In addition to the standard correlation coefficient (CC) and RMS error, non-standard metrics that have beneficial cross-validation properties include dynamic time warp (DTW), complexity invariant-distance (CID) see [2], and a CID-modified DTW. The link above describes my implementation of the DTW metric but I have yet to describe the CID metric. It’s essentially the CC multiplied by a factor that empirically adjusts the embedded summed distance between data points (i.e. the stretched length) of the time-series so that the signature or look of two time-series visually match in complexity.

   CID = CC * min(Length(Model, Data))/ max(Length(Model, Data))

The authors of the CID suggest that it’s a metric based on “an invariance that the community seems to have missed”.

And a CID-modified DTW is thus:

CID = DTW * min(Length(Model, Data))/ max(Length(Model, Data))

I have tried this on the QBO model with good cross-validation results featuring up to-data data from https://www.atmohub.kit.edu/data/qbo.dat

These have similar tidal factor compositions and differ mainly in the LTE modulation and phase delay. As discussed earlier, any anomalies in the QBO behavior are likely the outcome of an erratic periodicity caused by incommensurate annual and draconic cycles and exaggerated by LTE.

from https://gist.github.com/pukpr/e562138af3a9da937a3fb6955685c98f

REFERENCES

[1] Batista, Gustavo EAPA, et al. “CID: an efficient complexity-invariant distance for time series.” Data Mining and Knowledge Discovery 28 (2014): 634-669.R
https://link.springer.com/article/10.1007/s10618-013-0312-3