Editing Propagation of uncertainty

{{Short description|Effect of variables' uncertainties on the uncertainty of a function based on them}}
{{For|the propagation of uncertainty through time|Chaos theory#Sensitivity to initial conditions}}

In [[statistics]], '''propagation of uncertainty''' (or '''propagation of error''') is the effect of [[Variable (mathematics)|variables]]' [[uncertainty|uncertainties]] (or [[Errors and residuals in statistics|errors]], more specifically [[random error]]s) on the uncertainty of a [[function (mathematics)|function]] based on them. When the variables are the values of experimental measurements they have [[Observational error|uncertainties due to measurement limitations]] (e.g., instrument [[Accuracy and precision|precision]]) which propagate due to the combination of variables in the function.

The uncertainty ''u'' can be expressed in a number of ways.
It may be defined by the [[absolute error]] {{math|Δ''x''}}. Uncertainties can also be defined by the [[relative error]] {{math|(Δ''x'')/''x''}}, which is usually written as a percentage.
Most commonly, the uncertainty on a quantity is quantified in terms of the [[standard deviation]], {{mvar|σ}}, which is the positive square root of the [[variance]]. The value of a quantity and its error are then expressed as an interval {{math|''x'' ± ''u''}}. 
However, the most general way of characterizing uncertainty is by specifying its [[probability distribution]].
If the [[probability distribution]] of the variable is known or can be assumed, in theory it is possible to get any of its statistics. In particular, 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 approximately ± one standard deviation {{math|''σ''}} from the central value {{math|''x''}}, which means that the region {{math|''x'' ± ''σ''}} will cover the true value in roughly 68% of cases.

If the uncertainties are [[correlated]] then [[covariance]] must be taken into account. Correlation can arise from two different sources. First, the ''measurement errors'' may be correlated. Second, when the underlying values are correlated across a population, the ''uncertainties in the group averages'' will be correlated.<ref>{{cite web| last1=Kirchner | first1=James | title=Data Analysis Toolkit #5: Uncertainty Analysis and Error Propagation | url=http://seismo.berkeley.edu/~kirchner/eps_120/Toolkits/Toolkit_05.pdf|website=Berkeley Seismology Laboratory|publisher=University of California | access-date=22 April 2016}}</ref>

In a general context where a nonlinear function modifies the uncertain parameters (correlated or not), the standard tools to propagate uncertainty, and infer resulting quantity probability distribution/statistics, are sampling techniques from the [[Monte Carlo method]] family.<ref name="kr11">{{cite book|author-last1=Kroese |author-first1=D. P. |author-last2=Taimre |author-first2=T. |author-last3=Botev |author-first3=Z. I. |title=Handbook of Monte Carlo Methods |year=2011 |publisher=John Wiley & Sons}}</ref> For very large datasets or complex functions, the calculation of the error propagation may be very expensive so that a [[surrogate model]]<ref>{{Cite journal|last1=Ranftl|first1=Sascha|last2=von der Linden|first2=Wolfgang|date=2021-11-13|title=Bayesian Surrogate Analysis and Uncertainty Propagation|journal=Physical Sciences Forum|volume=3|issue=1|pages=6|doi=10.3390/psf2021003006|issn=2673-9984|doi-access=free |arxiv=2101.04038}}</ref> or a [[parallel computing]] strategy<ref>{{cite journal|author-last1=Atanassova |author-first1=E. |author-last2=Gurov |author-first2=T. |author-last3=Karaivanova |author-first3=A. |author-last4=Ivanovska |author-first4=S. |author-last5=Durchova |author-first5=M. |author-last6=Dimitrov |author-first6=D. |year=2016 |title=On the parallelization approaches for Intel MIC architecture |journal=AIP Conference Proceedings |volume=1773 |issue=1 |pages=070001 |doi=10.1063/1.4964983 |bibcode=2016AIPC.1773g0001A}}</ref><ref>{{cite journal|author-last1=Cunha Jr |author-first1=A. |author-last2=Nasser |author-first2=R. |author-last3=Sampaio |author-first3=R. |author-last4=Lopes |author-first4=H. |author-last5=Breitman |author-first5=K. |year=2014 |title=Uncertainty quantification through the Monte Carlo method in a cloud computing setting |journal=Computer Physics Communications |volume=185 |issue=5 |pages=1355–1363 |doi=10.1016/j.cpc.2014.01.006 |arxiv=2105.09512 |bibcode=2014CoPhC.185.1355C |s2cid=32376269}}</ref><ref>{{cite journal|author-last1=Lin |author-first1=Y. |author-last2=Wang |author-first2=F. |author-last3=Liu |author-first3=B. |year=2018 |title=Random number generators for large-scale parallel Monte Carlo simulations on FPGA |journal = Journal of Computational Physics |volume=360 |pages=93–103 |doi=10.1016/j.jcp.2018.01.029 |bibcode=2018JCoPh.360...93L}}</ref> may be necessary.

In some particular cases, the uncertainty propagation calculation can be done through simplistic algebraic procedures. Some of these scenarios are described below.

==Linear combinations==
Let <math>\{f_k(x_1, x_2, \dots, x_n)\}</math> be a set of ''m'' functions, which are linear combinations of <math>n</math> variables <math>x_1, x_2, \dots, x_n</math> with combination coefficients <math>A_{k1}, A_{k2}, \dots,A_{kn}, (k = 1, \dots, m)</math>:
<math display="block">f_k = \sum_{i=1}^n A_{ki} x_i,</math>
or in matrix notation,
<math display="block">\mathbf{f} = \mathbf{A} \mathbf{x}.</math>

Also let the [[variance–covariance matrix]] of {{math|1=''x'' = (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}} be denoted by <math>\boldsymbol\Sigma^x</math> and let the mean value be denoted by <math>\boldsymbol{\mu}</math>:
<math display="block">\begin{align}
\boldsymbol\Sigma^x = \operatorname{E}[(\mathbf{x}-\boldsymbol\mu)\otimes (\mathbf{x}-\boldsymbol\mu)]
&=
\begin{pmatrix}
   \sigma^2_1 & \sigma_{12} & \sigma_{13} & \cdots \\
   \sigma_{21} & \sigma^2_2 & \sigma_{23} & \cdots\\
   \sigma_{31} & \sigma_{32} & \sigma^2_3 & \cdots \\
   \vdots & \vdots & \vdots & \ddots
\end{pmatrix} \\[1ex]
&=
\begin{pmatrix}
   {\Sigma}^x_{11} & {\Sigma}^x_{12} & {\Sigma}^x_{13} & \cdots \\
   {\Sigma}^x_{21} & {\Sigma}^x_{22} & {\Sigma}^x_{23} & \cdots \\
   {\Sigma}^x_{31} & {\Sigma}^x_{32} & {\Sigma}^x_{33} & \cdots \\
   \vdots & \vdots & \vdots & \ddots
\end{pmatrix}.
\end{align}
</math>
<math>\otimes</math> is the [[outer product]].

Then, the variance–covariance matrix <math>\boldsymbol\Sigma^f</math> of ''f'' is given by
<math display="block">\begin{align}
\boldsymbol\Sigma^f
&= \operatorname{E}\left[(\mathbf{f} - \operatorname{E}[\mathbf{f}]) \otimes (\mathbf{f} - \operatorname{E}[\mathbf{f}])\right]
= \operatorname{E}\left[\mathbf{A}(\mathbf{x}-\boldsymbol\mu) \otimes \mathbf{A}(\mathbf{x}-\boldsymbol\mu)\right] \\[1ex]
&= \mathbf{A} \operatorname{E}\left[(\mathbf{x}-\boldsymbol\mu) \otimes (\mathbf{x}-\boldsymbol\mu)\right] \mathbf{A}^\mathrm{T}
= \mathbf{A} \boldsymbol\Sigma^x \mathbf{A}^\mathrm{T}.
\end{align}</math>

In component notation, the equation
<math display="block">\boldsymbol\Sigma^f = \mathbf{A} \boldsymbol\Sigma^x \mathbf{A}^\mathrm{T}</math>
reads
<math display="block">\Sigma^f_{ij} = \sum_k^n \sum_l^n A_{ik} {\Sigma}^x_{kl} A_{jl}.</math>

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
<math display="block">\Sigma^f_{ij} = \sum_k^n A_{ik} \Sigma^x_k A_{jk},</math>
where <math>\Sigma^x_k = \sigma^2_{x_k}</math> 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 <math>\boldsymbol\Sigma^x</math> is a diagonal matrix, <math>\boldsymbol\Sigma^f</math> is in general a full matrix.

The general expressions for a scalar-valued function ''f'' are a little simpler (here '''a''' is a row vector):
<math display="block">f = \sum_i^n a_i x_i = \mathbf{a x},</math>
<math display="block">\sigma^2_f = \sum_i^n \sum_j^n a_i \Sigma^x_{ij} a_j = \mathbf{a} \boldsymbol\Sigma^x \mathbf{a}^\mathrm{T}.</math>

Each covariance term <math>\sigma_{ij}</math> can be expressed in terms of the [[Pearson product-moment correlation coefficient|correlation coefficient]] <math>\rho_{ij}</math> by <math>\sigma_{ij} = \rho_{ij} \sigma_i \sigma_j</math>, so that an alternative expression for the variance of ''f'' is
<math display="block">\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.</math>

In the case that the variables in ''x'' are uncorrelated, this simplifies further to
<math display="block">\sigma^2_f = \sum_i^n a_i^2 \sigma^2_i.</math>

In the simple case of identical coefficients and variances, we find
<math display="block">\sigma_f = \sqrt{n}\, |a| \sigma.</math>

For the arithmetic mean, <math>a=1/n</math>, the result is the [[standard error of the mean]]:
<math display="block">\sigma_f = \frac{\sigma} {\sqrt{n}}.</math>

== 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 linearised by approximation to a first-order [[Taylor series]] expansion, though in some cases, exact formulae can be derived that do not depend on the expansion as is the case for the exact variance of products.<ref name="Goodman1960">{{Cite journal
 | last = Goodman
 | first= Leo
 | author-link = Leo Goodman
 | title = On the Exact Variance of Products
 | journal = Journal of the American Statistical Association
 | year = 1960
 | volume = 55
 | issue = 292
 | pages = 708–713
 | doi = 10.2307/2281592
 | jstor=2281592
}}</ref> The Taylor expansion would be:
<math display="block">f_k \approx f^0_k+  \sum_i^n \frac{\partial f_k}{\partial {x_i}} x_i </math>
where <math>\partial f_k/\partial x_i</math> denotes the [[partial derivative]] of ''f<sub>k</sub>'' with respect to the ''i''-th variable, evaluated at the mean value of all components of vector ''x''. Or in [[matrix notation]],
<math display="block">\mathrm{f} \approx \mathrm{f}^0 + \mathrm{J} \mathrm{x}\,</math>
where J is the [[Jacobian matrix]]. Since f<sup>0</sup> 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<sub>ki</sub>'' and ''A<sub>kj</sub>'' by the partial derivatives, <math>\frac{\partial f_k}{\partial x_i}</math> and <math>\frac{\partial f_k}{\partial x_j}</math>. In matrix notation,<ref>Ochoa1, Benjamin; Belongie, Serge [http://vision.ucsd.edu/sites/default/files/ochoa06.pdf "Covariance Propagation for Guided Matching"] {{Webarchive|url=https://web.archive.org/web/20110720080130/http://vision.ucsd.edu/sites/default/files/ochoa06.pdf |date=2011-07-20 }}</ref>
<math display="block">\mathrm{\Sigma}^\mathrm{f} = \mathrm{J} \mathrm{\Sigma}^\mathrm{x} \mathrm{J}^\top.</math>

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 <math>\mathrm{J = A}</math>.

=== Simplification ===
Neglecting correlations or assuming independent variables yields a common formula among engineers and experimental scientists to calculate error propagation, the variance formula:<ref>{{cite journal|last=Ku|first=H. H.|date=October 1966 |title=Notes on the use of propagation of error formulas|url=http://nistdigitalarchives.contentdm.oclc.org/cdm/compoundobject/collection/p16009coll6/id/99848/rec/1|journal=Journal of Research of the National Bureau of Standards | volume=70C|issue=4|page=262|doi=10.6028/jres.070c.025|issn=0022-4316|access-date=3 October 2012|doi-access=free}}</ref>
<math display="block">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 + \cdots}</math>
where <math>s_f</math> represents the standard deviation of the function <math>f</math>, <math>s_x</math> represents the standard deviation of <math>x</math>, <math>s_y</math> represents the standard deviation of <math>y</math>, and so forth.

This formula is based on the linear characteristics of the gradient of <math>f</math> and therefore it is a good estimation for the standard deviation of <math>f</math> as long as <math>s_x, s_y, s_z,\ldots</math> are small enough. Specifically, the linear approximation of <math>f</math> has to be close to <math>f</math> inside a neighbourhood of radius <math>s_x, s_y, s_z,\ldots</math>.<ref>{{Cite book |last=Clifford |first=A. A. |title=Multivariate error analysis: a handbook of error propagation and calculation in many-parameter systems |publisher=John Wiley & Sons |year=1973 |isbn=978-0470160558 }}{{page needed|date=October 2012}}</ref>

=== Example ===
Any non-linear differentiable function, <math>f(a,b)</math>, of two variables, <math>a</math> and <math>b</math>, can be expanded as
<math display="block">f\approx f^0+\frac{\partial f}{\partial a}a+\frac{\partial f}{\partial b}b.</math>
If we take the variance on both sides and use the formula<ref>{{Cite web| last=Soch|first=Joram| date=2020-07-07| title=Variance of the linear combination of two random variables|url=https://statproofbook.github.io/P/var-lincomb.html| access-date=2022-01-29| website=The Book of Statistical Proofs|language=en}}</ref> for the variance of a linear combination of variables
<math display="block">\operatorname{Var}(aX + bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) + 2ab \operatorname{Cov}(X, Y),</math>
then we obtain
<math display="block">\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},</math>
where <math>\sigma_{f}</math> is the standard deviation of the function <math>f</math>, <math>\sigma_{a}</math> is the standard deviation of <math>a</math>, <math>\sigma_{b}</math> is the standard deviation of <math>b</math> and <math>\sigma_{ab} = \sigma_{a}\sigma_{b} \rho_{ab}</math> is the covariance between <math>a</math> and <math>b</math>.

In the particular case that {{nowrap|<math>f = ab</math>,}} {{nowrap|<math>\frac{\partial f}{\partial a} = b</math>,}} {{nowrap|<math>\frac{\partial f}{\partial b} = a</math>.}} Then
<math display="block">\sigma^2_f \approx b^2\sigma^2_a+a^2 \sigma_b^2+2ab\,\sigma_{ab}</math>
or
<math display="block">\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}</math>
where <math>\rho_{ab}</math> is the correlation between <math>a</math> and <math>b</math>.

When the variables <math>a</math> and <math>b</math> are uncorrelated, <math>\rho_{ab}=0</math>. Then
<math display="block">\left(\frac{\sigma_f}{f}\right)^2 \approx \left(\frac{\sigma_a}{a} \right)^2 + \left(\frac{\sigma_b}{b}\right)^2.</math>

===Caveats and warnings===
Error estimates for non-linear functions are [[Bias of an estimator|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(1+''x'') increases as ''x'' increases, since the expansion to ''x'' is a good approximation only when ''x'' is near zero.

For highly non-linear functions, there exist five categories of probabilistic approaches for uncertainty propagation;<ref>{{cite journal |first1=S. H. |last1=Lee |first2=W. |last2=Chen |title=A comparative study of uncertainty propagation methods for black-box-type problems |journal=Structural and Multidisciplinary Optimization |volume=37 |issue=3 |year=2009 |pages=239–253 |doi=10.1007/s00158-008-0234-7 |s2cid=119988015 }}</ref> see [[Uncertainty quantification#Forward propagation|Uncertainty quantification]] for details.

====Reciprocal and shifted reciprocal====
{{main|Reciprocal normal distribution}}
In the special case of the inverse or reciprocal <math>1/B</math>, where <math>B=N(0,1)</math> follows a [[standard normal distribution]], the resulting distribution is a reciprocal standard normal distribution, and there is no definable variance.<ref name=Johnson>{{cite book
  | last1 = Johnson | first1 = Norman L.
  | last2 = Kotz    | first2 = Samuel
  | last3 = Balakrishnan | first3 = Narayanaswamy
  | title = Continuous Univariate Distributions, Volume 1
  | year = 1994
  | publisher = Wiley
  | isbn=0-471-58495-9
  | pages = 171
  }}</ref>

However, in the slightly more general case of a shifted reciprocal function <math>1/(p-B)</math> for <math>B=N(\mu,\sigma)</math> following a general normal distribution, then mean and variance statistics do exist in a [[principal value]] sense, if the difference between the pole <math>p</math> and the mean <math>\mu</math> is real-valued.<ref name=lecomte2013exact>{{Cite journal
| last1= Lecomte | first1 = Christophe
| title = Exact statistics of systems with uncertainties: an analytical theory of rank-one stochastic dynamic systems
| journal = Journal of Sound and Vibration
| volume = 332
| issue =  11
| date = May 2013
| pages = 2750–2776
| doi = 10.1016/j.jsv.2012.12.009
| bibcode = 2013JSV...332.2750L
}}</ref>

====Ratios====
{{main|Normal ratio distribution}}
Ratios are also problematic; normal approximations exist under certain conditions.

==Example formulae==
This table shows the variances and standard deviations of simple functions of the real variables <math>A, B</math> with standard deviations <math>\sigma_A, \sigma_B,</math> [[Covariance and correlation|covariance]] <math>\sigma_{AB} = \rho_{AB} \sigma_A \sigma_B,</math> and correlation <math>\rho_{AB}.</math> The real-valued coefficients <math>a</math> and <math>b</math> are assumed exactly known (deterministic), i.e., <math>\sigma_a = \sigma_b = 0.</math>

In the right-hand columns of the table, <math>A</math> and <math>B</math> are [[expected value|expectation values]], and <math>f</math> is the value of the function calculated at those values.

{| class="wikitable"
! Function !! Variance !! Standard deviation
|-
| <math>f = aA\,</math>
| <math>\sigma_f^2 = a^2\sigma_A^2</math>
| <math>\sigma_f = |a|\sigma_A</math>
|-
| <math>f = A + B</math>
| <math>\sigma_f^2 = \sigma_A^2 + \sigma_B^2 + 2\sigma_{AB}</math>
| <math>\sigma_f = \sqrt{\sigma_A^2 + \sigma_B^2 + 2\sigma_{AB}}</math>
|-
| <math>f = A - B</math>
| <math>\sigma_f^2 = \sigma_A^2 + \sigma_B^2 - 2\sigma_{AB}</math>
| <math>\sigma_f = \sqrt{\sigma_A^2 + \sigma_B^2 - 2\sigma_{AB}}</math>
|-
| <math>f = aA + bB</math>
| <math>\sigma_f^2 = a^2\sigma_A^2 + b^2\sigma_B^2 + 2ab\,\sigma_{AB}</math>
| <math>\sigma_f = \sqrt{a^2\sigma_A^2 + b^2\sigma_B^2 + 2ab\,\sigma_{AB}}</math>
|-
| <math>f = aA - bB</math>
| <math>\sigma_f^2 = a^2\sigma_A^2 + b^2\sigma_B^2 - 2ab\,\sigma_{AB}</math>
| <math>\sigma_f = \sqrt{a^2\sigma_A^2 + b^2\sigma_B^2 - 2ab\,\sigma_{AB}}</math>
|-
| <math>f = AB</math>
| <math>\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]</math><ref>{{cite web |url=http://ipl.physics.harvard.edu/wp-uploads/2013/03/PS3_Error_Propagation_sp13.pdf |title=A Summary of Error Propagation |page=2 |access-date=2016-04-04 |archive-url = https://web.archive.org/web/20161213135602/http://ipl.physics.harvard.edu/wp-uploads/2013/03/PS3_Error_Propagation_sp13.pdf |archive-date=2016-12-13 |url-status=dead }}</ref><ref>{{cite web |url=http://web.mit.edu/fluids-modules/www/exper_techniques/2.Propagation_of_Uncertaint.pdf |title=Propagation of Uncertainty through Mathematical Operations |page=5 |access-date=2016-04-04}}</ref>
| <math>\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} }</math>
|-
| <math>f = \frac{A}{B}</math>
| <math>\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]</math><ref>{{cite web |url=http://www.sagepub.com/upm-data/6427_Chapter_4__Lee_%28Analyzing%29_I_PDF_6.pdf |title=Strategies for Variance Estimation |page=37 |access-date=2013-01-18}}</ref>
| <math>\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} }</math>
|-
| <math>f = \frac{A}{A+B}</math>
| <math>\sigma_f^2 \approx \frac{f^2}{\left(A+B\right)^2} \left(\frac{B^2}{A^2}\sigma_A^2  +\sigma_B^2 - 2\frac{B}{A} \sigma_{AB} \right)</math>
| <math>\sigma_f \approx \left|\frac{f}{A+B}\right| \sqrt{\frac{B^2}{A^2}\sigma_A^2  +\sigma_B^2 - 2\frac{B}{A} \sigma_{AB}  }</math>
|-
| <math>f = a A^b</math>
| <math>\sigma_f^2 \approx \left( {a}{b}{A}^{b-1}{\sigma_A} \right)^2 = \left( \frac{{f}{b}{\sigma_A}}{A} \right)^2 </math>
| <math>\sigma_f \approx \left| {a}{b}{A}^{b-1}{\sigma_A} \right| = \left| \frac{{f}{b}{\sigma_A}}{A} \right| </math>
|-
| <math>f = a \ln(bA)</math>
| <math>\sigma_f^2 \approx \left(a \frac{\sigma_A}{A} \right)^2</math><ref name=harris2003>{{citation |first1=Daniel C. |last1=Harris | title=Quantitative chemical analysis |edition=6th |publisher=Macmillan |year=2003 |isbn=978-0-7167-4464-1 |page=56 |url=https://books.google.com/books?id=csTsQr-v0d0C&pg=PA56 }}</ref>
| <math>\sigma_f \approx \left| a \frac{\sigma_A}{A}\right|</math>
|-
| <math>f = a \log_{10}(bA)</math>
| <math>\sigma_f^2 \approx \left(a \frac{\sigma_A}{A \ln(10)} \right)^2</math><ref name=harris2003/>
| <math>\sigma_f \approx \left| a \frac{\sigma_A}{A \ln(10)} \right|</math>
|-
| <math>f = a e^{bA}</math>
| <math>\sigma_f^2 \approx f^2 \left( b\sigma_A \right)^2</math><ref>{{cite web|url=http://www.foothill.edu/psme/daley/tutorials_files/10.%20Error%20Propagation.pdf|date=October 9, 2009|title=Error Propagation tutorial|work=Foothill College | access-date=2012-03-01}}</ref>
| <math>\sigma_f \approx \left| f \right| \left| \left( b\sigma_A \right) \right| </math>
|-
| <math>f = a^{bA}</math>
| <math>\sigma_f^2 \approx f^2 (b\ln(a)\sigma_A)^2</math>
| <math>\sigma_f \approx \left| f \right| \left| b \ln(a) \sigma_A \right|</math>
|-
| <math>f = a \sin(bA)</math>
| <math>\sigma_f^2 \approx \left[ a b \cos(b A) \sigma_A \right]^2</math>
| <math>\sigma_f \approx \left| a b \cos(b A) \sigma_A \right|</math>
|-
| <math>f = a \cos \left( b A \right)\,</math>
| <math>\sigma_f^2 \approx \left[ a b \sin(b A) \sigma_A \right]^2</math>
| <math>\sigma_f \approx \left| a b \sin(b A) \sigma_A \right|</math>
|-
|<math>f = a \tan \left( b A \right)\,</math>
|<math>\sigma_f^2 \approx \left[ a b \sec^2(b A) \sigma_A \right]^2</math>
|<math>\sigma_f \approx \left| a b \sec^2(b A) \sigma_A \right|</math>
|-
| <math>f = A^B</math>
| <math>\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]</math>
| <math>\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} } </math>
|-
| <math>f = \sqrt{aA^2 \pm bB^2}</math>
| <math>\sigma_f^2 \approx \left(\frac{A}{f}\right)^2 a^2\sigma_A^2 + \left(\frac{B}{f}\right)^2 b^2\sigma_B^2 \pm 2ab\frac{AB}{f^2}\,\sigma_{AB}</math>
| <math>\sigma_f \approx \sqrt{\left(\frac{A}{f}\right)^2 a^2\sigma_A^2 + \left(\frac{B}{f}\right)^2 b^2\sigma_B^2 \pm 2ab\frac{AB}{f^2}\,\sigma_{AB}}</math>
|}

For uncorrelated variables (<math>\rho_{AB} = 0</math>, <math>\sigma_{AB} = 0</math>) expressions for more complicated functions can be derived by combining simpler functions. For example, repeated multiplication, assuming no correlation, gives
<math display="block">f = ABC; \qquad \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.</math>

For the case <math>f = AB </math> we also have Goodman's expression<ref name="Goodman1960"/> for the exact variance: for the uncorrelated case it is
<math display="block">\operatorname{V}[XY] = \operatorname{E}[X]^2 \operatorname{V}[Y] + \operatorname{E}[Y]^2 \operatorname{V}[X] + \operatorname{E}\left[\left(X - \operatorname{E}(X)\right)^2 \left(Y - \operatorname{E}(Y)\right)^2\right],</math>
and therefore we have
<math display="block">\sigma_f^2 = A^2\sigma_B^2 + B^2\sigma_A^2 +  \sigma_A^2\sigma_B^2.</math>

===Effect of correlation on differences===
If ''A'' and ''B'' are uncorrelated, their difference ''A'' − ''B'' will have more variance than either of them. An increasing positive correlation (<math>\rho_{AB} \to 1</math>) will decrease the variance of the difference, converging to zero variance for perfectly correlated variables with the [[homoscedastic|same variance]]. On the other hand, a negative correlation (<math>\rho_{AB} \to -1</math>) will further increase the variance of the difference, compared to the uncorrelated case.

For example, the self-subtraction ''f'' = ''A'' − ''A'' has zero variance <math>\sigma_f^2 = 0</math> only if the variate is perfectly [[autocorrelation|autocorrelated]] (<math>\rho_A = 1</math>). If ''A'' is uncorrelated, <math>\rho_A = 0,</math> then the output variance is twice the input variance, <math>\sigma_f^2 = 2\sigma^2_A.</math> And if ''A'' is perfectly anticorrelated, <math>\rho_A = -1,</math> then the input variance is quadrupled in the output, <math>\sigma_f^2 = 4 \sigma^2_A</math> (notice <math>1 - \rho_A = 2</math> for ''f'' = ''aA'' − ''aA'' in the table above).

==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
<math display="block">f(x) = \arctan(x),</math>
where <math>\Delta_x</math> is the absolute uncertainty on our measurement of {{mvar|x}}. The derivative of {{math|''f''(''x'')}} with respect to {{mvar|x}} is
<math display="block">\frac{d f}{d x} = \frac{1}{1+x^2}.</math>

Therefore, our propagated uncertainty is
<math display="block">\Delta_{f} \approx \frac{\Delta_x}{1+x^2},</math>
where <math>\Delta_f</math> is the absolute propagated uncertainty.

===Resistance measurement===
A practical application is an [[experiment]] in which one measures [[current (electricity)|current]], {{mvar|I}}, and [[voltage]], {{mvar|V}}, on a [[resistor]] in order to determine the [[electrical resistance|resistance]], {{mvar|R}}, using [[Ohm's law]], {{math|1=''R'' = ''V'' / ''I''}}.

Given the measured variables with uncertainties, {{math|''I'' ± ''σ''<sub>''I''</sub>}} and {{math|''V'' ± ''σ''<sub>''V''</sub>}}, and neglecting their possible correlation, the uncertainty in the computed quantity, {{math|''σ''<sub>''R''</sub>}}, is:

<math display="block">\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 }.</math>

==See also==
{{div col}}
* [[Accuracy and precision]]
* [[Automatic differentiation]]
* [[Bienaymé's identity]]
* [[Delta method]]
* [[Dilution of precision (navigation)]]
* [[Errors and residuals in statistics]]
* [[Experimental uncertainty analysis]]
* [[Interval finite element]]
* [[Measurement uncertainty]]
* [[Numerical stability]]
* [[Probability bounds analysis]]
* [[Uncertainty quantification]]
* [[Random-fuzzy variable]]
* {{slink|Variance#Propagation}}
{{div col end}}

== References ==
{{reflist|30em}}

==Further reading==
* {{Citation |last1=Bevington |first1=Philip R. |last2=Robinson |first2=D. Keith |year=2002 |title=Data Reduction and Error Analysis for the Physical Sciences |edition=3rd |publisher=McGraw-Hill |isbn=978-0-07-119926-1 }}
* {{citation | first1=Paolo | last1=Fornasini | title=The uncertainty in physical measurements: an introduction to data analysis in the physics laboratory | publisher=Springer | year=2008 | isbn=978-0-387-78649-0 | page=161 | url=https://books.google.com/books?id=PBJgvPgf2NkC&pg=PA161 }}
* {{Citation |last=Meyer |first=Stuart L. |year=1975 |title=Data Analysis for Scientists and Engineers |publisher=Wiley |isbn=978-0-471-59995-1 }}
* {{Citation |last=Peralta |first=M. |date=2012 |title=Propagation Of Errors: How To Mathematically Predict Measurement Errors |publisher=CreateSpace }}
* {{Citation |last=Rouaud |first=M. |date=2013 |url=http://www.incertitudes.fr/book.pdf |title=Probability, Statistics and Estimation: Propagation of Uncertainties in Experimental Measurement |edition=short }}
* {{Citation |last=Taylor |first=J. R. |date=1997 |title=An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements |edition=2nd |publisher=University Science Books }}
* {{Cite journal |last1=Wang |first1=C. M. |last2=Iyer |first2=Hari K. |date=2005-09-07 |title=On higher-order corrections for propagating uncertainties |journal=Metrologia |volume=42 |issue=5 |pages=406–410 |doi=10.1088/0026-1394/42/5/011 |bibcode=2005Metro..42..406W |s2cid=122841691 |issn=0026-1394}}

==External links==
*[http://www.av8n.com/physics/uncertainty.htm 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
*[https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf/cb0ef43f-baa5-11cf-3f85-4dcd86f77bd6 GUM], Guide to the Expression of Uncertainty in Measurement
*[http://srl.informatik.uni-freiburg.de/papers/arrasTR98.pdf EPFL An Introduction to Error Propagation], Derivation, Meaning and Examples of Cy = Fx Cx Fx'
*[https://pythonhosted.org/uncertainties/ uncertainties package], a program/library for transparently performing calculations with uncertainties (and error correlations).
*[https://pypi.org/project/soerp/ soerp package], a Python program/library for transparently performing *second-order* calculations with uncertainties (and error correlations).
*{{cite tech report| author=Joint Committee for Guides in Metrology| title=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| year=2011| institution=JCGM| url=http://www.bipm.org/utils/common/documents/jcgm/JCGM_102_2011_E.pdf| access-date=13 February 2013}}
*[https://uncertaintycalculator.com/ Uncertainty Calculator] Propagate uncertainty for any expression

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[[Category:Algebra of random variables]]
[[Category:Numerical analysis]]
[[Category:Statistical approximations]]
[[Category:Statistical deviation and dispersion]]