Multilayer perceptron is a typical artificial neural network that can segment complex spaces and fit strongly nonlinear data by combining multiple multilayer perceptrons [17]. It is the most common structure in artificial neural networks, which has a great advantage in analyzing data sets with a large number of continuous numerical descriptors. The MLP model structure is shown in Fig. 1.

In this model, the input values are connected to the hidden layer through the input layer. The hidden layer accepts all inputs and performs the appropriate actions. The resulting data is fitted as input to the next layer. These layers are completely connected. The calculation is as follows:(2) yi=f(ui)=f(∑i=1nwixi+bi) where xi represents the input vector of node i or the calculated value of previous layer, wi and bi represent the coefficients learned at node i of the MLP model, and f(⋅) represents a non-linear activation function ReLU.

Long short-term memory networks (LSTM) are widely used in many related domains and are capable of capturing the long-term correlation of temporal signals [18]. Nevertheless, a typical LSTM can only “listen” to temporal dependencies in one direction, either backwards or forwards. In order to explore the bidirectional dependency of time series data, LSTM is combined with a bidirectional recursive network structure to construct a model.

Fig. 2 depicts a typical BiLSTM structure, including an input layer, hidden layer, and output layer. The input layer of the network receives the time series signals, which are then extracted by the hidden layer. The hidden layer also predicts future time series signals and outputs them through the output layer. Every hidden layer, as seen in Fig. 2, consists of two common hidden layers that process the input both forwards and backwards (the symbol represents the sum) in order to capture the features that exist before and after. The following is a description of the hidden layer for forward processing output:

(3) {i→t=σ(W→xixt+W→hih→t−1+b→i)f→t=σ(W→xfxt+W→hfh→t−1+b→f)c^→t=φ(W→xcxt+W→hch→t−1+b→c)c→t=f→t⊙c→t−1+i→t⊙c^→to→t=σ(W→xoxt+W→hoh→t−1+b→o)h→t=o→t⊙φ(c→t)

The output of each hidden layer of backward processing can be described as:(4) {i←t=σ(W←xixt+W←hih←t+1+b←i)f←t=σ(W←xfxt+W←hfh←t+1+b←f)c^←t=φ(W←xcxt+W←hch←t+1+b←c)c←t=f←t⊙c←t+1+i←t⊙c^←to←t=σ(W←xoxt+W←hoh←t+1+b←o)h←t=o←t⊙φ(c←t) where h→t and h←t represent the forward and backward outputs of neurons, i→t , f→t , o→t and i←t , f←t , o←t are the forward and backward processing input gate, forgetting gate, and output gate, respectively. W→xi , W→hi , W→xf , W→hf , W→xc , W→hc , W→xo , W→ho and W←xi , W←hi , Wxf→ , W←hf , W←xc , W←hc , W←xo , W←ho represent the weight matrix for input data and recursive data. b→i , b→f , b→c , b→o and b←i , b←f , b←c , b←o represent forward and backward deviations, and σ represent the sigmoid and tanh activation functions, ⊙ represents Hadamard products.

As indicated in Table 1, many features are extracted using time-domain (TD), frequency-domain (FD), and time-frequency domain (TFD) techniques in order to create a thorough description of the rolling bearings’ degrading characteristics. Time-domain features are represented by TD1 through TD16, frequency-domain features by FD1 through FD16, maximum value, root mean square error, and signal-to-noise ratio (RMS) of the second frequency band signal obtained by three-layer wavelet packet transform, respectively, and inherent energy of the first component after VMD decomposition by TFD1~TFD3. Selecting so many statistical features to provide a more comprehensive description of the bearing degradation process is beneficial in avoiding the influence of manual experience on HI construction.

However, these features have characteristics such as different ranges, high dimensionality, and high nonlinearity. As a result, a suitable technique must be created to extract the most important data from these original traits. KPCA is a simple and efficient technique for reducing nonlinear dimensionality. It involves two steps to complete:

(1) Projecting the initial features onto a new high-dimensional feature space using nonlinear kernel functions.

(2) Principal components are extracted from a high-dimensional feature space using linear PCA techniques.

The first principal component was selected as HI to characterise the degradation process. Although HI maintains global monotonicity, it typically contains a large number of local random fluctuations, which may interfere with the life prediction of rolling bearings. As a result, using certain technologies to improve HI is meaningful.

For smoothing online bearing HI, the EWMA approach based on historical data time series smoothing is adopted. The following is how its mechanism is explained:(5) ft=α(pt+βpt−1+β2pt−2+⋯+βt−1p1) where α represents the smoothing parameter ranging from 0 and 1, and the value of β is (1−α) ; pt stands for the current observation, ft stands for its estimated value, which is the value of modified HI at time t ; pt−1,  pt−2, ⋯, p1 are the historical observations. EWMA can eliminate short-term fluctuations, maintain long-term trends accurately, and effectively capture abrupt changes. As a result, EWMA is an appropriate method to change HI. In addition, the simplified calculation formula for reducing the computational demand of EWMA can be expressed as follows:(6) ft=αpt+(1−α)ft−1

On this basis, an MLP model is constructed to calculate HI. The MLP model can automatically extract representative features from the original vibration signal without the need for manual feature extraction. the framework of the HI calculation method based on the MLP model is shown in Fig. 4. The proposed method consists of three stages: (1) Extract HI for training rolling bearings, as mentioned in the previous section. (2) Train the MLP model. Train the MLP model using the original vibration signal as input and HI value as the target output. Typical characteristics are captured from the input original vibration signal through two full connection layers and the ReLU activation function. Then, the fully connected layer is utilized as the regression layer to generate predictive output (HI label). (3) Calculate and test the HI value of rolling bearings. After the training process is completed, the online vibration signal of the test bearing can be input into the learned MLP model. The MLP model determines the current HI value by directly capturing typical festures from the original vibration signal.

In this section, the BiLSTM model is employed to predict the future HI of rolling bearings, and the model parameters are optimized through PSO.

The BiLSTM model consists of H hidden layers and K units in each hidden layer and includes training and testing steps. In the training steps, the history HI values of the bearing are utilized to construct the training dataset. The construction of the training dataset adopts sliding time window technology. The sliding time window technology process is shown in Fig. 5, which uses a fixed length time window to sample HI values continuously. Then, the training dataset can be formed as {Xt, yt}t=1T , where Xt=[ht, ht+1,⋯, ht+w] is the i-th training sample vector, where ht represents the HI value of the training rolling bearing at time t, and w is the length of the time window. yt=ht+w+1 is the corresponding label. Train BiLSTM by minimizing the mean square error (MSE) function, which can be represented as:

(7) MSE=1T∑t=1T(yt−y^t)2 where yt and y^t are actual labels and predicted labels. T represents the total number of training samples. In order to quickly converge the training process, the BiLSTM model is equipped with an Adam optimizer.

The XJTU-SY rolling bearing dataset is a set of vibration state signals for the entire life cycle of rolling bearings [19]. The sampling interval is set to 1 minute, with each sampling lasting 1.28 s, and each sample contains 32,768 data points. The horizontal vibration signals are employed in this study as they contain more useful information. The dataset includes 15 sets of bearing data, which were obtained under three different operating conditions. Table 2 shows the total number of samples and failure locations for all bearings. When training MLP, the following four bearings are selected as the training set: bearings 1_2, 1_4, 3_2, and 3_3. When training BiLSTM, the data of the health stage and the initial degradation stage of the remaining bearings are used as the training set, and the data of the degradation stage is utilized as the test set.

According to the expression in Table 1, 36 statistical features are extracted from the time domain, frequency domain, and time-frequency domain to constitute the original feature set. Then, the KPCA is applied to combine these features, and the first principal component after fusion is used as a HI to describe the degradation process of bearing. It is worth noting that the Gaussian kernel function is employed as the kernel function, the kernel radius is chosen as 0.01, and the number of principal components to retain is set to 3. Then, EWMA was employed to further modify HI, with a smoothing parameter selection of 1/50. Fig. 6 shows the raw vibration data and health indicators of bearings 1_2 and 3_2 used to construct the HI. HI is relatively stable in the initial phase, starts to decline in the intermediate stage, and rises rapidly towards the end of the bearing’s service life. The results show that the HI can effectively capture the degradation trend, and the actual degradation process of the rolling bearing is described. The HI of bearing 3_2 fluctuates repeatedly in the rapid rise stage. Still, the deterioration process of bearing is irreversible and should not have a downward trend, which reflects the limitations of artificial feature extraction.

An MLP model was constructed for the HI calculation of online rolling bearings. The structure of the MLP model greatly affects the performance of the network. The 2560 central data points from the original vibration signal of the bearing are employed as input to the MLP model. The HI calculated by KPCA-EWMA is selected as the training label of the MLP model. The ReLU activation function is used between fully connected layers. During training, the MSE function is used as the loss function of MLP. The optimal model parameters are obtained after 50 epochs using the Adagrad optimizer.

To effectively evaluate HI and screen suitable network hyperparameters, an effective evaluation method is needed. However, most existing methods directly assess HI itself. Since the performance degradation of rolling bearings is a stochastic process, HI can be divided into trend indicators and residual parts to better measure the degradation of rolling bearings. Therefore, a rolling bearing HI evaluation criterion that integrates trend and robustness is proposed, which can be expressed as:(8) {Y(tk)=YT(tk)+XR(tk)Vcorr(Y(tk),T(tk))=|∑k=1K(Y(tk)−Y~)(T(tk)−T~)|∑k=1K(Y(tk)−Y~)2∑k=1K(T(tk)−T~)2Vrob(Y(tk))=exp⁡(−std(XR(tk))mean|Y(t1)−Y(tK)|)V=ω1Vcorr(Y(tk),T(tk))+ω2Vrob(Y(tk))

The HI curve is decomposed into the trend part YT(tk) and residual part XR(tk) , k is the length of the time series, Y(tk) is the HI. Vcorr(Y(tk), T(tk)) is the trend value, Y~ and T~ are the mean values of health index Y(tk) and time vector T(tk) , Vrob(Y(tk)) is the robustness value, Y(t1) and Y(tK) are the starting and ending values of the HI Y(tk) . ω1 and ω2 are two weight coefficients greater than zero and with a sum of 1. According to the effect of HI on the prediction of remaining life, the trend and robustness of HI were weighted by 0.5 and 0.5.

Taking bearing 1 as an example, experimental verification of HI and MLP model construction was carried out. The comparison of HI constructed by KPCA-EWMA and HI constructed by MLP models with different parameters are shown in Table 3 and Fig. 7. The results indicate that the HI constructed by the MLP model has signifcant advantages in trend and robustness, and the HI score of the MLP model is the highest when using the 2560-160-16-1 parameter. Therefore, the model trained with this parameter is employed to carry out subsequent calculations.

Using the remaining 11 bearing vibration data as the online data, each sampled vibration data is sequentially input into the trained MLP model and then smoothed by EWMA to obtain the following results. As shown in Table 4, all bearings have high HI scores, which fully reflects the effectiveness of the model. Fig. 8 shows the HI results of the three bearings under condition 1. f It can be seen that HI is relatively stable in the healthy state of the bearings and gradually increases after degradation begins, showing a monotonic upward trend overall, effectively capturing its degradation trend.

A BILSTM model for predicting future HI has been established. In order to comprehensively evaluate the performance of the proposed method, three metrics are adopted, including normalized mean square error (NMSE), root mean square error (RMSE), and maximum absolute error (MaxAE), defined as follows:(9) NMSE=∑t=1T(yt−y^t)2∑t=1Ty^t2 (10) RMSE=1T−1∑t=1T(yt−y^t)2

(11) MaxAE=max(|yt−y^t| where y^t and yt represent actual and predicted values, respectively.