492 R. Luck, J. W. Stevens / ISA Transactions 43 (2004) 491–497pressure, and density are commonly used to ﬁx the and higher terms of the Taylor approximation arethermodynamic state and to calculate other prop- not negligible relative to the second term. Theerties such as enthalpy, internal energy, and en- Taylor series for a function of one variable, R ( x ) ,tropy. Propagation of uncertainty through such expanded around x 0 is ͯcalculations can readily lead to nonlinear effects inthe neighborhood of the critical point. dR R ͑ x ͒ ϷR ͑ x 0 ͒ ϩ ͑ xϪx 0 ͒ Unfortunately, for many engineering applica- dx x0tions, the inversions of the analysis equations re-quired for a proper analytical approach are eitherunwieldy or impossible to obtain. Monte Carlocomputer simulations can yield satisfactory re- ϩ 1 d 2R 2! dx 2 ͯ x0 ͑ xϪx 0 ͒ 2 ϩ¯ . ͑1͒sults, but typically require very large numbers of Then, the condition for the linear approximation tofunction evaluations. This paper presents a direct be valid isapproach to estimate nonlinear uncertainty propa-gation by using a piecewise linear approximationto the analysis function and a normal distributionfor the input variable. The approach can be used dR dx ͯ x0 ӷ 1 d 2R 2! dx 2 ͯ x0 ͑ xϪx 0 ͒ ͯwith any function including tabular data. Using the 1 d 3Rnormal distribution, only the error function ͑erf͒ ϩ ͑ xϪx 0 ͒ 2 ϩ¯ . ͑2͒needs to be evaluated. 3! dx 3 x0 This condition tends to be violated by large higher2. Background derivatives or large uncertainty ( xϪx 0 ) . Thus large uncertainty simply implies that the linear ap- The topic of estimation of experimental uncer- proximation of the truncated Taylor series is inad-tainty is covered in a wide variety of forums. The equate to represent the data reduction equation.American Society of Mechanical Engineers pub- An analytical approach for estimating nonlinearlishes an uncertainty standard as part of the per- uncertainty propagation is to transform the prob-formance test codes: ASME PTC 19.1-1998, Test ability density function ͑p.d.f.͒ of the measuredUncertainty ͓1͔. The International Organization variable through the data reduction equation to getfor Standardization ͑ISO͒ also publishes a guide a probability density function for the result. Theon uncertainty calculation and terminology en- expected mean and conﬁdence intervals can thentitled ‘‘Guide to the Expression of Uncertainty in be calculated from the p.d.f. of the result. If x isMeasurement’’ ͓2͔. These two approaches are the measured variable, and f x ( x ) is the p.d.f. ofcompared by Steele et al. ͓3͔. Most textbooks on that variable, and yϭh ( x ) is the data reductionexperimental measurements include a section on equation, then the p.d.f. for the result can be cal-uncertainty propagation as well ͑for example, culated byRefs. ͓4 – 6͔͒. Some textbooks specialize in uncer-tainty ͓7,8͔. The technical literature also has nu-merous treatments of uncertainty estimation andpropagation in speciﬁc applications ͑for example, f y ͑ y ͒ ϭ f x „h Ϫ1 ͑ y ͒ … ͯ dh Ϫ1 ͑ y ͒ dy ͯ. ͑3͒Refs. ͓9–12͔͒. While this works easily for simple functions, the While estimation of measurement uncertainty is inversion of the analysis equation can becometreated in great detail in these references, the very awkward or impossible for complicated func-propagation of measurement uncertainty in data tions or tabular data.reduction or data analysis equations is invariably The approach demonstrated in this paper is tohandled by a truncated Taylor expansion. This ap- approximate the analysis function with a series ofproach to propagation assumes that the uncertainty straight line segments. The transformation for thein a measured variable is not too large for the data line segments is straightforward and an approxi-reduction equation to be represented well locally mate p.d.f. for the dependent variable can be con-by a straight line. More precisely in this context, structed. From that p.d.f., an estimate of the mean‘‘large’’ perturbations simply means that the third and conﬁdence intervals can be obtained. For the
R. Luck, J. W. Stevens / ISA Transactions 43 (2004) 491–497 493 tities of primary interest. The mean can be ex- pressed in terms of the probability density func- tion by yϭ ͵ ϱ Ϫϱ y f y dy. ͑7͒ For an N segment, piecewise linear approximation to the function this becomes ͵ NϪ1 x kϩ1 yϭ ͚ ͑ a k ϩb k ͒ f x ͑ ͒ d kϭ0 xk ͵ NϪ1 x kϩ1 ϭ ͚ kϭ0 ak xk ͑ ͒ f x͑ ͒ d Fig. 1. Nomenclature of line segments. ϩb k ͵ xk x kϩ1 f x͑ ͒ d , ͑8͒special case of a normal distribution in the mea-sured variable, the mean and conﬁdence intervals where is the dummy variable of integration be-will be shown to depend only on evaluation of the tween limits x k and x kϩ1 .error function ͑erf͒. Using integration by parts on the ﬁrst term and using the cumulative distribution function deﬁned3. Development as Consider a linear function of the form: yϭaxϩb, with a probability density function on the F x͑ x ͒ ϭ ͵ x Ϫϱ f x͑ ͒ d ͑9͒independent variable of f x ( x ) . Then from Eq. ͑3͒, then Eq. ͑8͒ can be writtenthe probability density function of the dependent ͫ ͵ ͬ NϪ1variable is given by x kϩ1 y ϭ ͚ a k F x ͑ ͒ ͉ x kϩ1 Ϫ x F x͑ ͒ d f y͑ y ͒ϭ 1 ͉a͉ ͩ ͪ fx yϪb a . ͑4͒ kϭ0 x ϩb k F x ͑ ͒ ͉ x kϩ1 . k xk ͑10͒ kOver a range of interest, an analysis function ortabular data may be approximated by a series of 3.2. Conﬁdence intervalline segments of arbitrary length ͑see Fig. 1͒: yϭa k xϩb k , x k ϽxϽx kϩ1 . ͑5͒ Approximate conﬁdence intervals can be taken directly from the cumulative distribution functionFor each segment, the probability density function of the dependent variable. For the N-segmentof the dependent variable is given by the piece- piecewise linear approximation this iswise version of Eq. ͑4͒: ͵ ͩ ͪ P Ϫb k ͩ ͪ y kϩ1 1 1 yϪb k F y͑ y ͒ϭ ͚ f d, ͑11͒ f y͑ y ͒ϭ f , y k ϽyϽy kϩ1 . ͑6͒ kϭ0 yk ͉ a k͉ x a k ͉ a k͉ x a k where P is the interval containing the argument of3.1. Mean the cumulative distribution function. Normally, the probability density function tends 3.3. Normal distributionto be of lesser direct interest for the purposes ofengineering uncertainty analysis. Instead, the Eqs. ͑10͒ and ͑11͒ are valid for any probabilitymean and conﬁdence intervals are often the quan- density function of the independent variable. For
494 R. Luck, J. W. Stevens / ISA Transactions 43 (2004) 491–497the special ͑and most common͒ case of a normaldistribution, a further simpliﬁcation for the evalu-ation of Eqs. ͑10͒ and ͑11͒ is possible. However, itmust be emphasized that Eqs. ͑10͒ and ͑11͒ aregeneral in nature and are applicable to any prob-ability density function. The additional steps de-scribed below render the evaluation of Eqs. ͑10͒and ͑11͒ very convenient for a single special case;that of a normal distribution. For a normal distribution with mean x andstandard deviation , the probability density func-tion can be written Fig. 2. Example function with data samples. 1 2 /2 2 f x͑ z ͒ ϭ e Ϫ ͑ zϪ x ͒ . ͑12͒ 4. Simple example ͱ2 A simple example will be used to illustrate thisThe corresponding cumulative distribution func- approach to uncertainty estimation. This rathertion for a normal distribution is trivial one-dimensional mathematical function was F͑ x ͒ϭ ˆ ͵Ϫϱ x f 1 1 ˆ x ͑ ͒ d ϭ ϩ erf 2 2 xϪ x & , ͩ ͪ chosen so that the exact analytical solution is available for comparison in order to better illus- trate the method. While many real one- ͑13͒ dimensional uncertainty problems exist ͑thermo-where erf is the error function deﬁned as couple voltage-temperature curves, ideal gas pressure-temperature relation at ﬁxed volume, erf͑ x ͒ ϭ 2 ͱ ͵e 0 x Ϫt 2 dt. ͑14͒ etc.͒, few are represented by such a conveniently simple function. Hence this mathematical function was chosen primarily for clarity in illustration.After some algebra it can be shown that for the Assume that the data reduction equation is a pa-normal distribution ͓Eq. ͑12͔͒, rabola, ͵ x yϭx 2 ϩ0.5, ͑18͒ F x͑ ͒ d ϭ ͑ Ϫ x ͒ F x͑ ͒ ϩ f x͑ ͒ ϩ . ˆ ˆ 2ˆ 2 and that the uncertainty is to be evaluated around ͑15͒ xϭ2. The independent variable is distributed nor-With this result, Eq. ͑10͒ can be expressed mally with a standard deviation of 0.65. Fig. 2 shows the parabola ͓Eq. ͑18͔͒ with 21 ͯͫ ͩ ͪ ͬ NϪ1 1 data samples spaced, for this example, at even in- y ϭ ͚ a k x F x͑ ͒ Ϫ ˆ ˆ Ϫ 2 ˆ x͑ ͒ f tervals of probability in a normal distribution. The kϭ0 2 sample spacing is not critical to the analysis. ϩb k F x ͑ ͒ ˆ ͯ ϭx kϩ1 ϭx k . ͑16͒ Fig. 3 shows the approximate probability den- sity function for y calculated according to Eq. ͑6͒. For comparison, the dashed line in Fig. 3 shows the exact probability density function from Eq. ͑4͒For evaluating the conﬁdence interval ͓Eq. ͑11͔͒, it which is easy to calculate for this simple case of acan be shown that for the special case of the nor- parabola. The triangles in Fig. 3 shows the resultsmal distribution, of the probability density function calculated from ͵ ˆf ͩ yϪb ͪ dyϭaͫ F ͩ yϪb ͪ Ϫ 2 ͬ . x a ˆ a 1 x ͑17͒ a Monte Carlo analysis with one million function evaluations. By comparison, the approximate probability density function calculated from Eq.Thus, for this special case, both the mean and con- ͑6͒ required only 21 function evaluations.ﬁdence interval can be evaluated using only the The mean for y, y , calculated from Eq. ͑16͒error function. with the 21 data points shown in Fig. 2, is 4.9122
R. Luck, J. W. Stevens / ISA Transactions 43 (2004) 491–497 495 solution. Also as expected, the error in the ﬁrst- order analysis increases rapidly with the level of uncertainty in the independent variable, rising from less than 0.5% at ϭ0.15 to nearly 17% at ϭ0.95. This increase is approximately propor- tional to the square of the standard deviation for this particular case. The error from Eq. ͑16͒, on the other hand, decreases slightly with increasing independent variable uncertainty and is less than 0.3% over the range shown in Fig. 4. Using Eq. ͑11͒ with the simpliﬁcation of Eq. ͑17͒, the 5% and 95% conﬁdence limits are calcu- Fig. 3. Probability density function. lated at 1.382 and 9.987, respectively for the case of ϭ0.65. For this simple case, the conﬁdencewhile the exact mean would be 4.9225, or a dif- limits calculated from the linear analysis are quiteference of 0.2%. For comparison, the mean value close ͑1.366 and 9.92, respectively͒, but in gen-calculated from a one-million-point Monte Carlo eral, they may differ signiﬁcantly due to the non-analysis is 4.9226, a difference of 0.002%. For a linearity. The differences in conﬁdence intervalssimple ﬁrst-order uncertainty analysis the mean between those calculated by Eq. ͑11͒ and thosewould be 4.5, or a difference of 8.6% from the from a ﬁrst-order analysis increase with increasingexact result. Thus, even on this simple equation standard deviation in a manner similar to y , butwith relatively gentle curvature, there is a signiﬁ- with smaller magnitude.cant effect of the nonlinearities on the uncertaintypropagation. 5. Variations This effect is explored more fully in Fig. 4where the percent error in y with respect to the While one of the most attractive features of theexact answer is plotted as a function of the stan- method outlined above lies in its simplicity, thedard deviation in the independent variable x. The overall efﬁciency and accuracy of the method canstandard deviation is representative of the uncer- be improved slightly at the expense of somewhattainty in the independent variable. The errors for greater complexity. Two such variations are ex-the one-million-point Monte Carlo analysis and plained below.for Eq. ͑16͒ are plotted with the left vertical scale,while the error in the ﬁrst-order analysis is plotted 5.1. Sample distributionwith the right vertical scale. As would be expected, the one-million-point The probability density function from Eq. ͑6͒Monte Carlo analysis is nearly exact, with very shown in Fig. 3 exhibits increasing instability as xsmall random errors around the exact analytical approaches zero. This is because the transforma- tion of the linear pieces becomes undeﬁned at zero slope. Furthermore, as the slope gets small, the inverse of the slope ͓see Eq. ͑4͔͒ gets large, and small changes in the slope are ampliﬁed. Thus there are large changes between each piece of the approximation. A possible improvement would be for the sample points to be distributed more closely in areas where the function slope is small. For a ﬁxed number of total sample points, this can be accomplished relatively easily, but at the cost of making two passes of function evaluations: one to calculate the slopes, and another to redistribute the sample points. This results in approximately Fig. 4. Error in calculated mean values y . twice as many function evaluations as the original
496 R. Luck, J. W. Stevens / ISA Transactions 43 (2004) 491–497 Fig. 5. Variations for calculating the p.d.f.method. The result of such an approach is shown 6. Conclusionin Fig. 5. This ﬁgure contains the original approxi-mate and exact p.d.f.’s from Fig. 3, along with the A novel and simple method of estimating non-p.d.f. resulting from the redistributed sample linear uncertainty propagation has been describedpoints. The sample point distribution is shown on and illustrated. The method is derived by using athe small inset plot on the left. The redistribution piecewise linear approximation to the data reduc-smooths out the worst of the error, but at a cost of tion equation and transforming the probabilitydoubling the number of function evaluations. A density function of the linear pieces, then deter-similar reduction in error at similar cost could be mining the estimated mean and conﬁdence inter-achieved by simply doubling the number of vals from the result. This general method is inde-sample points. pendent of the particular probability density function employed. In addition, for the special5.2. Cubic spline function case of a normal distribution in the independent variable it is shown that only the error function In place of a piecewise linear approximation to ͑erf͒ is required for the analysis.the original function, a cubic spline can be usedwhich matches the function at the sample pointsand also has continuous ﬁrst and second deriva- Referencestives at the sample points. While this leads to an ͓1͔ ASME PTC 19.1-1998, Test Uncertainty. Americanincrease in the complexity of the equations, the Society of Mechanical Engineers, 1998.concept is identical to that outlined above for the ͓2͔ Guide to the Expression of Uncertainty in Measure- ment. ISO, 1993.linear case. In addition, far fewer sample points ͓3͔ Steele, W. G., et al., Comparison of ANSI/ASME andare needed to give excellent results. Fig. 5 shows ISO models for calculation of uncertainty. ISA Trans.the result of using a cubic spline with only six 33, 339–352 ͑1994͒.sample points on the simple parabola used earlier. ͓4͔ Dally, J. W., Riley, W. F., and McConnell, K. G., In-The sample points are shown on the inset plot on strumentation for Engineering Measurements, 2nd ed. Wiley, New York, 1993.the right. The probability density function matches ͓5͔ Figliola, R. S. and Beasley, D. E., Theory and Designthe exact result in this case, since six sample for Mechanical Measurements, 2nd ed. Wiley, Newpoints are sufﬁcient to exactly ﬁt the parabola. York, 1995.
R. Luck, J. W. Stevens / ISA Transactions 43 (2004) 491–497 497͓6͔ Holman, J. P., Experimental Methods for Engineers, ͓10͔ Kline, S. J. and McClintock, F. A., Describing uncer- 5th ed. McGraw-Hill, New York, 1989. tainties in single sample experiments. Mech. Eng.͓7͔ Barry, B. A., Engineering Measurements. Wiley, New ͑Am. Soc. Mech. Eng.͒ 75, 3– 8 ͑1953͒. York, 1964. ͓11͔ Coleman, H. W. and Steele, W. G., Engineering appli-͓8͔ Coleman, H. W. and Steele, W. G., Experimentation cation of experimental uncertainty analysis. AIAA J. and Uncertainty Analysis for Engineers. Wiley, New 33, 1888 –1896 ͑1995͒. York, 1989. ͓12͔ Wyler, J. S., Estimating the uncertainty of spacial and͓9͔ Moffat, R. J., Contributions to the theory of a single- time average measurements. J. Eng. Power October sample uncertainty analysis. J. Fluids Eng. 104, 250– 97, 473– 476 ͑1975͒. 260 ͑1982͒.