Machine learning for understanding pulsating stars I: the non-linear phenomenon in δ Scuti stars

$δ$ Scuti stars are pulsating variable stars that exhibit both radial and non-radial pulsations, making them key objects for understanding stellar evolution and internal structures. The current classification of $δ$ Scuti stars into High-Amplitude $δ$ Scuti (HADS) and Low-Amplitude $δ$ Scuti (LADS) stars is based on the peak-to-peak amplitude of their light curves (>0.3 mag). Nevertheless, this classification may not fully capture the complexity of their pulsation mechanisms and non-linear effects, leading to possible misclassifications. This investigation aims to challenge the existing classification of $δ$ Scuti stars according to amplitude, employing the exploration of frequency domain features and non-linear mechanisms in order to identify intrinsic subgroups. The objective is to get a deeper understanding of the properties of $δ$ Scuti stars. We use machine learning clustering techniques, specifically hierarchical clustering (HC) with Ward’s linkage, to analyze a sample of 142 $δ$ Scuti stars observed by space telescopes such as CoRoT, Kepler, and TESS. We focus on frequency-domain features, including fundamental and overtone modes, as well as non-linear features such as harmonic, sums, and subtraction frequencies, to uncover intrinsic subgroups within $δ$ Scuti stars. The results of the clustering process indicate that the present amplitude-based classification (HADS/LADS) exhibits partial alignment with the clusters identified by using features from the frequency-domain. However, the study identified additional sub-groups, suggesting a greater variety of nonlinear effects that are not captured by the amplitude alone. It highlights the importance of non-linear features, such as the number of subtraction combinations, which may be indicative of resonance effects or other internal physical mechanisms.

💡 Research Summary

The paper challenges the long‑standing classification of δ Scuti pulsating stars into High‑Amplitude (HADS) and Low‑Amplitude (LADS) groups, which relies solely on a 0.3 mag peak‑to‑peak light‑curve threshold. Recognizing that this amplitude‑only scheme may overlook the rich non‑linear dynamics inherent to these stars, the authors employ a data‑driven approach using hierarchical clustering to uncover intrinsic sub‑populations based on frequency‑domain and non‑linear features.

Data and Feature Extraction
A sample of 142 δ Scuti stars observed by the space missions CoRoT, Kepler (short‑cadence), and TESS forms the basis of the study. Light curves are processed with the Best Parent Method (BPM), which iteratively identifies “parent” frequencies (the fundamental mode f₁ and, when present, the first overtone f₂) and their “child” combination frequencies. From each star the authors extract nine quantitative descriptors: f₁, its amplitude A₁, f₂, its amplitude A₂ (set to zero for monoperiodic stars), the number of harmonics of the fundamental (Harm1) and overtone (Harm2), and the counts of additive (AddComb) and subtractive (SubComb) combination frequencies. These variables are chosen to capture both the linear pulsation content and the degree of non‑linear interaction among modes.

Pre‑processing and Outlier Handling
Standard outlier detection (z‑score, IQR) proved ineffective due to the highly skewed, non‑Gaussian distributions of many features. Log‑transformations improved normality but still left ambiguous cases. Density‑based clustering (DBSCAN) was explored for robust outlier identification, yet its performance depended sensitively on the choice of ε and minPts. Ultimately, the authors integrated outlier handling into the clustering pipeline itself, selecting Ward’s hierarchical clustering for its resilience to outliers and ability to respect the intrinsic geometry of the data.

Clustering Methodology
Multiple clustering paradigms were trialed: centroid‑based K‑means, distribution‑based Gaussian Mixture Models, density‑based DBSCAN, and hierarchical clustering. K‑means suffered from sensitivity to initialization and an inability to capture the irregular, elongated shapes of the data clouds. GMMs assumed Gaussianity that the data violated, leading to poor alignment with physical expectations. DBSCAN identified some dense groups but failed to accommodate varying densities across the sample. Hierarchical clustering with Ward’s linkage emerged as the most reliable, producing a dendrogram that suggested four natural clusters. The optimal number of clusters was corroborated by silhouette scores and visual inspection of the dendrogram.

Results and Physical Interpretation
Two of the four clusters map reasonably onto the traditional HADS/LADS dichotomy, confirming that amplitude still carries discriminative power. However, the remaining two clusters are defined primarily by non‑linear metrics, especially the number of subtractive combination frequencies (SubComb). Stars in one of these clusters exhibit moderate amplitudes but a high SubComb count, implying strong mode‑mode coupling or resonant interactions that are not reflected in the simple amplitude measure. The other cluster consists largely of monoperiodic, low‑amplitude stars with negligible non‑linear signatures, representing a “pure” pulsation regime.

The prominence of SubComb as a clustering driver suggests that subtractive combinations may trace underlying physical processes such as resonances, rotational splitting, or internal mixing that affect the energy exchange between modes. Consequently, the authors argue that a classification scheme incorporating non‑linear diagnostics would be more physically meaningful than the current amplitude‑only taxonomy.

Methodological Reflections
The study emphasizes careful feature engineering: the exclusion of phase P₂ (found to be non‑informative) avoided artificial correlations, while encoding absent overtone modes as zeros preserved the distinction between monoperiodic and multimode stars without inflating dimensionality. Dimensionality‑reduction techniques (PCA, MDS, t‑SNE) were evaluated for visualization but not adopted for feature selection, preserving interpretability of the original astrophysical parameters.

Conclusions and Future Directions
The authors conclude that (1) the traditional HADS/LADS classification is insufficient to capture the full phenomenology of δ Scuti stars, (2) non‑linear frequency combinations, particularly subtractive ones, provide a robust basis for defining new sub‑groups, and (3) hierarchical clustering offers a transparent, outlier‑robust framework for such analyses. They propose extending the work by (i) correlating the identified clusters with independent stellar parameters (mass, metallicity, rotation rate), (ii) expanding the sample size with upcoming TESS extended missions, and (iii) integrating physical non‑linear models (e.g., Volterra expansions) to translate statistical clusters into concrete stellar interior diagnostics.

In summary, this paper demonstrates that machine‑learning clustering, when fed with thoughtfully engineered frequency‑domain and non‑linear features, can reveal hidden structure in δ Scuti star populations, paving the way for a more nuanced, physics‑driven classification system.