Covariance Error Propagation
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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, consider covariance propagation and next best view planning for 3d reconstruction the simple example where we estimate the area of a rectangle from replicate measurements standard error covariance of length and width. The area $$ area = length \cdot width $$ can be computed from each replicate. The standard deviation
Error Propagation Standard 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
Error Propagation Formula
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 for error propagation calculator 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 formula assumes indpendence
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Error Propagation Physics
developed, and nurturedbyEricWeisstein at WolframResearch Probability and Statistics>Error Analysis> Interactive Entries>Interactive Demonstrations> error propagation chemistry Error Propagation Given a formula with an absolute error in of , the absolute error is . error propagation average The relative error is . If , then (1) where denotes the mean, so the sample variance is given by (2) (3) The definitions of variance and covariance then http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm give (4) (5) (6) (where ), so (7) If and are uncorrelated, then so (8) Now consider addition of quantities with errors. For , and , so (9) For division of quantities with , and , so (10) Dividing through by and rearranging then gives (11) For exponentiation of quantities with (12) and (13) so (14) (15) If , http://mathworld.wolfram.com/ErrorPropagation.html then (16) For logarithms of quantities with , , so (17) (18) For multiplication with , and , so (19) (20) (21) For powers, with , , so (22) (23) SEE ALSO: Absolute Error, Accuracy, Covariance, Percentage Error, Precision, Relative Error, Significant Digits, Variance REFERENCES: Abramowitz, M. and Stegun, I.A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p.14, 1972. Bevington, P.R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, pp.58-64, 1969. Referenced on Wolfram|Alpha: Error Propagation CITE THIS AS: Weisstein, Eric W. "Error Propagation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ErrorPropagation.html Wolfram Web Resources Mathematica» The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha» Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project» Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Computerbasedmath.org» Join the initiative for modernizing math education. Online Integral Calculator» Solve integrals with Wolfram|Alpha. Step-by-step Solutions» Walk through homework problems step-by-step from beginning to end. Hints help