Mathematica Error Propagation
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Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Wed Oct 19 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Probability and error propagation formula Statistics>Error Analysis> Interactive Entries>Interactive Demonstrations> Error Propagation Given a formula with an absolute
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error in of , the absolute error is . The relative error is . If , then (1) where error propagation formula calculator denotes the mean, so the sample variance is given by (2) (3) The definitions of variance and covariance then give (4) (5) (6) (where ), so (7) If and are uncorrelated, then
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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 , then (16) For logarithms of quantities with , , so (17) (18) For multiplication with , and , so error propagation physics (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 you try the next step on your own. Wolfram Problem Generator» Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Wolfram Education Portal» Collection of teaching and learning tools bui
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Dismiss Join the Stack Overflow Community Stack Overflow is a community of 6.2 million programmers, just like you, helping each other. Join them; it uncertainty calculator online only takes a minute: Sign up Uncertainty propagation formula in mathematica up vote 1 down vote favorite I'm trying to write a short piece of code that will perform propagation of errors. So far, I can get Mathematica to http://mathworld.wolfram.com/ErrorPropagation.html generate the formula for the error delta_f in a function f(x1,x2,...,xi,...,xn) with errors dx1,dx2,...,dxi,...dxn: fError[f_, xi__, dxi__] := Sum[(D[f[xi], xi[[i]]]*dxi[[i]])^2, {i, 1, Length[xi]}]^(1/2) where fError requires that the input function f has all of its variables surrounded by {...}. For example, d[{mv_, Mv_, Av_}] := 10^(1/5 (mv - Mv + 5 - Av)) FullSimplify[fError[d, {mv, Mv, Av}, {dmv, dMv, dAv}]] returns 2 Sqrt[10^(-(2/5) (Av - mv + Mv)) (dAv^2 + dmv^2 + dMv^2)] Log[10] My question is, how can I http://stackoverflow.com/questions/15169955/uncertainty-propagation-formula-in-mathematica evaluate this? Ideally I would like to modify fError to something like: fError[f_, xi__, nxi__, dxi__] where nxi is the list of actual values of xi (separated since setting the xi's to their numerical values will destroy the differentiation step above.) This function should find the general formula for the error delta_f and then evaluate it numerically, if possible. I think the solution should be as simple as a Hold[] or With[] or something like that, but I can't seem to get it. wolfram-mathematica uncertainty share|improve this question asked Mar 2 '13 at 2:17 user1748343 135 This may prove useful. –rcollyer Mar 2 '13 at 4:04 @rcollyer Those all seem a little more advanced than what I need. I'm really just looking for an elegant, one- or two-line solution that I can adapt to various calculations. I never have to deal with correlated errors, either. I have found, though, that if I just call fError[f,xi,dxi]/.{x1->nx1,...}, it works. That's good enough for my purposes. –user1748343 Mar 2 '13 at 7:37 add a comment| 1 Answer 1 active oldest votes up vote 1 down vote I'm not following everything that you've done, and since this was posted two years ago it's likely you aren't working on it anymore anyways. I'll give you my solution for error propagation in hopes that it will somehow help you or others. I tried to include the best documentation that
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business http://mathematica.stackexchange.com/questions/106048/error-propagation-code Learn more about hiring developers or posting ads with us Mathematica Questions Tags Users https://en.wikipedia.org/wiki/List_of_uncertainty_propagation_software Badges Unanswered Ask Question _ Mathematica Stack Exchange is a question and answer site for users of Mathematica. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Error propagation code up vote 3 down vote favorite error propagation I have a code for error propagation :- err[F_, w__] := {F[Map[First, List[w]] /. List -> Sequence], Block[{parms = Table[Unique[], {x, 1, Length[List[w]]}], values = Table[List[w][[i, 1]], {i, 1, Length[List[w]]}], errors = Table[List[w][[i, 2]], {i, 1, Length[List[w]]}]}, Sqrt[Total[ Table[(D[F[parms /. List -> Sequence], parms[[i]]] errors[[i]])^2, {i, 1, Length[values]}] /. Table[parms[[i]] -> values[[i]], {i, 1, Length[values]}]]]]} I am not able to propagate error in a complex function with this code.For example in case of real error propagation formula function which is defined as :- f[a_, b_] := a + b; err[f, {2, 0.1}, {3, 0.2}] The output for this function is :- {5, 0.223607}. But in case of complex function say :- f[a_, b_] := Abs[a + I b]; err[f, {2, 0.1}, {3, 0.2}] The output is `{Sqrt[13], Sqrt[(-0.03 + 0. I) Derivative[1][Abs][2 + 3 I]^2]}` In output both values should be a numerical number. How can I modify my code that it will give me a numerical number in output. Thanks. Please help me . programming probability-or-statistics share|improve this question edited Feb 11 at 5:56 asked Feb 10 at 4:10 Dinesh Kumar 314 1 Please provide a few sample arguments for err, both real and complex. –bbgodfrey Feb 10 at 4:27 You may be interested in this: with expr = ComplexExpand[Abs[a + I b]], PropagateCovariance[expr, {a, b}] // Simplify is (2 a b cov[a, b] + a^2 var[a] + b^2 var[b])/(a^2 + b^2). This is the (first-order approximation of the) variance in f as a function of the variances in a and b. –Oleksandr R. Feb 11 at 14:11 You could approximate the derivatives with finite differences. –Daniel Lichtblau Feb 11 at 15:26 add a comment| 1 Answer 1 active oldest votes up vote 3 down vote accepted The troublesome part her
See also 4 References Software[edit] ASUE This is a powerful web interface powered by webMathematica for evaluating uncertainty symbolically using GUM. Besides, the webpage also allows symbolic uncertainty evaluation using ASUE framework (with reference), which is an extension to GUM framework Dempster Shafer with Intervals (DSI) Toolbox is a MATLAB toolbox for verified computing under Dempster–Shafer theory. It provides aggregation rules, fast (non-)monotonic function propagation, plots of basic probability assignments, verified Fault tree analysis, and much more. EasyGraph is a graphing package that supports error propagation directly into the error bars. Epc is an open source script based tool that calculates the propagation of errors in variables. To quote the text on the Epc web page "This is done by repeated calculation of the expression using variable-values which are generated using a random number generator whose mean and standard-deviation match the values specified for the variable". Error Calculator Free/libre cross-platform calculator with minimalistic interface. Designed for use in practical courses at natural sciences. Exposes all formulae needed to calculate the results, interoperability with office, support for physical quantities with units. Error Propagation Calculator Free cross-platform calculator (OSX/Windows/Linux) written in Python. Essentially a GUI interface for the python Uncertainties library. Very easy to use and install. ErrorCalc is a scientific calculator app for iPhone or iPad that performs error propagation (Supports Algebraic and RPN modes of entry) FuncDesigner GUMsim is a Monte Carlo simulator and uncertainty estimator for Windows GUM Tree is a design pattern for propagating measurement uncertainty. There is an implementation in R and add-ons for Excel (real and complex numbers). GUM Tree Calculator is a programmable Windows command-line tool with full support for uncertainty calculations involving real and complex quantities. GUM Workbench implements a systematic way to analyze an uncertainty problem for single and multiple results. GUM + Monte Carlo. Free restricted educational version available. The Gustavus propagator is an open source calculator that supports error propagation developed by Thomas Huber. The laffers.net propagator is a web based tool for propagating errors in data. The tool uses the standard methods for propagation. Mathos Core Library Uncertainty package Open source (.NET targeting library). MC-Ed is a native Windows software to perform uncertainty calculations according to the Supplement 1