top down uncertainty estimation
DESCRIPTION
TOP DOWN Uncertainty EstimationTRANSCRIPT
TOP-DOWNUncertainty estimation
Detlef Schiel, Olaf Rienitz
13.-14. April 2008
Physikalisch-Technische Bundesanstalt Braunschweig
Two ways for determining the uncertainty
Top down
Bottom up
According to GUMstarting from the model equation and considering all individualuncertainty contributions
Phenomenological approach basing on validation and QC dataassuming that they - cover all influence factors and- that they are representative
for all measurements with this method
Top-down appraoch
Uncertainty
Bias(Trueness)
Reproducibility(Precision)
according to „Nordtest“:
By measurement of control samples
uR = sR uBias
Reference materials,Comparison measurement,
Standard addition
22BiasR uuu +=
ukU ∗=
Example : Uncertainty estimation using data (s , ∆) from a validation of the measurement procedure by means of a CRM
Precondition: The composition of the samples should be similar to that of the CRM in order to be able to apply the validation data for future measurements
Parameter: s precision, ∆ trueness
Top Down example
Certified reference value:
Mean value from n single measurement results xk underrepeatebility conditions, as those expected for further measurements.
xMeasurement result for the CRM:
Systematic deviation: refxx −=∆
Standard deviation: ∑ −−
==
n
kk xx
ns
1
2)(1
1
Validation by means of a CRM
xref
Significant
Uncertainty and Significance Test
Not significant
)(2)(2 ∆u∆∆u >≥
Measurements of the CRM Certified RM
)(2)()(2)(2 ref
22
ref
22 xunsxuxu∆u +=+=
Uncertainty of ∆:
Test criteria for the relevance of the deviation
xRef x
∆
xRef x
∆
a) ∆ is significant correction of y
∆yy −=korr
)()()( ref
22
222
korr xunss∆usyu ++=+=
Correction for ∆
Repeat measurements
of CRM
CRMRepeat
measurementsof the sample
b) ∆ significant but no correction of y: Instead of that consideration of the deviation in the uncertainty.
2222
korr
2 )()()( ∆∆us∆yuyu ++=+=
2
ref
22
2 )( ∆xunss +++=
Integration of ∆ in the uncertainty
Repeat measurements
of CRM
CRMRepeat
measurementsof the sample
Bias
c) ∆ is not significant no correction of y
2222
korr
2
korrnicht )()()( ∆∆us∆yuyu ++=+=
2
ref
22
2 )( ∆xunss +++=Same expression as for b)
Integration of ∆ in the uncertainty
Repeat measurements
of CRM
CRMRepeat
measurementsof the sample
Bias
∆ is proportional to the magnitude of the quantity
Recovery:refxxR =
ref
1x∆R =−and
Proportional deviation
Ryy 1
korr ∗=Multiplicative correction useful:
Is ∆ significant then R is also significant
a) y is corrected for R:
)()( 2
rel
2
relkorrrel Rusyu +=
)( ref
2
rel
2
rel2
rel xun
ss ++=
yss =reland
Correction for R
Repeat measurements
of CRM
CRMRepeat
measurementsof the sample
22
rel
2
relkorrnichtrel )1()()( −++= RRusyu
b) y needs not to be corrected because R don‘t differ enough from 1
2
ref
2
rel
2
rel2
rel )1()( −+++= Rxun
ss
Integration of R in the uncertainty
Repeat measurements
of CRM
CRMRepeat
measurementsof the sample
Bias
Uncertainty estimations by means of other data
Comparison measurements:R
iT
iR
R
R
nns
n∆s∆u
∑ ⋅+∑+= ,
2
,2
22
25,12)(2
Mean of the deviations
Mean standard uncertaity of the reference value
Standard addition: 2
tan
22
22)(2 dardSVol
R
R uun∆s∆u ++∑+=
Volume measurement
Added standard solution
Hg measurement (natural water sample)
)()()( ref
22
222
korr xunss∆usyu ++=+=
u (ykorr) = = 24urel(ykorr) = 4,1%U(Ykorr) = u (ykorr) *2 = 47y korr = xm- ∆
xref = 582 ng/l Y korr = (582 ± 47) ng/l
222 71020 ++
in ng/lx1 = 546 s (xm) = 20x2 = 546 u (xm) = 10x3 = 588 u (xref) = 7 x4 = 553 ∆ = -24xm = 558
)(2)(2 ref
22
xuns∆u +=
24710*2(2 22 =+ =)∆u
Hg measurement (natural water sample)
)()()( ref
22
222
korr xunss∆usyu ++=+=
Y korr= (582 ± 47) ng/l
2222 2471020 +++u (y) = = 34urel(y) = 5,9%
Y =(558 ± 67) ng/l
2222
korr
2 )()()( ∆∆us∆yuyu ++=+=
xref = 0,582 ng/l
Thanks for your attention
Basic equation for the statistical model
According to ISO/TS 21748:
y = µ + δ + B + Σcixi + e u2(y) = u2(δ) + sL2 + Σci
2u2(xi) + sr2
y observed result sL2 variance of B (certified)
µ expectation value of the resultδ intrinsic bias sr
2 variance of eB laboratory bias u2(δ) variance due to the
measurement of RMxi deviation from the nominal valueci sensitivity coefficiente random residual error u2(xi)