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

(Difference between revisions)
Jump to: navigation, search
 Revision as of 17:21, 6 June 2006 (view source)Michail (Talk | contribs) (→Estimators for averaged quantities)← Older edit Revision as of 06:00, 7 June 2006 (view source)Michail (Talk | contribs) (→Estimators for averaged quantities)Newer edit → Line 1: Line 1: == Estimators for averaged quantities == == Estimators for averaged quantities == - Since there can never an infinite number of realizations + 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 == == Bias and convergence of estimators ==

## 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.