The last set of cross-validation results are based on training of held-out data for intervals outside of 0.6-0.8 (i.e. training on t<0.6 and t>0.8 of the data, which extends from t=0.0 to t=1.0 normalized). This post considers training on intervals outside of 0.3-0.6 — a narrower training interval and correspondingly wider test interval.
Each fitted model result shows the cross-validation results based on training of held-out data — i.e. training on only the intervals outside of 0.6-0.8 (i.e. training on t<0.6 and t>0.8 of the data, which extends from t=0.0 to t=1.0 normalized). The best results are for time-series that have 100 years or more worth of monthly data, so the held-out data is typically 20 years. There is no selection bias trickery here, as this is a collection of independent sites and nothing in the MLR fitting process is specific to an individual time-series. In the following, the collection of results starts with the Stockholm site in Sweden, keeping in mind that the dashed line in the charts indicates the test or validation interval.
This article in MIT News:
https://news.mit.edu/2025/simpler-models-can-outperform-deep-learning-climate-prediction-0826
“New research shows the natural variability in climate data can cause AI models to struggle at predicting local temperature and rainfall.” … “While deep learning has become increasingly popular for emulation, few studies have explored whether these models perform better than tried-and-true approaches. The MIT researchers performed such a study. They compared a traditional technique called linear pattern scaling (LPS) with a deep-learning model using a common benchmark dataset for evaluating climate emulators. Their results showed that LPS outperformed deep-learning models on predicting nearly all parameters they tested, including temperature and precipitation.“
Machine learning and other AI approaches such as symbolic regression will figure out that natural climate variability can be done using multiple linear regression (MLR) with cross-validation (CV), which is an outgrowth or extension of linear pattern scaling (LPS).
https://pukpr.github.io/results/image_results.html
“When this was initially created on 9/1/2025, there were 3000 CV results on time-series
that averaged around 100 years (~1200 monthly readings/set) so over 3 million data points“
I developed this github model repository to make it easy to compare many different data sets, much better than using an image repository such as ImageShack.
There are about 130 sea-level height monitoring stations in the sites, which is relevant considering how much natural climate variation a la ENSO has an impact on monthly mean SLH measurements. See this paper Observing ENSO-modulated tides from space
“In this paper, we successfully quantify the influences of ENSO on tides from multi-satellite altimeters through a revised harmonic analysis (RHA) model which directly builds ENSO forcing into the basic functions of CHA. To eliminate mathematical artifacts caused by over-fitting, Lasso regularization is applied in the RHA model to replace widely-used ordinary least squares. “
On RealClimate.org
Paul Pukite (@whut) says
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“If so, do you have an explanation why the diurnal tides do not move the thermocline, whereas tides with longer periods do?”
The character of ENSO is that it shifts by varying amounts on an annual basis. Like any thermocline interface, it reaches the greatest metastability at a specific time of the year. I’m not making anything up here — the frequency spectrum of ENSO (pick any index NINO4, NINO34, NINO3) shows a well-defined mirror symmetry about the value 0.5/yr. Given that Incontrovertible observation, something is mixing with the annual impulse — and the only plausible candidate is a tidal force.
So the average force of the tides at this point is the important factor to consider. Given a very sharp annual impulse, the near daily tides alias against the monthly tides — that’s all part of mathematics of orbital cycles. So just pick the monthly tides as it’s convenient to deal with and is a more plausible match to a longer inertial push.
Sunspots are not a candidate here.
Some say wind is a candidate. Can’t be because wind actually lags the thermocline motion.
So the deal is, I can input the above as a prompt to ChatGPT and see what it responds with
https://chatgpt.com/share/68649088-5c48-8010-a767-4fe75ddfeffc
Chat GPT also produces a short Python script which generates the periodogram of expected spectral peaks.
I placed the results into a GitHub Gist here, with charts:
https://gist.github.com/pukpr/498dba4e518b35d78a8553e5f6ef8114
I made one change to the script (multiplying each tidal factor by its frequency to indicate its inertial potential, see the ## comment)
At the end of the Gist, I placed a representative power spectrum for the actual NINO4 and NINO34 data sets showing where the spectral peaks match. They all match. More positions match if you consider a biennial modulation as well.
Now, you might be saying — yes, but this all ChatGPT and I am likely coercing the output. Nothing of the sort. Like I said, I did the original work years ago and it was formally published as Mathematical Geoenergy (Wiley, 2018). This was long before LLMs such as ChatGPT came into prominence. ChatGPT is simply recreating the logical explanation that I had previously published. It is simply applying known signal processing techniques that are generic across all scientific and engineering domains and presenting what one would expect to observe.
In this case, it carries none of the baggage of climate science in terms of “you can’t do that, because that’s not the way things are done here”. ChatGPT doesn’t care about that prior baggage — it does the analysis the way that the research literature is pointing and how the calculation is statistically done across domains when confronted with the premise of an annual impulse combined with a tidal modulation. And it nailed it in 2025, just as I nailed it in 2018.
A climate science breakthrough likely won’t be on some massive computation but on a novel formulation that exposes some fundamental pattern (perhaps discovered by deep mining during a machine learning exercise). Over 10 years ago, I wrote on a blog post on how one can extract the ENSO signal by doing simple signal processing on a sea-level height (SLH) tidal time-series — in this case, at Fort Denison located in Sydney harbor.
The formulation/trick is to take the difference between the SLH reading and that from 2 years (24 months) prior, described here
Check the recent blog post Lunar Torque Controls All for context of how it fits in to the unified model.
The rationale for this 24 month difference is likely related to the sloshing of the ocean triggered on an annual basis. I think this is a pattern that any ML exercise would find with very little effort. After all, it didn’t take me that long to find it. But the point is that the ML configuration has to be open and flexible enough to be able to search, generate, and test for the same formulation. IOW, it may not find it if the configuration, perhaps focused on computationally massive PDEs, is too narrow. That was my comment to a RC post on applying machine learning to climate science, see the following link and subsequent quote:
Nick McGreivy commented on:
“ML-based parameterizations have to work well for thousands of years of simulations, and thus need to be very stable (no random glitches or periodic blow-ups) (harder than you might think). Bias corrections based on historical observations might not generalize correctly in the future.”
This same issue arises when using ML to simulate PDEs. The solution is to analytically calculate what the stability condition(s) is (are), then at each timestep to add some numerical diffusion that nudges the solution towards satisfying the stability condition(s). I imagine this same technique could be used for ML-based parametrizations.
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
Reminded by a 20-year anniversary post at RealClimate.org, that I’ve been blogging for 20 years + 6 months on topics of fossil fuel depletion + climate change. The starting point was at a BlogSpot blog I created in May 2004, where the first post set the stage:
Click on the above to go to the complete archives (almost daily posts) until I transitioned to WordPress and what became the present blog. After 2011, my blogging pace slowed down considerably as I started to write in more in more technical terms. Eventually the most interesting and novel posts were filtered down to a set that would eventually become the contents of Mathematical Geoenergy : Discovery, Depletion, and Renewal, published in late 2018/early 2019 by Wiley with an AGU imprint.
The arc that my BlogSpot/WordPress blogging activity followed occupies somewhat of a mirror universe to that of RealClimate. I initially started out with an oil depletion focus and by incrementally understanding the massive inertia that our FF-dependent society had developed, it placed the climate science aspect into a different perspective and context. After realizing that CO2 did not like to sequester, it became obvious that not much could be done to mitigate the impact of gradually increasing GHG levels, and that it would evolve into a slow-moving train wreck. That’s part of the reason why I focused more on research into natural climate variability. In contrast, RealClimate (and all the other climate blogs) continued to concentrate on man-made climate change. At this point, my climate fluid dynamics understanding is at some alternate reality level, see the last post, still very interesting but lacking any critical acceptance (no debunking either, which keeps it alive and potentially valid).
The oil depletion aspect more-or-less spun off into the PeakOilBarrel.com blog [*] maintained by my co-author Dennis Coyne. That’s like watching a slow-moving train wreck as well, but Dennis does an excellent job of keeping the suspense up with all the details in the technical modeling. Most of the predictions regarding peak oil that we published in 2018 are panning out.
As a parting thought, the RealClimate hindsight post touched on how AI will impact information flow going forward. Having worked on AI knowledgebases for environmental modeling during the LLM-precursor stage circa 2010-2013, I can attest that it will only get better. At the time, we were under the impression that knowledge used for modeling should be semantically correct and unambiguous (with potentially a formal representation and organization, see figure below), and so developed approaches for that here and here (long report form).
As it turned out, lack of correctness is just a stage, and AI users/customers are satisfied to get close-enough for many tasks. Eventually, the LLM robots will gradually clean up the sources of knowledge and converge more to semantic correctness. Same will happen with climate models as machine learning by the big guns at Google, NVIDIA, and Huawei will eventually discover what we have found in this blog over the course of 20+ years.
Note:
[*] In some ways the PeakOilBarrel.com blog is a continuation of the shuttered TheOilDrum.com blog, which closed shop in 2013 for mysterious reasons.
I’m looking at side-band variants of the lunisolar orbital forcing because that’s where the data is empirically taking us. I had originally proposed solving Laplace’s Tidal Equations (LTE) using a novel analytical derivation published several years ago (see Mathematical Geoenergy, Wiley/AG, 2019). The takeaway from the math results — given that LTEs form the primitive basis of the GCM-specific shallow-water approximation to oceanic fluid dynamics — was that my solution involved a specific type of non-linear modulation or amplification of the input tidal. However, this isn’t the typical diurnal/semi-diurnal tidal forcing, but because of the slower inertial response of the ocean volume, the targeted tidal cycles are the longer period monthly and annual. Moreover, as very few climate scientists are proficient at signal processing and all the details of aliasing and side-bands, this is an aspect that has remained hidden (again thank Richard Lindzen for opening the book on tidal influences and then slamming it shut for decades).
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With the recent total solar eclipse, it revived lots of thought of Earth’s ecliptic plane. In terms of forcing, having the Moon temporarily in the ecliptic plane and also blocking the sun is not only a rare and (to some people) an exciting event, it’s also an extreme regime wrt to the Earth as the combined reinforcement is maximized.
In fact this is not just any tidal forcing — rather it’s in the class of tidal forcing that has been overlooked over time in preference to the conventional diurnal tides. As many of those that tracked the eclipse as it traced a path from Texas to Nova Scotia, they may have noted that the moon covers lots of ground in a day. But that’s mainly because of the earth’s rotation. To remove that rotation and isolate the mean orbital path is tricky. And that’s the time-span duration where long-period tidal effects and inertial motion can build up and show extremes in sea-level change. Consider the 4.53 year extreme tidal cycle observed at the Bay of Fundy (see Desplanque et al) located in Nova Scotia. This is predicted if the long-period lunar perigee anomaly (27.554 days and the 8.85 absidal precessional return cycle) amplifies the long period lunar ecliptic nodal cycle, as every 9.3 years the lunar path intersects the ecliptic plane, one ascending and the other descending as the moon’s gravitational pull directly aligns with the sun’s. The predicted frequencies are 1/8.85 ± 2/18.6 = 1/4.53 & 1/182, the latter identified by Keeling in 2000. The other oft-mentioned tidal extreme is at 18.6 years, which is identified as the other long period extreme at the Bay of Fundy by Desplanque, and that was also identified by NASA as an extreme nuisance tide via a press release and a spate of “Moon wobble” news articles 3 years ago.
What I find troubling is that I can’t find a scholarly citation where the 4.53 year extreme tidal cycle is explained in this way. It’s only reported as an empirical observation by Desplanque in several articles studying the Bay of Fundy tides.
Continue readingBased on the previous post on applying Dynamic Time Warping as a metric for LTE modeling of oceanic indices, it makes sense to apply the metric to the QBO model of atmospheric winds. A characteristic of QBO data is the sharp transitions of wind reversals. As described previously, DTW allows a fit to adjust the alignment between model and data without incurring a potential over-fitting penalty that a conventional correlation coefficient will often lead to.
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