John wrote:

>Whoops, you’re right - I made a typo.

Yep, but I was assuming that you meant:

$$ \langle (A - \langle A \rangle ) \; (B - \langle B \rangle) \rangle $$

since you where talking about covariance. And so - I still think that it seems that for this case here:

$$ \langle (A - \langle A \rangle ) \; (B - \langle B \rangle) \rangle \neq \langle A B \rangle - \langle A \rangle \langle B \rangle $$

>using the principle that $\langle \langle X \rangle \rangle = \langle X \rangle$ - you can pull a number out of a mean.

As I wrote above, exactly this principle doesn't hold if you follow the definitions in the article and thats why I strongly suspect that the two terms are not equal for this case here or that the definition is different. (?)

David wrote:

>The result could be fluke resulting from an arbitrary conjecture whose parameters were optimized using too small a dataset. The training and predictions periods are each about thirty years long. I saw it stated that during the training period there were ten El Niño events. Can statistics be applied here, to help address this concern? (Though we’d have to view statistical reassurances also with a grain of salt.)

I agree the dataset seem too small for statistics. But it might be in principle be sufficient for determining a threshold, like as in the case that if you notice that if your car starts to sideslip in a certain curve at a certain velocity than you may not want to wait for a sample rate of 1000. In particular I have the slight suspicion - but this is still rather speculative - that the El Nino may be at least partially due to some kind of regular biannual temperature change, which is due to celestial mechanics. (Where I have to say that I have no idea where the biannual cycle should come from (something moon orbit and sun orbit ?) and that this suspicion is sofar quite on shaky grounds, in particular I looked only at a rather short temperature sample.) So it would be some kind of stochastic resonance/more-or-less-regular-reoccurring thing.

>Whoops, you’re right - I made a typo.

Yep, but I was assuming that you meant:

$$ \langle (A - \langle A \rangle ) \; (B - \langle B \rangle) \rangle $$

since you where talking about covariance. And so - I still think that it seems that for this case here:

$$ \langle (A - \langle A \rangle ) \; (B - \langle B \rangle) \rangle \neq \langle A B \rangle - \langle A \rangle \langle B \rangle $$

>using the principle that $\langle \langle X \rangle \rangle = \langle X \rangle$ - you can pull a number out of a mean.

As I wrote above, exactly this principle doesn't hold if you follow the definitions in the article and thats why I strongly suspect that the two terms are not equal for this case here or that the definition is different. (?)

David wrote:

>The result could be fluke resulting from an arbitrary conjecture whose parameters were optimized using too small a dataset. The training and predictions periods are each about thirty years long. I saw it stated that during the training period there were ten El Niño events. Can statistics be applied here, to help address this concern? (Though we’d have to view statistical reassurances also with a grain of salt.)

I agree the dataset seem too small for statistics. But it might be in principle be sufficient for determining a threshold, like as in the case that if you notice that if your car starts to sideslip in a certain curve at a certain velocity than you may not want to wait for a sample rate of 1000. In particular I have the slight suspicion - but this is still rather speculative - that the El Nino may be at least partially due to some kind of regular biannual temperature change, which is due to celestial mechanics. (Where I have to say that I have no idea where the biannual cycle should come from (something moon orbit and sun orbit ?) and that this suspicion is sofar quite on shaky grounds, in particular I looked only at a rather short temperature sample.) So it would be some kind of stochastic resonance/more-or-less-regular-reoccurring thing.