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

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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:

X_{N}=\frac{1}{N}\sum^{N}_{n=1} x_{n}

Now, as we observed in our simple coin-flipping experiment, since the x_{n} are random, so must be the value of the estimator X_{N}. For the estimator to be unbiased, the mean value of X_{N} must be true ensemble mean, X, i.e.

\lim_{N\rightarrow\infty} X_{N} = X

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

\left\langle X_{N} \right\rangle & = & \left\langle \frac{1}{N} \sum^{N}_{n=1} x_{n} \right\rangle \\
& = & \frac{1}{N} \sum^{N}_{n=1} \left\langle  x_{n} \right\rangle \\
& = & \frac{1}{N} NX = X \\

(Note that the expected value of each x_{n} is just X since the x_{n} are assumed identically distributed). Thus x_{N} 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 \epsilon^{2}_{X_{N}}, to be:

\epsilon^{2}_{X_{N}}\equiv \frac{var \left\{ X_{N} \right\} }{X^{2}} = \frac{\left\langle  \left( X_{N}- X \right)^{2} \right\rangle }{X^{2}}

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

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}\\

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

\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 \\
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