Propagation of uncertainty
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 This article deals with the propagation of uncertainty via algebraic manipulations. For the propagation of uncertainty through time, see Chaos theory#Sensitivity to initial conditions.
In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate to the combination of variables in the function.
The uncertainty is usually defined by the absolute error Δx. Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as a percentage.
Most commonly the error on a quantity, Δx, is given as the standard deviation, σ. Standard deviation is the positive square root of variance, σ^{2}. The value of a quantity and its error are often expressed as an interval x ± Δx. If the statistical probability distribution of the variable is known or can be assumed, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a onedimensional variable belonging to a normal distribution are ± one standard deviation from the value, that is, there is approximately a 68% probability that the true value lies in the region x ± σ.
If the variables are correlated, then covariance must be taken into account.
Contents
Linear combinations
Let be a set of m functions which are linear combinations of variables with combination coefficients .
 or
and let the variancecovariance matrix on x be denoted by .
Then, the variancecovariance matrix of f is given by
 .
This is the most general expression for the propagation of error from one set of variables onto another. When the errors on x are uncorrelated the general expression simplifies to
where the x superscript is merely notation, not exponentiation. Note that even though the errors on x may be uncorrelated, the errors on f are in general correlated; in other words, even if is a diagonal matrix, is in general a full matrix.
The general expressions for a single function, f, are a little simpler.
Each covariance term, can be expressed in terms of the correlation coefficient by , so that an alternative expression for the variance of f is
In the case that the variables x are uncorrelated this simplifies further to
Nonlinear combinations
When f is a set of nonlinear combination of the variables x, an interval propagation could be performed in order to compute intervals which contain all consistent values for the variables. In a probabilistic approach, the function f must usually be linearized by approximation to a firstorder Taylor series expansion, though in some cases, exact formulas can be derived that do not depend on the expansion as is the case for the exact variance of products.^{1} The Taylor expansion would be:
where denotes the partial derivative of f_{k} with respect to the ith variable. Or in matrix notation,
where J is the Jacobian matrix. Since f ^{0} is a constant it does not contribute to the error on f. Therefore, the propagation of error follows the linear case, above, but replacing the linear coefficients, A_{ik} and A_{jk} by the partial derivatives, and . In matrix notation, ^{2}
 .
That is, the Jacobian of the function is used to transform the rows and columns of the covariance of the argument.
Simplification
Neglecting correlations or for independent variables yields a common formula among engineers and experimental scientists to calculate error propagation, the variance formula:^{3}
where represents the standard deviation of the function , represents the standard deviation of , represents the standard deviation of , and so forth. One practical application of this formula in an engineering context is the evaluation of relative uncertainty of the insertion loss for power measurements of random fields.^{4}
It is important to note that this formula is based on the linear characteristics of the gradient of and therefore it is a good estimation for the standard deviation of as long as are small compared to the partial derivatives.^{5}
Example
Any nonlinear differentiable function, f(a,b), of two variables, a and b, can be expanded as
hence:
In the particular case that , . Then
or
Caveats and warnings
Error estimates for nonlinear functions are biased on account of using a truncated series expansion. The extent of this bias depends on the nature of the function. For example, the bias on the error calculated for log x increases as x increases since the expansion to 1+x is a good approximation only when x is small.
In the special case of the inverse where , the distribution is a reciprocal normal distribution and there is no definable variance. For such inverse distributions and for ratio distributions, there can be defined probabilities for intervals which can be computed either by Monte Carlo simulation, or, in some cases, by using the Geary–Hinkley transformation.^{6} The statistics, mean and variance, of the shifted reciprocal function, , where however exist in a principal value sense if the difference between the shift or pole, , and the mean is real. The mean of this transformed random variable is then indeed the scaled Dawson's function .^{7} In contrast, if the shift is purely complex, the mean exists and is a scaled Faddeeva function whose exact expression depends on the sign of the imaginary part, . In both cases, the variance is a simple function of the mean .^{8} Therefore, the variance has to be considered in a principal value sense if is real while it exists if the imaginary part of is nonzero. Note that these means and variances are exact as they do not recur to linearisation of the ratio. The exact covariance of two ratios with a pair of different poles and is similarly available .^{9} The case of the inverse of a complex normal variable , shifted or not, exhibits different characteristics.^{7}
For highly nonlinear functions, there exist five categories of probabilistic approaches for uncertainty propagation;^{10} see Uncertainty Quantification#Methodologies for forward uncertainty propagation for details.
Example formulas
This table shows the variances of simple functions of the real variables , with standard deviations , covariance and precisely known realvalued constants .

Function Variance ^{11} ^{12} ^{13} ^{13} ^{14}
For uncorrelated variables the covariance terms are zero. Expressions for more complicated functions can be derived by combining simpler functions. For example, repeated multiplication, assuming no correlation gives,
For the case we also have Goodman's expression^{1} for the exact variance: for the uncorrelated case it is
and therefore we have:
Partial derivatives
Given

Absolute Error Variance ^{15}
Example calculation: Inverse tangent function
We can calculate the uncertainty propagation for the inverse tangent function as an example of using partial derivatives to propagate error.
Define
where is the absolute uncertainty on our measurement of . The derivative of with respect to is
Therefore, our propagated uncertainty is
where is the absolute propagated uncertainty.
Example application: Resistance measurement
A practical application is an experiment in which one measures current, I, and voltage, V, on a resistor in order to determine the resistance, R, using Ohm's law,
Given the measured variables with uncertainties, I±σ_{I} and V±σ_{V}, the uncertainty in the computed quantity, σ_{R} is
See also
 Accuracy and precision
 Automatic differentiation
 Delta method
 Errors and residuals in statistics
 Experimental uncertainty analysis
 Interval finite element
 List of uncertainty propagation software
 Measurement uncertainty
 Significance arithmetic
 Uncertainty quantification
Notes
 ^ ^{a} ^{b} Goodman, Leo (1960). "On the Exact Variance of Products". Journal of the American Statistical Association 55 (292): 708–713. doi:10.2307/2281592. JSTOR 2281592.
 ^ Ochoa1,Benjamin; Belongie, Serge "Covariance Propagation for Guided Matching"
 ^ Ku, H. H. (October 1966). "Notes on the use of propagation of error formulas". Journal of Research of the National Bureau of Standards (National Bureau of Standards) 70C (4): 262. ISSN 00224316. Retrieved 3 October 2012.
 ^ Arnaut, L. R. (December 2008). "Measurement uncertainty in reverberation chambers  I. Sample statistics". NPL Technical Report TQE 2, 2nd. ed., sec. 4.1.2.2 (National Physical Laboratory) TQE (2): 52. ISSN 17542995.
 ^ Clifford, A. A. (1973). Multivariate error analysis: a handbook of error propagation and calculation in manyparameter systems. John Wiley & Sons. ISBN 0470160551.^{page needed}
 ^ Hayya, Jack; Armstrong, Donald; Gressis, Nicolas (July 1975). "A Note on the Ratio of Two Normally Distributed Variables". Management Science 21 (11): 1338–1341. doi:10.1287/mnsc.21.11.1338. JSTOR 2629897.
 ^ ^{a} ^{b} Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rankone stochastic dynamic systems". Journal of Sound and Vibrations 332 (11): 2750–2776. doi:10.1016/j.jsv.2012.12.009.
 ^ Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rankone stochastic dynamic systems". Journal of Sound and Vibrations 332 (11). Section (4.1.1). doi:10.1016/j.jsv.2012.12.009.
 ^ Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rankone stochastic dynamic systems". Journal of Sound and Vibrations 332 (11). Eq.(39)(40). doi:10.1016/j.jsv.2012.12.009.
 ^ S. H. Lee and W. Chen, A comparative study of uncertainty propagation methods for blackboxtype problems, Structural and Multidisciplinary Optimization Volume 37, Number 3 (2009), 239253, DOI: 10.1007/s0015800802347
 ^ "Strategies for Variance Estimation". p. 37. Retrieved 20130118.
 ^ Fornasini, Paolo (2008), The uncertainty in physical measurements: an introduction to data analysis in the physics laboratory, Springer, p. 161, ISBN 038778649X
 ^ ^{a} ^{b} Harris, Daniel C. (2003), Quantitative chemical analysis (6th ed.), Macmillan, p. 56, ISBN 0716744643
 ^ "Error Propagation tutorial". Foothill College. October 9, 2009. Retrieved 20120301.
 ^ Lindberg, Vern (20091005). "Uncertainties and Error Propagation". Uncertainties, Graphing, and the Vernier Caliper (in eng). Rochester Institute of Technology. p. 1. Archived from the original on 20041112. Retrieved 20070420. "The guiding principle in all cases is to consider the most pessimistic situation."
References
 Bevington, Philip R.; Robinson, D. Keith (2002), Data Reduction and Error Analysis for the Physical Sciences (3rd ed.), McGrawHill, ISBN 0071199268
 Meyer, Stuart L. (1975), Data Analysis for Scientists and Engineers, Wiley, ISBN 0471599956
External links
 A detailed discussion of measurements and the propagation of uncertainty explaining the benefits of using error propagation formulas and Monte Carlo simulations instead of simple significance arithmetic
 Uncertainties and Error Propagation, Vern Lindberg's Guide to Uncertainties and Error Propagation.
 GUM, Guide to the Expression of Uncertainty in Measurement
 EPFL An Introduction to Error Propagation, Derivation, Meaning and Examples of Cy = Fx Cx Fx'
 uncertainties package, a program/library for transparently performing calculations with uncertainties (and error correlations).
 soerp package, a python program/library for transparently performing *secondorder* calculations with uncertainties (and error correlations).
 Joint Committee for Guides in Metrology (2011). JCGM 102: Evaluation of Measurement Data  Supplement 2 to the "Guide to the Expression of Uncertainty in Measurement"  Extension to Any Number of Output Quantities (Technical report). JCGM. Retrieved 13 February 2013.
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