Uncertainty
Transcript of Uncertainty
Uncertainty Analysis
2141-375 Measurement and Instrumentation
Measurement Error
Mea
sure
d va
lue,
x
x
'xTrue data
Bias error
Precision error in xi
Measurement number
Uncertainty defines an interval about the measured v alue within which we suspect the true value must fallWe call the process of identifying and quantifying e rrors as uncertainty analysis.
Design-Stage Uncertainty Analysis
Design-stage uncertainty analysis refers to an initia l analysis performed prior to the measurement
Useful for selecting instruments, measurement techniq ues and to estimate the minimum uncertainty that would resu lt from the measurement .
Design-Stage Uncertainty Analysis
%)( 220 Puuu cd += RSS method for combining error
Design-state uncertainty22
0 cd uuu +=
Design-state uncertainty
Interpolation error
0uInstrument error
cu
Zero-Order Uncertainty (Interpolation Error)Even when all error are zero, the value of the measurand must be
affected by the ability to resolve the information pro vided by the instrument. This is called zero-order uncertainty. At z ero-order, we assume that the variation expected in the measurand will be less than that caused by the instrument resolution. And that al l other aspects of the measurement are perfectly controlled (ideal condi tions)
(95%) resolution 2/10 ±=u
Design-Stage Uncertainty Analysis
Instrument Uncertainty, uc
This information is available from the manufacturer ’s catalog
x
y
resolution
yo
uncertainty1/2 resolution
Design-Stage Uncertainty Analysis
0-1000 cm H2O±15 V dc0-5 V0-50oC nominal at 25oC
±0.5%FSOLess than ±0.15%FSO±0.25%of reading0.02%/oC of reading from 25oC0.02%/oC FSO from 25oC
OperationInput rangeExcitationOutput rangeTemperature rangePerformanceLinearity error eL
Hysteresis error eh
Sensitivity error eS
Thermal sensitivity error eST
Thermal zero drift eZT
Specifications: Typical Pressure Transducer
The root of sum square approach:
223
22
21 nrss eeeee L+++= (95%)
Example: Consider the force measuring instrument described by the catalog data that follows. Provide an estimate of the uncertainty attributable to this instrument and the instrument design state uncertainty.
Known: Instrument specifications
Solution:
Assume: Values representation of instrument 95% pro bability
Design-Stage Uncertainty Analysis
Force measuring instrumentResolution: 0.25 N Range: 0 - 100 NLinearity: within 0.20 N over range Repeatability: within 0.30 N over range
Design-state uncertainty22
0 cd uuu +=
Design-state uncertainty
0u cu
½ Resolution = 0.125 N N 36.03.02.0 2222 ±=+±=+ rl ee
N 38.036.0125.0 22 ±=+±=du
Example: A voltmeter is to be used to measure the output from a pressure transducer that outputs an electrical signal. The nominal pressure expected will be ~3 psi (3 lb/in2). Estimate the design-state uncertainty in this combination. The following information is available:
Known: Instrument specifications
Solution:
Assume: Values representation of instrument 95% pro bability
Design-Stage Uncertainty Analysis
VoltmeterResolution: 10 µV Accuracy: within 0.001% of reading
TransducerRange: ±5 psiSensitivity: 1 V/psiInput power: 10 Vdc ± 1%Output: ±5 VLinearity: within 2.5 mV/psi over rangeRepeatability: within 2 mV/psi over rangeResolution: negligible
Design-Stage Uncertainty Analysis
Design-state uncertainty
( ) ( )22
PdEdd uuu +=
Design-state uncertainty
Design-state uncertainty
( ) ( ) ( )220 EcEEd uuu +=
Design-state uncertainty Design-state uncertainty
( ) ( ) ( )220 PcPPd uuu +=
Design-state uncertainty
Error Propagation
Computation of the overall uncertainty for a measur ement system consisting of a chain of components or several instruments
Let R is a known function of the n independent variables xi1, xi2 , xi3, …, xiL
),,,( 21 LxxxfR K=
L is the number of independent variables. Each variab le contains some uncertainty ( ux1, ux2, ux3,…, uxL) that will affect the result R.
%)( ' PuRR R±=
Application of Taylor’s expansion gives, (neglect t he higher order term)
xLL
xx
LxLLxx
ux
fu
x
fu
x
f
xxxfuxuxuxfRR
∂∂
++∂∂
+∂∂
+≈±±±=∆±
...
),...,,(),...,,(
22
11
212211
The best estimate value, R’
Where ),...,,( 21 LxxxfR =
Error Propagation
( ) %)( 1
2
22
22
2
11
Pu
ux
fu
x
fu
x
fu
L
ixii
xLL
xxR
∑=
±=
∂∂
++
∂∂
+
∂∂
±=
θ
K
The combination of uncertainty of all variables (pr obable estimate of uR)
Where θθθθi is the sensitivity index relate to the uncertainty of xi
ii x
f
∂∂
=θ
Example: For a displacement transducer having a calibration curve y = KE, estimate the uncertainty in displacement y for E = 5.00 V, if K = 10.10 mm/V with uk = ±0.10 mm/V and uE = ±0.01 V at 95% confidence
Solution: Find uy
Error Propagation
Known: y = KEE = 5.00 V uE = 0.01 VK = 10.10 mm/V uk = 0.10 mm/V
( ) ( )22KKEEy uuu θθ +±=
KE
yE =
∂∂
=θ EK
yK =
∂∂
=θ
uE = 0.01 V uK = 0.10 mm/V
yy uKEuyy ±=±='
( ) ( )
( ) ( ) mm 51.0mm/V 10.0V 5V 01.0mm/V 10.10 22
22
±=×+×±=
+±= KEy EuKuu
Sequential Perturbation
A numerical approach can also be used to estimated the propagation of uncertainty. This refers to as sequential perturba tion. This method is straightforward and uses the finite difference to a pproximate the derivatives (sensitivity index)
1) Calculate the average result from the independen t variables
),...,,( 21 LxxxfR =
2) Increase the independent variables by their resp ect uncertainties and recalculate the result based on each of these n ew values. Call these values +
iR
),...,,(
),...,,(
),,...,,(
21
2212
2111
LLL
L
L
uxxxfR
xuxxfR
xxuxfR
+=
+=
+=
+
+
+
3) Decrease the independent variables by their resp ect uncertainties and recalculate the result based on each of these n ew values. Call these values −
iR
Sequential Perturbation
4) Calculate the difference for each element
RRR
RRR
ii
ii
−=
−=−−
++
δ
δ
5) Finally, evaluate the approximation of the uncer tainty contribution from each variables
ii
ii
i uRR
R θδδ
δ ≈+
=−+
2
The uncertainty in the result
( )2/1
1
2
±= ∑
=
L
iiR Ru δ
),...,,(
),...,,(
),,...,,(
21
2212
2111
LLL
L
L
uxxxfR
xuxxfR
xxuxfR
−=
−=
−=
−
−
−
Example: For a displacement transducer having a calibration curve y = KE, estimate the uncertainty in displacement y for E = 5.00 V, if K = 10.10 mm/V with uk = ±0.10 mm/V and uE = ±0.01 V at 95% confidence
Solution: Find uy
Error Propagation
Known: y = KEE = 5.00 V uE = 0.01 VK = 10.10 mm/V uk = 0.10 mm/V
i u i x i +u i x i -u i R i+ R i
- δδδδ R i+ δδδδ R i
- δδδδ R i
1 E 5 0.01 5.01 4.99 50.60 50.40 0.10 -0.10 0.10
2 K 10.1 0.1 10.20 10.00 51.00 50.00 0.50 -0.50 0.50
x i
( ) ( )22KEy RRu δδ +±=
yy uKEuyy ±=±='
( )( ) mm 50.50510.10 === KEy
Steps in measurement process1) Calibration2) Data-acquisition3) Data-reduction (Analysis)
Error Sources
Calibration error
K,12,11ee
Data-acquisition error
K,22,21ee
Data-reduction error
K,32,31ee
eij
i = Error source groupi = 1 for Calibration Errori = 2 for Data-acquisition Errori = 3 for Data-reduction Error
j = Elemental error
Calibration Error Source Group
Element (j) Error Source 1 Calibration curve fit 2 Truncation error
Etc.
Data-Acquisition Error Source Group
Data-Reduction Error Source Group
Element (j) Error Source 1 Measurement system operating conditions 2 Sensor-transducer stage (instrument error) 3 Signal conditioning stage (instrument error) 4 Output stage (instrument error) 5 Process operating conditions 6 Process installation effects 7 Environmental effects 8 Spatial variation error 9 Temporal variation error
Etc.
Element (j) Error Source 1 Primary to interlab standard 2 Interlab to transfer standard 3 Transfer to lab standard 4 Lab standard to measurement system 5 Calibration technique
Etc.
Multiple-Measurement Uncertainty Analysis
The procedure for a multiple-measurement uncertaint y analysis
Identify the elemental errors in each of the three source groups(calibration, data acquisition, and data reduction)
Estimate the magnitude of bias and precision error in each of the elemental errors
Estimate any propagation of uncertainty through to the result
Calibrate e11, e12 ,...
Data acquisition e21, e22 ,...
Data reduction e31, e32 ,...
e1j=P1j+B1j e2j=P2j+B2j e3j=P3j+B3j
This section develops a method for the estimate of the uncertainty in the value assigned to a measured variable based on repe ated measurements
Multiple-Measurement Uncertainty Analysis
Consider the measurement of variable, x which is subject to elemental precision errors, Pij and bias, Bij in each of three source groups. Let i = 1, 2, 3 refer to the error source groups ( calibration er ror i = 1, data acquisition error i = 2, data-reduction i = 3) and j = 1,2,…,K refer to each of up to any Kerror elements of error eij
Source Precision index Pi
[ ] 2/1222
21 ... ikiii PPPP +++=
Source Bias limit Bi
[ ] 2/1222
21 ... ikiii BBBB +++=
[ ] 2/123
22
21 BBBB ++=
Measurement Precision index P
[ ] 2/123
22
21 PPPP ++=
Measurement Bias limit B
3 ,2 ,1=i
3 ,2 ,1=i
Multiple-Measurement Uncertainty Analysis
The measurement uncertainty in x, ux
( ) (95%) , 295
2 PtBu vx +=
The degrees of freedom, v (Welch-Satterthwaite formula)
( )∑∑
∑∑
= =
= =
= 3
1 1
4
23
1 1
2
/i
K
jijij
i
K
jij
vP
P
v
Multiple-Measurement Uncertainty Analysis
Measurand, x
Identify elemtal errorsin measurement, eij
eij=Pij+Bij
Source precision index, Pi Source bias limit, Bi
Measurement bias limit, BMeasurement precision index, P
Measurement uncertainty, ux
[ ] 2/1222
21 ... ikiii BBBB +++=[ ] 2/122
221 ... ikiii PPPP +++=
[ ] 2/123
22
21 PPPP ++= [ ] 2/12
322
21 BBBB ++=
( )[ ] (95%) 2/12
95,2 PtBu vx +=
Measurement bias limit, BMeasurement precision index, P
Measurement uncertainty, ux
Example: After an experiment to measure stress in a load beam, an uncertainty analysis reveals the following source errors in stress measurement whose magnitude were computed from elemental errors
B1 = 1.0 N/cm2 B2 = 2.1 N/cm2 B3 = 0 N/cm2
P1 = 4.6 N/cm2 P2 = 10.3 N/cm2 P3 = 1.2 N/cm2
v1 = 14 v2 = 37 v3 = 8If the mean value of the stress in the measurement is 223.4 N/cm2, determine the best estimate of the stress
Solution: Find uσσσσ
Known: Experimental error source indices
Assume: All elemental error have been included
Multiple-Measurement Uncertainty Analysis
[ ] 2/123
22
21 PPPP ++= [ ] 2/12
322
21 BBBB ++=
( )[ ] (95%) 2/12
95,2 PtBu vx +=
Propagation Uncertainty Analysis to a result
The measurement uncertainty , uR
The degrees of freedom, v
[ ]
[ ]{ }∑
∑
=
=
= L
ixixii
L
ixii
vP
P
v
1
4
2
1
2
/θ
θ
( ) (95%) , 295
2RvRR PtBu +=
∑=
±=L
ixiiR PP
1
2][θ ∑=
±=L
ixiiR BB
1
2][θwhere
Consider the result, R which is determined from the function of the n independent variables xi1, xi2 , xi3, …, xiL
%)( ' PuRR R±=
Example: The density of a gas, ρρρρ, which is believed to follow the ideal gas equation of state, ρρρρ = p/RT, is to be estimated through separate measurements of pressure, p, and temperature, T. the gas is housed with in a rigid impermeable vessel. The literature accompanying the pressure measurement system states an accuracy to within 1% of the reading an that accompanying the temperature measuring system suggest 0.6oR. Twenty measurements of pressure, Np = 20, and ten measurements of temperature, NT = 10, are made with the following statistical outcome:
Where psfa refers to lb/ft2 absolute. Determine a best estimate of the density. The gas constant is R = 54.7 ft lb/lbm
oR
Solution: Find
Known:
Assume: Gas behaves as an ideal gas
Propagation Uncertainty Analysis to a result
psfa 91.2253=p psfa 21.167=pS
R4.560 o=T R0.3 o=TS
Tp STSp ,,,
R lb/lbft 7.54 / om== RRTPρ
ρρρ u+='
Propagation Uncertainty Analysis to a result
( )[ ] (95%) 2/12
95,2 PtBu v+=ρ
( ) ( )22TTpp BBB θθ +±= ( ) ( )22
TTpp PPP θθ +±=
( ) ( )[ ]( ) ( ) TTTppp
TTpp
vPvP
PPv
// 44
222
θθ
θθ
+
+=where
RTpp
1=
∂∂
=ρ
θ
R lb/lbft 7.54 / om== RRTPρ
2RT
p
TT −=∂∂
=ρ
θ
where