Variability based indicators

Spatial variance

Critical slowing down can lead to stronger fluctuations around the equilibrium state of the system. This causes spatial variance of the system to increase prior to a transition. Spatial variance is formally defined as the second moment around the spatial mean of the state variable.

(1)   \begin{equation*} \sigma^2 = \frac{1}{MN}\sum_{n=1}^N\sum_{m=1}^M (z[m,n]-\bar{z})^2 \end{equation*}

Spatial skewness

Fluctuations around the mean can become increasingly asymmetric as the system approaches a transition. This is because the fluctuations in the direction of the alternative stable state take longer to return back to the equilibrium than those in the opposite direction; this asymmetry can also arise due to local flickering events (i.e. occasional jumps of local units between their current and alternative state). The spatial asymmetry can be measured by spatial skewness, which is the third central moment scaled by the standard deviation.

(2)   \begin{equation*} \gamma = \frac{1}{MN}\sum_{n=1}^N\sum_{m=1}^M \frac{(z[m,n]-\bar{z})^3}{\sigma^3} \end{equation*}