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Sample mean

Now let xt be a time series made up of identically distributed random numbers: mx and $\sigma_x$ do not depend on time. Let us also suppose that they are independently chosen; this means in particular that for any different times t and s ($t\neq s$):
\begin{displaymath}
\E(x_t x_s) \eq \E(x_t) \E(x_s)\end{displaymath} (19)
Suppose we have a sample of n points of xt and are trying to determine the value of mx. We could make an estimate $\hat m_x$ of the mean mx with the formula  
 \begin{displaymath}
\hat m_x \eq {1 \over n} \sum^n_{t = 1} x_t\end{displaymath} (20)

A somewhat more elaborate method of estimating the mean would be to take a weighted average. Let wt define a set of weights normalized so that  
 \begin{displaymath}
\sum w_t \eq 1\end{displaymath} (21)
With these weights, the more elaborate estimate $\hat m$ of the mean is  
 \begin{displaymath}
\hat m_x \eq \sum w_t \, x_t\end{displaymath} (22)
Actually (20) is just a special case of (22); in (20) the weights are wt = 1/n.

Further, the weights could be convolved on the random time series, to compute local averages of this time series, thus smoothing it. The weights are simply a filter response where the filter coefficients happen to be positive and cluster together. Figure 6 shows an example: a random walk function with itself smoothed locally.

 
walk
walk
Figure 6
Random walk and itself smoothed (and shifted downward).


view


next up previous print clean
Next: Variance of the sample Up: TIME-STATISTICAL RESOLUTION Previous: Probability and independence
Stanford Exploration Project
10/21/1998