Gaussian Error Propagation
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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 error propagation calculator measurement limitations (e.g., instrument precision) which propagate to the combination of variables in
Error Propagation Physics
the function. The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error error propagation chemistry 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,
Error Propagation Square Root
the positive square root of variance, σ2. The value of a quantity and its error are then expressed as an interval x ± u. 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 error propagation inverse 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 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] Contents 1 Linear combinations 2 Non-linear combinations 2.1 Simplification 2.2 Example 2.3 Caveats and warnings 2.3.1 Reciprocal 2.3.2 Shifted reciprocal 3 Example formulas 4 Example calculations 4.1 Inverse tangent function 4.2 Resistance measurement 5 See also 6 References 7 Further reading 8 External links Linear combinations[edit] Let { f k ( x 1 , x 2 , … , x n ) } {\displaystyle \ ρ 5(x_ ρ 4,x_ ρ 3,\dots ,x_ ρ 2)\}} be a set of m functions which are linear combinations of n {\displaystyle n} variables x 1 , x 2 , … , x n {\displaystyle x_ σ 7,x_ σ 6,\dots ,x_ σ 5} with combination coefficients A k 1 , A k 2 , … , A k n , (
The approach to uncertainty analysis that has been followed up to this point in the discussion has been what is called a top-down approach. Uncertainty components are estimated from direct repetitions of the measurement result. To contrast this with a propagation of error approach,
Error Propagation Definition
consider the simple example where we estimate the area of a rectangle from replicate measurements
Error Propagation Reciprocal
of length and width. The area $$ area = length \cdot width $$ can be computed from each replicate. The standard error propagation average deviation of the reported area is estimated directly from the replicates of area. Advantages of top-down approach This approach has the following advantages: proper treatment of covariances between measurements of length and width proper treatment of https://en.wikipedia.org/wiki/Propagation_of_uncertainty unsuspected sources of error that would emerge if measurements covered a range of operating conditions and a sufficiently long time period independence from propagation of error model Propagation of error approach combines estimates from individual auxiliary measurements The formal propagation of error approach is to compute: standard deviation from the length measurements standard deviation from the width measurements and combine the two into a standard deviation for area using the approximation http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm for products of two variables (ignoring a possible covariance between length and width), $$ s_{area} = \sqrt{width^2 \cdot s_{length}^2 + length^2 \cdot s_{width}^2} $$ Exact formula Goodman (1960) derived an exact formula for the variance between two products. Given two random variables, \(x\) and \(y\) (correspond to width and length in the above approximate formula), the exact formula for the variance is: $$ V(\bar{x} \bar{y}) = \frac{1}{n} \left[ X^2 V(y) + Y^2 V(x) + 2XYE_{11} + 2X\frac{E_{12}}{n} + 2Y\frac{E_{21}}{n} + \frac{V(x) V(y)}{n} + \frac{Cov((\Delta x)^2, (\Delta y)^2) -E_{11}^2 }{n^2} \right] $$ with \(X = E(x)\) and \(Y = E(y)\) (corresponds to width and length, respectively, in the approximate formula) \(V(x)\) is the variance of \(x\) and \(V(y)\) is the variance \(y\) (corresponds to \(s^2\) for width and length, respectively, in the approximate formula) \( E_{ij} = {(\Delta x)^i, (\Delta y)^j}\) where \( \Delta x = x - X \) and \( \Delta y = y - Y \) \( Cov((\Delta x)^2, (\Delta y)^2) = E_{22} - V(x) V(y) \) To obtain the standard deviation, simply take the square root of the above formula. Also, an estimate of the statistic is obtained by substituting sample estimates for the corresponding population values on the right hand side of the equation. Approximate for
same results as Monte Carlo http://www.openlca.org/documentation/index.php/Gaussian_error_propagation_formulas. Simulation provided that the relative error (i.e. the variation coefficient, standard deviation divided by the mean) is below 20% (Ciroth 2001). This condition is checked during the calculation; if the variation coefficient is error propagation found to be higher during the calculation, a warning message is issued together with the calculation results. To use approximation for uncertainty calculation, you can edit settings for the calculation in the "Preferences" under gaussian error propagation "Settings for calculation methods", and change the default entry use uncertainty calculation to true. Note that you can use only the sequential methods for the calculation. As default uncertainty approximation is inactive. Retrieved from "http://www.openlca.org/documentation/index.php/Gaussian_error_propagation_formulas." Personal tools Log in Namespaces Page Discussion Variants Views Read View source View history Actions Search openLCA help menu Beginners guide Advanced users guide Case studies Developers guide openLCA Website openLCA Project Converter Network download Resources Forum Contact us Toolbox What links here Related changes Special pages Printable version Permanent link This page was last modified on 24 May 2013, at 14:21. Disclaimers
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