intro to uncertainty analysis
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Introdução à Análise de IncertezasTRANSCRIPT
Introduction toIntroduction toUncertaintyUncertainty AnalysisAnalysis
Howard Castrup, Ph.D.Suzanne Castrup, MSME
Integrated Sciences GroupBakersfield, CA 93306
www.isgmax.com
Introduction to Uncertainty Analysis 2 Integrated Sciences Group
Introduction to Uncertainty Analysis
Topic OutlineBasic ConceptsDirect MeasurementsUncertainty SidekickRecap
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Introduction to Uncertainty Analysis
Basic ConceptsFundamental Measurement ModelError DistributionUncertainty DefinitionVariance and UncertaintyVariance Addition RuleCorrelation CoefficientsUncertainty Sidekick DistributionsType A and Type B Estimates
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Basic ConceptsFundamental Measurement ModelThe difference between a measured value and the “true” value is the measurement error
measured true xx x ε= +
The true value is a fixed quantity
The measurement error is a variable
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Basic Concepts
Error Distribution
A relationship between the value of a measurement error and its probability of occurrence
εx0
f(εx)
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Basic Concepts
Uncertainty DefinitionMeasurement Uncertainty quantifies the “spread” of the measurement error distribution
0
SmallUncertainty
LargeUncertainty
0
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Basic Concepts
Uncertainty Definition (cont.)The spread of an error distribution is the distribution standard deviationThe standard deviation is the square root of the distribution variance
2var( ) var( )measured x xx ε ε= =
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Basic Concepts
Variance and Uncertainty
Measurement Uncertainty
var( ) var( )var( )
measured true x
x
x x εε
= +=
var( )xx xεσ σ ε= =
x xx xu uε εσ σ= = =
Distribution Variance
Distribution Standard Deviation
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Basic Concepts
Variance Addition RuleSuppose we have a variable zcomposed of variables x and y
z ax by= +
The variance of z is given by
2 2
var( ) var( )var( ) var( ) 2 cov( , )
z ax bya x b y ab x y
= +
= + +
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Basic Concepts
Variance Addition Rule (cont.)Variances and Covariance
2
2
var( ) var( )
var( ) var( )
cov( , ) cov( , )
x
y
x
y
x y
x u
y u
x y
ε
ε
ε
ε
ε ε
= =
= =
=
The Variance Addition Rule2 2var( ) var( ) var( ) 2 cov( , )x y x xz a b abε ε ε ε= + +
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Basic Concepts
Correlation CoefficientsCovariances can be expressed in terms of Correlation Coefficients
,
cov( , )x y
x y
x y
u uε εε ε
ε ερ =
The Variance Addition Rule Becomes2 2 2 2var( ) 2
x y x y x yax by a u b u ab u uε ε ε ε ε ερ+ = + +
Correlation Coefficients range from –1 to +1
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Basic Concepts
Uncertainty Sidekick DistributionsNormalUniformStudent’s t
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Uncertainty Sidekick Distributions
The Normal Distribution
( )2 2/ 21( )2
uf eu
ε εε µ
ε
επ
− −=
f(ε)
ε0- a a
f(ε)
ε0- a a
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The Normal Distribution
CommentsThe “workhorse” of statistics and probability.
Usually assumed to be the underlying distribution for errors.
Most uncertainty analysis tools are based on the assumption that measurement errors are normally distributed, regardless of the distributions used to estimate the uncertainties themselves.
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The Normal Distribution
Uncertainty EstimatesType A Estimates: Compute mean and standard deviation from a sample. Nearly always assume a normal distribution
1 12
aupε
−=
+⎛ ⎞Φ ⎜ ⎟⎝ ⎠
( )2
1
11
n
ii
u x xnε
=
= −− ∑
Type B Estimates: Work with error containment limits ± a and containment probability p
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Uncertainty Sidekick Distributions
The Uniform Distribution
1 ,( ) 2
0, otherwise ,
a af a
εε
⎧ − ≤ ≤⎪= ⎨⎪⎩
0
f (ε)
ε−a a0
f (ε)
ε−a a
3auε =
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The Uniform Distribution
ApplicabilityApplicable to
Digital Resolution ErrorQuantization ErrorRF Phase Angle
Criteria for useNeed minimum bounding limitsUniform probability within the limits100% containment probability
Very limited applicability
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The Uniform Distribution
CommentsArguments for Use
Lack of Knowledge – “Use in case you know nothing about the error other than its bounding limits.”Easy Out – Divide bounding limits by root 3
Equivalent to assuming a normal distribution with 91.67% in-tolerance probability. GUM, Sec 4.3.7: “When a component of uncertainty is determined in this manner contributes significantly to the uncertainty of a measurement result, it is prudent to obtain additional data for its further evaluation.”
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Uncertainty Sidekick Distributions
Student’s t Distribution
2 ( 1) / 2
12( ) (1 / )
2
f x x ν
ν
ννπν
− +
+⎛ ⎞Γ⎜ ⎟⎝ ⎠= +
⎛ ⎞Γ⎜ ⎟⎝ ⎠
f (ε)
ε0
ν = Degrees of Freedom
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Student’s t Distribution
ApplicabilityComputation of Confidence Limits for Normally Distributed Errors with Known Degrees of Freedom (ν )Statistical testing of hypotheses
Equivalence of laboratories (MAPs)Significance of curve fit linear slopeetc.
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Basic Concepts
Type A and Type B EstimatesType A
Estimate the standard deviation from a sample of data
2 2
1
1 ( )1
n
x ii
u x xn =
= −− ∑
Type BEstimate heuristically from Error Limits and a Containment Probability (Confidence Level)Choose the appropriate error distribution
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Introduction to Uncertainty Analysis
Direct MeasurementsDefinitionError SourcesThe Error ModelCombined UncertaintyError Source CorrelationsDegrees of FreedomError Source Uncertainties
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Direct Measurements
DefinitionThe value of an attribute is measured directly by comparison with a measurement reference (device)
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Direct Measurements
Error SourcesParameter BiasRandom Error (Repeatability)Resolution ErrorOperator Bias (Reproducibility)Environmental Factors
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Direct Measurements
The Error ModelFor a Direct Measurement, themeasurement error is the sum of the error sources:
, , , , ,x x bias x ran x res x op x envε ε ε ε ε ε= + + + + + ⋅⋅⋅
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Direct Measurements
Combined Uncertainty
The Uncertainty in εx
var( )x xu ε=
By the Variance Addition Rule
, , , , ,
, , , , , , , ,
, , , , ,
2 2 2 2 2
, ,
var( ) var( )
2 2x bias x ran x res x op x env
x bias x ran x bias x ran x bias x res x bias x res
x x bias x ran x res x op x env
u u u u u
u u u uε ε ε ε ε
ε ε ε ε ε ε ε ε
ε ε ε ε ε ε
ρ ρ
= + + + + + ⋅⋅⋅
= + + + + + ⋅⋅⋅
+ + + ⋅⋅⋅
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Direct Measurements
Error Source UncertaintiesError Source Variances:
, ,
, ,
,
2 2, ,
2 2, ,
2,
var( ) var( )
var( ) var( )
var( )
x bias x res
x ran x env
x op
x bias x res
x ran x env
x op
u u
u u
u
ε ε
ε ε
ε
ε ε
ε ε
ε
= =
= =
=
Error Source Correlations:Nearly always zero for direct measurementsMay be present in making environmental correctionsMay appear in the analysis of operator bias
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Direct Measurements
Degrees of FreedomThe amount of information used in obtaining an uncertainty estimateDetermined for each error source
,x biasενE.g., bias degrees of freedom =Estimated for the combined uncertainty
,
,
4
4 , , , ,x
x
x i
x ii
ui bias ran res
uε
εε
ε
ν
ν
= = ⋅⋅⋅
∑
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Introduction to Uncertainty Analysis 30 Integrated Sciences Group
Introduction to Uncertainty Analysis
Uncertainty SidekickInteractive Tool for Estimating Uncertainty in Measurement
Uncertainty Estimated for Direct MeasurementsUncertainty Estimated from Technical Data and User KnowledgeData and Technical Knowledge Entered in Special FormatsStatistics and other Math Performed in BackgroundBuilt-in Interface to Measurement Units DatabaseIncludes Bayesian Analysis of Measurement ResultsReport Preview, Export and PrintingAnalysis File Save / Open
Files can be opened in Sidekick, Sidekick Pro and UncertaintyAnalyzer
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Uncertainty SidekickUncertainty Sidekick
AnalysisAnalysisSetupSetup
What’s BeingWhat’s BeingMeasured?Measured?
Who’sWho’sMeasuring It?Measuring It?
Where’s itWhere’s itBeingBeing
Measured?Measured?
MeasurementConfiguration
SubjectParameterMeasurement
Area and Units
What’sWhat’sMeasuring It?Measuring It?
Nominal Value
MeasurementReference
Operator
Environment
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Uncertainty Sidekick
ExampleHP 973A Digital Multimeter Calibration
Accuracy: ± 0.1% of readingResolution: 1 mV
Measurement Reference: Fluke 732B DC Voltage Reference
Accuracy: ± 0.1 ppmLinear Stability: 2.0 ppm / year
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Uncertainty Sidekick Example
Error SourcesBias in the Fluke 732B ReferenceError in Stability of the Fluke 732BRepeatability ErrorHP732A DMM Resolution Error
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Uncertainty Sidekick Example
Analysis ProcedureSetup the AnalysisDefine the Subject Parameter (DMM)
Bias UncertaintyResolutionRepeatability
Define the Measurement ReferenceBias Uncertainty
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Uncertainty Sidekick Example
Setup the AnalysisMeasurement Configuration
Subject Parameter Measures the Value of the Measurement ReferencePassive Reference Configuration
Measurement Area: DC VoltageNominal Value: 10 VTolerance Units: mV
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Uncertainty Sidekick Example
The Subject ParameterHP 973A DMM10 V DC Nominal Value
Output by the Fluke 732BSpecs:
Accuracy: ± 0.1% of readingResolution: 1 mV
In-Tolerance Probability: 90%Measurement Sample
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Uncertainty Sidekick Example
The Subject Parameter (cont.)Measurement Sample Reading Voltage
1 10.001
2 10.003
3 9.999
4 10.003
5 10.001
6 10.000
7 9.999
8 10.001
9 10.001
10 10.002
11 9.998
12 9.999
13 10.002
14 9.998
15 10.001
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Uncertainty Sidekick Example
The Subject Parameter (cont.)Parameter Resolution
1 mVDigital Display
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Uncertainty Sidekick Example
The Reference ParameterFluke 732B Voltage Reference10 V DC Nominal ValueSpecs:
Accuracy: ± 0.1 ppm of readingStability: ± 2 ppm / year
In-Tolerance Probability: 99%Measured by the Subject Parameter
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Uncertainty Sidekick Example
The Analysis ResultsMeasured Mean Value
10.0005 V0.5 mV Above NominalWithin ± 10 mV Tolerance
Total Standard Uncertainty: 0.402 mVDegrees of Freedom: 15
Bayesian Analysis:0.5333 mV Estimated Bias6.1074 mV Bias Uncertainty100% In-Tolerance Probability
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Introduction to Uncertainty Analysis
RecapUncertainty = Error Standard DeviationError Sources
bias, random, resolution, operator, environment, etc.
Direct MeasurementsThe value of an attribute is measured directly by comparison with a measurement reference (device)
CorrelationsDependence of error sources on one another
Type A AnalysisMean, Standard Deviation and Degrees of Freedom estimated from Measured Values
Type B AnalysisStandard Deviation computed from Error Limits and Containment ProbabilityEstimate Degrees of FreedomSelect Appropriate Distribution