Bias-Adjusted Augmented Functional Random Forests for Interpretable Classification of Functional Trajectories

Abstract

INTRODUCTION
Functional data analysis treats time-indexed observations as curves rather than isolated measurements, allowing classification models to exploit signal level, local velocity and curvature. Random Forests provide accurate non-parametric classifiers for complex predictors. When functional trajectories are augmented with first and second derivatives and summarized through functional principal component scores, the resulting predictors may be highly correlated. In this setting, standard variable-importance measures may overstate the role of components that mainly act as proxies for other functional information.
OBJECTIVES
This study proposes a bias-adjusted extension of Augmented Functional Random Forests for interpretable classification of functional trajectories. The aim is not only to combine the original curve, its first derivative and its second derivative within an ensemble classifier, but also to obtain importance measures that are less affected by redundancy among functional components. The central methodological contribution is a conditional randomization scheme for augmented functional principal components, combined with a stability criterion for identifying components that are both informative and robust.
METHODS
Observed trajectories are smoothed into continuous functions and augmented through successive derivatives. A separate Functional Principal Component Analysis is performed for each functional block, namely the original function, the first derivative and the second derivative. The resulting scores are used as predictors in an augmented Random Forest. Because components extracted from different blocks may describe related aspects of the same dynamic process, variable importance is assessed through Conditional Permutation Importance rather than marginal permutation alone. The randomization of each component is performed conditionally on correlated components belonging to the same functional rank, following the logic of conditional variable importance for Random Forests. The conditional loss in predictive accuracy is interpreted as the component-specific contribution not already explained by alternative functional representations. This measure is then combined with a Conditional Importance Stability Index based on repeated refitting of the forest.
RESULTS
The proposed approach separates marginal relevance from conditional relevance in augmented functional components. Components that appear important under ordinary permutation may lose relevance after conditioning, suggesting an inflation effect due to functional redundancy. Conversely, components that preserve high conditional importance and stable behaviour across refitted forests provide stronger evidence of a genuine predictive role. The resulting rankings are more selective, less driven by proxy effects and more coherent with the dynamic interpretation of the trajectories. This distinction is especially relevant in biomedical applications, where signal level, velocity and curvature may contain overlapping but non-equivalent information.
CONCLUSIONS
Bias-adjusted Augmented Functional Random Forests preserve the predictive flexibility of ensemble learning while improving the interpretability of functional classification. Conditional randomization reduces the risk of attributing importance to redundant components, whereas the stability index favours predictors whose contribution is robust across repeated model fits. The framework therefore links classification accuracy with a more controlled assessment of variable relevance, supporting a clearer interpretation of the dynamic information contained in functional trajectories.