Table 3 Formulas for moving window statistics
From: Forecasting infectious disease emergence subject to seasonal forcing
| Statistic | Formula |
|---|---|
| Mean t | \(\sum _{t'=1}^{T} w_{t,t'} X_{t'}\) |
| Variance t | \(\sum _{t'=1}^{T} w_{t,t'} (X_{t'}- \text {mean}_{t'})^{2}\) |
| (Variance convexity) t | variance t −variance t−1 |
| Autocovariance t | \(\sum _{t'=(\text {lag} + 1)}^{T} w_{t,t'} (X_{t'} - \text {mean}_{t'}) (X_{t' - \text {lag}} - \text {mean}_{t' -\text {lag}})\) |
| Autocorrelation t | autocovariance t /(variance t ×variance t−lag)0.5 |
| (Decay time) t | −lag/(log min(max(autocorrelation t ,0),1)) |
| (Index of dispersion) t | variance t /mean t |
| (Coefficient of variation) t | (variance t )0.5/mean t |
| Skewness t | \(\sum _{t'=1}^{T} w_{t,t'} (X_{t'} - \text {mean}_{t'})^{3} / (\text {variance}_{t})^{1.5}\) |
| Kurtosis t | \(\sum _{t'=1}^{T} w_{t,t'} (X_{t'} - \text {mean}_{t'})^{4} / (\text {variance}_{t})^{2}\) |