Uncertainty Analysis

Measurement Error True data

Measured value, x

x'

Bias error

x

Precision error in xi

Measurement number

Uncertainty defines an interval about the measured value within which we suspect the true value must fall We call the process of identifying and quantifying errors as uncertainty analysis.

Design-Stage Uncertainty Analysis Design-stage uncertainty analysis refers to an initial analysis performed prior to the measurement Useful for selecting instruments, measurement techniques and to estimate the minimum uncertainty that would result from the measurement .

Design-Stage Uncertainty Analysis

u d = u02 + uc2 ( P %)

RSS method for combining error

Design-state uncertainty

ud = u02 + uc2

Interpolation error

u0

Instrument error

uc

Design-Stage Uncertainty Analysis 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 provided by the instrument. This is called zero-order uncertainty. At zero-order, we assume that the variation expected in the measurand will be less than that caused by the instrument resolution. And that all other aspects of the measurement are perfectly controlled (ideal conditions) y

u0 = ±1 / 2 resolution (95%) yo

Instrument Uncertainty, uc This information is available from the manufacturer’s catalog x resolution

uncertainty 1/2 resolution

Design-Stage Uncertainty Analysis Specifications: Typical Pressure Transducer Operation Input range Excitation Output range Temperature range Performance Linearity error eL Hysteresis error eh Sensitivity error eS Thermal sensitivity error eST Thermal zero drift eZT

0-1000 cm H2O ±15 V dc 0-5 V 0-50oC nominal at 25oC ±0.5%FSO Less than ±0.15%FSO ±0.25%of reading 0.02%/oC of reading from 25oC 0.02%/oC FSO from 25oC

The root of sum square approach:

erss = e12 + e22 + e32 + L en2

(95%)

Design-Stage Uncertainty Analysis 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. Force measuring instrument Resolution: 0.25 N Range: 0 - 100 N Linearity: within 0.20 N over range Repeatability: within 0.30 N over range

Known: Instrument specifications Assume: Values representation of instrument 95% probability Solution:

Design-state uncertainty

ud = u + u 2 0

u0 ½ Resolution = 0.125 N

2 c

u d = ± 0.1252 + 0.36 2 = ±0.38 N

uc el2 + er2 = ± 0.2 2 + 0.32 = ±0.36 N

Design-Stage Uncertainty Analysis 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 designstate uncertainty in this combination. The following information is available: Voltmeter Resolution: Accuracy: Transducer Range: Sensitivity: Input power: Output: Linearity: Repeatability: Resolution:

10 µV within 0.001% of reading ±5 psi 1 V/psi 10 Vdc ± 1% ±5 V within 2.5 mV/psi over range within 2 mV/psi over range negligible

Known: Instrument specifications Assume: Values representation of instrument 95% probability Solution:

Design-Stage Uncertainty Analysis Design-state uncertainty

ud =

(ud )2E + (ud )2P

Design-state uncertainty

Design-state uncertainty

(ud )E = (u )

(ud )P = (u0 )2P + (uc )2P

2 0 E

+ (u

)

2 c E

Error Propagation Computation of the overall uncertainty for a measurement 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

R = f ( x1 , x2 , K , xL ) L is the number of independent variables. Each variable contains some uncertainty (ux1, ux2, ux3,…, uxL) that will affect the result R. Application of Taylor’s expansion gives, (neglect the higher order term)

R ± ∆R = f ( x1 ± u x1 , x2 ± u x 2 ,..., xL ± u xL ) ≈ f ( x1 , x2 ,..., xL ) + ∂f ∂f ∂f u x1 + u x 2 + ... + u xL ∂x1 ∂x2 ∂xL The best estimate value, R’

R' = R ± u R ( P%) Where R = f ( x1 , x2 ,..., xL )

Error Propagation The combination of uncertainty of all variables (probable estimate of uR) 2

2

∂f ∂f ∂f u R = ± u x1 + u x 2 + K + u xL ∂x1 ∂x2 ∂xL =±

2

L

2 ( ) θ u ∑ i xi

( P %)

i =1

Where θi is the sensitivity index relate to the uncertainty of xi

θi =

∂f ∂xi

Error Propagation 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

Known: y = KE E = 5.00 V K = 10.10 mm/V

uE = 0.01 V uk = 0.10 mm/V

Solution: Find uy

y ' = y ± u y = KE ± u y u y = ± (θ E u E ) + (θ K u K ) 2

θE =

∂y =K ∂E

uE = 0.01 V uy = ±

2

θK =

∂y =E ∂K

uK = 0.10 mm/V

(Ku E )2 + (Eu K )2

= ± (10.10 mm/V × 0.01 V ) + (5 V × 0.10 mm/V ) = ±0.51 mm 2

2

Sequential Perturbation A numerical approach can also be used to estimated the propagation of uncertainty. This refers to as sequential perturbation. This method is straightforward and uses the finite difference to approximate the derivatives (sensitivity index) 1) Calculate the average result from the independent variables

R = f ( x1 , x2 ,..., xL ) 2) Increase the independent variables by their respect uncertainties and recalculate the result based on each of these new values. Call these values Ri+

R1+ = f ( x1 + u1 , x2 ,..., xL ),

R2+ = f ( x1 , x2 + u 2 ,..., xL ) RL+ = f ( x1 , x2 ,..., xL + u L ) 3) Decrease the independent variables by their respect uncertainties and recalculate the result based on each of these new values. Call these values Ri−

Sequential Perturbation R1− = f ( x1 − u1 , x2 ,..., xL ), R2− = f ( x1 , x2 − u2 ,..., xL ) RL− = f ( x1 , x2 ,..., xL − u L ) 4) Calculate the difference for each element

δRi+ = Ri+ − R δRi− = Ri− − R

5) Finally, evaluate the approximation of the uncertainty contribution from each variables

δRi =

δRi+ + δRi− 2

≈ θ i ui

The uncertainty in the result

2 u R = ± ∑ (δRi ) i =1 L

1/ 2

Error Propagation 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

Known: y = KE E = 5.00 V K = 10.10 mm/V

uE = 0.01 V uk = 0.10 mm/V

Solution: Find uy

y ' = y ± u y = KE ± u y u y = ± (δRE ) + (δRK ) 2

2

y = KE = (10.10 )(5) = 50.50 mm i 1 2

ui

x i +u i

x i -u i

Ri+

Ri-

δRi+

δRi-

δRi

5

0.01

5.01

4.99

50.60

50.40

0.10

-0.10

0.10

10.1

0.1

10.20

10.00

51.00

50.00

0.50

-0.50

0.50

xi E K

Error Sources Steps in measurement process 1) Calibration 2) Data-acquisition 3) Data-reduction (Analysis) Calibration error

e11, e12, K

Data-acquisition error

e21, e22, K

Data-reduction error

e31, e32, K

eij j = Elemental error i = Error source group i = 1 for Calibration Error i = 2 for Data-acquisition Error i = 3 for Data-reduction Error

Calibration Error Source Group Element (j) 1 2 3 4 5 Etc.

Error Source Primary to interlab standard Interlab to transfer standard Transfer to lab standard Lab standard to measurement system Calibration technique

Data-Acquisition Error Source Group Element (j) 1 2 3 4 5 6 7 8 9 Etc.

Error Source Measurement system operating conditions Sensor-transducer stage (instrument error) Signal conditioning stage (instrument error) Output stage (instrument error) Process operating conditions Process installation effects Environmental effects Spatial variation error Temporal variation error

Data-Reduction Error Source Group Element (j) 1 2 Etc.

Error Source Calibration curve fit Truncation error

Multiple-Measurement Uncertainty Analysis This section develops a method for the estimate of the uncertainty in the value assigned to a measured variable based on repeated measurements The procedure for a multiple-measurement uncertainty analysis e1j=P1j+B1j

e2j=P2j+B2j

e3j=P3j+B3j

Calibrate e11, e12 ,...

Data acquisition e21, e22 ,...

Data reduction e31, e32 ,...

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

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 error i = 1, data acquisition error i = 2, data-reduction i = 3) and j = 1,2,…,K refer to each of up to any K error elements of error eij Source Precision index Pi

[

Pi = Pi12 + Pi 22 + ... + Pik2

Measurement Precision index P

[

P = P12 + P22 + P32 Source Bias limit Bi

]

1/ 2

]

1/ 2

[

Bi = Bi21 + Bi22 + ... + Bik2 Measurement Bias limit B

[

B = B12 + B22 + B32

i = 1, 2, 3

]

1/ 2

]

1/ 2

i = 1, 2, 3

Multiple-Measurement Uncertainty Analysis The measurement uncertainty in x, ux u x = B 2 + (tv ,95 P )

2

(95%)

The degrees of freedom, v (Welch-Satterthwaite formula) 2

3 K 2 ∑∑ Pij i =1 j =1 v= 3 K ∑∑ Pij4 / vij

(

i =1 j =1

)

Multiple-Measurement Uncertainty Analysis Measurement uncertainty, ux

[

u x = B 2 + (tv ,95 P )

]

2 1/ 2

Measurement precision index, P

[

(95%)

Measurement bias limit, B

]

[

2 1/ 2 3

P = P12 + P22 + P

B = B12 + B22 + B32

Source precision index, Pi

[

1/ 2

Source bias limit, Bi

]

[

2 1/ 2 ik

Pi = Pi12 + Pi 22 + ... + P

]

Bi = Bi21 + Bi22 + ... + Bik2

eij=Pij+Bij

Identify elemtal errors in measurement, eij

Measurand, x

]

1/ 2

Multiple-Measurement Uncertainty Analysis 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 = 8 If the mean value of the stress in the measurement is 223.4 N/cm2, determine the best estimate of the stress

Known: Experimental error source indices Assume: All elemental error have been included Solution: Find uσ Measurement uncertainty, ux

[

u x = B 2 + (tv ,95 P )

Measurement precision index, P

[

]

2 1/ 2 3

P = P12 + P22 + P

]

2 1/ 2

(95%)

Measurement bias limit, B

[

B = B12 + B22 + B32

]

1/ 2

Propagation Uncertainty Analysis to a result Consider the result, R which is determined from the function of the n independent variables xi1, xi2 , xi3, …, xiL

R' = R ± u R ( P%) The measurement uncertainty, uR u R = BR2 + (tv ,95 PR )

2

where

PR = ±

(95%)

L

∑ [θi Pxi ]2

BR = ±

i =1

L

∑ [θ B i

i =1

The degrees of freedom, v 2

L 2 ∑ [θ i Pxi ] v = L i =1 4 ∑ [θ i Pxi ] / vxi

{

i =1

}

2 ] xi

Propagation Uncertainty Analysis to a result Example: The density of a gas, ρ, which is believed to follow the ideal gas equation of state, ρ = 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:

p = 2253.91 psfa

S p = 167.21 psfa

T = 560.4o R

ST = 3.0o R

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

Known:

p , S p , T , ST

ρ = P / RT R = 54.7 ft lb/lbm o R Assume: Gas behaves as an ideal gas Solution: Find

ρ ' = ρ + uρ

Propagation Uncertainty Analysis to a result [

u ρ = B 2 + (tv ,95 P )

]

[(θ P ) + (θ P ) ]

2 1/ 2

(95%)

where v =

p

(θ P )

4

p

B = ± (θ p B p ) + (θT BT ) 2

where

2

P = ± (θ p Pp ) + (θT PT ) 2

ρ = P / RT R = 54.7 ft lb/lb m o R θp =

∂ρ 1 = ∂p RT

2 2

2

θT =

∂ρ p =− ∂T RT 2

2

p

p

T

T

/ v p + (θ T PT ) / vT 4