Propagation of uncertainty

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, 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 $\mathrm{f}=\mathrm{Ax}\,$

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

$\mathrm{\Sigma^x} = \begin{pmatrix} \sigma^2_1 & \sigma_{12} & \sigma_{13} & \cdots \\ \sigma_{12} & \sigma^2_2 & \sigma_{23} & \cdots\\ \sigma_{13} & \sigma_{23} & \sigma^2_3 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix} = \begin{pmatrix} \mathit{\Sigma}^x_1 & \mathit{\Sigma}^x_{12} & \mathit{\Sigma}^x_{13} & \cdots \\ \mathit{\Sigma}^x_{12} & \mathit{\Sigma}^x_2 & \mathit{\Sigma}^x_{23} & \cdots\\ \mathit{\Sigma}^x_{13} & \mathit{\Sigma}^x_{23} & \mathit{\Sigma}^x_3 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}$

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

$\mathit{\Sigma}^f_{ij}= \sum_k^n \sum_\ell^n A_{ik} \mathit{\Sigma}^x_{k\ell} A_{j\ell}$,

or, in matrix notation:

$\mathrm{\Sigma^f= A \Sigma^x A^\top}$.

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

$\mathit{\Sigma}^f_{ij}= \sum_k^n A_{ik} \mathit{\Sigma}^x_k A_{jk}.$

where $\mathit{\Sigma}^x_k = \sigma^2_{x_k}$ is the variance of k-th element of the x vector. Note that even though the errors on x may be uncorrelated, the errors on f are in general correlated; in other words, even if $\mathrm{\Sigma^x}$ is a diagonal matrix, $\mathrm{\Sigma^f}$ is in general a full matrix.

The general expressions for a scalar-valued function, f, are a little simpler.

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

$\sigma^2_f= \sum_i^n \sum_j^n a_i \mathit{\Sigma}^x_{ij} a_j= \mathrm{a \Sigma^x a^\top}$

(where a is a row-vector).

Each covariance term, $\sigma_{ij}$ can be expressed in terms of the correlation coefficient $\rho_{ij}\,$ by $\sigma_{ij}=\rho_{ij}\sigma_i\sigma_j\,$, 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 in x are uncorrelated this simplifies further to

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

In the simplest case of identical coefficients and variances, we find

$\sigma_{f}= \sqrt{n} a \sigma.$

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, evaluated at the mean value of all components of vector x. Or in matrix notation,

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

where J is the Jacobian matrix. Since f⁰ 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,

$\mathrm{\Sigma}^\mathrm{f} = \mathrm{J} \mathrm{\Sigma}^\mathrm{x} \mathrm{J}^\top$.

That is, the Jacobian of the function is used to transform the rows and columns of the variance-covariance matrix of the argument. Note this is equivalent to the matrix expression for the linear case with $\mathrm{J = A}$.

Simplification

Neglecting correlations or assuming independent variables yields a common formula among engineers and experimental scientists to calculate error propagation, the variance formula:

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

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. 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.

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 differentiable 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:

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

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

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

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.

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.

Reciprocal

In the special case of the inverse or reciprocal $1/B$, where $B=N(0,1)$, 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.

Shifted reciprocal

The statistics, mean and variance, of the shifted reciprocal function $\frac{1}{p-B}$ for $B=N(\mu,\sigma)$, however, exist in a principal value sense if the difference between the shift or pole $p$ and the mean $\mu$ is real. The mean of this transformed random variable is then indeed the scaled Dawson's function $\frac{\sqrt{2}}{\sigma} F \left(\frac{p-\mu}{\sqrt{2}\sigma}\right)$. In contrast, if the shift $p-\mu$ is purely complex, the mean exists and is a scaled Faddeeva function, whose exact expression depends on the sign of the imaginary part, $\operatorname{Im}(p-\mu)$. In both cases, the variance is a simple function of the mean. Therefore, the variance has to be considered in a principal value sense if $p-\mu$ is real, while it exists if the imaginary part of $p-\mu$ is non-zero. 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 $p_1$ and $p_2$ is similarly available. The case of the inverse of a complex normal variable $B$, shifted or not, exhibits different characteristics.

Example formulas

This table shows the variances of simple functions of the real variables $A,B\!$, with standard deviations $\sigma_A, \sigma_B\,$, covariance $\sigma_{AB}$ and exactly known real-valued constants $a,b\,$ (i.e., $\sigma_a=\sigma_b=0$).

Function	Variance	Standard Deviation
$f = aA\,$	$\sigma_f^2 = a^2\sigma_A^2$	$\sigma_f = |a|\sigma_A$
$f = a A + bB\,$	$\sigma_f^2 = a^2\sigma_A^2 + b^2\sigma_B^2 + 2ab\,\sigma_{AB}$	$\sigma_f = \sqrt{a^2\sigma_A^2 + b^2\sigma_B^2 + 2ab\,\sigma_{AB}}$
$f = a A - bB\,$	$\sigma_f^2 = a^2\sigma_A^2 + b^2\sigma_B^2 - 2ab\,\sigma_{AB}$	$\sigma_f = \sqrt{a^2\sigma_A^2 + b^2\sigma_B^2 - 2ab\,\sigma_{AB}}$
$f = AB\,$	$\sigma^2_f \approx B^2\sigma^2_A+A^2 \sigma_B^2+2AB\,\sigma_{AB}$	$\sigma_f \approx \sqrt{ B^2\sigma^2_A+A^2 \sigma_B^2+2AB\,\sigma_{AB}}$
$f = \frac{A}{B}\,$	$\sigma_f^2 \approx f^2 \left[\left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 - 2\frac{\sigma_{AB}}{AB} \right]$	$\sigma_f \approx \left| f \right| \sqrt{ \left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 - 2\frac{\sigma_{AB}}{AB} }$
$f = a A^{b}\,$	$\sigma_f^2 \approx \left( {a}{b}{A}^{b-1}{\sigma_A} \right)^2 = \left( \frac{{f}{b}{\sigma_A}}{A} \right)^2$	$\sigma_f \approx \left| {a}{b}{A}^{b-1}{\sigma_A} \right| = \left| \frac{{f}{b}{\sigma_A}}{A} \right|$
$f = a \ln(bA)\,$	$\sigma_f^2 \approx \left(a \frac{\sigma_A}{A} \right)^2$	$\sigma_f \approx \left| a \frac{\sigma_A}{A}\right|$
$f = a \log_{10}(A)\,$	$\sigma_f^2 \approx \left(a \frac{\sigma_A}{A \ln(10)} \right)^2$	$\sigma_f \approx \left| a \frac{\sigma_A}{A \ln(10)} \right|$
$f = a e^{bA}\,$	$\sigma_f^2 \approx f^2 \left( b\sigma_A \right)^2$	$\sigma_f \approx \left| f \left( b\sigma_A \right) \right|$
$f = a^{bA}\,$	$\sigma_f^2 \approx f^2 \left( b\ln(a)\sigma_A \right)^2$	$\sigma_f \approx \left| f \left( b\ln(a)\sigma_A \right) \right|$
$f = A^B\,$	$\sigma_f^2 \approx f^2 \left[ \left( \frac{B}{A}\sigma_A \right)^2 +\left( \ln(A)\sigma_B \right)^2 + 2 \frac{B \ln(A)}{A} \sigma_{AB} \right]$	$\sigma_f \approx \left| f \right| \sqrt{ \left( \frac{B}{A}\sigma_A \right)^2 +\left( \ln(A)\sigma_B \right)^2 + 2 \frac{B \ln(A)}{A} \sigma_{AB} }$

For uncorrelated variables ($\rho_{AB}=0$) the covariance terms are also zero, as $\sigma_{AB}=\rho_{AB}\sigma_A\sigma_B\,$.

In this case, expressions for more complicated functions can be derived by combining simpler functions. For example, repeated multiplication, assuming no correlation gives,

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

For the case $f = AB$ we also have Goodman's expression for the exact variance: for the uncorrelated case it is

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

and therefore we have:

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

Example calculations

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 σ_x is the absolute uncertainty on our measurement of x. The derivative of f(x) with respect to x is

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

Therefore, our propagated uncertainty is

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

where σ_f is the absolute propagated uncertainty.

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, and neglecting their possible correlation, the uncertainty in the computed quantity, σ_R is

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

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

Further reading

• Taylor, J. R., 1997: An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. 2nd ed. University Science Books, 327 pp.
• Peralta, M, 2012: Propagation Of Errors: How To Mathematically Predict Measurement Errors, CreateSpace.

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 *second-order* calculations with uncertainties (and error correlations).