🌀 nonlinear sciences

Including the magnitude variability of a signal into the ordinal pattern analysis

This paper proposes a novel method that enhances traditional Ordinal Pattern analysis by incorporating discarded signal magnitude variability as a complementary variable to permutation entropy, thereby improving the characterization of dynamical behaviors in both synthetic maps and real-world EEG data for better feature engineering and AI classification.

Original authors: Melvyn Tyloo, Joaquín González, Nicolás Rubido

Published 2026-01-27

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Melvyn Tyloo, Joaquín González, Nicolás Rubido

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

Imagine you are trying to describe a song to someone who has never heard it.

The Old Way: The "Mood" Playlist
Traditionally, scientists have used a method called Ordinal Patterns to analyze signals (like brain waves or weather data). Think of this like creating a playlist based only on the order of the songs, ignoring how loud or quiet they are.

If the song gets louder, you write "Up."
If it gets quieter, you write "Down."
If it stays the same, you write "Flat."

This method is great because it's simple and ignores background noise. However, it has a big flaw: it throws away the volume. You know the song went from "soft" to "loud," but you don't know if it went from a whisper to a shout, or from a murmur to a slight increase in volume. You've lost the "magnitude" or the "intensity" of the signal.

The New Idea: Adding the "Volume Knob"
This paper proposes a simple but powerful fix: Keep the order, but also measure the volume.

The authors suggest that when we look at these "Up/Down/Flat" patterns, we should also calculate how much the signal actually varied in size during that moment. They call this the "magnitude variability."

Think of it like this:

Old Method: "The temperature went up, then down." (You know the direction, but not the scale).
New Method: "The temperature went up by 1 degree, then down by 20 degrees." (You know the direction and the intensity).

How They Tested It
The researchers tested this idea in two ways:

With Computer Simulations (The "Math Maps"):
They used complex mathematical formulas (called Logistic and Hénon maps) that generate chaotic, unpredictable numbers.
- The Result: Sometimes, two different chaotic patterns looked exactly the same when you only checked the "Up/Down" order. But when they added the "volume" check, the two patterns looked completely different. It was like realizing two people were walking in the same direction, but one was strolling and the other was sprinting. The new method could tell them apart.
With Rat Brain Waves (The "Sleep Study"):
They recorded brain activity from rats in three states: Awake, REM Sleep (dreaming), and Deep Sleep.
- The Problem: Using the old "Up/Down" method, it was very hard to tell the difference between REM sleep and Deep Sleep. They looked too similar.
- The Solution: When they added the "volume" check, the difference became obvious. The brain waves during REM sleep had a different "intensity" or variability compared to Deep Sleep, even though the order of the waves was similar.

Why This Matters
The paper concludes that by combining the order (the pattern) with the intensity (the magnitude), we get a much clearer picture of what is actually happening.

It helps distinguish between different types of chaos that look the same at first glance.
It helps tell apart different sleep stages in rats that were previously hard to distinguish.

The authors suggest this approach is useful for "feature engineering," which is a fancy way of saying: "Giving computers better clues so they can make smarter decisions." If you are building an AI to recognize patterns, giving it both the "shape" of the data and the "size" of the data makes it a much better detective.

In a Nutshell
Don't just look at the sequence of events; look at how big those events are. By adding the "volume" back into the analysis, we can see details that were previously invisible.

Technical Summary: Including the Magnitude Variability of a Signal into the Ordinal Pattern Analysis

Problem Statement
Ordinal Pattern (OP) analysis is a widely used method for characterizing signals by transforming them into symbolic sequences based on the relative ordering of signal magnitudes within embedded vectors. While OP analysis is conceptually clear, robust to noise, and effective for short signals, it possesses a significant limitation: the encoding process discards the actual magnitude of the signal at each timestamp, retaining only the ordinal relationships. Consequently, information regarding the amplitude variability of the signal is lost. The authors note that while the permutation entropy (a statistical quantification of the OP sequence) is useful, it may be insufficient to fully characterize complex dynamical systems, particularly higher-dimensional ones where the relationship between permutation entropy and the Kolmogorov-Sinai entropy is not as direct as in one-dimensional mappings.

Methodology
The authors propose a complementary approach that integrates the standard deviation of signal magnitudes within the embedded vectors into the OP analysis framework. The methodology involves the following steps:

Signal Encoding: Following the Bandt-Pompe method, univariate signals are divided into embedded vectors of dimension $D$ . Each vector is transformed into an Ordinal Pattern (OP) based on the number of permutations required to sort its elements.
Entropy Calculation: The authors calculate the Rényi min-entropy ( $H_\infty$ ) of the resulting OP sequence. This metric quantifies the average information content (uncertainty) of the symbolic sequence.
Magnitude Variability Quantification: For each unique OP symbol $\alpha$ , the authors calculate the standard deviation ( $\sigma_j$ ) of the signal magnitudes at the $j$ -th component of all embedded vectors corresponding to that symbol. They then compute the probability-weighted average of the logarithm of these standard deviations, denoted as $\langle \log_2 [\sigma_j] \rangle$ .
Combined Analysis: The characterization of the signal is performed by plotting the permutation entropy ( $H_\infty$ ) against the average magnitude variability ( $\text{avg}_j\{\langle \log_2 [\sigma_j] \rangle\}$ ).

The study validates this approach using three distinct datasets:

Synthetic Models: Trajectories from coupled identical logistic maps and the Hénon map, analyzed under various control parameters and coupling strengths. Both noiseless and noisy (white Gaussian noise) conditions were tested.
Experimental Data: Intracranial electroencephalographic (EEG) recordings from 11 freely moving Wistar rats, covering three sleep-wake states: Active Wakefulness (AW), Rapid Eye Movement (REM) sleep, and Non-REM (NREM) sleep.

Key Contributions

Novel Metric Integration: The paper introduces a method to retain magnitude information discarded by standard OP encoding by using the standard deviation of signal magnitudes as a complementary variable to permutation entropy.
Theoretical Justification: The approach is grounded in the work of Politi, which suggests that the logarithm of the standard deviation of signal magnitudes is necessary, alongside the information dimension, to make permutation entropy converge to the Kolmogorov-Sinai entropy.
Enhanced Differentiation: The study demonstrates that combining entropy with magnitude variability provides a more robust characterization of dynamical behaviors than using permutation entropy alone.

Results

Synthetic Maps: In the analysis of coupled logistic and Hénon maps, the authors observed that distinct dynamical regimes often yielded similar permutation entropy values ( $H_\infty$ ) but drastically different magnitude variability values. For instance, in coupled logistic maps, different control parameters ( $r$ ) produced similar $H_\infty$ values (e.g., ~1.92 vs. 1.95 bits) but magnitude variability values differing by an order of magnitude. Conversely, some regimes showed significant entropy differences but similar variability. The combined 2D plot of $H_\infty$ vs. variability allowed for clearer separation of these regimes than either metric alone.
Noise Robustness: The magnitude variability metric followed a power-law behavior as a function of the embedding dimension and noise strength, aiding in the distinction between chaotic and stochastic signals even under significant observational noise.
EEG Analysis: When applied to rat EEG data, permutation entropy ( $H_\infty$ ) successfully distinguished Active Wakefulness (highest entropy) from sleep states. However, it failed to significantly differentiate between REM and NREM sleep, as their entropy values were similar. In contrast, the magnitude variability metric showed a difference of approximately one order of magnitude between REM and NREM states across most electrodes. This aligns with the physiological characteristics of these states (low-voltage fast waves in REM vs. high-voltage slow waves in NREM). The simultaneous use of both metrics allowed for the clear differentiation of all three sleep-wake states.

Significance and Claims
The paper claims that incorporating signal magnitude variability into ordinal pattern analysis improves the characterization of dynamical behaviors in both synthetic and experimental time series. The authors assert that this approach is particularly useful for:

Feature Engineering: Providing complementary signal features that can improve the accuracy of AI classifiers and machine learning algorithms.
State Classification: Enhancing the ability to distinguish between complex physiological states (such as sleep stages) where traditional entropy measures may be insufficient.
Robustness: Offering a method that remains effective under observational noise and varying embedding dimensions.

The authors remain modest regarding the scope, noting that their conclusions are based on specific synthetic maps and a specific set of rat EEG recordings. They suggest that while the current method uses standard deviation, future applications involving short signals might benefit from using the inter-quartile range for robustness against outliers. They also propose that extending the analysis to a three-dimensional space including a complexity measure (e.g., Jensen-Shannon complexity) could further improve the characterization of non-chaotic dynamical regimes.

More like this