Here we will pick apart the following paper:

* Josef Ludescher, Avi Gozolchiani, Mikhail I. Bogachev, Armin Bunde, Shlomo Havlin, and Hans Joachim Schellnhuber, [Improved El Niño forecasting by cooperativity detection](, _Proceedings of the National Academy of Sciences_, 30 May 2013.

Note: this was originally in the Climate networks thread, but I'm moving it here into its own space.

For any $f(t)$, denote the moving average over the past year by:

$$\langle f(t) \rangle = \frac{1}{365} \sum_{d = 0}^{364} f(t - d) $$

Let $i$ be a node in the El Niño basin, and $j$ be a node outside of it.

Let $t$ range over every tenth day in the time span from 1950 to 2011.

Let $T_k(t)$ be the daily atmospheric temperature anomalies (actual temperature
value minus climatological average for each calendar day).

Define the time-delayed cross-covariance function by:

$$ C_{i,j}^{t}(-\tau) = \langle T_i(t) T_j(t - \tau) \rangle - \langle T_i(t) \rangle \langle T_j(t - \tau) \rangle $$

$$ C_{i,j}^{t}(\tau) = \langle T_i(t - \tau) T_j(t) \rangle - \langle T_i(t - \tau) \rangle \langle T_j(t) \rangle $$

They consider time lags $\tau$ between 0 and 200 d, where "a reliable estime of the backround noise level can be guaranteed."

Divide the cross-covariances by the standard deviations of $T_i$ and $T_j$ to obtain the cross-correlations.

Only temperature data from the past are considered when estimating the cross-correlation function at day $t$.