Error analysis (mathematics)

Error analysis is the study of kind and quantity of error that occurs, particularly in the fields of applied mathematics (particularly numerical analysis), applied linguistics and statistics.

Error analysis in numerical modeling

In numerical simulation or modeling of real systems, error analysis is concerned with the changes in the output of the model as the parameters to the model vary about a mean.

For instance, in a system modeled as a function of two variables $\scriptstyle z\,=\,f(x,y)$. Error analysis deals with the propagation of the numerical errors in $\scriptstyle x$ and $\scriptstyle y$ (around mean values $\scriptstyle {\bar {x}}$ and $\scriptstyle {\bar {y}}$) to error in $\scriptstyle z$ (around a mean $\scriptstyle {\bar {z}}$).

In numerical analysis, error analysis comprises both forward error analysis and backward error analysis. Forward error analysis involves the analysis of a function $\scriptstyle z'=f'(a_{0},\,a_{1},\,\dots ,\,a_{n})$ which is an approximation (usually a finite polynomial) to a function $\scriptstyle z\,=\,f(a_{0},a_{1},\dots ,a_{n})$ to determine the bounds on the error in the approximation; i.e., to find $\scriptstyle \epsilon$ such that $\scriptstyle 0\,\leq \,|z-z'|\,\leq \,\epsilon$. Backward error analysis involves the analysis of the approximation function $\scriptstyle z'\,=\,f'(a_{0},\,a_{1},\,\dots ,\,a_{n})$, to determine the bounds on the parameters $\scriptstyle a_{i}\,=\,{\bar {a_{i}}}\,\pm \,\epsilon _{i}$ such that the result $\scriptstyle z'\,=\,z$.

Applications

Global Positioning System

Main article: Global positioning system

The analysis of errors computed using the Global Positioning System is important for understanding how GPS works, and for knowing what magnitude errors should be expected. The Global Positioning System makes corrections for receiver clock errors and other effects but there are still residual errors which are not corrected. The Global Positioning System (GPS) was created by the United States Department of Defense (DOD) in the 1970s. It has come to be widely used for navigation both by the U.S. military and the general public.

Molecular dynamics simulation

In molecular dynamics (MD) simulations, there are errors due to inadequate sampling of the phase space or infrequently occurring events, these lead to the statistical error due to random fluctuation in the measurements.

For a series of M measurements of a fluctuating property A, the mean value is:

$\langle A\rangle ={\frac {1}{M}}\sum _{\mu =1}^{M}A_{\mu }.$

When these M measurements are independent, the variance of the mean <A> is:

$\sigma ^{2}(\langle A\rangle )={\frac {1}{M}}\sigma ^{2}(A),$

but in most MD simulations, there is correlation between quantity A at different time, so the variance of the mean <A> will be underestimated as the effective number of independent measurements is actually less than M. In such situations we rewrite the variance as :

$\sigma ^{2}(\langle A\rangle )={\frac {1}{M}}\sigma ^{2}A\left[1+2\sum _{\mu }\left(1-{\frac {\mu }{M}}\right)\phi _{\mu }\right],$

where $\phi _{\mu }$ is the autocorrelation function defined by

$\phi _{\mu }={\frac {\langle A_{\mu }A_{0}\rangle -\langle A\rangle ^{2}}{\langle A^{2}\rangle -\langle A\rangle ^{2}}}.$

We can then use the auto correlation function to estimate the error bar. Luckily, we have a much simpler method based on block averaging.

See also

• Error analysis (linguistics)
• Error bar
• Errors and residuals in statistics
• Propagation of uncertainty

External links

• [1] All about error analysis.