🌀 nonlinear sciences

Time Series Correlations and Kolmogorov Complexity: A Hausdorff Dimension Perspective

This paper proposes using Kolmogorov complexity and effective Hausdorff dimension to distinguish spurious correlations from genuine relationships in time series, demonstrating that high correlation between independent low-complexity series is likely accidental while introducing a joint complexity indicator ( $J_{\rm LZ}$ ) to rigorously filter false positives.

Original authors: Boumediene Hamzi, Marianne Clausel, Kamal Dingle, Marcus Hutter, Mohammed Terry-Jack

Published 2026-03-31

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Boumediene Hamzi, Marianne Clausel, Kamal Dingle, Marcus Hutter, Mohammed Terry-Jack

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: "Nonsense Correlations"

Imagine you are looking at two graphs.

Graph A: Shows the number of pirates on the high seas over the last 200 years (it's going down).
Graph B: Shows the average global temperature over the last 200 years (it's going up).

If you run a standard math test on these, you might find they are perfectly correlated (one goes up, the other goes down). But does that mean pirates are cooling the planet? Of course not! This is a spurious correlation.

The problem is that the world is full of simple, boring patterns (like straight lines going up or down). Because these simple patterns are so common, they often accidentally line up with each other, tricking us into thinking they are related when they aren't.

The Solution: Measuring "Chaos" (Complexity)

The authors of this paper propose a new way to filter out these fake connections. Instead of just asking, "Do these lines look similar?" they ask, "How complicated are these lines?"

They use a concept called Kolmogorov Complexity. Think of it as a "Compression Test."

Low Complexity (Simple): Imagine a wallpaper pattern that repeats Red-Blue-Red-Blue forever. You can describe this entire wall with just three words: "Red, Blue, Repeat." It is very easy to compress.
High Complexity (Chaotic): Imagine a static-filled TV screen or a truly random rock fall. To describe this, you have to list the color of every single pixel or the position of every single rock. You cannot compress it; it is "rough" and "messy."

The Core Idea:
If two simple lines (like a pirate count and a temperature trend) line up, it's probably a coincidence. But if two complex, chaotic lines line up, it is highly unlikely to be a coincidence. It's like two people walking randomly through a giant forest; if they bump into each other, it's a real event. If two people walking in a straight line down a hallway bump into each other, it's just because the hallway is narrow.

The "Noise" Problem

There is a catch. Real-world data is never perfectly clean; it has "noise" (measurement errors, static).

The Analogy: Imagine trying to compress a song. If the song is clear, it compresses well. If there is static on the recording, the file gets bigger because the computer has to describe the static too.
The Insight: The paper proves that even a "simple" signal with a little bit of noise looks slightly complex. So, we have to set a "noise floor." If a signal looks too simple, it might just be a simple trend. If it looks moderately complex, it might just be noise. We need a signal that is genuinely "rough" and chaotic to trust a correlation.

The New Tool: The "JLZ" Score

The authors created a new score called JLZ (Joint Complexity). Think of it as a "Trust Score" for a relationship between two data sets.

They use a simple rule: Both sides of the relationship must be complex for the relationship to be trustworthy.

Scenario A: Two complex, chaotic series correlate. -> High Trust. (They are likely actually related).
Scenario B: Two simple, smooth series correlate. -> Low Trust. (It's likely a coincidence).
Scenario C: One complex and one simple series correlate. -> Low Trust. (The simple one is dragging the whole thing down).

The Experiments: Testing the Theory

The authors tested this idea with two "toy models" (simulated worlds):

1. The Chaotic Pendulum (Logistic Maps)
They simulated two pendulums that swing in a chaotic, unpredictable way.

When the pendulums were uncoupled (not touching), they wandered randomly. Even though they were independent, they never accidentally lined up perfectly.
When they were coupled (connected), they started moving together.
The Surprise: The JLZ score could detect that the pendulums were about to lock into sync before they actually did. It saw their individual "chaos" collapse into a single pattern right before they synchronized. This is something standard math tools missed.

2. The Fractal Clouds (Fractional Brownian Motion)
They simulated "clouds" that can be either "smooth" (like a gentle hill) or "rough" (like jagged mountains).

Smooth Clouds: When they generated two smooth, rolling hills, they often looked like they were moving together by pure chance. (High false alarms).
Rough Clouds: When they generated two jagged, chaotic mountains, they almost never looked like they were moving together unless they were actually connected.
The Result: The "Rough" (high complexity) clouds were much better at avoiding fake correlations.

The Practical Advice: What Should You Do?

If you are analyzing data (in finance, science, or economics), the paper suggests a new two-step checklist:

Check for Stationarity: Make sure your data isn't just a straight line going up or down (like a stock market that only grows). If it is, take the "difference" (look at the daily changes, not the total price).
Check the Complexity Score (JLZ): Before you celebrate a strong correlation, ask: "Are both of these series complex and messy?"
- If YES: Great! The correlation is likely real.
- If NO (they are both smooth and simple): Be very skeptical. It's probably just a coincidence, like the pirates and the temperature.

Summary Metaphor

Imagine you are a detective trying to solve a crime.

Old Method: You see two suspects wearing the same red hat. You arrest them. (Correlation = Guilt).
New Method: You realize that everyone in town wears a red hat (Simple Pattern). You check if the suspects are actually doing something complex and unique, like speaking a secret language. If they are both speaking a secret, complex language at the same time, then you know they are working together. If they are just standing there wearing red hats, they are probably innocent.

The Bottom Line: Don't trust a connection just because the lines look similar. Trust it only if the lines are complicated enough that they couldn't possibly have lined up by accident.

1. Problem Statement

The paper addresses the persistent issue of spurious correlations in time series analysis. It is a well-known phenomenon that simple, low-complexity patterns (e.g., monotone trends, periodic sequences, or low-degree polynomials) frequently exhibit high Pearson correlation coefficients ( $\rho$ ) despite having no causal relationship.

The Core Issue: Standard statistical significance testing (e.g., p-values based on randomization) often fails to distinguish between genuine causal links and coincidental alignments of simple trends because these simple patterns are exponentially more probable in natural data than complex, random ones.
The Gap: Existing methods lack a principled, theoretical framework to filter out these false positives based on the intrinsic complexity of the data itself.

2. Theoretical Framework

The authors ground their approach in Algorithmic Information Theory (AIT) and Fractal Geometry, connecting three key concepts:

Kolmogorov Complexity ( $K(x)$ ): Defined as the length of the shortest program generating a string $x$ . High $K(x)$ implies randomness/complexity; low $K(x)$ implies compressibility/simplicity. Since $K(x)$ is uncomputable, the paper uses Lempel-Ziv (LZ) compression as a practical proxy.
Effective Hausdorff Dimension: The paper utilizes the Lutz-Mayordomo theorem, which links the normalized Kolmogorov complexity of a sequence to its effective Hausdorff dimension ( $\dim_H$ $dim_{H}$ ).
- Simple series (smooth, periodic) have low dimension ( $\dim_H \approx 1$ ).
- Complex series (chaotic, rough) have high dimension ( $\dim_H \approx 2$ ).
The Spurious Alignment Bound: The authors prove a formal bound showing that the probability of two independent series having a high correlation decays exponentially with their combined Kolmogorov complexity.
- Intuition: Simple series occupy a tiny, densely populated corner of the sequence space, making accidental alignment highly probable. Complex series "wander" over a high-dimensional space, making accidental alignment geometrically improbable.
Noise Inflation (Posobin-Shen Theorem): The paper acknowledges that measurement noise increases observed complexity. They derive a "noise floor" (approx. 0.15 for normalized LZ complexity) below which a series is effectively simple, regardless of its theoretical nature.

3. Methodology

A. Complexity Estimators

The authors propose three empirical measures to approximate Kolmogorov complexity:

Normalized LZ Complexity ( $\tilde{C}_{LZ}$ ): Based on zlib compression of raw IEEE-754 byte-serialized time series.
Spectral Entropy ( $\tilde{H}$ ): A Fourier-domain proxy measuring the distribution of energy across frequencies.
Joint Complexity Indicator ($JLZ$): A novel metric defined as the geometric mean of the normalized complexities of two series:
$JLZ(x, y) = \sqrt{\tilde{C}_{LZ}(x) \cdot \tilde{C}_{LZ}(y)}$
- Rationale: Unlike similarity metrics (which measure if complexities are equal), $JLZ$ measures if both series are individually complex. If either series is simple, $JLZ$ collapses toward zero.

B. Experimental Models

Two complementary toy models were used to validate the theory:

Coupled Logistic Maps:
- Swept from periodic ( $r=3.4$ , low complexity) to chaotic ( $r=3.9$ , high complexity) regimes.
- Coupling strength ( $\epsilon$ ) controlled the true correlation.
Multivariate Fractional Brownian Motion (mfBm):
- Used to link complexity directly to the Hurst parameter ( $H$ ) and Hausdorff dimension ( $\dim_H = 2 - H$ ).
- Tested both stationary increments (fGn) and non-stationary paths (fBm).

4. Key Results

A. False Positive Rates

Low Complexity = High False Positives: In the logistic map experiment, independent periodic series ( $r=3.4$ ) produced spurious correlations ( $|\rho| > 0.3$ ) in 100% of trials.
High Complexity = Low False Positives: Independent chaotic series ( $r=3.9$ ) produced 0% false positives.
mfBm Validation: For stationary fractional Gaussian noise, the false positive rate dropped below 5% when the normalized LZ complexity exceeded $\approx 0.4$ (corresponding to $H \lesssim 0.7$ ). Conversely, smooth series ( $H \approx 0.9$ ) showed false positive rates of 25–40%.

B. The JLZ Indicator Performance

Screening Power: $JLZ$ successfully separates data into four quadrants:
1. High $JLZ$, High $\rho$ : Genuine relationship (Strong evidence).
2. Low $JLZ$, High $\rho$ : Spurious correlation (Likely nonsense).
3. High $JLZ$, Low $\rho$ : Genuine independence.
4. Low $JLZ$, Low $\rho$ : Trivial independence.
Early Warning Signal: In the chaotic logistic map, $JLZ$ detected a sharp collapse in individual complexity just before the synchronization transition ( $\epsilon_c$ ). This phenomenon was invisible to similarity-based indicators (like $SLZ$) which only measure the difference in complexity.

C. Non-Stationarity Handling

The study confirmed that non-stationary integrated processes (like raw fBm paths) produce spurious correlations regardless of complexity (Yule's nonsense correlation). However, the complexity metric correctly identified these as "low complexity" due to the dominating trend.

Recommendation: A two-stage pipeline is required:
1. Ensure stationarity (differencing/detrending).
2. Apply the $JLZ$ complexity screen.

5. Significance and Contributions

Theoretical Unification: The paper provides a rigorous proof that spurious correlations are a geometric consequence of low Hausdorff dimension, formalized through the Solomonoff prior and effective Hausdorff dimension.
Practical Diagnostic Tool: It introduces $JLZ$ as a computationally cheap, principled filter. Unlike surrogate testing (which assumes a specific null model like i.i.d. noise), $JLZ$ evaluates the structural richness of the data itself.
Threshold Calibration: The authors provide a data-driven threshold ($JLZ > 0.3$) calibrated against the mfBm false-positive curve, offering a concrete rule for practitioners to avoid false positives.
Complementarity: The framework is designed to work alongside standard significance testing. A correlation is only considered "genuine" if it is both statistically significant (via surrogates) and structurally robust (via high $JLZ$).

6. Practical Recommendations

The authors propose the following workflow for time series analysis:

Preprocessing: Test for stationarity; difference or detrend if necessary.
Complexity Calculation: Compute $\tilde{C}_{LZ}$ for each series.
Joint Screening: Calculate $JLZ$.
- If $JLZ < 0.3$ (or domain-specific threshold), treat any high correlation with strong skepticism (likely spurious).
- If $JLZ > 0.3$ and $\rho$ is high, the relationship is a strong candidate for genuine causality.
Reporting: Always report complexity metrics alongside correlation coefficients.

Conclusion

This paper shifts the paradigm of spurious correlation detection from purely statistical hypothesis testing to algorithmic complexity analysis. By demonstrating that high correlations between simple, low-dimensional series are mathematically inevitable, it argues that complexity filtering is a necessary prerequisite for credible time series inference. The proposed $JLZ$ metric offers a robust, theoretically grounded solution to a century-old problem in data science.