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

### From CFD-Wiki

(→Bias and convergence of estimators) |
(→Bias and convergence of estimators) |
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var \left\{ X_{N} \right\} & = & \left\langle X_{N} - X^{2} \right\rangle \\ | var \left\{ X_{N} \right\} & = & \left\langle X_{N} - X^{2} \right\rangle \\ | ||

& = & \left\langle \left[ \lim_{N\rightarrow\infty} \frac{1}{N} \sum^{N}_{n=1} \left( x_{n} - X \right) \right]^{2} \right\rangle - X^{2}\\ | & = & \left\langle \left[ \lim_{N\rightarrow\infty} \frac{1}{N} \sum^{N}_{n=1} \left( x_{n} - X \right) \right]^{2} \right\rangle - X^{2}\\ | ||

+ | \end{matrix} | ||

+ | </math> | ||

+ | </td><td width="5%">(2)</td></tr></table> | ||

+ | |||

+ | since <math>\left\langle X_{N} \right\rangle = X</math> from equation 2.46. Using the fact that operations of averaging and summation commute, the squared summation can be expanded as follows: | ||

+ | |||

+ | <table width="100%"><tr><td> | ||

+ | :<math> | ||

+ | \begin{matrix} | ||

+ | \left\langle \left[ \lim_{N\rightarrow\infty} \sum^{N}_{n=1} \left( x_{n} - X \right) \right]^{2} \right\rangle & = & \lim_{N\rightarrow\infty}\frac{1}{N^{2}} \sum^{N}_{n=1} \sum^{N}_{m=1} \left\langle \left( x_{n} - X \right) \left( x_{m} - X \right) \right\rangle \\ | ||

+ | & = & dsdsaf \\ | ||

+ | & = & dsadsf \\ | ||

\end{matrix} | \end{matrix} | ||

</math> | </math> | ||

</td><td width="5%">(2)</td></tr></table> | </td><td width="5%">(2)</td></tr></table> |

## Revision as of 07:19, 9 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.

## 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, . For independent realizations where is finite, is given by:

| (2) |

Now, as we observed in our simple coin-flipping experiment, since the are random, so must be the value of the estimator . For the estimator to be *unbiased*, the mean value of must be true ensemble mean, , i.e.

| (2) |

It is easy to see that since the operations of averaging adding commute,

| (2) |

(Note that the expected value of each is just since the are assumed identically distributed). Thus is, in fact, an *unbiased estimator for the mean*.

The question of *convergence* of the estimator can be addressed by defining the square of **variability of the estimator**, say , to be:

| (2) |

Now we want to examine what happens to as the number of realizations increases. For the estimator to converge it is clear that should decrease as the number of sample increases. Obviously, we need to examine the variance of first. It is given by:

| (2) |

since from equation 2.46. Using the fact that operations of averaging and summation commute, the squared summation can be expanded as follows:

| (2) |