QBO: Pattern recognition and signal processing

/ Paul Pukite (@whut) / 2 Comments

TANSTAAFL: there ain’t no such thing as a free lunch … but there’s always crumbs for the taking.

Machine learning won’t necessarily make a complete discovery by uncovering some ground-breaking pattern in isolation, but more likely a fragment or clue or signature that could lead somewhere. I always remind myself that there are infinitely many more non-linear formulations than linear ones potentially lurking in nature, yet humans are poorly-equipped to solve most non-linear relationships. ML has started to look at the tip of the non-linear iceberg and humans have to be alert when it uncovers a crumb. Recall that pattern recognition and signal processing are well-established disciplines in their own right, yet consider the situation of searching for patterns in signals hiding in the data but unknown in structure. That’s often all we are looking for — some foot-hold to start from.

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Teleconnection vs Common-Mode

/ Paul Pukite (@whut) / 3 Comments

A climate teleconnection is understood as one behavior impacting another — for example NINOx => AMO, meaning the Pacific ocean ENSO impacting the Atlantic ocean AMO via a remote (i.e. tele) connectiion. On the other hand, a common-mode behavior is a result of a shared underlying cause impacting a response in a uniquely parameterized fashion — for example NINOx = g(F(t), {n1, n2, n3, ...}) and AMO = g(F(t), {a1, a2, a3, ...}), where the n's are a set of constant parameters for NINOx and the a's are for AMO.

In this formulation F(t) is a forcing and g() is a transformation. Perhaps the best example of a common-mode response to a forcing is in the regional tidal response in local sea-level height (SLH). Obviously, the lunisolar forcing is a common mode in different regions and subtle variations in the parametric responses is required to model SLH uniquely. Once the parameters are known, one can make practical predictions (subject to recalibration as necessary).

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Topology shapes climate dynamics

/ Paul Pukite (@whut) / 3 Comments

A paper from last week with high press visibility that makes claims to climate1 applicability is titled: Topology shapes dynamics of higher-order networks

The higher-order Topological Kuramoto dynamics, defined in Eq. (1), entails one linear transformation of the signal induced by a boundary operator, a non-linear transformation due to the application of the sine function, concatenated by another linear transformation induced by another boundary operator. These dynamical transformations are also at the basis of simplicial neural architectures, especially when weighted boundary matrices are adopted.

\dot{\theta}_i = \omega_i + \sum_{j} K_{ij} \sin(\theta_j - \theta_i) + F(t)

This may be a significant unifying model as it could resolve the mystery of why neural nets can fit fluid dynamic behaviors effectively without deeper understanding. In concise terms, a weighted sine function acts as a nonlinear mixing term in a NN and serves as the non-linear transformation in the Kuramoto model2.

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Subjectivity and Perception

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What really is the color GREEN? What does it mean to different people? We actually have no idea. Now, consider The Dress on its 10 year anniversary. One can either argue with others whether it is BLUE and BLACK or WHITE and GOLD, or objectively look at it’s composition from a technical standpoint. Below is a cross-section of the dress image using the online image-picker application showing a palette from selected pixels in the image

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Difference Model Fitting

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By applying an annual impulse sample-and-hold on a common-mode basis set of tidal factors, a wide range of climate indices can be modeled and cross-validated. Whether it is a biennial impulse or annual impulse, the slowly modulating envelope is roughly the same, thus models of multidecadal indices such as AMO and PDO show similar skill — with cross validation results evaluated here for a biennial impulse. Now we will evaluate for annual impulse.

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Google Gemini Deep Research on ENSO

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First evaluation of Gemini Advanced 1.5 Pro with Deep Research. Logged in with trial subscription and gave this as an initial prompt. Saved the results to a Google Docs file, and then created the following PDF. Note the top-level focus on this blog and published citations in the scope of Chapter 12 of Mathematical Geoenergy, even though the chapter was not directly cited — only via an embedded cite in a submitted ESD Ideas article, reference 1. Interesting that the Lin & Qian paper not cited.

Prompt: Explain tidal forcing behind ENSO using derivations based on reduced effective gravity on equatorial thermocline.

Tidal Forcing and ENSO Dynamics_Download

Tidal Gauge Differential

/ Paul Pukite (@whut) / 2 Comments

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

An ENSO Predictor Based on a Tide Gauge Data Model

Check the recent blog post Lunar Torque Controls All for context of how it fits in to the unified model.

https://geoenergymath.com/wp-content/uploads/2024/11/image-30.png

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:

¡AI Caramba!

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.

Bluesky

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In a #Geophysics feed on Bsky.app, the latest “post” after 2 days is still this:

Instead, this is what they should be discussing [1,2]:

Possible breakthroughs in understanding the inverse energy cascade behavior of fluid dynamics.bpb-us-e1.wpmucdn.com/wp.nyu.edu/d…arxiv.org/pdf/2406.00264www.realclimate.org/index.php/ar…

— Puͣkiͧte̍ (@pukite.com) 2024-12-20T00:39:51.019Z

Also discussed at https://www.realclimate.org/index.php/archives/2024/11/twenty-years-of-blogging-in-hindsight/

yet responses are not on point

Typical response totally ignores the climate science aspect.Always gotchas .. so reciprocated.

— Puͣkiͧte̍ (@pukite.com) 2024-12-24T04:12:35.056Z

CITES

  1. Shavit, Michal, Oliver Bühler, and Jalal Shatah. “Sign-indefinite invariants shape turbulent cascades.” Physical Review Letters 133.1 (2024): 014001. https://bpb-us-e1.wpmucdn.com/wp.nyu.edu/dist/0/18842/files/2024/09/PhysRevLett.133.014001.pdf
  2. Vivanco, Isis, et al. “A synchrotron-like pumped ring resonator for water waves.” arXiv preprint arXiv:2406.00264 (2024). https://arxiv.org/pdf/2406.00264

QBO Metrics

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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

Low Dimensions

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“A key enabling assumption, sometimes called the manifold hypothesis [14], is that the data lie on or near a low-dimensional manifold; for physical systems with dissipation, such manifolds can often be rigorously shown to exist [15–18]. These manifolds enable a low-dimensional latent state representation, and hence, low-dimensional dynamical models. Linear manifold learning techniques, such as principal component analysis, cannot learn the nonlinear manifolds that represent most systems in nature. To do so, we require nonlinear methods, some of which are developed in [19–25] and reviewed in [26].“

Floryan, Daniel, and Michael D. Graham. “Data-driven discovery of intrinsic dynamics.” Nature Machine Intelligence 4.12 (2022): 1113-1120.

From <https://www.nature.com/articles/s42256-022-00575-4>

GC22A-04 Can ML beats chaos?

Abstract

“Chaos is typically blamed for the lack of predictability beyond a forecasting time window, which is on the order of 10 days for weather forecasting. However, on the one hand, most turbulent and chaotic systems exhibit strong coherence in the flow, such as synoptic events in weather or coherent structures in turbulence. On the other hand, most physical model might have additional structural errors that limit their capacity to correctly forecast beyond a certain time horizon, independent of chaos.

We will show in this presentation that most chaotic and turbulent flows can be predicted on relatively long range, at least longer than expected with both physical models and standard deep learning, using a combination of a reduced order model (that captures the low-dimensional coherent structures in the flow) and generative AI to obtain the crisp results and details of the flow. We will conclude stating that current AI-based weather models might not have achieved a plateau in performance yet, especially at longer time scales, and that physics-based weather model still have room for improvements. Reduced order models might not be able to beat chaos but can lead to much longer-range prediction than currently expected.“

https://agu.confex.com/agu/agu24/meetingapp.cgi/Paper/1522150