2. 492 R. Luck, J. W. Stevens / ISA Transactions 43 (2004) 491–497
pressure, and density are commonly used to fix the and higher terms of the Taylor approximation are
thermodynamic state and to calculate other prop- not negligible relative to the second term. The
erties 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 in
the neighborhood of the critical point. dR
R ͑ x ͒ ϷR ͑ x 0 ͒ ϩ ͑ xϪx 0 ͒
Unfortunately, for many engineering applica- dx x0
tions, the inversions of the analysis equations re-
quired for a proper analytical approach are either
unwieldy or impossible to obtain. Monte Carlo
computer simulations can yield satisfactory re-
ϩ
1 d 2R
2! dx 2 ͯ x0
͑ xϪx 0 ͒ 2 ϩ¯ . ͑1͒
sults, but typically require very large numbers of
Then, the condition for the linear approximation to
function evaluations. This paper presents a direct
be valid is
approach to estimate nonlinear uncertainty propa-
gation by using a piecewise linear approximation
to the analysis function and a normal distribution
for 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 3R
normal distribution, only the error function ͑erf͒ ϩ ͑ xϪx 0 ͒ 2 ϩ¯ . ͑2͒
needs to be evaluated. 3! dx 3 x0
This condition tends to be violated by large higher
2. 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 nonlinear
lishes 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 measured
Uncertainty ͓1͔. The International Organization variable through the data reduction equation to get
for Standardization ͑ISO͒ also publishes a guide a probability density function for the result. The
on uncertainty calculation and terminology en- expected mean and confidence intervals can then
titled ‘‘Guide to the Expression of Uncertainty in be calculated from the p.d.f. of the result. If x is
Measurement’’ ͓2͔. These two approaches are the measured variable, and f x ( x ) is the p.d.f. of
compared by Steele et al. ͓3͔. Most textbooks on that variable, and yϭh ( x ) is the data reduction
experimental measurements include a section on equation, then the p.d.f. for the result can be cal-
uncertainty propagation as well ͑for example, culated by
Refs. ͓4 – 6͔͒. Some textbooks specialize in uncer-
tainty ͓7,8͔. The technical literature also has nu-
merous treatments of uncertainty estimation and
propagation in specific applications ͑for example,
f y ͑ y ͒ ϭ f x „h Ϫ1 ͑ y ͒ … ͯ dh Ϫ1 ͑ y ͒
dy ͯ. ͑3͒
Refs. ͓9–12͔͒. While this works easily for simple functions, the
While estimation of measurement uncertainty is inversion of the analysis equation can become
treated 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 to
handled by a truncated Taylor expansion. This ap- approximate the analysis function with a series of
proach to propagation assumes that the uncertainty straight line segments. The transformation for the
in 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
3. 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 , ͑8͒
special 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 defined
3. Development as
Consider a linear function of the form: yϭax
ϩb, with a probability density function on the
F x͑ x ͒ ϭ ͵ x
Ϫϱ
f x͑ ͒ d ͑9͒
independent variable of f x ( x ) . Then from Eq. ͑3͒, then Eq. ͑8͒ can be written
the probability density function of the dependent
ͫ ͵ ͬ
NϪ1
variable 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
k
Over a range of interest, an analysis function or
tabular data may be approximated by a series of 3.2. Confidence interval
line 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 function
For each segment, the probability density function of the dependent variable. For the N-segment
of the dependent variable is given by the piece- piecewise linear approximation this is
wise 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 of
3.1. Mean the cumulative distribution function.
Normally, the probability density function tends 3.3. Normal distribution
to be of lesser direct interest for the purposes of
engineering uncertainty analysis. Instead, the Eqs. ͑10͒ and ͑11͒ are valid for any probability
mean and confidence intervals are often the quan- density function of the independent variable. For
4. 494 R. Luck, J. W. Stevens / ISA Transactions 43 (2004) 491–497
the special ͑and most common͒ case of a normal
distribution, a further simplification for the evalu-
ation of Eqs. ͑10͒ and ͑11͒ is possible. However, it
must be emphasized that Eqs. ͑10͒ and ͑11͒ are
general in nature and are applicable to any prob-
ability density function. The additional steps de-
scribed below render the evaluation of Eqs. ͑10͒
and ͑11͒ very convenient for a single special case;
that of a normal distribution.
For a normal distribution with mean x and
standard 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 this
The corresponding cumulative distribution func-
approach to uncertainty estimation. This rather
tion 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. ͑4͒
For evaluating the confidence interval ͓Eq. ͑11͔͒, it
which is easy to calculate for this simple case of a
can be shown that for the special case of the nor-
parabola. The triangles in Fig. 3 shows the results
mal 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. ͑16͒
error function. with the 21 data points shown in Fig. 2, is 4.9122
5. 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 confidence
while the exact mean would be 4.9225, or a dif- limits calculated from the linear analysis are quite
ference 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 intervals
simple first-order uncertainty analysis the mean between those calculated by Eq. ͑11͒ and those
would be 4.5, or a difference of 8.6% from the from a first-order analysis increase with increasing
exact result. Thus, even on this simple equation standard deviation in a manner similar to y , but
with relatively gentle curvature, there is a signifi- with smaller magnitude.
cant effect of the nonlinearities on the uncertainty
propagation. 5. Variations
This effect is explored more fully in Fig. 4
where the percent error in y with respect to the While one of the most attractive features of the
exact answer is plotted as a function of the stan- method outlined above lies in its simplicity, the
dard deviation in the independent variable x. The overall efficiency and accuracy of the method can
standard deviation is representative of the uncer- be improved slightly at the expense of somewhat
tainty 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 distribution
with the right vertical scale.
As would be expected, the one-million-point The probability density function from Eq. ͑6͒
Monte Carlo analysis is nearly exact, with very shown in Fig. 3 exhibits increasing instability as x
small 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
6. 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. Conclusion
in 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 described
points. The sample point distribution is shown on and illustrated. The method is derived by using a
the 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 probability
doubling 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 special
5.2. 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 used
which matches the function at the sample points
and also has continuous first and second deriva- References
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increase 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.
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͓3͔ Steele, W. G., et al., Comparison of ANSI/ASME and
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the result of using a cubic spline with only six 33, 339–352 ͑1994͒.
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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 Design
the exact result in this case, since six sample for Mechanical Measurements, 2nd ed. Wiley, New
points are sufficient to exactly fit the parabola. York, 1995.
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5th ed. McGraw-Hill, New York, 1989. tainties in single sample experiments. Mech. Eng.
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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.
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York, 1989. ͓12͔ Wyler, J. S., Estimating the uncertainty of spacial and
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