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I have two time series and I am studying the mutual information between them in different parts of the year (I have calculated the mutual information in sliding windows). In january, the mutual information drops all the years. In my opinion, this is weird because if you plot the time series together, you can see that they are very correlated in january. That month, the time series are practically constant and they have low values compared to the rest of the year. However, some days there are peaks in both time series simultaneously. I though that the mutual information was dropping because the entropy was dropping, so I divided the mutual information by the entropy to remove that effect. However, I still get the same result.

Do you have any ideas about what can be happening?

I can provide more details, figures... if you want; just, tell me.

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In my opinion, this is weird because if you plot the time series together, you can see that they are very correlated in january.

This is not weird. The mutual information in not mutual correlation, it is the mutual entropy. So for 100% correlated data with zero entropy the mutual information will also be zero. See my detailed answer with examples.

I though that the mutual information was dropping because the entropy was dropping, so I divided the mutual information by the entropy to remove that effect. However, I still get the same result.

This is indeed strange, but difficult to explain without seeing an example with your actual data.

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