12° IMEKO YC1 & TC7 Joint Symposium on
Man Science and Measurement
September, 3-5, 2008, Annecy, France
ON SOME KEY CONCEPTS AND TERMS IN
MEASUREMENT HAVING A CROSSDISCIPLINARY IMPACT
Giovanni Battista Rossi
Università degli Studi di Genova, DIMEC
for questions, comments, please write to
[email protected]
Key concepts and terms in measurement
Some key concepts and terms
•
•
•
•
(Measure) Value
Measuring system
Measurement value
Measurement model
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
(Measure) value –
deterministic approach
• Measurement concerns the description of
characteristics of objects by numbers
• Such numbers are intended to reproduce in a
numerical domain empirical relations
a ³ b  m a   m b 
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
(measure) value…2
• Consider a finite set of objects A
• Fix by convention the degrees of freedom of the
scale under consideration so that the number
assignment is unique, then
• From the empirical relations that hold among the
elements of A and the conventional constraints we
obtain a system of inequalities…
a ³ b  m a   m b 
b  c  m  b   m c 
...
constraints
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
(measure) value…3
For any object a belonging to A,
The measure value is that value that fulfils all
the constraints resulting from the empirical
relations, once that they have been mapped
into the corresponding numerical ones, plus
the additional conventional constraints
If the hypotheses of the representation theorem
are fulfilled, the solution exists and is unique
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
A very simple example
• Suppose that we have only three elements, a,
b and c, and that the following relations hold:
abc
and the conventional constraint is (informally):
use only the first integers: 1, 2, …
• Then the solution is
m a   2
m  b   m c   1
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Need for a probabilistic
representation
• If a and b are very “close” to each other, their
“difference” is comparable with the
“repeatability” of the comparator, and we
repeat the comparison more times, we may
observe sometimes a  b , some other
a  b and some other even a  b
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
need for a probabilistic
representation…2
• If we have two equally reliable comparators,
C and D, it may be that
with C we obtain
a C b
whilst with D we obtain
a D b
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
A very simple probabilistic
example
• Qualitatively, suppose that a empirically
results to be is fairy “distant” from b and c,
whilst c and b are very “close” to each other.
• Then a possible probabilistic description of
this empirical scenario may be
P  a  b   P  a  c   1.0
P  b  c   0.8
P  b  c   P  c  b   0.1
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Probabilistic representation for
order structures
A random variable xa is assigned to each
element a, belonging to A, in such a way that,
if a and b belong to A,
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Probability distributions for the random
variables xa, xb, xc in the example
Marginal distributions of xa, xb, xc
P(xc)
1
0.5
0
0
1
2
3
4
0
1
2
3
4
0
1
2
values of the variables
3
4
P(xb)
1
0.5
0
P(xa)
1
0.5
0
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Deterministic versus probabilistic
representations
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Measure value –
probabilistic approach
The (measure) value is a random variable
associated to each object, whose values are
the values satisfying the possible empirical
structures; each value has a probability
related to the probabilities of the possible
empirical structures
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Some applications
• Uncertainty of primary standards,
“nanoscale”?
• “Scaling” of quantities related to human
perception
• Intrinsic or “definition” uncertainty (VIM 3)
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Measuring system
How do we measure?
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Construction of a reference scale
Reference scale
Calibration of a measuring system
Direct comparison with the
reference scale
Measurement process based on a
calibrated measuring system
Measurement value
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
A definition of measuring system
an empirical system that is able to interact with
objects carrying the quantity under
investigation and to produce, as result of the
interaction, an observable output, the
(instrument) indication, on the basis of which
it is possible to assign a value to the object to
measure
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
The measurement process
Measurement results from the concatenation of:
• observation: the measurand (object to be measured)
is inputted to the measuring system that produces an
indication and
• restitution: on the basis of the indication, thanks to
the calibration function, we obtain the measurement
value.
characteristic to
be measured
object
measurement
value
instrument
indication
observation
restitution
measuring system
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Deterministic model of the
measurement process
• Observation
y  f x
• Restitution
x̂  f 1  y 
• Measurement
x̂  f 1 f  x 
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
A simple example
20
20
18 y
18
16
y = kx
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2
1 2 3 4 5 6 7 8 9 10 x
y
y = kx
x̂
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 x
x̂  x
1 2 3 4 5 6 7 8 9 10 x
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Probabilistic model of the
measurement process
20
20
18
18
P(y|x)
16
16
14
14
12
12
10
10
8
P(y|x=6)
P(x|y=12)
x̂
10
9
8
7
6
5
4
3
2
1
8
6
6
4
4
2
2
1 2 3 4 5 6 7 8 9 10 x
P(y|x)
1 2 3 4 5 6 7 8 9 10 x
P  xˆ | x 
P  xˆ | x  6 
1 2 3 4 5 6 7 8 9 10 x
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Probabilistic model of the
measurement process
• Observation
P y | x
• Restitution
P x | y   P y | x
• Measurement P  xˆ | x    y   xˆ    x | y P  y | x 
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Measurement theory - synopsis
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Measurement value
• It differs from the measure value since it is
obtained by a measurement process, and not
directly from the empirical relations
• Its uncertainty is not less than that of the
measure value
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
A taxonomy of uncertainty
sources
• Intrinsic uncertainty: the minimum uncertainty
related to the empirical relations
• Uncertainty related to the measurement
process
Both of them may be related both to random
variations and to systematic effects.
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Measurement model
• Model
an abstract system intended to represent, to
some extent and from some standpoint, a
real system (or a class of real systems)
A theory may be somewhat regarded as a
very general model
• Measurement model
a model assumed for founding the
measurability of a characteristic or for
performing a measurement.
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Modelling for measurability
• Models based on the internal properties of a
characteristic
• Models dealing with influence quantities
• Models for derived measurement
Models that concerns the measurand.
The most general is the representational
framework itself.
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
Modelling for measurement
• Any proper implementation of the
observation/restitution model
• Generalisation are possible to include
indirect, vector and dynamic measurements.
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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Key concepts and terms in measurement
MINET
MEASURING the IMPOSSIBLE NETWORK
• EU Coordination Action
• Involves 21 partners
• Very different disciplines (Psychology, Physics,
Psychophysics, Biology, Social sciences,…..
• eMINET: a virtual community of researchers involved
in Measurement the Impossible themes.
• Register following the link at :
www.minet.wordpress.com
Giovanni Battista Rossi, Università degli Studi di Genova - Italy
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