A New Framework for Multi-Line Analysis Combined Kernel PCA and Kernel SHAP: A Case of NGC 1068 ALMA Band 3 Data

We present a new framework for multi-line analysis that combines kernel principal component analysis (Kernel PCA), an unsupervised machine-learning method, and Kernel SHapley Additive exPlanations (Kernel SHAP), an explainable artificial intelligence (XAI) technique. To enable a comparison with PCA-based studies, which have been widely used in multi-line analyses, we apply our framework to integrated intensity maps of 13 molecular lines from Atacama Large Millimeter/submillimeter Array (ALMA) Band 3 archival data of the nearby galaxy NGC 1068. Previous PCA-based studies of NGC 1068 reported that physically meaningful structures are mainly captured up to the second component. In contrast, our framework can interpret physically meaningful features up to the fourth component. Furthermore, by comparing the results obtained from our framework with molecular column densities derived from local thermodynamical equilibrium (LTE) analysis, we suggest that the abundance of HCO+ is relatively enhanced in the molecular outflow region extending to a radius of about 400 pc from the galactic center, likely due to the effects of ultraviolet radiation and highly dense gas. These results show that our framework can provide data-driven insights into physical and chemical features that have not been clearly identified in previous studies. It also provides an efficient tool for interpreting the rapidly increasing amount of multi-line observational data.

💡 Research Summary

This paper introduces a novel, data‑driven framework for the analysis of multi‑line molecular observations that combines kernel principal component analysis (Kernel PCA) with the explainable‑AI method Kernel SHAP. The authors apply the framework to integrated intensity maps of 13 molecular transitions observed with ALMA Band 3 toward the nearby Seyfert galaxy NGC 1068. The dataset consists of 683 spatial pixels (samples) each described by 13 standardized line intensities (features).

Traditional linear dimensionality‑reduction techniques such as principal component analysis (PCA) have been widely used in multi‑line studies, but they are limited to capturing only the dominant linear variance, typically yielding physically interpretable information in the first two components. The authors argue that the complex interplay of density, temperature, optical depth, and radiation fields in galactic nuclei creates intrinsically non‑linear relationships among line intensities, which linear PCA cannot fully represent.

Kernel PCA addresses this limitation by implicitly mapping the data into a high‑dimensional Hilbert space using a radial‑basis‑function (RBF) kernel, where linear PCA is performed. This non‑linear embedding preserves curvature and multi‑modal structures in the original space, allowing the extraction of higher‑order components that retain physical meaning. In the NGC 1068 analysis, the first four Kernel PCA components each reveal distinct morphological features: the starburst ring, the circumnuclear disk (CND) around the active galactic nucleus (AGN), and a molecular outflow extending to ~400 pc. Notably, components three and four, which are essentially noise in conventional PCA, contain coherent structures when examined through the kernel approach.

While Kernel PCA provides the transformed components, it does not indicate which molecular lines drive each component. To solve this “black‑box” problem, the authors employ Kernel SHAP, a model‑agnostic implementation of Shapley values that approximates the contribution of each input feature to a given output. By computing SHAP values for each Kernel PCA component, the study quantifies the importance of individual lines. The analysis shows that HCO⁺ (J=1–0) has the strongest positive SHAP contribution to the fourth component, which spatially coincides with the outflow region. This result is corroborated by an independent LTE column‑density analysis, which finds an enhanced HCO⁺ abundance in the outflow relative to the surrounding disk. The authors interpret the enhancement as a consequence of strong ultraviolet radiation and high gas density promoting ion‑molecule chemistry (e.g., CO + H₃⁺ → HCO⁺ + H₂).

The paper also presents Mahalanobis‑distance‑based two‑dimensional scatter plots for selected line pairs, illustrating that many pairs exhibit two distinct linear trends (shallow and steep slopes). Such bimodal behavior reflects underlying variations in physical conditions and creates the non‑linear manifolds that Kernel PCA successfully captures.

Methodologically, the authors emphasize the importance of standardizing features, regridding to a spatial scale (150 pc) that approximates independent giant molecular clouds, and ensuring that the number of samples exceeds the number of features to avoid statistical instability. They note that for future applications involving dozens of lines, high‑dimensional statistical techniques may be required.

In summary, the combined Kernel PCA + Kernel SHAP framework extends the interpretive power of multi‑line analyses beyond the linear regime, enabling the discovery of subtle chemical signatures such as the HCO⁺ enrichment in the NGC 1068 outflow. The approach is scalable to the rapidly growing ALMA line surveys and offers a systematic way to extract, visualize, and physically interpret non‑linear patterns in large spectroscopic datasets, paving the way for more detailed studies of AGN feedback, starburst activity, and molecular outflows across diverse galactic environments.