Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations
From CFD-Wiki
(Difference between revisions)
(→Estimators for averaged quantities) |
(→Estimators for averaged quantities) |
||
| 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 == | ||
Revision as of 06:00, 7 June 2006
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.
