Moving window statistics (see this, for example) are sample statistics calculated over moving/rolling windows over a time-series.
For example, given the time-series $\{x_1,x_2,\dots\}$ one can construct moving windows of width $W$ and calculate statistics $S(\mathrm{window})$ like this:
$$ \begin{matrix} x_1 & x_2 & x_3 & x_4 & & & & \Rightarrow S(\cdot) & \Rightarrow S_4\\ & x_2 & x_3 & x_4 & x_5 & & & \Rightarrow S(\cdot) & \Rightarrow S_5\\ & & x_3 & x_4 & x_5 & x_6 & & \Rightarrow S(\cdot) & \Rightarrow S_6\\ & & & x_4 & x_5 & x_6 & x_7 & \Rightarrow S(\cdot) & \Rightarrow S_7 \end{matrix} $$
The $S_t$ could be the moving average $\mu_t$ or the moving variance $\sigma_t^2$:
$$ \begin{aligned} \mu_t &= \frac1W \sum_{i=1}^W x_{t-i+1} = \mathrm{avg}\left\{ x_t, x_{t-1}, \dots, x_{t-W+1} \right\}\\ \sigma_t^2 &= \frac1W \sum_{i=1}^W (x_{t-i+1} - \mu_t)^2= \mathrm{var}\left\{ x_t, x_{t-1}, \dots, x_{t-W+1} \right\} \end{aligned} $$
My question is what do these moving statistics actually estimate?
- Does the moving average estimate the expected value of the corresponding stochastic process, $\mu_t = \mathbb{E}X_t$? Note that moving averages typically vary over time, so it would seem that they're supposed to estimate $\mu_t$, which varies in time as well.
- Does the moving variance estimate $\sigma_t^2 = \mathbb{V}X_t$?
- And so on. Do these kind of statistics estimate anything? Or are they just filtering out "noise"?
- Given that $x_t$ are not independent, why does it make sense to compute sample statistics for them?
I know that for ergodic processes the expanding (!) average, $\frac1T \int_0^T X(t)\mathrm{d}t$, estimates $\mathbb{E}X_t = \mu_t = \mu$, which doesn't change in time. However, for moving averages/variances/etc. we're constantly discarding the "oldest" observation and adding the most recent one to the window, so the estimate changes in time.
Are there any theoretical treatments of this issue? Do moving statistics have anything to do with ergodicity, stationarity etc?
