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

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(Bias and convergence of estimators)
(Bias and convergence of estimators)
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:<math>     
:<math>     
\lim_{N\rightarrow\infty} X_{N} = X
\lim_{N\rightarrow\infty} X_{N} = X
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</math> 
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</td><td width="5%">(2)</td></tr></table>
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It is easy to see that since the operations of averaging adding commute,
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:<math>   
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\begin{matrix}
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\left\langle X_{N} \right\rangle & = & \left\langle \frac{1}{N} \sum^{N}_{n=1} x_{n} \right\rangle \\
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& = & \frac{1}{N} \sum^{N}_{n=1} \left\langle  x_{n} \right\rangle \\
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& = & \frac{1}{N} NX = X \\
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\end{matrix}
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</math>   
</td><td width="5%">(2)</td></tr></table>
</td><td width="5%">(2)</td></tr></table>

Revision as of 10:18, 8 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, 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}
(2)

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
(2)

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

    
\begin{matrix}
\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 \\
\end{matrix}
(2)
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