Error analysis

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.

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Error analysis in second language acquisition

Main article: Error analysis (linguistics)

In second language acquisition, error analysis studies the types and causes of language errors. Errors are classified according to:

• modality (i.e., level of proficiency in speaking, writing, reading, listening)
• linguistic levels (i.e., pronunciation, grammar, vocabulary, style)
• form (e.g., omission, insertion, substitution)
• type (systematic errors/errors in competence vs. occasional errors/errors in performance)
• cause (e.g., interference, interlanguage)
• norm vs. system

Error analysis in SLA was established in the 1960s by Stephen Pit Corder and colleagues. Error analysis was an alternative to contrastive analysis, an approach influenced by behaviorism through which applied linguists sought to use the formal distinctions between the learners' first and second languages to predict errors. Error analysis showed that contrastive analysis was unable to predict a great majority of errors, although its more valuable aspects have been incorporated into the study of language transfer. A key finding of error analysis has been that many learner errors are produced by learners making faulty inferences about the rules of the new language.

Error analysis in 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:

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

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 autocorrelation function to estimate the error bar. Luckily, we have a much simpler method based on block averaging.

See also

• Error bar
• Errors and residuals in statistics
• For error analysis in applied linguistics, see contrastive analysis
• Propagation of uncertainty

External links

• [1] – Definitions and graphical explanation.
• [2] All about it.