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A book claims this as a motivation for introducing exponential averaging:

A disadvantage of ensemble averaging is that the resulting estimate cannot track dynamic changes occurring in the observed signal.

-- L. Sörnmo and P. Laguna, Bioelectrical signal processing in cardiac and neurological applications

The authors seem to use the term ensemble averaging to refer to averaging all observations of a stochastic process. Modifying this method to keep a moving average, i.e. averaging only the past $n$ observations instead of all of them, seems very close at hand.

Such a moving-average smoother would be able to track changes in the stochastic process being observed, just like the exponential-average smoother. Is this the difference in dynamic capabilities that the authors are referring to, or have I missed a deeper difference?

$$ \begin{align} Y_{\mathrm{ensemble,}N}(z) &= \frac{1}{N}\left(z^{0}+z^{-1}+z^{-2}+\cdots+z^{-(N-1)}\right)X(z) \\ Y_{\mathrm{exponential},\alpha}(z) &= \frac{\alpha}{1-(1-\alpha) z^{-1}}X(z) \end{align}$$

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  • $\begingroup$ It's difficult to tell the intent of the quote without some more context. Unfortunately I don't have the text available. $\endgroup$ – Jason R Dec 19 '11 at 12:54
  • $\begingroup$ In order to have a meaningful there is a need to specify a model for $X(z)$, e.g. random walk $\endgroup$ – rwong Dec 20 '11 at 5:31
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You're misinterpreting ensemble averages.... this is really in "batch" like after recording a few runs. You average functions on a time interval, e.g. that's where those doctor's charts for kids' height/weight mean & spread over time arise. Time-averaging is totally different...

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