Optimal window when the data is noisy

Question

You forecast a drifting quantity with the average of the last $W$ observations. Observation noise is $\sigma^2 = 108$ 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. When each observation is very noisy, the sampling-error term dominates, and averaging many points to beat it down is worth the extra staleness bias. So the optimal window grows. Minimize. The cube-root rule again: W^\star = \left(\frac{2\sigma^2}{\delta^2}ight)^{1/3}. Plug in \sigma^2 = 108, \delta = 1: W^\star = \left(\frac{2 	imes 108}{1}ight)^{1/3} = 216^{1/3} = 6. A 6-point average is best here, three times longer than the quiet-data case, purely because the noise is larger. The window widens to smooth out the heavier noise, accepting a bit more lag in exchange. Takeaway. N…

Optimal window when the data is noisy

Your answer

Discussion

Learn the concepts

Related questions