XStacking: Advanced Stacking Techniques

• XStacking is a dual methodology that enhances radio interferometry by processing visibilities in the uv-plane and augments ensemble ML through explanation-guided meta-learning.
• In radio astronomy, XStacking circumvents image-plane convolution to preserve calibration details, reduce flux bias, and improve uncertainty quantification.
• In ensemble ML, XStacking integrates model-agnostic explanation vectors to boost interpretability and accuracy across diverse datasets.

XStacking denotes two distinct but technically advanced methodologies—one in radio interferometric data analysis and one in ensemble machine learning—that both address inherent limitations of conventional stacking by introducing structural innovations. In radio astronomy, XStacking (technically, uv-plane stacking) directly operates in the measurement (visibility) domain to recover faint ensemble signals robustly, a method vital for upcoming Square Kilometer Array (SKA) applications. In ensemble learning, XStacking refers to a framework that fundamentally integrates explanation-based feature transformations into the meta-learning stage, rendering ensemble predictions inherently interpretable while enhancing predictive performance. Both approaches rigorously extend stacking beyond traditional paradigms by incorporating information otherwise marginalized or obfuscated by image-space or black-box aggregation.

1. XStacking in Radio Interferometry: Core Formulation and Theoretical Framework

XStacking in the context of radio interferometry refers to the ensemble combination of visibilities in the uv-plane—the Fourier domain where observed sky signals are recorded as complex correlations—rather than conventional stacking in image space. Let $V(u,v,w)$ represent calibrated visibility data from baseline $\mathbf{B} = (u,v,w)$, with $N$ target positions $\{S_k\}_{k=1}^N$ within a single pointing at phase center $S_0$. The XStacking operator $S$ acts on a set of visibilities $\{V_i(u,v,w)\}_i$, producing a stacked visibility

$$V_\text{stack}(u,v,w) = \frac{ \displaystyle\sum_{i} W_i V_i(u,v,w) \left[ \sum_{k=1}^N \frac{1}{A(S_k)} w_k e^{2\pi i\, \mathbf{B}_i \cdot (S_0 - S_k)/\lambda} \right] }{ \displaystyle\sum_{i} W_i \sum_{k=1}^N w_k }$$

where $A(S_k)$ is the primary-beam correction, $w_k$ is the stacking position weight (e.g., $w_k=1/\sigma_k^2$ for local rms), and $W_i$ is the per-visibility weight from calibration. The phase factor recenters each target's signal onto the common $S_0$. Calibration must have already corrected for instrumental response.

Key operational advantages include conservation of data volume (no replication), full parallelization, and preservation of the uv dataset structure, allowing downstream selection or masking by baseline, time, or frequency—critical for analyzing calibration artefacts and spatial filtering effects (Knudsen et al., 2015).

2. Contrasts with Image-Plane Stacking

In image-plane stacking, data are combined after imaging and deconvolution, typically by extracting and averaging cutouts around each target. The process implicitly convolves the stack with the synthesized beam $B(l,m)$ and entangles error propagation with correlated noise (e.g., sidelobes, deconvolution artefacts). Spatial filtering effects—such as the loss of sensitivity to extended emission or residuals from imperfect CLEANing—are baked into the result, complicating any per-baseline mitigation. XStacking circumvents these limitations by enabling baseline- and frequency-specific filtering and preserving the measurement domain’s linearity. Simulation studies consistently show reduced bias and more reliable uncertainty quantification for flux and size recovery in the uv-stacked regime (Knudsen et al., 2015).

3. Simulation Studies and Quantitative Evaluation

Robust benchmarking of XStacking versus image-plane stacking employs Monte Carlo simulations using JVLA and ALMA configurations. Targets include both unresolved and slightly resolved sources, overlaid with realistic foregrounds exhibiting diverse angular sizes and fluxes. Key metrics:

• Flux bias: Image-plane stacking can underestimate true fluxes by up to 10% when affected by residual foregrounds or CLEAN artefacts; XStacking remains unbiased (within ~1%).
• Recovered size: uv-plane model-fitting yields uncertainties $\sigma_{\theta,\mathrm{uv}} \sim 0.1''$ versus $\sim 0.2''$ for image-plane model-fitting.
• Signal-to-noise (S/N): Excluding problematic baselines in XStacking improves S/N by 10–20% post-filtering.
• Computational efficiency: XStacking scales linearly with the number of visibilities and is amenable to parallel architectures (Knudsen et al., 2015).

Empirical application to VLA and ALMA data confirms these trends: amplitude–baseline diagnostics expose mixed compact and extended source structure only accessible in the uv domain.

4. Implementation Strategies for SKA-Scale Data

Owing to the prohibitive data rates anticipated for SKA, strategies are needed to preserve XStacking’s benefits:

1. Real-time stacking queue: Users pre-commit stacking positions and weights, with the calibrated, phased stack generated in near-real-time during survey calibration (resulting in compact MeasurementSets).
2. uv-archive retention: Moderate averaging in time/frequency provides a reduced yet stacking-compatible calibrated uv archive (data volume ~10–20% of raw).

Best practices entail:

• Per-visibility retention of weight and phase center metadata,
• Rebinning to control image-plane smearing ($\Delta l_\text{smear} \leq 0.1''$ at high frequency),
• Avoidance of uv-cell averaging exceeding 10% of the smallest fringe spacing, and
• Full polarization and spectral channel fidelity.

A recommended software approach is to offer a user-accessible library or task (e.g., CASA task “xstack”) implementing the prescribed uv-stacking equation directly on such data (Knudsen et al., 2015).

5. XStacking in Ensemble Machine Learning: Explanation-Guided Stacked Learning

XStacking in machine learning denotes a principled augmentation of the two-stage stacking architecture. Standard stacking aggregates base model predictions $\hat{y}_i = (f_1(x_i), ..., f_K(x_i))$ as features for the meta-learner $E$, but this approach suffers from low meta-space diversity (when base predictions are correlated) and opacity of decision rationale.

XStacking addresses these limitations by concatenating model-agnostic Shapley value explanation vectors, $\phi_k(x_i)$, from each base learner, yielding the feature set

$$\hat{y}_i^* = [\phi_1(x_i) \| \cdots \| \phi_K(x_i)] \in \mathbb{R}^{K \cdot d}$$

for meta-learning. Shapley values $\phi_j$ are defined for set function $v:2^F\rightarrow\mathbb{R}$ (with $F$ the feature set of size $d$) as

$$\phi_j = \sum_{S\subseteq F\setminus \{j\}} \frac{|S|!(|F|-|S|-1)!}{|F|!}\left[v(S\cup\{j\})-v(S)\right]$$

encoding the marginal contribution of each feature $j$ across all orderings, computed for each base $f_k$ via a SHAP implementation (Garouani et al., 23 Jul 2025). The meta-learner $E$ then solves

$$\min_{\theta} \frac{1}{m} \sum_{i=1}^m \ell(E(\hat{y}_i^*; \theta), y_i) + \lambda \Omega(\theta)$$

where $\ell$ is the loss and $\Omega$ a regularizer.

6. Empirical Findings and Interpretability Trade-Offs

XStacking was evaluated across 29 datasets (17 classification, 12 regression) using a diverse set of base learners (decision trees, linear/logistic regression, multilayer perceptrons), with both SVM and XGBoost meta-learners. Key findings:

• Classification: XStacking matched or improved upon vanilla stacking on 16/17 datasets using an SVM meta-learner; notable accuracy increases include +2.6% (Adult) and +5.9% (Vehicle).
• Regression: Lower MSE observed on 11/12 tasks; e.g., cpu_small, MSE reduced from 22.4 (vanilla) to 11.3 (SVM XStacking) and 7.6 (XGBoost XStacking).
• Interpretability metrics: High local surrogate fidelity ($R^2>0.9$), meta-learner focus on a sparse set of SHAP features (3–5 per prediction), and stable top-k feature attributions (Jaccard $\sim$0.8 compared to $\sim$0.6 for single-model explanations).

The computational trade-off is the $O(mKd)$ cost of SHAP value computation, manageable for moderate-scale tasks and highly parallelizable (Garouani et al., 23 Jul 2025).

7. Implementation Recommendations and Limitations

Implementation guidelines for XStacking include:

• Assemble a diverse ensemble of base learners, trained using K-fold cross-validation,
• Compute Shapley (SHAP) explanations post-training for each base learner,
• Concatenate and cache all explanation vectors for meta-learner training,
• Utilize meta-learners amenable to high-dimensional, semantically organized features; tune regularization to avoid overfitting in the enriched space.

Key limitations:

• Increased memory and compute requirements for explanation storage,
• SHAP dependence (though alternative explanation methods may be explored in future work),
• Currently restricted to two-layer ensembles; generalizations to deeper stacks or sequence ensembles remain open research topics (Garouani et al., 23 Jul 2025).

A plausible implication is the flexibility of XStacking to be adapted for multi-modal input domains and for ensemble regularization via explicit Shapley-aligned constraints, as proposed for future developments.

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References:

• "Stacking of SKA data: comparing uv-plane and image-plane stacking" (Knudsen et al., 2015)
• "XStacking: Explanation-Guided Stacked Ensemble Learning" (Garouani et al., 23 Jul 2025)