Calculating the coherence (sometimes called magnitude squared coherence) between two signals indicates the presence or lack thereof of a linear transformation between the two signals. Is there an analogue for signals which are related by a non-linear transformation? Let's assume that the expected non-linear transformation is known. It would be even better if the calculation could identify the non-linear transformation within some bounds, but that is probably just wishful thinking.
An example of coherence between ocean water level (due to tides) and ground water level measured in a well, taken from the Wikipedia article linked above. This is an example of the coherence between two signals related by a linear transformation.