Propagation of uncertainty

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, σ². 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 one dimensional 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.

Linear combinations

Let $f_k(x_1,x_2,\dots,x_n)$ be a set of m functions which are linear combinations of $n$ variables $x_1,x_2,\dots,x_n$ with combination coefficients $A_{k1},A_{k2},\dots,A_{kn}, (k=1\dots m)$.

$f_k=\sum_i^n A_{ki} x_i$ or $\mathbf{f}=\mathbf{Ax}\,$

and let the variance-covariance matrix on x be denoted by $\Sigma^x\,$.

$\Sigma^x = \begin{pmatrix} \sigma^2_1 & \text{cov}_{12} & \text{cov}_{13} & \cdots \\ \text{cov}_{12} & \sigma^2_2 & \text{cov}_{23} & \cdots\\ \text{cov}_{13} & \text{cov}_{23} & \sigma^2_3 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}$

Then, the variance-covariance matrix $\Sigma^f\,$ of f is given by

$\Sigma^f_{ij}= \sum_k^n \sum_\ell^n A_{ik} \Sigma^x_{k\ell} A_{j\ell}: \Sigma^f=\mathbf{A} \Sigma^x \mathbf{A}^\top$.

This is the most general expression for the propagation of error from one set of variables onto another. When the errors on x are un-correlated the general expression simplifies to

$\Sigma^f_{ij}= \sum_k^n A_{ik} \left(\sigma^2_k \right)^x A_{jk}.$

Note that even though the errors on x may be un-correlated, their errors on f are always correlated.

The general expressions for a single function, f, are a little simpler.

$f=\sum_i^n a_i x_i: f=\mathbf {a x}\,$

$\sigma^2_f= \sum_i^n \sum_j^n a_i \Sigma^x_{ij} a_j= \mathbf{a \Sigma^x a^t}$

Each covariance term, $M_{ij}$ can be expressed in terms of the correlation coefficient $\rho_{ij}\,$ by $M_{ij}=\rho_{ij}\sigma_i\sigma_j\,$, so that an alternative expression for the variance of f is

$\sigma^2_f= \sum_i^n a_i^2\sigma^2_i+\sum_i^n \sum_{j (j \ne i)}^n a_i a_j\rho_{ij} \sigma_i\sigma_j.$

In the case that the variables x are uncorrelated this simplifies further to

$\sigma^{2}_{f}= \sum_i^n a_{i}^{2}\sigma^{2}_{i}.$

Non-linear combinations

See also: Taylor expansions for the moments of functions of random variables

When f is a set of non-linear 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 first-order 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. The Taylor expansion would be:

$f_k \approx f^0_k+ \sum_i^n \frac{\partial f_k}{\partial {x_i}} x_i$

where $\partial f_k/\partial x_i$ denotes the partial derivative of f_k with respect to the i-th variable. Or in matrix notation,

$\mathrm{f} \approx \mathrm{f}^0 + J \mathrm{x}\,$

where J is the Jacobian matrix. Since f⁰_k 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, $\frac{\partial f_k}{\partial x_i}$ and $\frac{\partial f_k}{\partial x_j}$. In matrix notation,

$\operatorname{cov}(\mathrm{f}) = J \operatorname{cov}(\mathrm{x}) J^\top$.

That is, the Jacobian of the function is used to transform the rows and columns of the covariance of the argument.

Nonetheless, the most common formula among engineers and experimental scientists to calculate error propagation for independent variables is the one proposed by the NIST:

$s_f = \sqrt{ \left(\frac{\partial f}{\partial {x} }\right)^2 s_x^2 + \left(\frac{\partial f}{\partial {y} }\right)^2 s_y^2 + \left(\frac{\partial f}{\partial {z} }\right)^2 s_z^2 + ...}$

where $s_f$ represents the standard deviation of the function $f$, $s_x$ represents the standard deviation of $x$, $s_y$ represents the standard deviation of $y$, and so forth.

It is important to note that this formula is based on the linear characteristics of the gradient of $f$ and therefore it is a good estimation for the standard deviation of $f$ as long as $s_x, s_y, s_z,...$ are small compared to the partial derivatives.

Example

Any non-linear function, f(a,b), of two variables, a and b, can be expanded as

$f\approx f^0+\frac{\partial f}{\partial a}a+\frac{\partial f}{\partial b}b$

hence:

$\sigma^2_f\approx\left| \frac{\partial f}{\partial a}\right| ^2\sigma^2_a+\left| \frac{\partial f}{\partial b}\right|^2\sigma^2_b+2\frac{\partial f}{\partial a}\frac{\partial f}{\partial b}\text{cov}_{ab}.$

In the particular case that $f=ab\!$, $\frac{\partial f}{\partial a}=b, \frac{\partial f}{\partial b}=a$. Then

$\sigma^2_f \approx b^2\sigma^2_a+a^2 \sigma_b^2+2ab\,\text{cov}_{ab}$

or

$\left(\frac{\sigma_f}{f}\right)^2 \approx \left(\frac{\sigma_a}{a}\right)^2+\left(\frac{\sigma_b}{b}\right)^2+2\left(\frac{\sigma_a}{a}\right)\left(\frac{\sigma_b}{b}\right)\rho_{ab}.$

Caveats and warnings

Error estimates for non-linear 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 $1/B$ where $B=N(0,1)$, the distribution is a Cauchy distribution and there is no definable variance. For such ratio distributions, there can be defined probabilities for intervals which can be defined either by Monte Carlo simulation, or, in some cases, by using the Geary-Hinkley transformation.

For highly non-linear functions, there exist five categories of probabilistic approaches for uncertainty propagation; see Uncertainty Quantification#Methodologies for forward uncertainty propagation for details.

Example formulas

This table shows the variances of simple functions of the real variables $A,B\!$, with standard deviations $\sigma_A, \sigma_B\,$, correlation coefficient $\rho_{AB}\,$ and precisely known real-valued constants $a,b\,$.

Function	Variance
$f = aA\,$	$\sigma_f^2 = a^2\sigma_A^2$
$f = a A \pm bB\,$	$\sigma_f^2 = a^2\sigma_A^2 + b^2\sigma_B^2\pm2ab\,\text{cov}_{AB}$
$f = AB\,$	$\left(\frac{\sigma_f}{f}\right)^2 \approx \left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 + 2\frac{\sigma_A\sigma_B}{AB}\rho_{AB}$
$f = \frac{A}{B}\,$	$\left(\frac{\sigma_f}{f}\right)^2 \approx \left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 - 2\frac{\sigma_A\sigma_B}{AB}\rho_{AB}$
$f = a A^{\pm b}\,$	$\frac{\sigma_f}{f} \approx b \frac{\sigma_A}{A}$
$f = a \ln(\pm bA)\,$	$\sigma_f \approx a \frac{\sigma_A}{A}$
$f = a \log(A)\,$	$\sigma_f \approx a \frac{\sigma_A}{A \ln(10)}$
$f = a e^{\pm bA}\,$	$\frac{\sigma_f}{f} \approx b\sigma_A$
$f = a^{\pm bA}\,$	$\frac{\sigma_f}{f} \approx b\ln(a)\sigma_A$

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,

$f = AB(C); \left(\frac{\sigma_f}{f}\right)^2 \approx \left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2+ \left(\frac{\sigma_C}{C}\right)^2.$

For the case $f = AB$ we also have an Goodman's expression to calculate its exact variance, for the uncorrelated case it would be:

$V(XY)= E(X)^2 V(Y) + E(Y)^2 V(X) + E((X-E(X))^2 (Y-E(Y))^2)^2$

and therefore we have:

$\sigma_f^2 = A^2\sigma_B^2 + B^2\sigma_A^2 + \sigma_A^2\sigma_B^2$

Partial derivatives

Given $X=f(A, B, C, \dots)$

Absolute Error	Variance
$\left |\Delta X\right |=\left |\frac{\partial f}{\partial A}\right |\cdot \left |\Delta A\right |+\left |\frac{\partial f}{\partial B}\right |\cdot \left |\Delta B\right |+\left |\frac{\partial f}{\partial C}\right |\cdot \left |\Delta C\right |+\cdots$	$\sigma_X^2=\left (\frac{\partial f}{\partial A}\sigma_A\right )^2+\left (\frac{\partial f}{\partial B}\sigma_B\right )^2+\left (\frac{\partial f}{\partial C}\sigma_C\right )^2+\cdots$

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

$f(x) = \arctan(x),$

where $\sigma_x$ is the absolute uncertainty on our measurement of $x$. The partial derivative of $f(x)$ with respect to $x$ is

$\frac{\partial f}{\partial x} = \frac{1}{1+x^2}.$

Therefore, our propagated uncertainty is

$\sigma_{f} \approx \frac{\sigma_x}{1+x^2},$

where $\sigma_f$ 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, $R = V / I.$

Given the measured variables with uncertainties, I±σ_I and V±σ_V, the uncertainty in the computed quantity, σ_R is

$\sigma_R \approx \sqrt{ \sigma_V^2 \left(\frac{1}{I}\right)^2 + \sigma_I^2 \left(\frac{-V}{I^2}\right)^2 }.$

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

References

• Bevington, Philip R.; Robinson, D. Keith (2002), Data Reduction and Error Analysis for the Physical Sciences (3rd ed.), McGraw-Hill, ISBN 0-07-119926-8 
• Meyer, Stuart L. (1975), Data Analysis for Scientists and Engineers, Wiley, ISBN 0-471-59995-6 

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).
• 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. JCGM. http://www.bipm.org/utils/common/documents/jcgm/JCGM_102_2011_E.pdf. Retrieved 13 February 2013.