estimating parameter bias uncertainty
TRANSCRIPT
-
8/8/2019 Estimating Parameter Bias Uncertainty
1/39
Estim
atingParameterBia
sUncertainty
HowardCastrup,
Ph.D.
President,IntegratedSciencesGroup
MeasurementS
cienceConferenc
e
TrackE,
Sessio
n6
UncertaintyTec
hniquesandMeth
ods
March3,
2006
-
8/8/2019 Estimating Parameter Bias Uncertainty
2/39
EstimatingP
aramete
rBias
Unc
ertainty
Ke
yConcepts
Ty
peAEstim
ation
Ty
peBEstim
ation
Ty
peBDegre
esofFree
dom
Ty
peAEstim
ation
Ba
yesianEstimation
-
8/8/2019 Estimating Parameter Bias Uncertainty
3/39
Estim
atingParameterBiasUncertainty
Key
Concepts
Measuremen
tsareAcc
ompanied
by
Measuremen
tError
Th
eFundam
entalMea
surement
Model
Th
eMeasurementErro
r
meas
true
meas
x
x
=
+
meas
bias
random
operator
=
+
+
+
-
8/8/2019 Estimating Parameter Bias Uncertainty
4/39
Estim
atingParameterBiasUncertainty
Typ
eAEstimation
Ex
ample:
UncertaintyDuetoRandomError:
Where
2
1
1
(
)
n
ran
dom
i
i
u
x
x
=
=
sampledv
alue
samplemean
samplesize
1
ix
ith
x n
n
= = = =
-
8/8/2019 Estimating Parameter Bias Uncertainty
5/39
KeyC
oncepts
Typ
eAEstimation(cont.)
Co
nfidenceL
imitsLforaConfide
nce
Le
velof1
t/2,
=t-statisticforasig
nificancele
vel
of
/2andde
greesoffreedom
/2,
random
L
t
u
=
-
8/8/2019 Estimating Parameter Bias Uncertainty
6/39
KeyC
oncepts
Typ
eBEstimation
Es
timationPr
ocedure
Co
ntainment
Limitsand
Probability
Un
certaintyD
efinition
BiasDistribut
ions
SeeEstimatingandCombiningUncertaintiesand
AC
omprehensiveCo
mparisonofUnce
rtaintyAnalysisT
oolsat
www.i
sgm
ax.c
om/articles_
papers.h
tm
.
-
8/8/2019 Estimating Parameter Bias Uncertainty
7/39
Type
BEstimation
Estimation
Procedu
re
Reversethe
TypeAEstimation
Procedure
EstimateorDevelopCon
tainmentLim
its
ParameterToleranceLimits
Other
EstimateaC
ontainmentProbability
PercentIn-tolerance
Confidence
Level
-
8/8/2019 Estimating Parameter Bias Uncertainty
8/39
Type
BEstimation
Con
tainmentLimitsand
Probability
C
ontainment
Limits
0
L1
L2
f()
Containm
ent
Probabilityp
TheParameterBiasD
istribution
-
8/8/2019 Estimating Parameter Bias Uncertainty
9/39
Type
BEstimation
Unc
ertaintyD
efinition
Th
estandard
uncertaintyinparameter
bia
sisthesta
ndarddeviationofthe
bias
dis
tribution.
Ap
plytheapp
ropriatedistribution.
Es
timatefr
omL1,L2andp.
2
var()
()
f
d
=
var()
u
= =
-
8/8/2019 Estimating Parameter Bias Uncertainty
10/39
Type
BEstimation
Bias
Distributions
Normal
Log
normal
Uniform(Rectang
ular)
Triangular
Tra
pezoidal
Quadratic
Cosine
Utility
U-S
haped
SeeDistribu
tionsforUncertai
ntyAnalysisat
www.i
sgm
ax.c
om/articles_papers.h
tm.
-
8/8/2019 Estimating Parameter Bias Uncertainty
11/39
BiasD
istributions
NormalDistribution
2
2
/2
1
()
2
u
f
eu
=
1
2
1
L
L
p
u
u
=
+
1
12
L
u
p
=
+
thepdf
solveforufrom
ifL1=L2then
0
L1
L2
f()
p
-
8/8/2019 Estimating Parameter Bias Uncertainty
12/39
BiasD
istributions
LognormalDistribution
Us
efulforParametersw
ithAsymmetric
To
lerances
2
2
1
(
)
exp
ln
2
(
)
2
q
f
qm
q
=
q
f
()
f
()
2
2
3
(
)
(
1)
u
qmm
me
=
=
Solve
fornumerica
lly
L1
+L2
p
-
8/8/2019 Estimating Parameter Bias Uncertainty
13/39
BiasD
istributions
Unif
orm(Rectangular)Distributio
n
1
,
()
2 0,
otherwise,
a
a
f
a
=
3a
u=
,
La
L
a
p=
Not
recommendedforestimatingbiasu
ncertainty
-L
L
p
f()
-a
a
0
-
8/8/2019 Estimating Parameter Bias Uncertainty
14/39
BiasD
istributions
TriangularDis
tribution
2 2
(
)/
,
0
()
(
)/
,
0
0,
othe
rwise.
a
a
a
f
a
a
a
+
=
6a
u=
,
1
1
L
a
La
p
=
Not
recommendedforestimatingbiasu
ncertainty
-L
L
p
f()
-a
a
0
-
8/8/2019 Estimating Parameter Bias Uncertainty
15/39
BiasD
istributions
Trap
ezoidalD
istribution
,
(
)(
)
1
,
()
,
(
)(
)
0.
o
therwise
b
b
a
a
bba
a
a
f
a
bb
a
b
a
bba
+
+
=
+
+
2
2
3
a
b
u
+
=
2
0
2
L
a
aL
a
b
p
ab
b
a
+
+
=
+
S
olveforafrom
Not
recommendedforestimatingbiasu
ncertainty
f()
-a
a
0
-L
L
p-b
b
-
8/8/2019 Estimating Parameter Bias Uncertainty
16/39
BiasD
istributions
Qua
draticDis
tribution
2
3
1
(
/
)
()
,
4 0,
otherwise
a
f
a
a
a
=
,5a
u=
3
3
2
0,
/
1
x
xp
xL
a
+
=
=
Solveforafrom
f()
-a
a
0
-L
L
p
-
8/8/2019 Estimating Parameter Bias Uncertainty
17/39
BiasD
istributions
Cos
ineDistrib
ution
1
1
cos
()
,
2 0,
oth
erwise
f
a
a
a
a
+
=
26
1
3a
u
=
sin(
/
)
0,
a
La
ap
L
L
a
+
=
Solveforafrom
a
-a
a
-a
-L
L
p
0f()
-
8/8/2019 Estimating Parameter Bias Uncertainty
18/39
BiasD
istributions
UtilityDistribu
tion
2
1
,
|
|
1
(|
|
)
()
cos
,
|
|
2(
)
0,
|
|
,a
ab
a
f
a
b
a
b
ba
b
+
=
+
(
)
3
3
2
2
2
1
2
3a
b
u
ba
ab
+
=
+
1
(
)
sin
0
ba
L
a
L
p
ab
ba
+
=
+
Solveforafrom
f()
-a
a
-b
b
-L
L
0
p
-
8/8/2019 Estimating Parameter Bias Uncertainty
19/39
BiasD
istributions
U-ShapedDis
tribution
2
2
1
,
()
0,
otherwise,
a
a
f
a