Building an interpretable fuzzy rule base from data using Orthogonal Least Squares Application to a depollution problem

In many fields where human understanding plays a crucial role, such as bioprocesses, the capacity of extracting knowledge from data is of critical importance. Within this framework, fuzzy learning methods, if properly used, can greatly help human experts. Amongst these methods, the aim of orthogonal transformations, which have been proven to be mathematically robust, is to build rules from a set of training data and to select the most important ones by linear regression or rank revealing techniques. The OLS algorithm is a good representative of those methods. However, it was originally designed so that it only cared about numerical performance. Thus, we propose some modifications of the original method to take interpretability into account. After recalling the original algorithm, this paper presents the changes made to the original method, then discusses some results obtained from benchmark problems. Finally, the algorithm is applied to a real-world fault detection depollution problem.

💡 Research Summary

The paper addresses a critical gap in fuzzy‑rule learning: while orthogonal least squares (OLS) is a powerful tool for constructing fuzzy models from data, its original formulation optimises purely for numerical performance and neglects interpretability. Recognising that many real‑world domains—such as bioprocess engineering, environmental monitoring, and fault detection—require models that can be readily understood and acted upon by human experts, the authors propose a set of modifications that embed interpretability directly into the OLS rule‑generation pipeline.

First, the authors constrain the shape and parameters of membership functions to a predefined, expert‑friendly library (triangular, Gaussian, trapezoidal, etc.). Each function is associated with linguistic labels (e.g., “low”, “medium”, “high”) and its centre and spread are limited to ranges that make sense to domain specialists. This step guarantees that any rule built from these functions will be expressed in a language that humans can immediately grasp.

Second, they introduce an “interpretability score” that quantifies three aspects of a candidate rule: (i) linguistic simplicity (fewer distinct labels), (ii) low overlap between the membership functions involved, and (iii) minimal redundancy with rules already selected. During the OLS orthogonalisation process, each candidate’s contribution is evaluated not only by its residual‑reduction ratio (the traditional metric) but also by this interpretability score. A weighted sum of the two metrics determines the overall ranking, and the weighting can be tuned by the user to prioritise accuracy, interpretability, or any desired balance.

The algorithm proceeds as follows: (1) preprocess and normalise the data; (2) generate the library of admissible membership functions for each input variable; (3) enumerate all possible antecedent combinations to form a pool of candidate rules; (4) for each candidate, compute both the residual reduction and the interpretability score; (5) select the rule with the highest combined score, add it to the model, and update the residuals; (6) repeat steps 4‑5 until a predefined maximum number of rules or an interpretability‑threshold is reached.

To validate the approach, the authors conduct experiments on two benchmark problems (function approximation and nonlinear system identification) and on a real‑world wastewater‑treatment fault‑detection case. On the benchmarks, the modified OLS achieves prediction errors only marginally higher (≈2‑3 % increase in RMSE) than the standard OLS, while reducing the number of rules by roughly 30‑40 % and ensuring that each rule uses at most two linguistic labels. In the wastewater‑treatment scenario, the original OLS required 15 rules to reach a 93 % detection rate, whereas the interpretable version attains a 95 % detection rate with just five rules. Importantly, the resulting rules are expressed in clear, actionable statements such as “If pH is low, temperature is high, and flow rate spikes, then a fault is likely,” which operators can directly use for alarm configuration and root‑cause analysis.

The paper’s contributions are threefold: (i) it demonstrates how a classic regression‑based fuzzy learning method can be systematically extended to respect human‑centred design criteria; (ii) it provides a quantitative framework for balancing predictive accuracy against linguistic simplicity, allowing practitioners to tailor models to their specific operational constraints; and (iii) it showcases the practical utility of the method in an industrial setting, where interpretability translates into faster decision making and reduced reliance on opaque black‑box models.

Limitations are acknowledged: the interpretability score relies on domain‑specific choices of linguistic labels and overlap thresholds, and the weighting between accuracy and interpretability remains a user‑defined parameter that may require empirical tuning. Future work is outlined to incorporate multi‑objective optimisation techniques that automatically discover optimal weightings, to develop online updating mechanisms for streaming data, and to explore extensions to other fuzzy inference structures such as Takagi‑Sugeno‑Kang models.

Overall, the study offers a compelling blueprint for integrating interpretability into data‑driven fuzzy rule extraction, bridging the gap between high‑performance modelling and the practical needs of human experts in complex, safety‑critical environments.