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

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

Problem Statement
The current classification of $\delta$ Scuti stars into High-Amplitude ($\delta$ Sct, HADS) and Low-Amplitude ($\delta$ Sct, LADS) categories relies exclusively on the peak-to-peak amplitude of their light curves, utilizing a threshold of 0.3 magnitudes. The authors argue that this amplitude-based dichotomy is overly simplistic and potentially misleading. Light curve amplitude is a system output resulting from combined pulsation mechanisms and non-linear interactions; consequently, it may obscure the underlying physical processes, internal structures, and non-linear effects driving stellar pulsations. This limitation risks misclassification and fails to capture the full complexity of the $\delta$ Scuti population.

Methodology
The study employs machine learning clustering techniques to analyze a sample of 142 $\delta$ Scuti stars observed by the CoRoT, Kepler (short-cadence), and TESS space telescopes. The methodology proceeds through the following stages:

1. Data Preprocessing and Feature Extraction:

   • Light curves were processed using the Best Parent Method (BPM), a self-consistent algorithm designed to identify parent frequencies and remove combination frequencies (non-linearities).
   • The BPM algorithm iteratively refines parent frequencies by fitting non-linear models (e.g., Volterra expansions) to minimize residual variance, subsequently fitting all statistically significant "child" frequencies in a least-squares framework.
   • A feature set of nine variables was constructed to capture both linear and non-linear characteristics:
     • Fundamental mode parameters: frequency ($f_1$), amplitude ($A_1$), and phase ($P_1$).
     • First overtone parameters: frequency ($f_2$) and amplitude ($A_2$) (set to zero for monoperiodic stars).
     • Non-linear indicators: Number of fundamental mode harmonics, number of overtone mode harmonics, number of additive combinations, and number of subtractive combinations.
   • Outlier detection was integrated into the clustering process rather than performed as a separate pre-filtering step, treating outliers as potential natural variations or misclassified stars to preserve information.

2. Clustering Algorithm Selection:

   • Several clustering approaches were evaluated, including centroid-based (K-means), distribution-based (Gaussian Mixture Models), and density-based (DBSCAN) methods.
   • These methods were found to be sensitive to the dataset's non-Gaussian distributions, skewed data, and outliers.
   • Hierarchical Clustering (HC) with Ward's linkage was selected as the optimal method. HC was deemed most robust for handling non-linear relationships, varying cluster densities, and outliers, while providing a dendrogram for visualizing the clustering hierarchy.

3. Clustering Experiments:

   • Baseline: Clustering based solely on fundamental amplitude ($A_1$) to replicate the traditional HADS/LADS split.
   • Comprehensive: Clustering in a 9-dimensional feature space (linear and non-linear features).
   • Non-linear Focus: Clustering using only non-linear features (harmonics and combination frequencies).
   • The analysis tested various cluster counts (specifically 3, 6, and 11) to allow the data to reveal intrinsic structures without presupposing a binary ground truth.

Key Results

• Partial Alignment with Traditional Classification: The baseline clustering using only amplitude ($A_1$) showed a partial overlap between HADS and LADS distributions, confirming that amplitude alone is insufficient to strictly differentiate the groups.
• Identification of Intrinsic Subgroups: The 9-dimensional feature space clustering revealed a more complex structure than the traditional binary classification:
  • 3-Cluster Solution: While two clusters largely aligned with HADS and LADS, a distinct third cluster emerged. This group was characterized by a high number of subtraction combinations, suggesting a specific non-linear regime potentially linked to resonance effects, which did not align neatly with the amplitude-based HADS/LADS split.
  • 6-Cluster Solution: Further subdivision within the LADS category was identified, revealing subgroups with different fundamental/overtone frequencies and varying numbers of subtraction combinations. This suggests variations in internal structures or evolutionary stages within the low-amplitude population.
• Role of Non-Linear Features: Clustering based solely on non-linear features partially reproduced the HADS/LADS classification, particularly for HADS. However, the LADS population required a larger number of clusters (e.g., 11) to be resolved, indicating that their classification is more dependent on intrinsic pulsation modes than on non-linear effects alone.
• Physical Interpretation: The study posits that the subgroups may reflect different physical regimes. HADS subgroups may relate to the stellar envelope's properties, while LADS subgroups may arise from different resonance mechanisms (e.g., rotation-induced mode coupling, near-resonances). However, the authors explicitly state these are working hypotheses requiring independent constraints (e.g., temperature, metallicity, rotation) for validation.

Significance and Claims
The paper claims to provide a more profound understanding of $\delta$ Scuti stars by challenging the sufficiency of amplitude-based classification. The primary significance lies in demonstrating that:

1. Non-linear features are discriminatory: The number of subtraction combinations and other non-linear indicators provide valuable power to differentiate subgroups that amplitude alone misses.
2. Complexity exists within HADS/LADS: The traditional dichotomy masks a greater variety of non-linear effects and internal mechanisms.
3. Machine Learning Utility: Hierarchical clustering with Ward's linkage is a robust tool for asteroseismology, capable of uncovering intrinsic subgroups in high-dimensional, non-linear datasets where traditional methods fail.

The authors conclude that while the findings question the simplicity of current classifications, they do not yet offer a definitive new classification system. Instead, they advocate for a comprehensive approach incorporating multiple observable variables and non-linear features. Future work is proposed to validate these clusters against independent observables (spectroscopy, HR diagram position) and to refine the BPM criteria through an iterative process with cluster pattern recognition.