Error Propagation Matrix Multiplication
Contents |
propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. error propagation multiplication and division When the variables are the values of experimental measurements they have error propagation multiplication by a constant uncertainties due to measurement limitations (e.g., instrument precision) which propagate to the combination of variables in the
Error Propagation Addition And Multiplication
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
Multiplying Error Propagation
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, 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 error propagation multiplication 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 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[edi
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
Error Propagation Calculator
policies of this site About Us Learn more about Stack Overflow the error propagation physics company Business Learn more about hiring developers or posting ads with us Cross Validated Questions Tags Users Badges Unanswered error propagation chemistry 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 https://en.wikipedia.org/wiki/Propagation_of_uncertainty 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 error in matrix multiplication involving an inversion up vote 0 down vote favorite Hi I have a system $a$ = $B^{-1}x$ where $a$ and $x$ are 9x1 vectors and $B$ is a 9x9 http://stats.stackexchange.com/questions/185733/propagation-of-error-in-matrix-multiplication-involving-an-inversion matrix to be inverted Each element in $B$ is a product of two values with their own uncertainties and the vectors $a$ and $x$ have uncertainties as well How can I propagate the errors through this matrix inversion (in say, MatLab)? matrix error-propagation share|improve this question asked Dec 8 '15 at 17:16 Sarah 1 2 Do you know the distributions for the "input" uncertainties, including the dependencies among them? Are you trying to determine the distribution (or samples) for output "a", given random instances of B and x? The answers to the preceding may determine whether it is possible to exactly or approximately determine the uncertainty in the output a via analytical calculations. The easiest and maybe best thing to do is to do Monte Carlo (stochastic) simulation in which you generate random instances of B and x, and for each such combination, find a, for instance via MATLAB's B\x . –Mark L. Stone Dec 8 '15 at 17:26 add a comment| active oldest votes Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook. Your
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 http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm with a propagation of error approach, consider the simple example where we estimate the area of a rectangle from replicate measurements of length and width. The area $$ area = length \cdot width $$ can be computed from each replicate. The 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: error propagation proper treatment of covariances between measurements of length and width proper treatment of 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 error propagation multiplication deviation from the width measurements and combine the two into a standard deviation for area using the approximation 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, s
be down. Please try the request again. Your cache administrator is webmaster. Generated Fri, 14 Oct 2016 15:15:56 GMT by s_ac15 (squid/3.5.20)