed lines; / i m i n and /i m a x were taken to be 0.1 and 3.0 in both cases. Peelle's measurements are marked by arrows and, to aid comparison with Fig. 1, all the pdfs have been scaled vertically so that Pr(/i|x, I) has a maximum value of unity.
117 5. Conclusions We have used Peelle's pertinent puzzle as a simple example to illustrate how the analysis of data is a dynamic process akin to holding a conversation. When the initial least-squares analysis of Section 3.1 led to results that seemed 'wrong', we reacted by looking more carefully at the validity of the assumptions that underlie that procedure. This prompted us to formulate a different question, addressed in Section 4.1, denned by an alternative interpretation of the information provided. In the absence of experimental details regarding the nature of the uncertainties associated with the given measurements, we again turned to probability theory to ask, in Section 4.2, what we could conclude in face of the ambiguity. To avoid any confusion, let us clarify further a few points regarding what we have done in this analysis of Peelle's pertinent puzzle and about our Bayesian viewpoint in general. We have not said that the least-squares analysis was wrong. Indeed, in Section 3.2, we have explained why the counter-intuitive result could actually be quite reasonable. We simply asked a series of questions, defined by alternative assumptions, and addressed them through probability theory — it was just a dialogue with the data. The Bayesian viewpoint expounded here follows the approach of mathematical physicists such as Laplace 1 , Jeffreys2 and Jaynes 3 , and is still not widely taught to science and engineering undergraduates today. It differs markedly in its accessibility for scientists from the works of many statisticians engaged in Bayesian field; the latter carry over much of the vocabulary and mind-set of their classical frequentist training, which we believe to be neither necessary nor helpful. We refer the reader to some recent textbooks, such as Jaynes 3 , Sivia4, MacKay 5 and Gregory 6 , for a good introduction to our viewpoint. To conclude, a black-box approach to the subject of data analysis, even with useful guidelines, is best avoided because it can be both limiting and misleading. All analyses are conditional on assumptions and approximations, and it's important to understand and state them clearly. While the evaluation of an arithmetic mean might seem objective and incontrovertible, for example, its status as a crucial number requires some qualified justification. We believe that an understanding of the principles underlying data analysis, along the lines outlined here, is at least as important as formal prescriptions of best practice.
118 Acknowledgments I a m grateful to Soo-youl Oh for bringing this fun little problem to my attention, and to Stephen Gull and David Waymont for giving me useful feedback on my analysis of it.
References 1. P. S. de Laplace, Theorie analytique des probabilites, Courcier Imprimeur, Paris (1812). 2. H. Jeffreys, Theory of probability, Clarendon Press, Oxford (1939). 3. E. T. Jaynes, Probability theory: the logic of science, edited by G. L. Bretthorst, Cambridge University Press, Cambridge (2003). 4. D. S. Sivia, Data analysis - a Bayesian tutorial, Oxford University Press, Oxford (1996). 5. D. J. C. MacKay Information Theory, Inference, and Learning Algorithms, Cambridge University Press, Cambridge (2003). 6. P. Gregory, Bayesian Logical Data Analysis for the Physical Sciences, Cambridge University Press, Cambridge (2005). 7. S.-Y. Oh and C.-G. Seo, PHYSOR 2004, American Nuclear Society, Lagrange Park, Illinois (2004). 8. S. Chiba and D. L. Smith, ANL/NDM-121, Argonne National Laboratory, Chicago (1991). 9. K. Hanson, T. Kawano and P. Talou, AIP Conf. Proc. 769, 304-307.
Advanced Mathematical and Computational Tools in Metrology VII Edited by P. Ciarlini, E. Filipe, A. B. Forbes, F. Pavese, C. Perruchet & B. Siebert © 2006 World Scientific Publishing Co. (pp. 119-129)
A VIRTUAL INSTRUMENT TO EVALUATE THE UNCERTAINTY OF MEASUREMENT IN THE CALIBRATION OF SOUND CALIBRATORS GUILLERMO DE ARCAS, MARIANO RUIZ, JUAN MANUEL LOPEZ, MANUEL RECUERO, RODOLFO FRAILE Grupo de Investigation en Instrumentation y Acustica Aplicada (I2A2), I.N.S.I.A., Universidad Politecnica de Madrid, Ctra. Valencia Km. 7, Madrid 28010, Spain A software tool to automate the calibration of sound calibrators is presented which includes the automatic estimation of measurement uncertainty. The tool is a virtual instrument, developed in LabVIEW, which uses several databases to store all the information needed to compute the uncertainty conforming to the Guide to the Expression of Uncertainty in Measurement. This information includes instruments specifications and calibration data.
1. Calibration of sound calibrators According to IEC 60942, sound calibrators* are designed to produce one or more known sound pressure levels at one or more specified frequencies, when coupled to specified models of microphone in specified configurations. They are calibrated in accordance with IEC 60942 and during their calibration the following magnitudes are determined: generated sound pressure level (SPL), short-term level fluctuation, frequency and total distortion of sound generated by the sound calibrator. These instruments are mainly used to check or adjust the overall sensitivity of acoustical measuring devices or systems. So, as they are only used as a transfer standard for sound pressure level, this is the most important magnitude to be determined during calibration. Therefore, although the application here presented includes the rest of the tests, the discussion will be focused in the determination of the SPL. The SPL generated by a sound calibrator can be determined using two methods: the comparison method or the insert voltage method. According to [1] the uncertainties obtained with the former are bigger, so the latter is preferred.
* In this paper the term sound calibrator is used for all devices covered by IEC 60942:2003 including pistonphones and multi-frequency calibrators.
120 In the insert voltage method, a measurement microphone, whose opencircuit sensitivity is known (.?dB), i s u s e ^ to determine the SPL generated by a sound calibrator. This can be done by measuring the open-circuit voltage (v0c) generated by the microphone, when coupled to the sound calibrator. Applying the definition of the SPL, or sound level (Lp), the following equation is obtained: Lp = 20 • log
20/iPa
*dB
(1)
As microphones are high output impedance transducers it is not easy to measure their open-circuit voltage directly, so the insert voltage method proposes an indirect way of determining this magnitude. A signal generator is inserted in series with the microphone as shown in Fig.l. First, the sound calibrator is switched on, the signal generator is switched off, and the voltage at the output of the preamplifier is measured. Then, the sound calibrator is switched off, the signal generator is switched on and its amplitude is adjusted to produce the same output voltage at the preamplifier's output, as the one produced when the sound calibrator was switched on. Last, the open circuit voltage at the microphone's output, voc» can be determined by measuring the open-circuit voltage at the output of the signal generator. Preamplifier
Microphone
Signal Generator
4= 0
Figure 1. Simplified schematic of the insert voltage method for determining the open-circuit voltage generated by a microphone on known sensitivity.
2. Software architecture A virtual instrument (VI) has been developed in Lab VIEW to automate the calibration of sound calibrators using the insert voltage method. 2.1. System architecture Figure 2 shows the instruments needed to calibrate a sound calibrator. The virtual instrument has been developed according to this system architecture, and assuming that all instruments except the microphone itself, its power supply and
121
the preamplifier, can be controlled from the computer trough a GPIB or similar interface. r-1
S
,=, Environmental Conditions Unit
Signal Generator
Microphone Power Supply
i
IF
Insert Voltage Unit
Microphone and preamplifier
->
4 Digital Multimeter Figure 2. System architecture for calibrating sound calibrators.
2.2. Software architecture The VI has a modular architecture that can be divided in two levels. High level modules, shown with rectangular shapes in Fig. 3, deal with the behavior of the application, while low level modules, represented with oval shapes, provide all the services needed to perform the basic operations: • Report Manager. Provides the functions needed to generate the calibration report annexes directly in a Microsoft Word file. It uses the National Instruments Report Generation for Microsoft Office Toolkit. • Error Manager. It manages all the possible errors that can occur in the application. • DDBB Manager. All the information needed in the application is stored in databases. This module provides all the functions needed to retrieve the information from the databases. It uses the National Instruments Database Connectivity Toolkit functions. • Uncertainty Manager. Provides the uncertainty calculation functions for each type of measurement. • Hardware managers: Generator Manager, DMM Manager and Environm. Manager. They provide the functions needed to control the hardware: configure the instruments, take measurements, etc. • Message Manager. Provides communication between modules (refer to section 2.3).
122 Main
Sequencer
/ Report \ V JManager,/*v / \
Error v \ Manager^/
y^'\ Config
+
f DDBB \ i f *• Manager J
DDBB Access
Lp Test
/'Me \ Manager /
X
f Generator \ Manager y
Test
DMM \ Manager J
hv E
Environm. \ Manager J
Uncertainty ^ Manager _/'
Figure 3. Software architecture of the virtual instrument.
These services are used by the high level modules to perform the following operations: • Main. Initializes the resources (queues, globals, instruments, etc) and launches the rest of the modules. • Sequencer. It a simple state machine that sequences the operation of the applications trough the following phases: configuration, database access, sound pressure level test, frequency test, and distortion test. • Configuration. Enables the user to choose the sound calibrator model to be calibrated, the microphone that will be used, and the calibration information (standard to be used, serial number of the calibrator, tests to be performed). • DDBB Access. Provides the user with access to databases content. It is only executed if the user needs to access the databases to enter a new type of sound calibrator. • Lp Test, Frequency test, and Distortion Test. These modules implement the algorithms of the three tests performed during calibration, and include the user interface management for each of them. 2.3. Implementation details The software architecture has been implemented following a message based structure, [2]. Low level modules are always running during execution. This means that high level modules run in parallel with them, and they send them messages to execute their services. Messages are multi-line strings sent through Lab VIEW queues, and have the following structure: the first line contains the destination manager; the second one the message, or command; and the third one, when applicable, the
123
command option or parameter value. This permits to embed the interface to low level modules in a component that can send all the possible commands to that module, as shown in Fig. 4 for the DMM Manager.
1^^
Commands Init Config DC ConFig AC Meas Close Exit
Command -
• Voltage
error in K =* error out DMM Command.vi
L_—p^
Format the message
Send the message
Wait For answer
Figure 4. Example of a VI used to send a command to a low level module.
Low level modules have all a similar structure. They are always waiting for commands to appear at their input queue. Once a command is received, it is executed and a response is sent informing of the result as shown in Fig. 5 for the DMM Manager. H No Error
|[ 0 that is, if the null hypothesis is true, all factors effects are "equal" and each observation is made up of the overall mean plus the random error £jj~N(0, a2). The total sum of squares, which is a measure of total variability in the data, may be expressed by:
154
tt(yy-y)2 =t£fa-y)+(y,-y,)\ = a
,,,
n
«Za -yf + H(X)-y,f +2 Z E a -Myt-y,) As the cross-product is zero [4], the total variability of data (SST) can be separated into: the sum of squares of differences between factor-levels averages and the grand average (SSFactor), a measure of the differences between factorlevels, and the sum of squares of the differences of observations within a factorlevels from the factor-levels average (SSE), due to the random error. Dividing each sum of squares by the respectively degrees of freedom, we obtain the corresponding mean squares (MS):
(5) a
n
EEC?,-?,)
2
'-i ;='
MSError -
a(n-\)
The mean square between factor-levels (MSFactor) [7] is an unbiased estimate of the variance a2, if H0 is true, or a surestimate of a2 (see Eq. 7), if it is false. The mean square within factor (error) (MSEn-or) is a n unbiased estimate of the variance a2. In order to test the hypotheses, we use the statistic: _
F
0
_
MS
Fac,or l *c ^Error
M
p-
(6) '
v
or,a-l,a(»-l)
where F a is the Fisher-Snedcor sampling distribution with a and ax(n- 1) degrees of freedom. If F0 > F^ a.i a(„.i), we reject the null hypothesis and conclude that the variance a2 is significantly different of zero, for a significance level a. The expected value of the MSFactor is [4]: E{MSFaMr) = E
a-llT
•a2+na2
(7)
The variance component of the factor is then obtained by:
F distribution - Sampling distribution. If %u2 and Xv2 are two independent chi-square random variables with u and v degrees of freedom, then its ratio /•"„ v is distributed as F with u numerator and v denominator degrees of freedom.
155 2
or,
_
E{MSFactor)-a
(8)
2.1.2. Model for three factors Considering now the three "stages" nested design of Fig. 1, the mathematical model is: yr*m=M+np+Ad+T,
+ £rtm
(9)
where yrd,m is the (rdtmf1 observation, // the overall mean, n^ the P-th random level effect, A,/ the D-th random level effect, T, the T-th random level effect and Eftdm the random error component. The errors and the level effects are assumed to be normally and independently distributed, respectively with mean zero and variance a2 or £j;~N(Q, a2) and with mean zero and variances ap2, erf and a2. The variance of any observation is composed by the sum of the variance components and the total number of measurements, N, is obtained by the product of the dimensions of the factors (N=R xDxTxM). The total variability of the data [8, 9] can be expressed by: p
d
t
m
p
P
d
2 p
d
t
(10)
m
or
This total variability of the data is the sum of squares of factor P (SSp), the P - factor effect, plus the sum of squares of factor D for the same P (SSD\P), plus the sum of squares of factor T for the same D and the same P (SST\DP) and finally SSE, the residual variation. Dividing by the respective degrees of freedom, P - 1, P x (D - 1), P x D x (T- 1) and PxDxTx {M- 1) we obtain the mean squares of the nested factors, which are estimates of a2, if there were no variation due to the factors. The estimates of the components of the variance, are obtained by equating the mean squares to their expected values and solving the resulting equations:
156
E(MSP) = E
SSP
E(MSDlP) = E
SS D\P P(D-\)
E(MSADP) = E E(MSE) = E
•• crz +MaT +TMaDi
SS,T\DP PD(T-l)
+DTMa/
• crz+Mcr/ + TMaD
(11)
= (T1 + MaT
SS,
PDT(M-l)
3. Example of the comparison of two thermometric water triple point cells in a three-stage nested experiment 3.1. Short description of the laboratorial work In the comparison of the two water cells (Fig. 2), JA and HS, we used two standard platinum resistance thermometers (SPRTs) A and B. After the preparation of the ice mantles, the cells were maintained in a thermo regulated water bath at a temperature of 0.007 °C. This bath can host until four cells and is able to maintain them at the triple point of water (t= 0.01 °C) during several weeks.
Figure 2. Triple point of water cell; A - Water vapour; B - Water in the liquid phase; C - Ice mantle; D -Thermometer (SPRT) well.
The ice mantle in the cells was prepared according to the laboratory procedure, 48 hours before beginning the measurements. Four measurement differences were obtained daily with the two SPRTs and this set of measurements was repeated during ten consecutive days. Two weeks after, a second ice mantle was prepared. A complete run was then repeated (Run/Plateau 2).
157 3.2. Measurement Differences Analysis Consider the data of table 1 represented schematically at figure 3. In this nested experiment, are considered the effects of Factor-P from the Plateaus (P = 2), the effects of the Factor-Z) from the Days (D= 10) for the same Plateau, the effects of the Factor-r from the Thermometers (T = 2) for the same Day and for the same Run and the variation between Measurements (M= 2) for the same Thermometer, the same Day and for the same Plateau or the residual variation. Table 1. Measurement results of the three-stage nested experiment
2 3 4
,__ Plate
3 cd
5 6 7 8 9 10
All Bll A12 B12 A13 B13 A14 B14 A15 B15 A16 B16 A17 B17 A18 B18 A19 B19 AlO BIO
Measurements (uK.) 1 2 103 93 93 93 73 78 118 68 91 126 130 96 93 88 78 118 117 97 118 118 108 98 80 80 105 100 128 108 80 110 67 77 97 92 104 84 106 96 63 68
a
X
Measurements (\iK) 1 2 107 117 74 54 123 48 103 83 114 64 102 72 74 119 70 105 68 58 104 104 93 83 100 70 77 89 84 104 62 112 91 51 68 63 69 68 88 95 78 113
Days SPRTs A21 B21 A22 B22 A23 B23 A24 B24 A25 B25 A26 B26 A27 B27 A28 B28 A29 B29 A20 B20
1 2 3 4 IN
Plateau
Days SPRTs
5 6 7 8 9 10
• • •
S
2< m
8 9 10 1 1 2 3 4 5 6 Pla tea u1
•
J!
n
a
Q
n
•
3 4 5 6 Pla tea u 2
\l
Figure 3. Schematic representation of the observed temperatures differences
9 10
158 The variance analysis is usually drawn in the ANOVA Table, displaying the sums of squares, the degrees of freedom, the mean squares, the expected values of the mean squares and the statistics F0 obtained calculating the ratios of subsequent levels mean squares. Table 2. Analysis of variance table, example of a comparison of two triple point of water cells in a three-level nested design Source of variation
Sum of squares
Plateaus Days Thermometers Measurements Total
2187.96 6873.64 7613.19 14762.50 31437.29
Degrees of freedom 1 18 20 40 79
Mean square 2187.96 381.87 380.66 369.06
Fo 5.7296 1.0032 1.0314
Expected value of mean square o2 + 2at2 + 4aD2+ 40
