I added the following to the wiki. I am trying to understand what the link strengths used by Ludescher actaully measure. I think its pretty weird!

#### Understanding the link strength

The link strength defined above by Yamasaki *et al* has some surprising behaviour. The following graphs are all based on simulated data. There are two time series of length 565, called "signal 1" and "signal 2" in the graphs, which consist of quadratics $q_1$ and $q_2$ plus independent gaussian noise. The noise has the same amplitude (standard deviation) in all cases, but $q_1$ and $q_2$ are multiplied by 1000 (leftmost column), 9 (second column), 3 (third column) and 1 (fourth column).

Examples of the signals themselves are shown in the top two rows, the value of $ c^{(t)}_{i,j}(\tau) $ is in the third row, and the fourth row shows an estimated density of the link strength derived from 100 replicates (different samplings of noise).

In the first column, the $q_1$ and $q_2$ overwhelm the guassian noise, so you can see their shapes. In particular, note that have positive correlation for all delays: it varies between about 0.87 and 0.97. The other three columns are intended to be more realistic signals which roughly resemble climate data. One would expect that as the multiplier for $q_1$ and $q_2$ decreases, the link strength would also decrease, but the opposite is the case.

#### Understanding the link strength

The link strength defined above by Yamasaki *et al* has some surprising behaviour. The following graphs are all based on simulated data. There are two time series of length 565, called "signal 1" and "signal 2" in the graphs, which consist of quadratics $q_1$ and $q_2$ plus independent gaussian noise. The noise has the same amplitude (standard deviation) in all cases, but $q_1$ and $q_2$ are multiplied by 1000 (leftmost column), 9 (second column), 3 (third column) and 1 (fourth column).

Examples of the signals themselves are shown in the top two rows, the value of $ c^{(t)}_{i,j}(\tau) $ is in the third row, and the fourth row shows an estimated density of the link strength derived from 100 replicates (different samplings of noise).

In the first column, the $q_1$ and $q_2$ overwhelm the guassian noise, so you can see their shapes. In particular, note that have positive correlation for all delays: it varies between about 0.87 and 0.97. The other three columns are intended to be more realistic signals which roughly resemble climate data. One would expect that as the multiplier for $q_1$ and $q_2$ decreases, the link strength would also decrease, but the opposite is the case.