Forskjell mellom versjoner av «Propagation of uncertainties for a function of observables»
| (2 mellomliggende revisjoner av samme bruker vises ikke) | |||
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| + | ''This is a personal derivation of a familiar result!'' Alex Read 14.05.2011 | ||
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We want to estimate the uncertainty of a function <math>f</math> of a set of <math>n</math> '''uncorrelated''' 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>\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 | + | 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 thus approximate the function with its Taylor-series expansion, keeping only the lowest-order derivatives |
<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>. | <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>. | ||
| Linje 24: | Linje 26: | ||
of a product of an uncorrelated pair of Gaussian distributed variables is the product of their averages, | 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 | 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'''. | + | <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 of a function of observables are to be added in quadrature'''. |
Nåværende revisjon fra 14. mai 2011 kl. 09:43
This is a personal derivation of a familiar result! Alex Read 14.05.2011
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 thus 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 of a function of observables are to be added in quadrature.
