Anomaly detection in rotating machinery via active learning of unlabeled vibration samples

2.1. Batch active learning

The crux of active learning algorithms lies in the selection of the active learning strategy, denoted by Q, with the more commonly employed active learning strategies comprising random sampling (RS), uncertainty sampling (US), and committee-based querying (QBC) [22]. Among these, the QBC strategy has been shown to consider the prediction results of multiple models and exhibits consistent performance [23,24]. The EQB [25] strategy is the most widely used QBC strategy. It is independent of learning algorithms, relying solely on classifier outputs. Consequently, it can be integrated with any learning algorithm.

The uncertainty of an unlabeled sample $u_i$ is quantified by the predictive entropy of the ensemble-averaged predictive distribution. Let $L$ denote the number of fault categories, and let ${\left\{{\theta }_k\right\}}_{k=1}^K$ denote the parameters of the $K$ committee models trained via bootstrap resampling. For each model, the class posterior is obtained from the logits $f_{{\theta }_k}\left(x\right)$ by temperature-scaled softmax [26, 27]:

Averaging the posteriors across the committee yields the predictive distribution:

and the predictive entropy of$u_i$ is computed as:

where $\epsilon =$10^-12 is a small constant for numerical stability, and log denotes the natural logarithm (uncertainty measured in Nats). A larger $H\left(u_i\right)$ indicates greater disagreement in the probability mass assigned by the committee and thus a more informative candidate for annotation. In contrast to vote entropy that relies on hard votes, Eq. (3) exploits soft probabilities and is therefore sensitive to calibration.

The committee is constructed by applying the bootstrap resampling technique to the current labeled set ${\mathcal{D}}_L$. Specifically, for each of the $K$ committee members, a class-stratified bootstrap subset ${{\mathcal{D}}_L}^{\left(k\right)}$ of size $\left|{\mathcal{D}}_L\right|$ is drawn with replacement (independent draws). Each member uses the same LSTM-FCN backbone but is trained from an independent random initialization on ${{\mathcal{D}}_L}^{\left(k\right)}$ with the schedule described in Section 2.2; out-of-bag examples ${{{\mathcal{D}}_L\backslash \mathcal{D}}_L}^{\left(k\right)}$ are used only to monitor early stopping and are never used for training. This procedure introduces the necessary diversity among the $K$ models (default $K=$ 8), enabling a committee-based estimate of predictive uncertainty without additional annotation cost. Unlabeled samples are ranked by $H\left(u_i\right)$; the integration with the deep clustering stage is described in Section 3. The computational complexity of the uncertainty calculation is O($NKL$) for $N$ unlabeled samples and $L$ classes.

Traditional active learning can only select one unlabeled sample for labeling at a time and retrain the classification after each labeling. However, frequent model training is time-consuming. Batch mode active learning (BMAL) [28] addresses the inefficiency of conventional active learning sample selection by attempting to select multiple unlabeled samples simultaneously. The BMAL algorithm enhances the efficiency of active learning in the context of vast data. However, it still lacks a more effective approach to address the redundancy of information among batch samples [29]. There is currently no superior approach to address this issue. In particular, the efficacy of hard clustering in enhancing the sample representativeness is compromised by the background noise interfering with the monitoring data of mechanical equipment. Consequently, this study proposes a combination of the entropy bagging method to investigate the batch active learning algorithm, to reduce the redundancy of information among unlabeled samples of machinery and equipment monitoring data.

2.2. Deep clustering algorithm based on LSTM-FCN

The development of active learning has resulted in the gradual integration of the BMAL algorithm with the clustering algorithm. Initially, clustering was performed on batch samples, and subsequently, a second screening was conducted between different clusters. This approach ensures that the obtained samples exhibit greater variation and are more representative [30]. Conventional clustering methodologies employed in the context of mechanical equipment monitoring signals are hindered by the presence of noise components, which impedes the attainment of precise clustering outcomes. Consequently, the extraction of time-domain data features, among other approaches, is typically used for clustering. To better adapt to the anomaly detection algorithm model and complete the automation of intelligent detection, the BMAL algorithm requires a clustering algorithm that uses the original monitoring signals and has good anti-jamming properties. This study proposes a deep clustering algorithm based on the LSTM-FCN, as shown in the network structure in Fig. 1. The deep-clustering module follows an LSTM-FCN architecture. It consists of two parallel branches. One is an FCN branch with three stacked 1-D convolutional blocks (128/256/128 filters). Each block is followed by batch normalization and ReLU. Global average pooling is applied at the end of the FCN branch. The other branch is a unidirectional LSTM with 128 hidden units. The two outputs are concatenated and projected to a $d$-dimensional embedding used for clustering (default $d=$ 10). Unless otherwise stated, optimization uses Adam (initial learning rate 10^-3, weight decay 10^-5), batch size 64, dropout 0.2 after the LSTM, early stopping with a patience of 10 epochs, and a maximum of 100 epochs.

Fig. 1LSTM-FCN clustering network structure.

The LSTM-FCN network is used to learn compact temporal embeddings from the vibration time series. It combines a 1-D convolutional branch to extract multi-scale local patterns with a unidirectional LSTM branch to capture long-range temporal dependencies. The outputs of the two branches are fused and then projected into a low-dimensional embedded feature of size 1×10, which is used for subsequent clustering and sample selection. Thereafter, the two components of the encoder are connected through the intermediate embedding layer. Finally, the embedded feature of each input sample is mapped to a label $q$. The loss function of this clustering algorithm is defined as follows:

where ${\mathcal{L}}_r$ is the reconstruction loss (L2 norm between the input $x$ and the reconstruction $\widehat{x}$ ); $\lambda$ is the trade-off coefficient; and ${\mathcal{L}}_c$ is the clustering loss defined as the Kullback-Leibler divergence $D_{KL}\left(p\right|\left|q\right)$ between the soft assignment $q$ and the target distribution $p$. The soft assignment $q$ is computed with a Student-t kernel (DEC-style) over the distances between the embedding $z_i=g_{\varphi }\left(x_i\right)$ and the cluster centers ${\{\mu }_j\}$ (degrees of freedom $\alpha =$ 1). The target distribution $p$ is obtained by sharpening $q$ (element-wise squaring followed by normalization) to emphasize confident assignments and mitigate class-imbalance. Cluster centers ${\{\mu }_j\}$ are initialized by $K$-means on the embeddings and are subsequently updated jointly with the network parameters through gradient-based backpropagation. In the pre-training stage, $\lambda =0$ is used to train the autoencoder and obtain a stable mapping from the data space to the embedding space. During fine-tuning, $\lambda =$ 0.1 is employed; the target distribution $p$ is refreshed periodically. Optimization uses Adam (as in Section 2.2). Training stops when the average change of $q$ between two consecutive updates of $p$ falls below a threshold $\delta$.

4.1. Rolling bearing full life dataset validation

The rolling bearing accelerated life test rig of Xi’an Jiaotong University comprises an alternating current motor, a motor speed controller, a rotating shaft, support bearings, a hydraulic loading system, and the test bearing [31]. The test bench provides adjustable working conditions, mainly the radial load and rotational speed. The hydraulic loading system applies the radial load to the bearing seat of the test bearing, while the rotational speed is set and regulated by the AC motor speed controller. The test bearing is an LDK UER204 rolling bearing, and its key parameters are listed in Table 1. The experimental settings and test information are summarized in Table 2.

The vibration dataset under consideration contains a total of 491 samples, with each sample comprising 32,768 points. The time interval between each acquisition is 1 min, the sampling frequency is 25.6 kHz, and the length of each sample is 1.28 s. Fig. 4 shows the waveforms of the normal and abnormal samples in the time and frequency domains.

Fig. 4Time and frequency-domain waveforms

a) Normal sample after 10 min run

b) Normal sample after 100 min run

c) Abnormal sample 40 min before failure

For the active-learning evaluation, an instance-level pool was further constructed from the original acquisitions to obtain a sufficiently large candidate set under a controlled label-scarcity protocol. Specifically, an instance-level pool of 4,500 samples (4,000 normal and 500 abnormal) was constructed from the 491 acquisitions for active-learning evaluation, where “normal” instances were drawn from the early healthy stage and “abnormal” instances were drawn from the pre-failure stage. Of these, 70 % are used for training set $X_{train}$ and the remaining 30 % for testing $X_{test}$, resulting in 2,800 normal samples $X_{train}^n$ and 350 abnormal samples $X_{train}^a$ in the training set $X_{train}=[X_{train}^n,X_{train}^a]$, totaling 3,150 samples. Subsequently, a proportion of the abnormal samples is assigned labels to form the labeled sample set $X_{label}=[X_{label}^n,X_{label}^a]$. Simply, there are 280 normal samples, 35 labeled abnormal samples $X_{label}^a$, and the remaining samples are the unlabeled samples $X_{unlabel}$.

4.2. Analysis of rolling bearing dataset validation results

To verify the accuracy of the proposed algorithm, a secondary sample screening strategy based on LSTM-FCN deep clustering is compared with a random strategy and the state-of-the-art (SOTA) baseline, EQB. The algorithms under evaluation consists of a committee of seven adaptive noise-reduction anomaly-detection classifiers, each trained on 75 % of the samples from the training set. Here $K=$ 7 is adopted as a trade-off between computational cost (which scales approximately as O($NKL$)) and the ensemble diversity required for a stable predictive-entropy ranking on this dataset; larger $K$ showed diminishing returns under the same runtime/memory budget. The LSTM-FCN strategy is predicated on the assumption that the primary candidate pool is $m=$ 20 and the secondary candidate pool is $b=$ 5. In essence, five samples are selected for labeling in each generation, and the total number of iterations is 20. The average value is obtained by repeating the experiment 10 times. In the experimental stage, the manual expert labeling method involves querying the real labels of the samples. The experimental results are shown in Fig. 5.

Fig. 5Comparison of performance improvement of sample screening methods

The accuracy of LSTM-FCN reaches 92.02 % after 20 iterations, while the accuracy obtained using all training samples to train the adaptive noise reduction classifier is 93 %. Random label selection achieves only 87.53 %, whereas the active learning algorithm attains comparable performance to the adaptive noise-reduction classifier using just 33 % of the training samples. This demonstrates the superiority of active learning in selecting samples. This finding underscores the efficacy of active learning in sample selection.

As presented in Table 3, Table 4, and Fig. 5, the EQB algorithm demonstrates enhanced accuracy compared to the original algorithm, despite employing an equivalent number of iterations, after the incorporation of the LSTM-FCN deep-clustering screening process. The LSTM-FCN algorithm differs from the original algorithm from the outset, showing a marked improvement in accuracy, with the largest increase of 4.39 % observed at the 5th iteration. The accuracy of the original EQB algorithm improves after incorporating the LSTM-FCN filtering process, likely owing to its ability to enhance balanced class sampling, ensuring that each class of samples is sampled as neutrally as possible in each selection of valuable samples. The efficacy of the algorithm becomes more evident as model evaluation is refined through iteration. As demonstrated in Tables 3-4, the model requires 7.8 % fewer training samples than the EQB algorithm to reach an accuracy of 85 %, thereby confirming the efficacy of the proposed methodology.

To validate the robustness of the proposed two-stage selection strategy, the impact of the primary screening size $m$ and the query batch size $b$ on detection performance was investigated. First, regarding the primary pool size $m$, values in the range of {10, 20, 30, 40} were evaluated. Results indicated that setting $m=$ 10 limited the diversity of the candidate pool, causing the omission of some informative boundary samples. Conversely, increasing $m$ beyond 30 introduced excessive low-uncertainty samples into the clustering stage, which diluted the density of high-value candidates and slightly degraded the clustering purity without improving accuracy. Thus, $m=$ 20 was adopted as the optimal threshold to balance candidate coverage and screening quality. Second, for the query batch size $b$, sizes of {5, 10, 15} were tested. Smaller batches ($b=$ 5) allowed for more frequent model updates, enabling the classifier to correct its decision boundary promptly after observing a few key samples. Larger batches ($b\ge$ 10) reduced the retraining frequency but led to the selection of redundant information within a single batch, resulting in “diminishing returns” for annotation efforts. Therefore, $b=$ 5 was selected to maximize the performance gain per labeled sample while maintaining a reasonable annotation workload.

4.3. Centrifugal compressor test bench dataset

The dataset under consideration was derived from a centrifugal compressor test bench belonging to China Liaohe Petroleum Engineering Co. The monitoring objects comprise a compressor, booster box, and motor, as well as the bearings that support these rotating bodies. The arrangement of test points on the test bench is presented in Table 5, with each point equipped with three directional sensors, positioned horizontally, vertically, and axially. These sensors can measure the vibration velocity signal in the specified directions.

The sampling frequency was set to 2,560 Hz, the sampling time to 1.6 s, and each data set contained 4,096 sampling points. As summarized in Table 6, comprehensive data of the monitoring process are available for review. The types of data from the monitoring equipment include normal and abnormal data. Abnormal data are further classified into the following categories: unbalance, misalignment, and looseness. The time and frequency domain waveforms are illustrated in Fig. 6.

Fig. 6Time and frequency-domain waveforms

a) Normal sample

b) Misaligned sample

c) Unbalanced sample

d) Looseness sample

Given the modest number of sample points, amounting to 4,096, neither segmentation nor sliding window sampling operations are prerequisites in the data preprocessing stage. Each sample is defined as a one-dimensional vector $X=[x_1,x_2,x_3,\dots ，x_{4096}]$. The dataset contains 23,560 normal samples and 5,420 abnormal samples, totaling 28,980 samples. Of these, 70 % are used for training set $X_{train}$ and the remaining 30 % for testing set $X_{test}$, resulting in 16,492 normal samples $X_{train}^n$ and 3,794 abnormal samples $X_{train}^a$ in the training set $X_{train}=[X_{train}^n,X_{train}^a]$, totaling 20,286 samples. Subsequently, a proportion of the abnormal samples is assigned labels to form the labelled sample set $X_{label}=[X_{label}^n,X_{label}^a]$. Therefore, there are 1,650 normal samples$X_{label}^n$, 379 labeled abnormal samples $X_{label}^a$, and the remaining samples are the unlabeled samples $X_{unlabel}$.

4.4. Analysis of compressor dataset validation results

First, 152 misaligned anomaly samples with labels from the training set are used, along with an equal number of randomly selected normal samples, while the remaining samples are treated as unlabeled. The test set consists of 654 misaligned anomaly samples and 7,068 normal samples. For initialization, 50 normal and 50 anomalous samples are used to train the noise-containing data anomaly detection model in a supervised manner. The primary screening sample pool is set to 20, and the secondary screening sample pool is set to six. In this setup, six samples are labeled per iteration, resulting in a total of 30 iterations. The experimental phase involves querying the real labels of the samples rather than manual expert labeling. The number of anomaly detection models serving as committee members is eight. For the centrifugal-compressor dataset, $K=$ 8 is used to accommodate the higher signal complexity and noise level, providing a slightly more diverse ensemble and more stable committee-averaged predictive distributions within the same experimental budget. Concurrently, active learning algorithms under various strategies are employed to ascertain the efficacy of the secondary sample screening strategy based on LSTM-FCN deep clustering. The comparison strategy includes random screening and the proposed screening strategy. The experimental results are presented in Fig. 7 and Table 7, which demonstrate that the accuracy of the secondary sample screening strategy is enhanced compared to both the random and entropy bagging strategies.

Fig. 7Comparison of the performance improvement of sample screening methods

After 30 iterations, the model achieves an anomaly detection accuracy of 93.05 % on the test set of 280 labeled samples. In comparison, the anomaly detection model based on all the training samples without the screening method achieves an accuracy of 94.73 % with 1,802 labeled samples. The anomaly detection algorithm based on the screening method achieves a comparable detection performance using 15.54 % of the labeling cost, demonstrating the superiority of the proposed method in terms of reducing labeling costs.

To contextualize performance, the proposed framework is evaluated against a widely used state-of-the-art uncertainty-based active-learning baseline, EQB, under identical backbone, label budgets, and training schedule. As shown in Table 7 and Fig. 7, the redundancy-aware two-stage strategy attains higher final accuracy, smoother learning curves with lower variance, and faster convergence than EQB. For reference, a fully supervised LSTM-FCN trained on 100 % labeled data serves as an upper bound. The method reaches >93 % of this upper-bound accuracy while using ≈15 % of the labels, indicating strong label-efficiency for label-scarce industrial scenarios.