Source-linked AI summary
Discovery of fully efficient fault indicators along a data-based diagnosis process
Igor Bezmaternykh, Louise Travé-Massuyès, Elodie Chanthery
TL;DR
Fault-diagnosis trees need interpretable multivariate relations, but DT4X optimizes target-pair separation without controlling how other classes split. DT4X+ changes training-set construction and the loss to preserve non-target class coherence; results show dynamic-dataset gains, while static outcomes vary.
Problem
DT4X optimizes each split for two target classes, leaving other classes unconstrained and potentially fragmented, which can hinder interpretability and diagnostic performance.
Source: §4.1 · p. 3Method
DT4X+ includes all node classes in symbolic regression and modifies the training data and fitness loss to penalize fragmentation of non-target classes.
Source: §4.3 · p. 4, §6 · p. 6Results
DT4X+ gives consistent improvements on dynamic datasets, while static results range from identical polybox scores to slight subtractor decreases.
Source: §5.2 · p. 5Takeaways & Limitations
For dynamic systems, performance gains can justify DT4X+’s added training cost, while original DT4X remains competitive for simpler static systems.
Source: §5.3 · p. 6Training time increases significantly with DT4X+, reaching nearly five times DT4X on the two-tank system because fragmentation loss evaluates non-target class distributions.
Source: §5.3 · p. 5
Abstract
from arXiv · showhide
The integration of model-based and data-driven paradigms provides a powerful framework for fault diagnosis by combining the interpretability of analytical redundancy relations, i.e., input-output relations that are used as diagnosis indicators in model-based diagnosis, with the adaptability of learning techniques. DT4X is a recent diagnosis algorithm that uses symbolic regression to generate multivariate relations leveraging some properties of analytical redundancy relations and uses them as split functions in a decision tree. However, its symbolic regression procedure optimizes only the separation between two selected classes at each node, often fragmenting the remaining classes and degrading both interpretability and diagnosis performance. This paper introduces DT4X+, an enhanced version of DT4X that modifies the construction of training sets and the symbolic-regression loss so that expressions separate the target classes while preserving the coherence of non-target classes. The resulting relations become fully consistent with ARR properties and lead to more informative splits, improved robustness, and better performance on dynamic-system datasets. Experiments conducted on several benchmark systems demonstrate the benefits of this enhanced formulation.
1. INTRODUCTION
Data-based diagnosis can adapt when expert knowledge is limited, but fault diagnosis still needs interpretable indicators that capture multivariate behavior. DT4X+ addresses DT4X’s fragmentation of non-target classes by incorporating them into symbolic regression, yielding relations more consistent with ARR properties.
1. INTRODUCTION
- Traditional univariate trees cannot capture the multivariate relations used to distinguish nominal and faulty behavior in model-based diagnosis.Analytical redundancy relations (ARRs) capture such relations when an analytical model is known.Source: §1 · p. 1
- DT4X uses symbolic regression to generate expressions that leverage ARR properties as decision-tree split functions.This reconnects data-driven learning with concepts from model-based diagnosis.Source: §1 · p. 1
- DT4X optimizes separation between two target classes but leaves non-target behavior unconstrained, so those classes can fragment across branches.Because ARR behavior is consistent within each class, this can make splits less informative and require additional relations deeper in the tree.Source: §1 · p. 1
- DT4X+ includes all classes in symbolic regression and penalizes non-target fragmentation, producing relations consistent with ARR properties and more coherent, interpretable trees.The paper also describes these relations as fully efficient fault indicators.Source: §1 · pp. 1–2
- The paper reviews related work, presents DT4X and its DT4X+ extension, evaluates both on static and dynamic systems, and concludes with future perspectives.Source: §1 · p. 2
2. RELATED WORK
Prior multivariate-tree approaches increase split expressiveness through linear combinations, engineered features, predefined functions, or embedded nonlinear classifiers, but each has constraints. Symbolic-regression trees can discover nonlinear relations; DT4X further incorporates ARR properties, and DT4X+ addresses its treatment of non-target classes.
2. RELATED WORK
- Linear oblique splits are more expressive than univariate splits, but their linearity limits their ability to capture nonlinear fault-indicator relations.Examples include OC1 and its variants (Murthy et al. (1993); Norouzi et al. (2015)) and continuous-optimization methods (Utgoff (1989); Wickramarachchi et al. (2016)).Source: §2 · p. 2
- Synthetic feature expansion can capture nonlinear relationships but scales poorly, while predefined nonlinear splits depend heavily on prior domain knowledge.The cited approaches include algebraic feature combinations (Kou et al. (2020)) and a limited set of predefined split functions (Hutchison et al. (2024)).Source: §2 · p. 2
- Kernel and neural tree splits provide flexible nonlinear boundaries, but opaque internal parametrization reduces interpretability for fault diagnosis.The cited examples include support-vector approaches (Bennett and Blue (1998); Montañana et al. (2021)) and neural or perceptron trees (Balestriero (2017); Yıldız et al. (2001)).Source: §2 · p. 2
- SREDT uses genetic programming and split impurity to select nonlinear expressions, but its arbitrary threshold does not make them ARRs (Fong and Motani (2024)).Source: §2 · p. 2
- DT4X learns symbolic relations leveraging ARR properties, including zero values for the nominal class, and uses them as decision-tree splits (Goupil et al. (2024)).Its expressions are automatically derived from data and serve as interpretable split functions.Source: §2 · p. 2
- DT4X focuses each split on a target pair, so other classes may be arbitrarily divided, contrary to the ARR requirement that samples within a class behave consistently.DT4X+ incorporates all classes and penalizes non-target fragmentation to preserve ARR properties and produce coherent splits.Source: §2 · p. 2
3. THE TREE BASED DIAGNOSIS METHOD DT4X
DT4X recursively builds a diagnosis tree by using symbolic classification to find multivariate expressions that separate class pairs through behavior around zero. Successful expressions split a node; if no suitable expression is found, the node becomes a majority-class leaf.
3. THE TREE BASED DIAGNOSIS METHOD DT4X
- At each node, DT4X considers sufficiently represented class pairs, prioritizing pairs of the nominal class with another class when the nominal class is sufficiently represented.Otherwise, it considers all ordered pairs among the sufficiently represented classes.Source: §3 · p. 2
- For each pair, DT4X samples the two target classes, adds nominal samples when a non-nominal class is involved, and balances the resulting sets.Symbolic classification then uses a threshold around zero and log loss as its fitness criterion.Source: §3 · p. 2
- The symbolic relation aims to evaluate close to zero for one target class and non-zero for the other; a sufficiently separating expression splits the node recursively.If no candidate expression reaches the required separation, DT4X labels the leaf with the majority class.Source: §3 · p. 3
- DT4X grows the tree breadth-first, placing a symbolic expression as a fault indicator at each internal node and fault classes at the leaves.Source: §3 · p. 3
4. ENHANCING DT4X WITH DT4X+
DT4X+ addresses non-target class fragmentation by changing both symbolic-regression training sets and the loss function. Its revised objective favors target-class separation while preserving the coherence of non-target classes.
4.1 Identification of the problem
- DT4X fragments non-target classes because its split loss considers only the selected target pair, potentially forcing deeper trees with additional relations.The problem is more pronounced with many fault classes or heterogeneous fault signatures.Source: §4.1 · p. 3, Figure 1 · p. 3
4.2 DT4X+ revised loss
- DT4X+ adds a fragmentation loss based on normalized, per-class Gini impurity to the target-pair discrimination loss.A non-target class scores zero when kept in one branch and one when split evenly; the class scores are weighted by relative frequency.Source: §4.2 · pp. 3–4
- The fragmentation penalty affects the total loss only when target-pair accuracy exceeds 0.97.The passage derives this threshold from the target-class log-loss.Source: §4.2 · p. 4
4.3 DT4X+ algorithm
- DT4X+ includes every class in the training set and combines target-class discrimination with a penalty for fragmenting non-target classes.Target classes are balanced as before, while other classes retain their original proportions; pair selection, recursive tree construction, and stopping criteria remain as in DT4X.Source: §4.3 · p. 4
- During training, DT4X+ favors expressions that separate the target pair while keeping non-target classes coherent.The candidate-expression fitness is the sum of the DT4X target-class classification loss and the non-target fragmentation loss.Source: §4.3 · p. 4
5. EXPERIMENTS
Experiments compare DT4X+ with DT4X on static and dynamic fault-diagnosis datasets, assessing diagnostic performance and computational cost. DT4X+ improves results on dynamic systems, with higher training time but nearly unchanged inference latency.
5.1 Experimental Setup
- DT4X+ is evaluated on four fault-diagnosis datasets, with DT4X as the baseline and accuracy, F1-score, training time, and inference latency as metrics.The datasets comprise static polybox and subtractor systems and dynamic two-tank and superheater systems; temporal features were generated for the dynamic datasets.Source: §5.1 · p. 5
5.2 Evaluation Results
- DT4X+ improves accuracy and F1-score on dynamic datasets, while polybox results are identical and subtractor global metrics decline slightly.On the subtractor circuit, accuracy decreases by 0.012 and F1-score by 0.058, despite improved isolation of a subset of class 1 without false positives.Source: Table 1 · p. 5, Table 2 · p. 5, §5.2 · p. 5
5.3 Impact on computation time
- Training time increases with class and sample counts, reaching nearly five times DT4X on two-tank and roughly double on subtractor and superheater, while inference latency remains almost identical.On polybox, the training-time impact remains below 20%.Source: §5.3 · pp. 5–6
- For training-time-sensitive use, DT4X+ is particularly advantageous on dynamic systems, while DT4X remains competitive on simpler static systems.The recommendation reflects the dynamic-system performance gains relative to the additional training cost.Source: §5.3 · p. 6
6. CONCLUSION
DT4X+ modifies training-set construction and the symbolic-regression loss to include non-target classes and penalize their fragmentation. Experiments show improved diagnostic performance, especially on dynamic systems, at the cost of increased training time.
6. CONCLUSION
- DT4X+ addresses DT4X’s non-target-class fragmentation by including all classes in symbolic regression and penalizing dispersion across branches.The resulting relations preserve ARR principles and yield more coherent, informative class separation.Source: §6 · p. 6
- Experiments on static and dynamic systems show improved diagnostic performance, especially on dynamic datasets with more complex class interactions and temporal behaviors.DT4X+ relations also have defined sensitivity for every class, unlike some DT4X fault indicators.Source: §6 · p. 6
- DT4X+ requires more training time, with the increase depending on the number of classes and samples; future work will integrate model-based constraints more deeply.The authors state this could strengthen the connection between data-driven learning and diagnosis meta-knowledge.Source: §6 · p. 6