Forskjell mellom versjoner av «Propagation of uncertainties for a function of observables»
(Ny side: We want to estimate the uncertainty of a function <math>f</math> of a set of <math>n</math> observables <math>\vec{x}</math> with associated uncertainties <math>\sigma_i</math>. We assume t...) |
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| − | We want to estimate the uncertainty of a function <math>f</math> of a set of <math>n</math> observables | + | We want to estimate the uncertainty of a function <math>f</math> of a set of <math>n</math> '''uncorrelated''' observables |
<math>\vec{x}</math> | <math>\vec{x}</math> | ||
| − | with associated uncertainties <math>\ | + | with associated uncertainties <math>\vec{\sigma}</math>. We assume that the observables <math>\vec{x}</math> are sampled from normal distributions which are narrow relative to the form of the function and approximate the function with its Taylor-series expansion, keeping only the lowest-order derivatives |
| − | <math>f(\vec{x}+\vec{\delta_x}) | + | <math>f(\vec{x}+\vec{\delta_x}) \simeq f(\vec{x}) + {df(\vec{x}) \over d{\vec{x}}}\cdot \vec{\delta_x} + h.o.</math>, where <math>\delta_{x_i}</math> represents the Gaussian-distributed deviation from the true mean (which is '''unknown''', but estimated by the observed value) <math>x_i</math>. In other words <math><\delta_{x_i}>=0</math> and <math><\delta_{x_i}^2>=\sigma_{x_i}^2</math>. |
The uncertainty in <math>f</math> is estimated as the square root of its mean variance | The uncertainty in <math>f</math> is estimated as the square root of its mean variance | ||
| − | <math>\sqrt{\sigma^2_f}</math> where <math>\sigma^2_f=<(f-<f>)^2>.</math> | + | <math>\sqrt{\sigma^2_f}</math> where <math>\sigma^2_f=<(f-<f>)^2></math>. |
| + | We replace <math>f</math> with the series approximation above and take advantage of that | ||
| + | in this approximation <math><f> = f(\vec{x})</math>. In addition we write the dot product as an explicit sum, taking care to have separate indices for the terms in the square of the dot product, obtaining | ||
| + | <math>\sigma^2_f = < | ||
| + | (\Sigma_{i=1}^n {df(\vec{x}) \over dx_i} \delta_{x_i}) | ||
| + | (\Sigma_{j=1}^n {df(\vec{x}) \over dx_j} \delta_{x_j}) | ||
| + | ></math>. The product of these two sums can be re-written as the sum of 2 terms: | ||
| + | <math> | ||
| + | \sigma^2_f = | ||
| + | < | ||
| + | \Sigma_{i=1}^n ({df(\vec{x}) \over dx_i} \delta_{x_i})^2 | ||
| + | + | ||
| + | \Sigma_{i=1}^n \Sigma_{j=1}^n (1-\delta_{ij}){df(\vec{x}) \over dx_i} {df(\vec{x}) \over dx_j}\delta_{x_i}\delta_{x_j} | ||
| + | > | ||
| + | </math> | ||
| + | |||
| + | Since the sub-terms in the second term above are only non-zero for <math>i\neq j</math>, and since <math>x_i</math> and <math>x_{j\neq i}</math> are uncorrelated, and since the average | ||
| + | of a product of an uncorrelated pair of Gaussian distributed variables is the product of their averages, | ||
| + | the second term vanishes due to the fact that <math><\delta_{x_i}>=0</math>. Since the variance of a <math>\delta_{x_i}</math> is <math>\sigma_{x_i}^2</math>, the final and familiar result is | ||
| + | <math>\sigma^2_f = \Sigma_{i=1}^n ({df(\vec{x}) \over dx_i} \sigma_{x_i})^2</math>. In other words, '''uncorrelated contributions to the total uncertainty are to be added in quadrature'''. | ||
Revisjonen fra 14. mai 2011 kl. 09:27
We want to estimate the uncertainty of a function of a set of uncorrelated observables with associated uncertainties . We assume that the observables are sampled from normal distributions which are narrow relative to the form of the function and approximate the function with its Taylor-series expansion, keeping only the lowest-order derivatives , where represents the Gaussian-distributed deviation from the true mean (which is unknown, but estimated by the observed value) . In other words and .
The uncertainty in is estimated as the square root of its mean variance where . We replace with the series approximation above and take advantage of that in this approximation . In addition we write the dot product as an explicit sum, taking care to have separate indices for the terms in the square of the dot product, obtaining . The product of these two sums can be re-written as the sum of 2 terms:
Since the sub-terms in the second term above are only non-zero for , and since and are uncorrelated, and since the average of a product of an uncorrelated pair of Gaussian distributed variables is the product of their averages, the second term vanishes due to the fact that . Since the variance of a is , the final and familiar result is . In other words, uncorrelated contributions to the total uncertainty are to be added in quadrature.
