Coefficient Of Variation 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 measurement limitations (e.g., instrument precision) which propagate to the combination of variables in the function. standard error and coefficient of variation The uncertainty u can be expressed in a number of ways. It may be defined by the
Coefficient Of Variation Standard Deviation
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 coefficient of variation confidence interval a quantity is quantified in terms of the standard deviation, σ, 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
Correlation Coefficient Standard Error
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 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 error propagation formula 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 4(x_ ρ 3,x_ ρ 2,\dots ,x_ ρ 1)\}} be a set of m functions which are linear combinations of n {\displaystyle n} variables x 1 , x 2 , … , x n {\displaystyle x_ σ 6,x_ σ 5,\dots ,x_ σ 4} with combination coefficients A k 1 , A k 2 , … , A k n , ( k = 1 … m ) {\displaystyle A_ σ 0,A_ ρ 9,\dots ,A_ ρ 8,(k=1\dots m)} . f k = ∑ i n A k i x i or f = A x {\displaystyle f_ ρ 4=\sum _ ρ 3^ ρ 2A_ ρ 1x_ ρ 0{\text{ or }}\mathrm σ 9 =\mathrm σ 8 \,} and let the variance-covariance matrix on x be denoted by Σ x {\displaystyle \mathrm {\Sigma ^ σ 0} \,} . Σ x = ( σ 1 2 σ 12 σ 13 ⋯ σ 12 σ 2 2
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in Sign in Registration Forgot password Expand/collapse global hierarchy Home Core Analytical Chemistry Quantifying Nature Expand/collapse global error propagation chemistry location Propagation of Error Last updated 20:33, 14 May 2016 Save as PDF Share Share Share Tweet Share IntroductionDerivation of Exact FormulaDerivation of Arithmetic ExampleCaveats and WarningsDisadvantages of Propagation https://en.wikipedia.org/wiki/Propagation_of_uncertainty of Error ApproachTreatment of Covariance TermsReferencesContributors Propagation of Error (or Propagation of Uncertainty) is defined as the effects on a function by a variable's uncertainty. It is a calculus derived statistical calculation designed to combine uncertainties from multiple variables, in order to provide an accurate measurement of uncertainty. Introduction Every measurement has an air of uncertainty about it, http://chem.libretexts.org/Core/Analytical_Chemistry/Quantifying_Nature/Significant_Digits/Propagation_of_Error and not all uncertainties are equal. Therefore, the ability to properly combine uncertainties from different measurements is crucial. Uncertainty in measurement comes about in a variety of ways: instrument variability, different observers, sample differences, time of day, etc. Typically, error is given by the standard deviation (\(\sigma_x\)) of a measurement. Anytime a calculation requires more than one variable to solve, propagation of error is necessary to properly determine the uncertainty. For example, lets say we are using a UV-Vis Spectrophotometer to determine the molar absorptivity of a molecule via Beer's Law: A = ε l c. Since at least two of the variables have an uncertainty based on the equipment used, a propagation of error formula must be applied to measure a more exact uncertainty of the molar absorptivity. This example will be continued below, after the derivation (see Example Calculation). Derivation of Exact Formula Suppose a certain experiment requires multiple instruments to carry out. These instruments each have different variability in their measurements. The results of each instrument are given
Tour Start 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 Learn more about hiring developers or http://stats.stackexchange.com/questions/12054/propagation-of-polynomial-coefficient-errors-in-fit posting ads with us Cross Validated Questions Tags Users Badges Unanswered Ask Question _ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. 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 Propagation of polynomial coefficient errors in fit up vote 5 down vote favorite I error propagation fit a cubic function (in mathematica) $$ y(x) = a + b x + c x^2 + d x^3 $$ to my data and obtained a function. I have the error in each coefficient ($\sigma_a$, $\sigma_b$, $\sigma_c$, $\sigma_d$). Now since I need the model to predict $y$ values for a given $x$ value, I also need a corresponding error for the $y$ value (coming from the errors in the coefficients). The error in the $x$ value (from our measurements) is negligible. I coefficient of variation can't figure out the formula I would use when plugging in an $x$ value to determine its corresponding error in $y$. I've exhausted my googling capabilities and my textbooks in error analysis. Any help is greatly appreciated. regression error-propagation share|improve this question edited Jun 18 '11 at 9:49 mbq 17.7k849103 asked Jun 18 '11 at 1:13 Illy 262 you will also need to know the pairwise covariances between each of the coefficient estimates to get a variance for the predicted value of $y$. Assuming the predictor $x$ is not symmetric around 0, the covariance between, $x$ and $x^{2}$, for example, will be non-zero. From there (conditioned on the value of $x$), you just need to use the formula for the variance of a sum. –Macro Jun 18 '11 at 1:51 1 Polynomial regression seems like the obvious choice here, given that you are willing to consider you $x$ measurement error negligible. A "quick and dirty" approach (the first that came to my mind) is to do ordinary least squares of $y$ on $(1,x,x^2,x^3)$. Then just use standard prediction formula from that. Perhaps I am missing something? –probabilityislogic Jun 18 '11 at 3:15 add a comment| 1 Answer 1 active oldest votes up vote 5 down vote The question appears to ask for a predicted value and for a prediction interval about that value. The predicted value is obtained by means of the formula using the estimated parameters and a specified valu