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Keysight Technologies
Uncertainty Analysis for
Uncorrelated Input Quantities
and a Generalization of the
Welch-Satterthwaite Formula
which handles Correlated
Input Quantities Abstract--The Guide to the Expression
of Uncertainty in Measurement
(GUM) has been widely adopted in
White Paper the different fields of the industry
and science. This guide established
general rules for evaluating and
expressing uncertainty in the
measurements. In this paper we will
give an overview on how to use it for
uncorrelated input quantities. We will
also introduce correlated magnitudes
and correlation types due to the
important issue in the evaluation
of measurement uncertainty as
a consequence of the correlation
between quantities. We will identify
situations not included into the
GUM, when the measurand can be
expressed as a function of quantities
with common sources. So the issue
appears when we use the typical
Welch-Satterthwaite formula used
to calculate the effective number
of degrees of freedom when the
measurement errors are not with finite
degrees of freedom and uncorrelated.
We will introduce a generalization
of the Welch-Satterthwaite formula
for correlated components with finite
degrees of freedom.
This paper will also include other
methods for computing confidence
limits and expanded uncertainties
such as using Convolution based on
mathematical methods or evaluating
the measurement uncertainty based
on the propagation of distributions
using Monte Carlo simulation.
introdu
Introduction (34)91-631-3155Rozas, Madrid 28230, Spain,
Phone:
Ctra N-VI km 18.200 Las
Fax: (34)91-631-3001
E-mail: [email protected] and cor
Phone: (34)91-631-3155 Fax: (34)91-631-3001 importa
In general aE-mail: [email protected] but is determined
measurement is not measured directly,
of meas
from n other quantities through a functional relationship:
as a co
Introduction
Introduction Y = f ( X 1 , X 2 , X 3 ... X n )
correla
We will
Speaker: Alberto Campillo Introduction
In general a In general a measurement is not measured directly, but is determinedinclude
measurement is not measured directly, but is determined
Keysight Technologies In cases where the input quantities are independent, the combined
from n otherfrom n other quantities through a functional relationship:
quantities through a functional relationship: the me
standard uncertainty is the positive square root of the combined variance
In general a measurement is not measured directly, but is determined as a fun
Madrid, Spain In cases where the input quantities are independent, the combined
which is given by: other quantities through a functional relationship:
standard uncertainty is the positiveY = f ( rootX 2 , X 3combined variance 2 X 1 , of the ... X n )
from n commo
n square
f 2 appear
which is given by: uC 2 ( y ) = Y= f X 1 , Xxi ,)X 3 ... X n )
In cases where the input x
( u ( are independent, the combined Welch-
quantities 2
Mutual dependences uncertaintyi =1 about the input root of the can be i
standard in the knowledge positive square quantities combined variance to
is the used
expressed as In a covariance theainput quantities are independent, the combined
cases where or correlation coefficient and can be used numbe
Mutual the propagation.theby: is the positive square root of the combined variance
which is given knowledge about the input quantities can be
during dependencesuncertainty
standard in 2 when th
expressed as a covarianceby: a correlation coefficient 2can be used
which is given or
n
f and
during the propagation. uC ( y ) = n
2
fxi 2
u ( xi )
2 are not
freedom
uC 2 ( y ) = n-1 n u ( xi )
i =1
n n
f f n
2
f 2 i =1 x f f introdu
uC 2 ( y ) = u ( xi , x j ) = theknowledge iabout the inputi ,quantities canWelch-
Mutual dependences in x u ( xi ) 2 + u ( x xj ) be
= 1 = 1 xi x j
i j i =1 i i = 1 j = i +1 xi x j
This paper will also include other
This paper will also include other expressed as a covariance or a correlation coefficient and can be used correla
The degree ofof correlation between xand x jxis characterized inputthe esti- can be
correlation between x
The degreeMutual dependences in theand is is characterized by quantities knowledge about the by the esti-
j characterized by the
methods for computing confidence other The degree expressed as abetween i i or a correlation coefficient and can be used degree
of correlation
during the propagation. and
methods for computing confidence
This paper will also include other estimated correlation coefficient.x covariance x
This paper will also include mated The degree of correlation between i and
correlation coefficient.
mated correlation coefficient.
This paper will also include other between x and xj is characterized by the esti- x
limits and expanded uncertainties confidence include The degree of correlation n f The idegree of correlation x by thei esti-1 n j isfcharacterized by the es
limits and expanded uncertainties will also
methods for computing also include other computing2confidence of correlation between xi and between x and by thef
methods for computing confidence
This paper other during j is characterized 2
methods mated correlationthe propagation.
Thecoefficient. f f esti-
n n n-
for
This paper will computing confidencecorrelationcoefficient.
(( ( ( )) ) )
such asas using limits and expanded for computing confidence
such using Convolution based on
limits and expanded uncertainties
Convolution based on
methods for
uncertainties
mated uC (degree of correlation u ( correlationandx x j juis(characterizedthe esti- u ( xi , x j )
x ) x ,
y ) degree
=
Themated correlation mated xi , ujuxi xcoefficient. 2 xi ) + 2by
between = ,x is characterized
limits and expanded uncertainties n x x coefficient.
( (( ))) )
methods and expanded uncertainties i ni =1 jxi 2
n -1 1 j = i +1 xi x j
mathematical methodsusingConvolution baseduncertainties Convolution)based onxixixjx jju ( = ,u xx=xxj j f u 2 ( x ) + 2 i = u x fx f u ( x , x )
mathematicalsuch as usingevaluatingevaluating as using
such as ororConvolution basedon
limits i j
mated correlationf, i, f = u i ,,
r rj coefficient.
r
ni 1 = 1 n
methodslimits andevaluating such on
( )
=
such as using Convolution based on uC ( y = xi xj = xi x ) i
mathematical methodsor Convolution based onmethods or evaluating xi,,x xj = u ujx x * 1 ux u r xix x x = j1 ix jx
mathematical methodsexpanded ,
( ( ) ()() ()(( )( () ) ) )
2
the measurement uncertaintyas using methods or evaluating
based
the measurement uncertainty based based or evaluating
mathematical r
= 1 = 1 xi j ( i u
r i i)ii,*u j *j j jx
)
, i
u ((xxx)* xui =xxu xix,jx j i , j j
i j
( ( )
such uncertainty
the measurement uncertainty based
mathematical = i ( )
i j i= 1 = i+
ux , x = u ( xi ) * u x j
i j
the measurement
( () )
the measurement uncertainty based r i j u ( xi ) * u x j
on the propagation ofof distributionsmethods or evaluating
on the will mathematical
on the distributions
on the propagationpropagation of distributionsuncertainty based
This paperpropagationmeasurement
alsothe of distributions
ncharacterized by xi ) * u x j
u ( the esti-
include other
the on confidence on The propagation of distributions x
the degree of
using Monte Carlosimulation. uncertainty based 2correlationnbetween 22i and j is --n -n n n
measurement nnn x nn 111 n1
-
using Monte Carlo simulation. propagation of using Monte 2 u simulation.2 22 2( x ) + 2
methodsMonte Carlo the propagation of distributions 2
using Monte Carlo simulation.
( ((( ) )))(((( ) ))()( ( ) ( ) ) ) ( ) (
for computing simulation.
limits and expanded uncertainties Carlodistributions CarloCy (y )) ci i u xxi n)+ + u2 2ci c j)n -1ci -uunux ux )xix2x j rx xc,cxu ( x ) u x r x , x
simulation. = ci cu u ( iixi + 2 2 ( iy jiu((cxi xi j( ux+i ,j r ix j j j i
n
mated correlation 2(
( () ) =ci2ujn)) i 2 2j c r xx xi , r x , x
( ( ) )
using 2 n n -1 n
on the Monte
using = y ic 2u =
uC C u coefficient.
u y
= =1 i 2 C
2 c c cu jxxi 1u x ri j , x j
n
)
= ( xii= =j1)ixji1=i )i1+i121 jciic1j u (jxi ) ui x 1 j =rij+1xi , x ij j
such as using Convolution based on simulation.
using Monte Carlo =u x2 ,ic21 u=j =C+( x ) + i =1 ci u ( )iu
i 2u ( y ) c i u=i (1= i +j+ i +
( ) ( )
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i =1 =1C1 (Cy )
i u= =
r ( xi , x j ) =
=
mathematical methods or evaluating ii=1 i= 1 = + j
the measurement uncertainty based The expanded uncertainty ofof (=xi ) * u ( x j is obtainedby multiplying the stan-
The expanded uncertaintyofmeasurement)is obtainedi +bymultiplying the stan- obtained by multiplying the
The expanded uncertainty measurement
u imeasurement iis1 obtained by multiplying
1 = j= 1
The expanded uncertainty of measurement is
on the propagation of distributions expanded n Thethe output1measurement is obtainedcoverage by a the stan-
dard uncertainty of expanded estimate by aacoverage factor k multiplying
of nmeasurement coverage by is k estimate multiplying stan-
is obtained
The expanded uncertaintyoutputuncertainty ofestimatefactor byby multiplying the thefactor k which is
The standard uncertainty of-thedard uncertainty of the output which is chosecoverage stan-
uncertainty of estimate by measurement a obtained by chose
the dard uncertainty of the uncertainty of measurement is obtained by multiplying the stan-
n output which is factor
using Monte Carlo simulation. The expanded
uncertaintyc dard x + 2 oftheuoutputato belevelx j ) level confidencewhich associated
dard=chosethedesiredlevelthecconfidence (of jthebyxaby ofcoveragethewhich beis chose
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defined by: iU = k the basis i of jthe
defined the onk* u of the = i +defined by: of confidence to be
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on the basisby: U thedesired U =ofofuconfidencektoube y
=1
desired
the internal k *
=1
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The expandedauncertainty u by: U = k * is ( y )
by: k of
When Normaldistribution ( () )
defined by:aU defined cuc y y can u attributed multiplying the stan- and the stan-
definedWhenUNormalkdistributioncan bec obtained byto the measurand, and the stan-
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Whenattributedwhich is chose
When a Normal distribution cancoverageNormalkdistributionmeasurand, and the stan-
be a to the can
dard uncertainty of the output estimate bydistributionfactorbe attributed to the reliability, and the
dard uncertainty associated with a theoutput can
dard of the desired a Normal withthetooutputestimate has sufficient measurand, the stan-
When a Normal
on the basis uncertainty associated distribution can be attributed tointernal the output estimate has sufficient reliab
When levelfactor of confidence uncertainty associated with reliability,
estimate has sufficient
dard be associated with the the measurand, and
standard uncertaintyuncertainty= 2with theused. output estimate hassufficient stan-
the standard coverage k = shall with
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When Normal * coverage factorcantheattributed toto the measurand,sufficient reliability,
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shall
dard The assumptionstandard coverage factor= 2 shallshall has sufficient
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uncertainty is that the with the error estimateused. used.
associated combined output a be has
When a Normal distribution thatbe attributed to assumption aand the stan-
The assumption is can the combined the measurand, normal (infinite degrees of
error follows
the standard coverage withdistributionestimatedegrees is that the combinedfromthe
the standardassociated assumptionkisThe2 the degreesof freedom) results from the
coverage the k thatshall be used. 2 shall sufficient
dard assumption-Studentfactor = =(finiteerror used. reliability, a (infinite
factor
freedom) or tt -Student distribution(finite combined error follows be of freedom) results error(infinite degrees offollows a normal (infinite degr
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freedom) is that is-Student distribution (finite a normal (infinite degrees of
assumptionshall be used. or followsdegrees of freedom) results from the
combined
that the combined error follows a normal (finite
the standard coverageTheorem.= or t
Central Limit Theorem. 2
freedom)
Central Limit factor k t-Student distribution (finite degrees of results from the
degrees of freedom) or Limit Theorem. Limit Theorem. Central
freedom) or -Student distribution (finite degrees of freedom) freedom)
Central
The assumptionthatis that the combinedaerror follows normal (infinite degrees ofof
Theassumption is Centralcombined errorTheorem. follows a a normal (infinite degrees
is that the combined error
The assumption Central Limit follows normal (infinite degrees converges
This theorem the Limit Theorem.
results from thedemonstrates that the combined error distribution of
This t t -Student distribution (finite degrees freedom) combined
freedom)or theorem demonstrates that the combined demonstrateserrorsthe results from the
freedom) tor -Student theorem demonstrates thaterrorcombinedthat increases, converges
distribution converges
freedom) or -Student distribution (finite degrees of freedom) results of freedom) resultserror distribution converge
toward the normal distribution as the number of the of fromerrors increases, from the
distribution (finite degrees
This theorem the
This the number ofconstituent error distribution
Centraltoward the This theorem demonstrates that the combined error distribution converges
normal distribution astoward the normal distribution as the number of constituent errors increa
Limit Theorem. constituent
Centralregardless Theorem. the normal distribution as the numberdistribution con- increases,
Limit Theorem.underlyingdistributions (Figure 1).
Central Limit of their underlying that the combined error of constituent errors
toward
This theorem demonstrates distributions (Figure 1).
regardless of their regardless of their underlying distributions (Figure 1).
toward the normal distribution as the number of constituent errors increases,
This theorem demonstrates that theof their underlying distributions (Figure 1).
regardless distribution as the converges
verges towardregardless of their underlying distributionnumber of constituent
the normal combined error distributions (Figure 1).
This theorem demonstrates that the constituent errorsdistributions (Figure 1).
This theorem distribution as the that ofcombined error distribution converges
toward the normal demonstratesnumberthe combined error distribution converges
errors increases, regardless of their underlying increases,
regardless of their underlying distributions (Figure 1).
toward the normal distribution asas the number of constituent errors increases,
toward the normal distribution the number of constituent errors increases,
regardless their underlying distributions (Figure 1).
regardless ofof their underlying distributions (Figure 1).
ff((TT))
f ( T ) f ( T )
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Figure 1. Combined error distribution1 i
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Figure 1. Combined error distribution a 1.i Combined error distribution
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2011 NCSL International Workshop and Figure 1. Combined error distribution
Figure 1. Combined error1. Combined error distribution
Figure
distribution
Figure 1. Combined error distribution
T T
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Symposium
- 2
A first approach A first approach to determine the expandedauncertainty for a confidenc
to determine the expanded uncertainty for confidence
coverage use a expanded distribution, k
is to use a level is to factor coverage factor of a normal distribution, k :
level approach to determine theof a normal uncertainty for:a confi-
A first
AA first approach to determine the expanded uncertainty confidence
first approach to determine the expanded uncertainty for a for a confidence
A first approach to determine the expanded1uncertainty for a confidence
level isis to useis to use factor of aof a factor of a normal distribution, k :
level tolevel coverage a coverage normal distribution, k 2 1
dence use a a coverage factor normal distribution, k : 1 : 1
U = U + normal = 2u U k2 2k
level is to use a coverage factor2of aUU 2 distribution, + :u 2 = u 2 + u
A 1
2
= Uk + 2
B
1 A A B 1B
1 A B
2
2
U U UA 2 2+2U BU B 21= =A 22 A 2B+u B12 2
= + 2 2kk uu + u 2 2
2
2
U = UU A+ U B 2 2= k u+ uB 2
=
A
A
random readings is small, so the is small, so u A can be the
If the number of If the number of random readings value of the the value ofnot u A can
u A can not
If the number ofof random readings is small, so of the randomAthe ube
IfIf correct, andrandom readings is ofis small, sovalueof theof the of becanto represent
the number of the correct, and small, so the component is u component is better to repre
the number random readings the random
distribution the distribution the value can be A
u better
If the number of random readings is small, so thethe valuethe to represent not not
value of
correct, andtthe distributiona the random distribution, random Awe could overvalue the uncerta
oftthe distribution we the but now component is
component is better
correct,aand the distribution of the but nowcomponent overvalue the uncertainty,
canby and -Student distribution, random of couldbetter to represent
it be not correct,by of -Student component is is better to represent
it and the random
it better-Studentdistribution buttnow we could overvalue the uncertainty, could
correct,
by a t the distribution,
it by aa t tto theif the especially but nowis could ofisovervalue theuuncertainty,u B the u A and u B
by -Student distribution,abutmeasurements measurements is A and
it especially number of measurements wedistribution, the uncertainty, and
especially if
-Student distribution, if-Student could small but now u small
represent it by of now we
number the number overvalue and the we
small and the u and
especiallyififthenumber ofof areespecially ifis small and the u A and u B
overvalue the uncertainty, similar in size and the A of and B
values
especially are similar
values are similar number measurements is small number A measurements
the in sizein size
values measurements the u uB
values aresimilar in u size
values are and the in A and
is small similar size uB values are similar in size
2 2 2
= U 2* u A 2 2+* u2 u2B 2 22u B* A + k * u B
= t 2 k 2 *2 + k 2 * t22 2 u 2 2 2
U t2 = A U
= 2 + + * B2
= tt* u Au A 2 k k u*
UU * uB
So the the best to solve this problemproblemthe usingproblem is using the approach of the Wel
So best way way So solve this is using is approach of the Welchof
to the best way to solve this the approach Welch
Satterwaite formula.solve this problem is usingis using the approach Welch
So the best way tototo solve this problem thethe approach of Welch the
So the best way solve this problem is using approach of the the of
So the best way Satterwaite formula.
Satterwaite formula.
Satterwaite formula. formula.
Satterwaite formula.
Welch Satterwaite
If a Normal distribution can be assumed, but the standard uncertainty asso-
ciatedawith the distributionbebe withassumed,the standard uncertainty asso- asso-
If Normal outputIf can can be assumed, but the standardbut
Normal distribution can distribution but the uncertainty asso-
estimate assumed, but the reliability and it is uncertainty
IfIfaaNormal distribution a Normalassumed, but can be assumed, not the standard uncertainty a
If aNormal distribution can is be insufficientstandard standard uncertainty
ciated
associated with
ciated with output number estimate measurements, we willinsufficient
to the the the estimate withis with estimate is reliability is and
possible with the output outputis repeatedinsufficient reliability and itthenot is not
ciated with is with is insufficient and is