An Explainable Pipeline for Machine Learning with Functional Data

TL;DR

The paper presents VEESA, an explainable ML pipeline for functional data that jointly accounts for vertical and horizontal variability via elastic functional PCA and provides interpretable explanations in the original data space using $PFI$ and visualizations. By smoothing, separating amplitude/phase, and projecting into efPCs, VEESA enables model training with uncorrelated predictors and yields interpretable variability directions linked to predictions. The approach is demonstrated on high-consequence tasks—H-CT material classification and inkjet printer identification from Raman spectra—showing interpretable, competitive performance and actionable insights for domain experts. The work highlights the importance of combining statistical description of functional structure with machine learning and outlines avenues for extending explainability to decision-makers and broader methodological refinements.

Abstract

Machine learning (ML) models have shown success in applications with an objective of prediction, but the algorithmic complexity of some models makes them difficult to interpret. Methods have been proposed to provide insight into these "black-box" models, but there is little research that focuses on supervised ML when the model inputs are functional data. In this work, we consider two applications from high-consequence spaces with objectives of making predictions using functional data inputs. One application aims to classify material types to identify explosive materials given hyperspectral computed tomography scans of the materials. The other application considers the forensics science task of connecting an inkjet printed document to the source printer using color signatures extracted by Raman spectroscopy. An instinctive route to consider for analyzing these data is a data driven ML model for classification, but due to the high consequence nature of the applications, we argue it is important to appropriately account for the nature of the data in the analysis to not obscure or misrepresent patterns. As such, we propose the Variable importance Explainable Elastic Shape Analysis (VEESA) pipeline for training ML models with functional data that (1) accounts for the vertical and horizontal variability in the functional data and (2) provides an explanation in the original data space of how the model uses variability in the functional data for prediction. The pipeline makes use of elastic functional principal components analysis (efPCA) to generate uncorrelated model inputs and permutation feature importance (PFI) to identify the principal components important for prediction. The variability captured by the important principal components in visualized the original data space. We ultimately discuss ideas for natural extensions of the VEESA pipeline and challenges for future research.

Figures (22)

• Figure 1: (Top Left) Training data functions from the shifted peak simulated data. (Top Right) The true, cross-sectional, and aligned functional means. (Bottom Left) The aligned functions. (Bottom Right) The warping functions from the alignment of the functions.
• Figure 2: Plot of the principal directions for interpreting the functional variability captured by jfPC 1 from the shifted peaks data shown in Figure \ref{['fig:fig1']}.
• Figure 3: (Top) Proportion of variation explained by the jfPCs computed from the shifted peaks data. (Middle) Boxplots of PFI values across replicates associated with each jfPC from the shifted peaks data random forest computed on the training data. The blue diamonds represent the PFI value averaged over replicates. (Bottom) Same as middle plot but computed on the shifted peaks test data.
• Figure 4: Plot of the principal directions for interpreting the functional variability captured by jfPC 2 from the shifted peaks data shown in Figure \ref{['fig:fig1']}.
• Figure 5: Observed (top row), smoothed and aligned (middle row), and warping functions (bottom row) of a subset of 1,000 H-CT signatures for each material.