ME120-11 Uncertainty Analysis
Uncertainty AnalysisAnanda MysoreSJSUSan José State University A. Mysore Spring 2009
Error Error is the difference between the measuredvalue and the true value, and everymeasurement is subject to error. The error can not actually be known until afterthe measurement, and—depending on whetheror not the true value is actually known—it maynever be known exactly.San José State University A. Mysore Spring 20092
Uncertainty Uncertainty is an estimate of the magnitude oferror, typically expressed in terms of aconfidence interval within which the error lies. “An uncertainty statement assigns crediblelimits to the accuracy of a reported value,stating to what extent that value may differfrom its reference pc/section5/mpc52.htm#ISO, September 2008] Uncertainty analysis considers both systematicerror and random error.San José State University A. Mysore Spring 20093
Propagation of Uncertainties When a result y is a function of variables xi, a first-ordervariation equation can be used to estimate a change y interms of small changes in each of the variables xi.y f {x1, x2 ,K xn } f f f y x1 x2 L xn x1 x2 xnHere the change y in output is expressed as a sum ofcontributing sources of uncertainty xi, weighted by sensitivitycoefficients.A “worst-case” uncertainty u from multiple uncertainties uicould be computed by:nu i 1 fuidxiIs there a better way to express the combined uncertainty?San José State University A. Mysore Spring 20094
Square Root of Sum-of-Squares Taking the square root of the sum-of-squares is an effectiveway to combine uncertainties into one value, and squaringeach contributing term before taking the sum has someimportant advantages: Positive and negative contributors to the uncertainty do notaccidentally “cancel out”. Larger error sources are magnified compared to smallerones, and this is desirable for identifying severe problems. Sum-of-squares does not over-estimate uncertainty as anextreme worst-case scenario.22 f f f u u1 u2 L un x1 x2 xn 2San José State University A. Mysore Spring 20095
Why Not Sum of Absolute Differences? The sum of absolute differences would be meaningful asa worst-case scenario in which all contributors werepositive or all were negative, but in general it severelyoverestimates the error.San José State University A. Mysore Spring 20096
Variant on Textbook Example 7.1 (In class)San José State University A. Mysore Spring 20097
Questions for Conducting Uncertainty Analysis Is the evaluation applied to random errors orsystematic errors? Can the uncertainty be based on statisticalprobability distributions or not? Is the uncertainty being estimated for a singlemeasurement or a sample mean? For more comprehensive discussion (as of September 2008), ction5/mpc5.htm]San José State University A. Mysore Spring 20098
Random and Systematic Uncertainties Quantifying uncertainty differs for single measurements versus sample means.Systematic (or bias B) uncertainty is the same in both cases, but random (orprecision P) uncertainty is reduced by increased sample size.Random uncertainty for a sample mean is estimated from the standard deviation,scaled by the t-distribution and the sample size.Px tsxFor large sample size (n 30), t 2.nImage(s) from Introduction to Engineering Experimentation by A. J. Wheeler and A. R. Ganji, ISBN 0-13-065844-8 2004 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.San José State University A. Mysore Spring 20099
Methodology for Uncertainty Analysis Define the relevant variables and exact method ofmeasurement. List all contributing elemental sources of systematic error andrandom error, and estimate their respective magnitudes. Quantify standard deviations Sx for random uncertainties. Forcomplex or single-value measurements, Sx is not obvious andmay need to come from auxiliary measurements. Calculate the systematic uncertainty B and random uncertaintyP separately, then combine to calculate the total uncertainty.kBx i 1Bi2mSx Si2Px tSxi 1Px tS xnu x Bx2 Px2u x Bx2 Px2San José State University A. Mysore Spring 200910
Which Errors are Systematic vs. Random? In general, any randomuncertainties assume largesample size (n 30). If in doubt, for the purposesof uncertainty analysisassume systematic error. To combine randomuncertainties, the sameconfidence level must applyto each elementaluncertainty.Table from Introduction to Engineering Experimentation by A. J. Wheeler and A. R. Ganji, ISBN 0-13-065844-8 2004 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.San José State University A. Mysore Spring 200911
Variant on Textbook Example 7.7 (In class)San José State University A. Mysore Spring 200912
Systematic Error and Random Error (Review) Systematic error (or “bias” error) is repeatable. e.g. imperfect calibration, residual loading, intrusivemeasurements, spatial bias Random error (or “precision” error) is not predictable. e.g. environmental variability, noise, vibrationImage(s) from Introduction to Engineering Experimentation by A. J. Wheeler and A. R. Ganji, ISBN 0-13-065844-8 2004 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.San José State University A. Mysore Spring 200913
Example What is the uncertainty in the P iv power of aresistive circuit, if the voltage is measured tobe v 100 1 V and the current is measured tobe i 10 0.1 A? How much difference is there between “worstcase scenario” and “best estimate”?San José State University A. Mysore Spring 200914
Example What is the uncertainty in the P iv power of a resistivecircuit, if the voltage is measured to be v 100 1 V and thecurrent is measured to be i 10 0.1 A? How much difference is there between “worst-case scenario”and “best estimate”? P i 10 Advuv 1 V P Pu uv uidvdiu 10(1) 100(0.1) W 20 W P v 100 Vdi2uv 0.1 V P P u u v ui dv di 2u (10·1) 2 (100·0.1) 2 W 14 WSan José State University A. Mysore Spring 200915
Example A pressure transducer has full-scale (FS) range 1000 kPa. Linearity uncertainty is 0.2% FS. Hysteresis uncertainty is 0.1% FS. The repeatability uncertainty, expressed in this case asstandard deviation over a large number of repeatedmeasurements at a fixed typical setting is 10 kPa. The transducer is subject to uncertainties fromtemperature, that affects measurements with a standarddeviation of 3 kPa. What is the total uncertainty of pressure measurementwith this transducer?San José State University A. Mysore Spring 200916
Example A pressure transducer has full-scale(FS) range 1000 kPa.Linearity uncertainty is 0.2% FS.Hysteresis uncertainty is 0.1% FS.The repeatability uncertainty,expressed in this case as standarddeviation over a large number ofrepeated measurements at a fixedtypical setting is 10 kPa.The transducer is subject touncertainties from temperature, thataffects measurements with astandard deviation of 3 kPa.What is the total uncertainty ofpressure measurement with thistransducer?BL 0.002(1000) kPa 2 kPaBH 0.001(1000) kPa 1 kPaBx (BL )2 (BH )2 5 kPaSx (S R )2 (ST )2 109 kPaPx tS x 2 109 kPau x Bx2 Px2 5 4(109) kPa 21 kPaSan José State University A. Mysore Spring 200917
Questions for Conducting Uncertainty Analysis Is the evaluation applied to random errors or systematic errors? Can the uncertainty be based on statistical probability distributions or not? Is the uncertainty being estimated for a single measurement or a sample mean ? For more comprehensive discussion (as of September 2008), see
1.1 Measurement Uncertainty 2 1.2 Test Uncertainty Ratio (TUR) 3 1.3 Test Uncertainty 4 1.4 Objective of this research 5 CHAPTER 2: MEASUREMENT UNCERTAINTY 7 2.1 Uncertainty Contributors 9 2.2 Definitions 13 2.3 Task Specific Uncertainty 19 CHAPTER 3: TERMS AND DEFINITIONS 21 3.1 Definition of terms 22 CHAPTER 4: CURRENT US AND ISO STANDARDS 33
73.2 cm if you are using a ruler that measures mm? 0.00007 Step 1 : Find Absolute Uncertainty ½ * 1mm 0.5 mm absolute uncertainty Step 2 convert uncertainty to same units as measurement (cm): x 0.05 cm Step 3: Calculate Relative Uncertainty Absolute Uncertainty Measurement Relative Uncertainty 1
fractional uncertainty or, when appropriate, the percent uncertainty. Example 2. In the example above the fractional uncertainty is 12 0.036 3.6% 330 Vml Vml (0.13) Reducing random uncertainty by repeated observation By taking a large number of individual measurements, we can use statistics to reduce the random uncertainty of a quantity.
Uncertainty in volume: DVm 001. 3 or 001 668 100 0 1497006 0 1 3 3. %. % .% m m ª Uncertainty in density is the sum of the uncertainty percentage of mass and volume, but the volume is one-tenth that of the mass, so we just keep the resultant uncertainty at 1%. r 186 1.%kgm-3 (for a percentage of uncertainty) Where 1% of the density is .
economic policy uncertainty index, while the trade policy uncertainty index is used in Section 5. Concluding remarks follow. 2. Literature on policy uncertainty and trade There is a large body of theoretical and empirical work that studies the impact of uncertainty, and of policy
Measurement Uncertainty Approach 10 ISOGUM -ISO Guide to the Expression of Uncertainty Determine the uncertainty components based on the model: Isolate each component in the model that is anticipated to cause variation in the measured results. Calculate the sensitivity coefficients: Sensitivity coefficients are multipliers that are used to convert uncertainty
Dealing with Uncertainty: A Survey of Theories and Practices Yiping Li, Jianwen Chen, and Ling Feng,Member, IEEE Abstract—Uncertainty accompanies our life processes and covers almost all fields of scientific studies. Two general categories of uncertainty, namely, aleatory uncertainty and epistemic uncertainty, exist in the world.
The American Revolution had both long-term origins and short-term causes. In this section, we will look broadly at some of the long-term political, intellectual, cultural, and economic developments in the eigh-teenth century that set the context for the crisis of the 1760s and 1770s. Between the Glorious Revolution of 1688 and the middle of the eigh- teenth century, Britain had largely failed .
