# A Divisive Unsupervised Feature Selection Approach for Explainable Remaining Useful Life Prediction

Mouhamadou Lamine Ndao, Genane Youness, Ndèye Niang, Gilbert Saporta

► To cite this version:

Mouhamadou Lamine Ndao, Genane Youness, Ndèye Niang, Gilbert Saporta. A Divisive Unsupervised Feature Selection Approach for Explainable Remaining Useful Life Prediction. Database and Expert Systems Applications. DEXA 2025, Aug 2025, Bangkok, Thailand. pp.270-286, 10.1007/978-3-032-02088-8_19. hal-05229009

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Submitted on 8 Sep 2025

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# A Divisive Unsupervised Feature Selection Approach for Explainable Remaining Useful Life Prediction

Mouhamadou Lamine Ndao¹,²[0009-0009-9507-1955], Genane Youness¹,²[0009-0003-9038-3489], Ndèye Niang²[0000-0002-6109-9935], and Gilbert Saporta²[0000-0002-3406-5887]

¹ Laboratoire LINEACT CESI {mlndao, gyouness}@cesi.fr

² Laboratoire CEDRIC MSDMA {n-deye.niang-keita, gilbert.saporta}@cnam.fr

**Abstract.** Predicting the Remaining Useful Life (RUL) in maintenance often encounters challenges such as high dimensionality, feature redundancy, and limited explainability. This paper presents a novel approach that combines Interpretable Divisive Feature Clustering (IDFC) with Long Short-Term Memory (LSTM) networks. The IDFC algorithm leverages the strengths of variable clustering methods (VARCLUS) and the Clustering of Variables around Latent Components (CLV) to identify significant features and non-orthogonal latent components. This method enables effective dimensionality reduction by selecting key features rather than combining them. Integrating IDFC with a single-layer LSTM and Shapley Additive Explanations (SHAP) results in a robust and interpretable framework for RUL prediction, achieving a balance between accuracy and transparency. Experimental results on a bearing dataset show that the IDFC + LSTM model outperforms traditional methods while enhancing interpretability through the identification of key energy-related features which influence the RUL prediction more.

**Keywords:** Dimensionality reduction · XAI · LSTM · variable classification · Predictive Maintenance.

# 1 Introduction and Related Works

## 1.1 Introduction

One key challenge in predictive maintenance is dimension reduction due to data availability in the industry, particularly with the rise of the Internet of Things (IoT) connected objects (16) and the effectiveness of Deep Learning models such

**Preprint notice.** This is the submitted version of the paper "A Divisive Unsupervised Feature Selection Approach for Explainable Remaining Useful Life Prediction". It has not undergone peer review or post-submission improvements. The Version of Record will appear in the DEXA 2025 proceedings (Springer Nature) and will be available at: https://link.springer.com/chapter/10.1007/978-3-032-02088-8_19.
as Long Short-Term Memory (LSTM, (6)) for Remaining Useful life (RUL) pre-
diction. This is due to the ability of models such as LSTM to tackle temporal
dependencies and handle complex datasets, such as multivariate time series sig-
nals, as shown in the literature (23; 25). Additionally, LSTM models address
the issue of vanishing gradients, which occurs when the hidden layer is unrolled
multiple times ((6)). However, the complexity of these models does not allow
one to understand the results they provide. They are often called black boxes
because their outputs are uninterpretable. This lack of transparency hinders
stakeholders' ability to assess their reliability.

Regarding the dimension reduction challenge, integrating multiple data sources
in industrial settings often leads to the inclusion of irrelevant features for RUL
prediction and the presence of highly correlated or redundant features. This
complicates the modeling process and hinders the interpretation of results by in-
troducing bias through multicollinearity and the presence of irrelevant features
(14; 19). This challenge is critical because high-dimensional data can lead to
problems such as overfitting, increased computational complexity, and reduced
interpretability. To address this challenge, numerous studies have explored vari-
ous techniques to reduce the number of features while preserving critical informa-
tion, which combines original features into synthetic features through techniques
such as PCA (e.g. Principal Component Analysis (1)), Autoencoder (EA) (24).
Even though these approaches perfectly correct feature redundancy, they are
not suitable when the aim is to use XAI method to explain the result of models.
These methods create a latent space represented as a combination of features,
making them inherently uninterpretable. As a result, even with XAI methods,
it is difficult to explain which feature most directly influences degradation, as it
lacks direct correspondence to the original features.

To address the lack of interpretability associated with synthetic features gen-
erated through dimension reduction, one solution is to employ a feature cluster-
ing method followed by data fusion within each feature cluster (4). This approach
reduces feature redundancy by generating sparse synthetic features, which ag-
gregate only features within relevant clusters. However, this method has several
limitations. Many approaches in the literature require predefined numbers of
clusters, and it is often noted that some clusters are atypical and require special
consideration. Furthermore, despite generating sparser synthetic features, the
literature highlights persistent challenges in interpreting results derived from
feature clustering methods, as the relationships between synthetic features and
original features often remain uninterpretable.

This study addresses two key challenges in predicting the RUL of mechanical
components, such as bearing elements: first, dimensionality reduction, which ad-
dresses feature redundancy and high dimensionality, and mitigates bias caused
by multicollinearity (14; 19) and second, the lack of explainability inherent in
RUL prediction models. We propose an Interpretable Divisive Feature Cluster-
ing (IDFC) approach to bridge the gap between predictive performance and
interpretability in industrial prognostics.
The paper is organized as follows: Section 1 presents an overview of the research problem and highlights key challenges in RUL prediction, particularly those related to dimensionality reduction and explainability. Section 2 presents the materials and methods, including the theoretical background (Subsection 2.1), the proposed methodology (Subsection 2.2), and the experimental setup (Subsection 2.7), which covers the dataset description, preprocessing steps, and hyperparameter configuration. Section 3 presents the results and discussion, comparing the proposed model with traditional approaches, analyzing its performance on benchmark datasets, and explaining its prediction using the SHAP method.

## 1.2 Related Works

Explainability and dimensionality reduction are two significant challenges in predicting Remaining Useful Life (RUL). Utilizing Explainable Artificial Intelligence (XAI) methods is essential for enhancing transparency in RUL prediction (7). In the context of RUL prediction, recent studies have employed dimensionality reduction (DR) techniques to manage the complexity of high-dimensional data. Methods such as Principal Component Analysis (PCA) effectively reduce data dimensions and improve model efficiency; however, they often compromise interpretability by transforming features into latent spaces. Some researchers, such as (20), have combined PCA with models like Random Forest (RF) or Support Vector Machine (SVM) to achieve reduced dimensionality while accurately predicting bearing RUL. Even when using less complex models, the dimensionality reduction approach does not always yield interpretable RUL predictions. Hybrid models that incorporate unsupervised dimensionality reduction techniques, such as EA-DNN (Deep Neural Networks) (17), PCA-CNN (Convolutional Neural Network)-LSTM (9), self-organizing map (SOM)-PCA-BiLSTM (10), and PCA-GCN (Graph Convolutional Network) (8), show promise in predicting RUL. However, these methods also face similar limitations in transparency. Advanced techniques like fractal dimensions and spatiotemporal GCNs (5) enhance accuracy by extracting spatial characteristics from signals but still lack transparency. Recent studies have integrated XAI methods to address this gap. For example, (3) and (25) utilize feature selection layers to rank contributions, while (26) combines feature clustering with SHAP values. However, this approach requires a predefined number of clusters and produces indirect explanations based on latent features (components). This underscores the critical need for explainability in dimensionality reduction, emphasizing the importance of providing synthetic features that align with domain knowledge and avoid opaque latent spaces.

To overcome these limitations and accurately predict bearing RUL using an unsupervised dimensionality reduction approach, we propose an LSTM-based model integrated with our Interpretable Divisive Feature Clustering (IDFC) method. Unlike traditional approaches, this novel framework automatically identifies the optimal number of features to retain, ensuring that predictions are grounded in domain-relevant features while enhancing both model efficiency and explainability.
## 2 Materials and methods

We denote $X = \{x^1, x^2, \dots, x^p\}$ a set of $p$ features $x_j$ describing $N$ instances; $\tilde{X} = \{c_1, c_2, \dots, c_K\}$ a projection of $X$ on a smaller latent space of dimension $K < p$, $\mathcal{P}_K = \{C_1, \dots, C_K\}$ is a partition of the features into $K$ clusters where each cluster $C_k$ is represented by the component $c_k$.

### 2.1 Background

There are several ways to reduce the dimension of $X$: by projection onto a low-dimensional $K < p$ space ($\tilde{X}$), resulting in a combination of features into components (PCA), by feature selection, or a combination of both: clustering and latent feature. The third approach identifies clusters of features and defines their latent components. Each cluster $C_k$ is represented by a latent feature (synthetic feature $c_k$) that summarizes, in the best way, its corresponding features, resulting in sparser and more interpretable components ((22; 4)). In the literature, two main algorithms are often used to perform feature clustering: The VARCLUS procedure of SAS software, which has never been the subject of scientific articles and clustering of variables around latent components (CLV) ((22), (21)).

VARCLUS is a divisive, top-down hierarchical algorithm. It starts by dividing the features into two clusters based on their correlation with the first two principal components obtained from a PCA of the entire set of features. The algorithm then iteratively refines these clusters until each cluster contains only one eigenvalue that is larger than one ($\lambda_1 > 1$) when performing PCA on that specific feature cluster. The condition on the eigenvalues gives VARCLUS two important advantages: the number of clusters $K$ is naturally obtained, and the resulting clusters, which are associated with a first large eigenvalue, are somewhat unidimensional. Clustering features into unidimensional blocks is a simple and efficient way to search for the so-called factor. Each block can then be represented by a single component combining only its features, which is necessarily sparse relative to the total number of features. When the cluster size is still too large, further simplification is needed. VARCLUS does not force orthogonality between cluster components and does not require the number of clusters $K$ to be specified in advance. However, the method can lead to many small clusters.

CLV algorithm (22) is an alternative to the VARCLUS approach. It is a k-means-like algorithm that clusters features around latent components by both identifying clusters of features and a latent component associated with each cluster. For a given number of clusters, $K$, CLV algorithm seeks a partition $\mathcal{P}_K$ of the features and $\tilde{X} = \{c_1, \dots, c_K\}$ of $K$ latent features, each associated with one cluster $k$, to maximize the internal coherence within the clusters. When examining the relationship between features using their covariance or correlation coefficients, the clusters found are called directional groups, regardless of the sign of the coefficients. In this case, the aim is to define the partition $\mathcal{P}_K$ and the group's latent features $\tilde{X}$ to obtain the optimal value $T^*$ of the CLV criterion,
so that:

$$T^* = \max_{(\delta_{k,j}, c_K)} \sum_{k=1}^{K} \sum_{j=1}^{p} \delta_{kj} \text{Cov}(x_j, c_k)^2, \quad (1)$$

under the constraint $||c_k|| = 1$ with $\delta_{kj} = 1$ if the j-th feature belongs to the cluster $C_k$, and $\delta_{kj} = 0$ otherwise. For each cluster $k$, $c_k$ is the first principal component of its features PCA. A comparison of the two described algorithms is given by Table 1. Unlike the VARCLUS algorithm, CLV reduces the number of clusters and does not produce a unidimensional feature cluster.

Both algorithms provide $\tilde{X}$ with latent features. Therefore, when used in conjunction with an XAI method, it may not be possible to directly explain the results provided by XAI method, even if all potential biases due to multicollinearity and redundancy are potentially reduced. If the explanation is unnecessary, it is generally also advantageous to concentrate on the most significant components, as it captures the essential patterns in the data. To correct all these issues, we propose the feature clustering approach described in the section below.

## 2.2 Interpretable Divisive Feature Clustering (IDFC)

Our proposed method integrates the strengths of both VARCLUS and CLV algorithms. IDFC approach involves three steps: an initialization step based on the VARCLUS algorithm, a clustering step based on the CLV algorithm, and a final step to extract interpretable features instead of latent features.

### Initialization step :

This step involves determining an initial partition $P_K$ using VARCLUS. However, VARCLUS may produce a large number of small clusters. To address this, clusters with only one feature are considered noise clusters and are merged into a single cluster. This initial partition is defined in the Algorithm 1 and illustrated in Fig. 1.

<figure><img src="image_1.png" /><figcaption>Fig. 1. Initialization of the partition using an extension of VARCLUS</figcaption></figure>

This algorithm provides an initial partition. The next step consists of using an approach based on the CLV algorithm to consolidate the initial partition and consider the positive and negative correlations between features.
**Algorithm 1 Initial Partition**

Input: $X = \{x^1, x^2, \dots, x^p\}$.

▷ Set of all features

Output: $\mathcal{P}_{\mathcal{K}'}$

▷ $K' = K + 1$ cluster merging all small clusters.

1: Perform PCA on $X$:

2: if $\lambda_2 > 1$ then:

Divide the $x^j$ of $X$ based on the first two components, denoted $c_1$ and $c_2$, obtained from a PCA on $X$.

3: end if

4: for $C_k \in \{C_1, C_2\}$ do

Repeat 1 and 2 on $X_1_{[\delta_{k,j}, j \in [1:p]]}$.

▷ Splitting

5: for $x^j \in X$ do:

Reassign it to the $C_k$ such that

$$ \max_{x^j \in C_k} r^2(x^j, c_k) $$

Stop the division when $|C_k| = 1$ or if $(\lambda_2^k < 1)$.

▷ Stopping Criterion

6: end for

7: end for

8: Obtain $\mathcal{P}_{\mathcal{K}}$ with a potentially high number of single-feature clusters.

9: Return $\mathcal{P}_{\mathcal{K}'}$ with $K' = K + 1$ cluster merging all small clusters.

**Refinement of the partition based on CLV algorithm with a K + 1 partition :**

In this step, the partition $\mathcal{P}_{\mathcal{K}}$ identified in the first step serves as the initialization partition. Then, a CLV algorithm is fitted using the 'K+1' approach proposed by (21), as illustrated in Fig. 2. It consists of finding $T_{\text{new}}$ given by :

$$ T_{\text{new}} = \max_{\delta_{jk}, c_k} \sum_{k=1}^{K} \sum_{j=1}^{p} \delta_{kj} \text{cov}^2(x_j, c_k) + \sum_{j=1}^{p} \left( 1 - \sum_{k=1}^{K} \delta_{kj} \right) \rho^2 \text{var}(x_j) \quad (2) $$

For each cluster $C_k$, $c_k$ is its first principal component. In Equation 2, an additional noise cluster can be represented by a prototypal feature that is expected to have the same correlation with all the observed features. This noise cluster is controlled with a tuning parameter $\rho$ to be chosen between 0 and 1. If $\rho = 0$, Equation 1 will be considered with no noise cluster. To find the refined partition, Algorithm 2 is considered.

**Feature extraction step :** This step involves selecting features according to Equation 3. To do this, the first principal component is computed for each cluster. Then, in each cluster, the most correlated feature is chosen as a prototype for each latent feature, given by :

$$ x^k = \arg \max_{x_j \in C_k} r^2(x_j, c_k) \quad (3) $$

Table 1 compares the proposed interpretable Feature Clustering with two of the most used feature clustering algorithms (CLV, VARCLUS). Unlike CLV,
**Algorithm 2 Refinement using CLV**

**Input:** $X = \{x^1, x^2, \dots, x^p\}; \mathcal{P}_{K'+1}$.

**Output:** the refined partition $\mathcal{P}_{K''+1}$ and the components of each cluster.

1: **for** each $C_k$ **do** ▷ (step 1)

2: Compute the first principal component $c_k$.

3: **end for**

4: **for** each $x^j$ **do**

5: Assign $x^j$ to the cluster that maximizes the internal homogeneity criterion $T_{new}$. ▷ (step 2)

6: **end for**

7: Repeat steps (1) and (2) until convergence.

8: **Return** $\mathcal{P}_{K''+1}$ and $c_1, \dots, c_{K''+1}$. ▷ The refined partition and the components of each cluster

<figure><img src="image_2.png" /><figcaption>**Fig. 2.** Partition refinement using CLV algorithm with a $K + 1$ strategy</figcaption></figure>

which requires the number of clusters to be specified beforehand, VARCLUS and the proposed method determine clusters without prior constraints. Additionally, while only CLV and the proposed method can distinguish positive and negative correlations, VARCLUS and the proposed method ensure unidimensional clusters, a feature absent in CLV. The proposed method also offers optional sparse components and is not sensitive to initialisation, similar to VARCLUS. Most importantly, it is the only method, compared to the other two, that produces interpretable results, making it a significant improvement over CLV and VAR-CLUS in terms of flexibility, handling of correlations, and overall interpretability.

<table><caption>**Table 1.** Comparison between CLV, VARCLUS and proposed method (IDFC)</caption><thead><tr><th></th><th>CLV</th><th>VARCLUS</th><th>IDFC</th></tr></thead><tbody><tr><td>Choice of Number of clusters K</td><td>✗</td><td>✓</td><td>✓</td></tr><tr><td>Unidimensionality</td><td>✗</td><td>✓</td><td>✓</td></tr><tr><td>Differentiate positive and negative correlations.</td><td>✓</td><td>✗</td><td>✓</td></tr><tr><td>Sparse components</td><td>✓</td><td>✗</td><td>✓</td></tr><tr><td>Initialization-sensitive</td><td>✗</td><td>✓</td><td>✓</td></tr><tr><td>Isolate noise features</td><td>✓</td><td>✗</td><td>✓</td></tr><tr><td>Interpretable result</td><td>✗</td><td>✗</td><td>✓</td></tr></tbody></table>
## 2.3 The RUL Prediction problem

The main objective of this analysis is to provide a parsimonious model to accurately predict the RUL and allow the use of XAI method to understand the model's result. Therefore, the first step will entail reducing the dataset dimension. Then, an optimal predictive model, denoted as $f$, will be identified to achieve the highest accuracy in RUL predictions. This is an optimization problem given by :

$$ \arg\min_{f \in \mathcal{F}} (\|y - f(x)\|) \qquad (4) $$

$\mathcal{F}$ is the space of all possible models, and $y$ the response feature (RUL) to predict. The RMSE (Root-Mean-Square Error) metric is used to choose the optimal prediction model $f$ on $\mathcal{F}$.

After predicting the RUL, an optimal local post-hoc XAI method, based on a surrogate model $g$, is identified. This method provides the optimal $\xi = g(x, f)$ to explain the feature importance corresponding to $RUL'_i = f(x_i)$.

## 2.4 Explainable Artificial Intelligence (XAI) in RUL prediction

In this study, our aim is to identify which part of the bearing impacts the RUL more. So, we use one of the most used, reliable and robust XAI post-hoc methods, the SHAP method (18; 11; 12). SHAP is a local post-hoc XAI method that identifies the most relevant features according to their feature importance.

For the SHAP method, the prediction $f(x)$ is considered as the final payoff in a game where the players are the features $j$. The Shapley value $\phi_j$ corresponds to the marginal contribution of feature $j$ in the prediction of $f(x)$, considering all possible combinations. According to theoretical considerations, Shapley values are the only values that meet three essential properties: Local Accuracy, which requires the explanation model to accurately reflect the behavior of the complex model in specific contexts; Missingness, which asserts that features designated as missing should not impact predictions; and Consistency, which states that if a feature's contribution increases, its assigned importance must not decrease.

## 2.5 Proposed Feature clustering-LSTM Network Model:
Experimental setting

The proposed pipeline for explainable RUL prediction in the context of predictive maintenance is illustrated in Figure 3. The process begins with data acquisition, during which raw sensor measurements are collected. In the feature extraction step, these measurements are transformed into meaningful features. A preprocessing phase follows, which includes normalization and data smoothing to reduce noise. A fixed-length sliding time window (TW) is then applied to emphasize the degradation phase during model training. The proposed IDFC algorithm is then applied for dimensionality reduction, grouping features into homogeneous clusters and selecting interpretable representatives from each cluster to minimize redundancy. The preprocessed data are used to train a single-layer
LSTM model. This architecture was chosen based on previous works (26; 11). Finally, the SHAP method provides clear insights into the impact of features on the model's predictions, ensuring a transparent interpretation of the results.

<figure><img src="image_3.png" /><figcaption>Fig. 3. Flowchart of the proposed method</figcaption></figure>

## 2.6 FEMTO Bearing Dataset

The dataset used is the bearing operation data collected by FEMTO-ST on the PRONOSTIA platform (13). Rolling bearings are crucial components in rotating machinery, and their failure can result in the malfunction of adjacent components and even the entire machine.

<figure><img src="image_4.png" /><figcaption>Fig. 4. (a) Normal and degraded bearing; (b) Overview of PRONOSTIA platform; (c) A vibration raw signal</figcaption></figure>

The FEMTO bearing dataset was created by accelerating the degradation of bearings on the PRONOSTIA experimental platform (cf. Fig 4) (a). This dataset
includes acceleration signals recorded in both horizontal and vertical directions. The sampling frequency is 25.6 kHz, with each sampling lasting 0.1 seconds, resulting in 2,560 data points recorded per sampling. The sampling interval is set at 10 seconds. Bearing RUL prediction consists of two stages. The first one involves extracting the degradation features from the measured vibration data of the target bearing and the second stage involves predicting the RUL according to the degradation features. According to (2), a total of 111 features were manually extracted from each of the signals in two orientations: horizontal and vertical. This study utilizes bearing data under three different operating conditions. The training and test sets used for this study are presented in Table 2. The target feature is the Time (or cycles) until bearing failure, calculated as follows: $RUL_{i,t} = T_i - t$, where: $RUL_{i,t}$ is the Remaining Useful Life of bearing $i$ at time step $t$, $T_i$ represents the failure time (i.e., the last cycle) of bearing $i$, $t$ is the current time step.

<table><caption>**Table 2.** Description of the PRONOSTIA Dataset: Bearing(sample size); Training and Test Sets under Different Conditions</caption><thead><tr><th></th><th>Condition 1</th><th>Condition 2</th><th>Condition 3</th></tr></thead><tbody><tr><th>Training sets</th><td>B 1-1 (2803)<br>B 1-2 (871)</td><td>B 2-1 (911)<br>B 2-2 (797)</td><td>B 3-1 (515)<br>B 3-2 (1637)</td></tr><tr><th>Test sets</th><td>B 1-3 (1802)<br>B 1-4 (1139)<br>B 1-5 (2302)<br>B 1-6 (2302)<br>B 1-7 (1502)</td><td>B 2-3 (1202)<br>B 2-4 (612)<br>B 2-5 (2002)<br>B 2-6 (572)<br>B 2-7 (172)</td><td>B 3-3 (352)</td></tr></tbody></table>

## 2.7 Experimental setting

Standardization is used to scale feature data to have a zero mean and unit variance, which helps facilitate model convergence. Additionally, data smoothing is performed using exponential smoothing, where the parameter $\alpha$ adjusts the weight assigned to recent versus historical observations. This effectively reduces noise in the sensor signals. A fixed-length sliding time window (TW) is then applied to transform the multivariate time series data. Table 3 summarizes the preprocessing parameters, as well as the hyperparameters, for the LSTM-based model applied to the bearing dataset.

## 2.8 Evaluation Metrics: RMSE and MAE

To evaluate the performance of our predictive models, we use two standard error metrics: the Root Mean Squared Error (RMSE) and the Mean Absolute Error (MAE). The RMSE is defined as follows: $RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (\hat{y}_i - y_i)^2}$; The MAE is defined as: $MAE = \frac{1}{n} \sum_{i=1}^{n} |\hat{y}_i - y_i|$. In these formulas, $\hat{y}_i$ represents the predicted values, and $y_i$ represents the true values for a set of $n$ observations. Both RMSE and MAE quantify the difference between predicted and actual values. Lower values for both metrics indicate better predictive performance.
<table><caption>**Table 3.** Hyperparameters used for the LSTM model</caption><thead><tr><th>Hyperparameters</th><th>Condition 1</th><th>Condition 2</th><th>Condition 3</th></tr></thead><tbody><tr><td>TW</td><td>45</td><td>20</td><td>50</td></tr><tr><td>Alpha</td><td>0.01</td><td>0.05</td><td>0.01</td></tr><tr><td>Nb layers</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Nodes</td><td>256</td><td>64</td><td>128</td></tr><tr><td>Dropout</td><td>0.3</td><td>0.3</td><td>0.3</td></tr><tr><td>Activation</td><td>tanh</td><td>relu</td><td>relu</td></tr><tr><td>Batch size</td><td>64</td><td>32</td><td>64</td></tr><tr><td>Learning rate</td><td>0.001</td><td>0.0001</td><td>0.001</td></tr><tr><td>Optimizer</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Nb features</td><td>33</td><td>33</td><td>31</td></tr><tr><td>Epochs</td><td>50</td><td>50</td><td>50</td></tr><tr><td>Standardisation</td><td colspan="3">Z-score</td></tr><tr><td>Data smoothing</td><td colspan="3">exponential smoothing</td></tr></tbody></table>

# 3 Results and discussion

## 3.1 Interpretable Divisive Feature Clustering results

Table 4 presents the clusters of features obtained, along with the number of features in each cluster (|C<sub>k</sub>|). Cluster C<sub>0</sub> corresponds to the noise cluster. In Condition 1, 17 features are identified as noise, while there are 22 noise features for Condition 2 and 20 for Condition 3. The mean number of features is shown in Fig. 5.

The results from the IDFC analysis differ depending on the experiment. For instance, both Condition 1 and Condition 2 yield 33 feature clusters in addition to one noise cluster, whereas Condition 3 produces 31 feature clusters along with a single noise cluster.

When analyzing the size of the obtained clusters, the minimum cluster size is 2 (specifically C<sub>3</sub>, C<sub>30</sub>, C<sub>31</sub>, and C<sub>32</sub>). Notably, some clusters contain a greater number of features. For example, in Condition 1, cluster C<sub>3</sub> includes 15 features, while cluster C<sub>24</sub> contains 12 features. This indicates that 15 features are represented by a single feature within cluster C<sub>3</sub>, and 12 features are represented by one feature in cluster C<sub>24</sub>.

<table><caption>**Table 4.** Distribution of features by clusters: C<sub>k</sub> represents cluster k and |C<sub>k</sub>| represents the number of features in cluster C<sub>k</sub></caption><tbody><tr><td rowspan="2">Condition 1</td><td>C<sub>k</sub></td><td>C<sub>0</sub></td><td>C<sub>1</sub></td><td>C<sub>2</sub></td><td>C<sub>3</sub></td><td>C<sub>4</sub></td><td>C<sub>5</sub></td><td>C<sub>6</sub></td><td>C<sub>7</sub></td><td>C<sub>8</sub></td><td>C<sub>9</sub></td><td>C<sub>10</sub></td><td>C<sub>11</sub></td><td>C<sub>12</sub></td><td>C<sub>13</sub></td><td>C<sub>14</sub></td><td>C<sub>15</sub></td><td>C<sub>16</sub></td></tr><tr><td>|C<sub>k</sub>|</td><td>17</td><td>3</td><td>7</td><td>15</td><td>6</td><td>6</td><td>5</td><td>9</td><td>4</td><td>2</td><td>6</td><td>2</td><td>13</td><td>4</td><td>3</td><td>6</td><td>3</td></tr><tr><td rowspan="2">Condition 2</td><td>C<sub>k</sub></td><td>C<sub>17</sub></td><td>C<sub>18</sub></td><td>C<sub>19</sub></td><td>C<sub>20</sub></td><td>C<sub>21</sub></td><td>C<sub>22</sub></td><td>C<sub>23</sub></td><td>C<sub>24</sub></td><td>C<sub>25</sub></td><td>C<sub>26</sub></td><td>C<sub>27</sub></td><td>C<sub>28</sub></td><td>C<sub>29</sub></td><td>C<sub>30</sub></td><td>C<sub>31</sub></td><td>C<sub>32</sub></td><td>C<sub>33</sub></td></tr><tr><td>|C<sub>k</sub>|</td><td>9</td><td>4</td><td>6</td><td>10</td><td>11</td><td>10</td><td>11</td><td>12</td><td>5</td><td>6</td><td>9</td><td>5</td><td>3</td><td>2</td><td>2</td><td>2</td><td>4</td></tr><tr><td rowspan="2">Condition 3</td><td>C<sub>k</sub></td><td>C<sub>0</sub></td><td>C<sub>1</sub></td><td>C<sub>2</sub></td><td>C<sub>3</sub></td><td>C<sub>4</sub></td><td>C<sub>5</sub></td><td>C<sub>6</sub></td><td>C<sub>7</sub></td><td>C<sub>8</sub></td><td>C<sub>9</sub></td><td>C<sub>10</sub></td><td>C<sub>11</sub></td><td>C<sub>12</sub></td><td>C<sub>13</sub></td><td>C<sub>14</sub></td><td>C<sub>15</sub></td><td>C<sub>16</sub></td></tr><tr><td>|C<sub>k</sub>|</td><td>20</td><td>5</td><td>3</td><td>13</td><td>10</td><td>3</td><td>10</td><td>6</td><td>4</td><td>7</td><td>4</td><td>16</td><td>5</td><td>7</td><td>4</td><td>6</td><td>12</td></tr><tr><td></td><td>C<sub>k</sub></td><td>C<sub>17</sub></td><td>C<sub>18</sub></td><td>C<sub>19</sub></td><td>C<sub>20</sub></td><td>C<sub>21</sub></td><td>C<sub>22</sub></td><td>C<sub>23</sub></td><td>C<sub>24</sub></td><td>C<sub>25</sub></td><td>C<sub>26</sub></td><td>C<sub>27</sub></td><td>C<sub>28</sub></td><td>C<sub>29</sub></td><td>C<sub>30</sub></td><td>C<sub>31</sub></td><td>C<sub>32</sub></td><td>C<sub>33</sub></td></tr><tr><td></td><td>|C<sub>k</sub>|</td><td>18</td><td>4</td><td>10</td><td>8</td><td>6</td><td>6</td><td>3</td><td>2</td><td>2</td><td>2</td><td>5</td><td>3</td><td>1</td><td>15</td><td>2</td><td></td><td></td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>Fig. 5. Distribution of the Number of features per Cluster across Conditions</figcaption></figure>

Table 5 shows the 33 features extracted and grouped according to the type of feature selected on the subset Condition 1.

<table><caption>Table 5. Remaining features grouped by their type on the Condition 1</caption><thead><tr><th>Type</th><th>Features</th></tr></thead><tbody><tr><td>Energy</td><td>energy_band_13_freq_h, energy_band_22_freq_h<br>energy_band_6_freq_v, energy_band_13_freq_v,<br>energy_band_16_freq_v, energy_band_20_freq_v, energy_h, energy_v</td></tr><tr><td>Statistical Moments</td><td>moment_7_h, moment_8_v, derivative_moment_9_v</td></tr><tr><td>Derivative Features</td><td>derivative peak to peak_v, derivative_std_h,<br>derivativeMargin_factor_h, derivativeMargin_factor_v, derivative_moment_9_v</td></tr><tr><td>Integral Features</td><td>integral_std_h, integralMargin_factor_h, integral_moment_6_h,<br>integral_max_abs_v, integralImpulse_factor_v,<br>integral_moment_7_v, integral_moment_11_v, integral_moment_6_v</td></tr><tr><td>RMS & IMF</td><td>imf_entropy_3_tf_h, imf_rms_1_tf_h, imf_rms_0_tf_v,<br>imf_rms_1_tf_v, imf_rms_2_tf_v, rms_freq_freq_h</td></tr><tr><td>Skewness</td><td>form_factor_freq_freq_h, skewness_freq_freq_v</td></tr><tr><td>Mean and frequency</td><td>mean_freq_freq_v, mean_v</td></tr></tbody></table>

## 3.2 Ablation Study

To evaluate the effectiveness of the proposed feature clustering approach for RUL prediction, we conducted an ablation study. This involved comparing our model (IDFC+LSTM) with a classical alternative, PCA+LSTM, using *K* components, where *K* represents the number of features selected by IDFC for each subset. Table 6 presents the results of our model in terms of accuracy, utilizing the Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) metrics. Analyzing the results shown in the table, we observe that our proposed model outperforms the classical PCA+LSTM method. For instance, in Condition 1, our model predicts RUL with an RMSE of 0.084 and an MAE of 0.064, while the PCA+LSTM model yields an RMSE of 0.098 and an MAE of 0.074. In Condition 2, the proposed algorithm achieves even more precise predictions, with an RMSE
of 0.126 and an MAE of 0.103, compared to the PCA+LSTM model's RMSE of 0.127 and MAE of 0.104.

These results demonstrate that our model outperforms the classical method combining PCA and LSTM across both conditions. The effectiveness of the pro-

<table><caption>**Table 6.** Performance by bearing and method, with averages including other state-of-the-art approaches. mean(std)</caption><thead><tr><th>Cond Bearing</th><th>Methods</th><th>RMSE</th><th>MAE</th></tr></thead><tbody><tr><td rowspan="2">1-3</td><td>IDFC+LSTM</td><td>0,063</td><td>0,051</td></tr><tr><td>PCA+LSTM</td><td>0,075</td><td>0,064</td></tr><tr><td rowspan="2">1-4</td><td>IDFC+LSTM</td><td>0,071</td><td>0,056</td></tr><tr><td>PCA+LSTM</td><td>0,059</td><td>0,050</td></tr><tr><td rowspan="2">1-5</td><td>IDFC+LSTM</td><td>0,151</td><td>0,121</td></tr><tr><td>PCA+LSTM</td><td>0,165</td><td>0,124</td></tr><tr><td rowspan="2">1-6</td><td>IDFC+LSTM</td><td>0,005</td><td>0,004</td></tr><tr><td>PCA+LSTM</td><td>0,012</td><td>0,010</td></tr><tr><td rowspan="2">1-7</td><td>IDFC+LSTM</td><td>0,133</td><td>0,088</td></tr><tr><td>PCA+LSTM</td><td>0,178</td><td>0,123</td></tr><tr><td rowspan="5">Moyenne</td><td>IDFC+LSTM</td><td>0,084 (0,052)</td><td>0,064 (0,039)</td></tr><tr><td>PCA+LSTM</td><td>0,098 (0,064)</td><td>0,074 (0,044)</td></tr><tr><td>LSTM (15)</td><td>0,152</td><td>0,120</td></tr><tr><td>GRU (10)</td><td>0,181</td><td>0,147</td></tr><tr><td>HDCN (27)</td><td>0,232</td><td>0,198</td></tr><tr><td></td><td>SOM+PCA+BiLSTM (10)</td><td>0,128</td><td>0,105</td></tr><tr><td rowspan="2">2-3</td><td>IDFC+LSTM</td><td>0,039</td><td>0,030</td></tr><tr><td>PCA+LSTM</td><td>0,039</td><td>0,030</td></tr><tr><td rowspan="2">2-4</td><td>IDFC+LSTM</td><td>0,037</td><td>0,031</td></tr><tr><td>PCA+LSTM</td><td>0,035</td><td>0,030</td></tr><tr><td rowspan="2">2-5</td><td>IDFC+LSTM</td><td>0,243</td><td>0,195</td></tr><tr><td>PCA+LSTM</td><td>0,247</td><td>0,199</td></tr><tr><td rowspan="2">2-6</td><td>IDFC+LSTM</td><td>0,110</td><td>0,088</td></tr><tr><td>PCA+LSTM</td><td>0,113</td><td>0,090</td></tr><tr><td rowspan="2">2-7</td><td>IDFC+LSTM</td><td>0,200</td><td>0,172</td></tr><tr><td>PCA+LSTM</td><td>0,198</td><td>0,169</td></tr><tr><td rowspan="5">Moyenne</td><td>IDFC+LSTM</td><td>0,126 (0,084)</td><td>0,103 (0,069)</td></tr><tr><td>PCA+LSTM</td><td>0,127 (0,085)</td><td>0,104 (0,070)</td></tr><tr><td>LSTM (15)</td><td>0,270</td><td>0,219</td></tr><tr><td>GRU (10)</td><td>0,226</td><td>0,171</td></tr><tr><td>HDCN (27)</td><td>0,283</td><td>0,244</td></tr><tr><td></td><td>SOM+PCA+BiLSTM (10)</td><td>0,164</td><td>0,121</td></tr><tr><td rowspan="2">3-3</td><td>IDFC+LSTM</td><td>0,029</td><td>0,001</td></tr><tr><td>PCA+LSTM</td><td>0,028</td><td>0,022</td></tr></tbody></table>

posed model is demonstrated in Figure 6, which compares the predicted RUL of bearing 1_3 to its actual RUL. It is evident that the predicted RUL aligns closely with the actual RUL, with both trends reaching zero at the end of the RUL period. To enhance transparency, it is essential to identify the features that most significantly influence the predicted RUL. To accomplish this, we employed the SHAP method.

## 3.3 Prediction explanation using SHAP XAI method

The proposed dimensionality reduction method allows for the application of Explainable Artificial Intelligence (XAI) techniques to clarify the RUL predictions made by an LSTM model. Unlike traditional methods, this new approach produces interpretable features that yield meaningful results, regardless of the components involved. By using SHAP, we can evaluate which features have the most significant impact on RUL predictions.
Figure 7 illustrates the feature importance determined by the SHAP method for the RUL prediction of bearing 1_3. It shows that the most relevant features for RUL prediction are energy_ban_6_freq_v and energy_ban_16_freq_v, which relate to the energy contained in the 6th and 16th frequency bands of the vertical signal component. Among the other features, the least relevant ones are moment_8 and energy_v. In contrast, classical methods like PCA and t-SNE do not provide such explanations, as they generate synthetic features that are not interpretable in terms of original features.

<figure><img src="image_6.png" /><figcaption>Fig. 6. Proposed Method: Prediction results on Bearing 1_3</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 7. Feature importance provided by the SHAP method for bearing 1_3.</figcaption></figure>
## 3.4 Comparison with other studies

We have compared the proposed model with several studies that used the same dataset for dimensionality reduction. Table 6 presents a performance comparison between our model and these studies, focusing on RMSE and MAE. The results indicate that the proposed model outperforms existing methods, achieving lower RMSE and MAE values. Specifically, for condition 1, the RMSE is 0.084, and the MAE is 0.064; for condition 3, the RMSE is 0.126, and the MAE is 0.103. These findings demonstrate the effectiveness of the proposed model using IDFC. Furthermore, our model is the only one among the three that permits the use of XAI methods to offer genuine explanatory features, in contrast to the studies against which we compared our results with.

# 4 Conclusion

This study introduces the Interpretable Divisive Feature Clustering (IDFC) method for predicting Remaining Useful Life (RUL), emphasizing dimensionality reduction and interpretability. IDFC effectively reduces features and provides transparent insights in contrast to PCA-based methods.

Experimental results from the FEMTO bearing dataset demonstrate that IDFC, when combined with Long Short-Term Memory (LSTM) networks, outperforms the PCA + LSTM approach. For Bearing 1, the Root Mean Square Error (RMSE) decreases from 0.98 to 0.084, and the Mean Absolute Error (MAE) drops from 0.74 to 0.064. For Bearing 2, the MAE significantly improves from 0.022 to 0.001. Furthermore, SHAP analysis reveals that specific energy-related features have a strong influence on RUL prediction, confirming the interpretability of the method.

As part of future work, additional empirical studies will be conducted using other datasets to compare the clustering methods of the IDFC algorithm, rigorously evaluating clustering quality and assessing the robustness of our approach. We will also explore alternative dimensionality reduction techniques, such as sparse PCA, kernel PCA, and autoencoder-based methods, to analyze their effects on both the quality and interpretability of the resulting clusters. These extensions will contribute to a deeper understanding of the methodological choices behind IDFC and refine its applicability to real-world scenarios.
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