Data-driven, non-Markovian modelling of weather in the presence of non-stationary, non-Gaussian, and heteroskedastic climate dynamics

This paper introduces a data-driven protocol that segments non-stationary, non-Gaussian, and heteroskedastic weather data into local pseudo-equilibrium seasons to construct accurate low-dimensional models using state-based generalized master equations, thereby overcoming the limitations of standard generalized Langevin equations in driven, dissipative systems.

The Gist

Imagine you are trying to predict the weather in Boulder, Colorado. You have a massive notebook full of temperature readings going back decades. You want to build a computer model that can say, "If it's cold today, here's what the temperature will likely be tomorrow."

The problem is that weather is messy. It's not just a simple, repeating loop. It changes with the seasons, it has wild swings, and it doesn't follow the neat, bell-curve statistics that most physics textbooks assume.

This paper is like a new, smarter recipe for cooking up a weather model. The authors, Thomas Sayer and Andrés Montoya-Castillo, realized that the old "standard recipe" (called the Generalized Langevin Equation, or GLE) breaks down when the ingredients are too chaotic. So, they invented a new method that treats the weather like a seasonal dance rather than a single, continuous song.

Here is the breakdown of their approach using simple analogies:

1. The Problem: The "Noisy Radio"

Think of the temperature data as a radio playing a song.

• The Song: The predictable part is the "seasonal song." It's the slow, steady rhythm of winter getting colder and summer getting hotter.
• The Static: The unpredictable part is the "static." These are the daily surprises: a sudden snowstorm in April or a heatwave in October.

The Old Way: Previous scientists tried to turn down the volume on the "seasonal song" (filtering it out) to study just the "static." In some places (like Berlin), this worked perfectly. The static looked like normal, random noise (Gaussian). They could build a simple model to predict it.

The Boulder Problem: When the authors tried this in Boulder, the "static" didn't look normal.

• In Winter, the noise was wild and lopsided (it got very cold very often, but rarely got super hot).
• In Summer, the noise was calmer and more symmetrical.
• The "old recipe" failed because it tried to force this messy, changing noise into a single, simple box. It was like trying to fit a square peg in a round hole.

2. The Solution: The "Seasonal Club"

Instead of trying to model the whole year at once, the authors decided to split the year into different "clubs" or "seasons" based on how the weather behaves.

• Step 1: The Filter (The DJ): They used a mathematical filter to separate the "seasonal song" (the baseline temperature) from the "daily static" (the fluctuations).
• Step 2: The Map (The Dance Floor): They mapped the baseline temperature and how fast it was changing onto a circle (like a clock face). This let them see exactly where they were in the yearly cycle.
• Step 3: The Clustering (The VIP Sections): They realized that even though the calendar says "March," the statistical behavior of the weather might be more like "late autumn." They used a smart algorithm to group days that behave statistically the same.
  • Club 1 (Winter): Wild, asymmetric swings.
  • Club 2 (Summer): Calm, symmetric swings.
  • Club 3 (Equinox): The messy transition between the two.

3. The New Engine: The "State-Based" Model

Once they had these three distinct "clubs," they built a specific model for each one.

• The Old Engine (GLE): This was like a complex, heavy machine that tried to remember everything that happened in the past to predict the future. It was slow and got confused by the weird shapes of the data.
• The New Engine (TPM-GME): This is like a board game.
  • Imagine the temperature is a token on a board.
  • In the "Winter Club," the rules of the game are different than in the "Summer Club."
  • The model calculates: "If we are in the 'Cold' square, what is the probability we move to the 'Very Cold' square tomorrow?"
  • Because they grouped the days into these specific "clubs," the rules became simple. The complex, long-term memory of the old machine turned out to be unnecessary. The weather in Boulder is actually Markovian (it only cares about right now, not what happened three weeks ago) within each specific season.

4. The Result: A Better Forecast

By using this "Seasonal Club" approach, they could generate fake weather data that looked and felt exactly like the real Boulder weather.

• It captured the wild winter swings.
• It captured the calm summer days.
• It knew exactly when to switch from one "club" to another as the year progressed.

The Big Takeaway

The authors showed that when nature is messy, non-stationary, and weird (non-Gaussian), you shouldn't try to force it into a single, perfect mathematical formula. Instead, you should slice the problem into smaller, manageable pieces where the rules are simple, and then stitch those pieces back together.

It's like trying to describe a chaotic jazz band. Instead of trying to write one sheet of music for the whole hour, you realize the drummer changes his rhythm every 10 minutes. If you write three separate mini-scores for those three 10-minute chunks, you can predict the music perfectly.

In short: They stopped trying to predict the "whole year" with one rulebook and started using three different rulebooks for three different "seasons of behavior," making their weather model accurate, efficient, and surprisingly simple.

Technical Summary

1. Problem Statement

The paper addresses the challenge of modeling complex, driven, dissipative systems (specifically weather and climate) using data-driven approaches. While the Generalized Langevin Equation (GLE) is a standard tool for such systems, its application to non-equilibrium, driven systems faces three critical limitations:

1. Non-Stationarity: External periodic driving (e.g., seasonal cycles) breaks time-translation invariance, rendering standard equilibrium GLE construction workflows invalid.
2. Non-Gaussianity: Residual fluctuations after filtering often exhibit non-Gaussian statistics (e.g., asymmetric tails), violating the common assumption that noise is Gaussian.
3. Heteroskedasticity: The variance of fluctuations changes depending on the state of the system (e.g., temperature fluctuations are larger in winter than in summer), meaning the noise amplitude is position-dependent.

Previous successful applications of GLEs to weather data (e.g., Berlin-Tegel) relied on filtering that resulted in stationary, Gaussian residuals. This paper investigates whether these methods hold for more complex, real-world scenarios like Boulder, Colorado, where filtered data remains non-Gaussian and heteroskedastic.

2. Methodology

The authors propose a multi-stage protocol to construct an accurate, low-dimensional description of weather fluctuations without relying on standard equilibrium assumptions.

A. Data Preprocessing and Filtering

• Dataset: Daily average temperature data from Boulder, CO (1992–2025).
• Filtering: The authors apply Fourier filtering to remove the deterministic annual cycle and zero-frequency trends.
• Observation: Unlike previous studies, the filtered Boulder data remains non-Gaussian (bistable/asymmetric histograms) and heteroskedastic (fluctuation magnitude varies by season).

B. Theoretical Framework: From GLE to TPM-GME

The authors argue that a standard GLE is inefficient for this data because:

• The memory kernel becomes position-dependent (friction varies with temperature).
• Constructing a position-dependent GLE requires converging an infinite-dimensional matrix of transitions.

To solve this, they introduce a State-Based Generalized Master Equation (TPM-GME) approach:

1. Floquet Theory & Baseline Extraction: They define a "baseline" trajectory ($T_b$) representing the deterministic drive. By mapping the baseline and its derivative to polar coordinates, they identify the system's position within the annual cycle.
2. Homoskedastic Projection (Season Clustering):
   • The trajectory is divided into "microstates" based on the baseline phase.
   • These microstates are clustered into "macrostates" (seasons) based on the statistical properties of their fluctuations (specifically, fitting the histogram to an asymmetric potential function).
   • This results in three distinct seasons: Summer, Winter, and an "Equinoctial" season. Within each season, the system is treated as pseudo-stationary and homoskedastic (constant variance).
3. Transition Probability Matrix (TPM):
   • For each season, a Transition Probability Matrix is constructed, quantifying the probability of moving between temperature bins.
   • The memory kernel is extracted via numerical inversion of the correlation functions.
   • Key Finding: The memory kernel for each season is found to be delta-like (short-lived), allowing the system to be modeled as a Markov State Model (MSM) rather than a non-Markovian GLE.

C. Hierarchical Reconstruction

To generate high-resolution predictions:

1. Coarse-Grained Step: A Markov chain is unraveled using the TPM to determine the sequence of temperature bins (2°F resolution).
2. Fine-Grained Step: Within each bin, specific temperature values are drawn stochastically from a fitted asymmetric distribution (stretched exponential) derived from the original data. This preserves the correct non-Gaussian tails without requiring high-resolution transition statistics.

3. Key Contributions

• Identification of Limitations in Standard GLEs: The paper demonstrates that standard equilibrium GLE workflows fail for Boulder temperature data due to persistent non-Gaussianity and heteroskedasticity, even after filtering.
• Introduction of "Local Homoskedasticity": A novel metric to identify time spans (seasons) where stationarity holds, enabling the application of equilibrium-like tools to non-equilibrium data.
• TPM-GME as a Superior Alternative: The authors show that a state-based Generalized Master Equation (TPM-GME) is more efficient than a GLE for position-dependent systems. It naturally encodes position-dependent friction through state aggregation and reduces memory lifetimes, often collapsing the dynamics into a Markovian process.
• Validation of the Protocol: The method successfully reproduces the evolving fluctuations of the Boulder temperature time series, including the correct magnitude of extreme events (non-Gaussian tails) and the seasonal variation in volatility.

4. Results

• Seasonal Clustering: The data naturally clusters into three seasons with distinct statistical behaviors:
  • Summer: Symmetric, near-Gaussian fluctuations.
  • Winter: Highly asymmetric with a "thick" cold tail (large fluctuations).
  • Equinoctial: Intermediate behavior.
• Memory Kernel Analysis:
  • Direct GLE application yields long-lived, oscillatory memory kernels due to residual periodicity.
  • The TPM-GME approach yields short-lived, delta-like memory kernels (Markovian) for each season.
  • This confirms that the non-Markovian nature of the global system arises from the switching between different stationary regimes (seasons), not from long-term memory within a single regime.
• Prediction Accuracy: The generated synthetic time series accurately match the historical data's 1-point histograms (distributions) and 2-point correlation functions. The model captures "colored, non-Gaussian noise" and the specific variance changes between seasons.

5. Significance

• Generalizability: The protocol offers a general framework for modeling driven, dissipative systems in physics, biology, and finance where data is non-stationary and non-Gaussian.
• Efficiency: By converting a complex non-Markovian problem into a set of simpler Markovian sub-problems (seasons), the method significantly reduces the data requirements for model construction compared to a full non-Markovian GLE.
• Climate Science: This work advances the "Hasselmann program" (separating deterministic climate signals from stochastic weather noise) by providing a rigorous, data-driven method to handle the non-Gaussian and heteroskedastic nature of real-world climate data. It bridges the gap between traditional statistical physics and modern machine learning approaches for weather prediction.

In summary, the paper provides a robust, data-driven workflow to model complex climate dynamics by decomposing the system into pseudo-equilibrium states, utilizing state-based master equations to handle non-Gaussianity and heteroskedasticity, and demonstrating that the resulting dynamics are effectively Markovian within these seasonal regimes.