# Retrieving interpretability to support vector machine regression models in dynamic system identification

## TL;DR

This paper introduces a method to make Support Vector Machine models more interpretable by revealing how each input affects the output in dynamic systems.

## Contribution

A post-hoc functional decomposition algorithm using Non-linear Oblique Subspace Projections (NObSP) is proposed for SVM regression interpretability.

## Key findings

- NObSP effectively retrieves partial nonlinear dynamics of each sub-system in benchmark simulations.
- The proposed method reduces computational complexity from 𝒪(N³) to 𝒪(Nd²).
- The approach decouples blended dynamics without losing predictive accuracy.

## Abstract

Black-box models, particularly Support Vector Machines (SVM), are widely employed for identifying dynamic systems due to their high predictive accuracy; however, their inherent lack of transparency hinders the understanding of how individual input variables contribute to the system output. Consequently, retrieving interpretability from these complex models has become a critical challenge in the control and identification community. This paper proposes a post-hoc functional decomposition algorithm based on Non-linear Oblique Subspace Projections (NObSP). The method decomposes the output of an already identified SVM regression model into a sum of partial (non)linear dynamic contributions associated with each input regressor. By operating in the non-linear feature space, NObSP utilizes oblique projections to mitigate cross-contributions from correlated regressors. Furthermore, an efficient out-of-sample extension is introduced to improve scalability. Numerical simulations performed on benchmark Wiener and Hammerstein structures demonstrate that the proposed method effectively retrieves the underlying partial nonlinear dynamics of each sub-system. Additionally, the computational analysis confirms that the proposed extension reduces the arithmetic complexity from 𝒪(N3) to 𝒪(Nd2), where d is the number of support vectors. These findings indicate that NObSP is a robust geometric framework for interpreting non-linear dynamic models, offering a scalable solution that successfully decouples blended dynamics without sacrificing the predictive power of the black-box model.