Effect of variables' uncertainties on the uncertainty of a function based on them
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, more specifically random 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 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 Δx. Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as a percentage.
Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, which is the positive square root of the variance. The value of a quantity and its error are then expressed as an interval 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 σ from the central value x, which means that the region 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.[1]
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.[2] For very expansive data or complex functions, the calculation of the error propagation may be very expansive so that a surrogate model[3] or a parallel computing strategy[4][5][6] 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.
^Kirchner, James. "Data Analysis Toolkit #5: Uncertainty Analysis and Error Propagation" (PDF). Berkeley Seismology Laboratory. University of California. Retrieved 22 April 2016.
^Kroese, D. P.; Taimre, T.; Botev, Z. I. (2011). Handbook of Monte Carlo Methods. John Wiley & Sons.
^Ranftl, Sascha; von der Linden, Wolfgang (2021-11-13). "Bayesian Surrogate Analysis and Uncertainty Propagation". Physical Sciences Forum. 3 (1): 6. arXiv:2101.04038. doi:10.3390/psf2021003006. ISSN 2673-9984.
^Atanassova, E.; Gurov, T.; Karaivanova, A.; Ivanovska, S.; Durchova, M.; Dimitrov, D. (2016). "On the parallelization approaches for Intel MIC architecture". AIP Conference Proceedings. 1773 (1): 070001. Bibcode:2016AIPC.1773g0001A. doi:10.1063/1.4964983.
^Cunha Jr, A.; Nasser, R.; Sampaio, R.; Lopes, H.; Breitman, K. (2014). "Uncertainty quantification through the Monte Carlo method in a cloud computing setting". Computer Physics Communications. 185 (5): 1355–1363. arXiv:2105.09512. Bibcode:2014CoPhC.185.1355C. doi:10.1016/j.cpc.2014.01.006. S2CID 32376269.
^Lin, Y.; Wang, F.; Liu, B. (2018). "Random number generators for large-scale parallel Monte Carlo simulations on FPGA". Journal of Computational Physics. 360: 93–103. Bibcode:2018JCoPh.360...93L. doi:10.1016/j.jcp.2018.01.029.
and 28 Related for: Propagation of uncertainty information
In statistics, propagationofuncertainty (or propagationof error) is the effect of variables' uncertainties (or errors, more specifically random errors)...
epistemic uncertainties. Accuracy and precision Confidence interval Experimental uncertainty analysis History of measurement Propagationofuncertainty Repeatability...
Uncertainty quantification (UQ) is the science of quantitative characterization and estimation ofuncertainties in both computational and real world applications...
calculations. More advanced scientific rules are known as the propagationofuncertainty. Radix 10 (base-10, decimal numbers) is assumed in the following...
sprint. To get more accurate results, most researchers use the propagationofuncertainty equation in order to reduce any error in experiments. σ Q = σ...
of uncertainty in its inputs. A related practice is uncertainty analysis, which has a greater focus on uncertainty quantification and propagationof uncertainty;...
the uncertaintyof the derived quantity. Uncertainty analysis is often called the "propagationof error." For example, an experimental uncertainty analysis...
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is...
condition number of the (local) inverse must be used. The condition number is derived from the theory ofpropagationofuncertainty, and is formally defined...
the basis of model outcomes. Interval finite element Uncertainty quantification Propagationofuncertainty Measurement uncertainty#Uncertainty evaluation...
despite not being an "error" in the statistical sense. Margin of error Propagationofuncertainty Ratio estimator Sampling (statistics) Wikimedia Commons has...
combination of a fixed collection of independent normal deviates is a normal deviate. Many results and methods, such as propagationofuncertainty and least...
Konstantinos G. (October 20, 2005). "Fourier Transform of the Gaussian" (PDF). Propagationofuncertainty Algebra of random variables Stable distribution Standard...
path would still give correct results. Propagationofuncertainty The degree to which the quality of the results of Spatial analysis methods and other processing...
Algorithms for calculating variance Stability theory Chaos theory Propagationofuncertainty This is a fixed point iteration for the equation x = ( x 2 − 2...
distance generalizing number of standard deviations to the mean Mean absolute error Pooled variance Propagationofuncertainty Percentile Raw data Reduced...
Distance covariance, or Brownian covariance. Law of total covariance Propagationofuncertainty Rice, John (2007). Mathematical Statistics and Data Analysis....
the error of the function. Let f {\displaystyle f} be a function with N {\displaystyle N} variables. Therefore, the propagationofuncertainty must be calculated...
also used in: propagationofuncertainty over time, regression analysis of gravitational orbits, ensemble forecasting of weather, dynamics of neural networks...
ShrinkCovMat), in Python (scikit-learn library [1]), and in MATLAB. Propagationofuncertainty Sample mean and sample covariance Variance components Smith, Steven...
and Σ X {\displaystyle \Sigma _{X}} is the covariance matrix of X. Propagationofuncertainty WKB approximation Delta method Haym Benaroya, Seon Mi Han,...
(first recording; coupled with music of Maurice Ravel and Lennox Berkeley) Arash Yazdani – PropagationofUncertainty (orchestral works) 1 January – Oldřich...
Global Information