Optimal window when the data is quiet

Question

You forecast a drifting quantity with the average of the last $W$ observations. Observation noise is $\sigma^2 = 4$ and the drift is $\delta = 1$ per step. The forecast error is

$\text{MSE}(W) \approx \frac{\sigma^2}{W} + \left(\frac{\delta W}{2}\right)^2.$

What window length $W$ minimizes the forecast MSE?

Accepted Answer

The idea. With little observation noise, the sampling-error term is already small, so there is less reason to average many points. A shorter, more responsive window keeps the stale-data bias down. Minimize. Setting the derivative to zero gives the same cube-root rule as always: W^\star = \left(\frac{2\sigma^2}{\delta^2}ight)^{1/3}. Plug in \sigma^2 = 4, \delta = 1: W^\star = \left(\frac{2 	imes 4}{1}ight)^{1/3} = 8^{1/3} = 2. Just a 2-point average is best. Because the data is quiet, averaging more would mostly reach back to stale points and add bias without buying much noise reduction. …

Optimal window when the data is quiet

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