# Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations

(Difference between revisions)
 Revision as of 06:00, 7 June 2006 (view source)Michail (Talk | contribs) (→Estimators for averaged quantities)← Older edit Revision as of 06:30, 7 June 2006 (view source)Michail (Talk | contribs) (→Bias and convergence of estimators)Newer edit → Line 10: Line 10: == Bias and convergence of estimators == == Bias and convergence of estimators == + + A procedure for answering these questions will be illustrated by considerind a simple '''estimator''' for the mean, the arithmetic mean considered above, [itex]X_{N}[/itex]. For [itex]N[/itex] independent realizations [itex]x_{n}, n=1,2,...,N[/itex] where [itex]N[/itex] is finite, [itex]X_{N}[/itex] is given by:

## Estimators for averaged quantities

Since there can never an infinite number of realizations from which ensemble averages (and probability densities) can be computed, it is essential to ask: How many realizations are enough? The answer to this question must be sought by looking at the statistical properties of estimators based on a finite number of realization. There are two questions which must be answered. The first one is:

• Is the expected value (or mean value) of the estimator equal to the true ensemble mean? Or in other words, is yje estimator unbiased?

The second question is

• Does the difference between the and that of the true mean decrease as the number of realizations increases? Or in other words, does the estimator converge in a statistical sense (or converge in probability). Figure 2.9 illustrates the problems which can arise.

## Bias and convergence of estimators

A procedure for answering these questions will be illustrated by considerind a simple estimator for the mean, the arithmetic mean considered above, $X_{N}$. For $N$ independent realizations $x_{n}, n=1,2,...,N$ where $N$ is finite, $X_{N}$ is given by: