State prediction study of vibration system based on time series entropy feature reconstruction-GRNN

2.1. Time series entropy

Entropy is a physical statistic used to measure the state of a material system. More precisely, it is the degree to which the system state may occur. It reflects the complexity of the time series [11, 12]. In general, the entropy of the time series includes SampEn, ApEn, PermEn, FuzzyEn, and ShannonEn. They can reflect the complexity or the irregularity of time series to varying degrees [13-15].

ApEn can describe the irregularity of time series. The larger its value, the more the time series is irregular [15, 16]. In general, the mode dimension ($m$) and similarity tolerance threshold ($r$) are the parameters that affect the ApEn of the time series, and $m$ is usually set to 2 or 3, while the value of $r$ is related to the standard deviation (std) of the time series. The literature [14] published by Pincus S. M. et al. suggests that when calculating ApEn, the value of $m$ is generally 2, and the range of $r$ is (0.1–0.2) $·$std. SampEn is an improved method of ApEn. However, its value is less affected by the length of the time series, and the changes of the mode dimension ($m$) and similarity tolerance threshold ($r$) show high consistency [16]. The mode dimension($m$) of SampEn can be 1 or 2, and the value range of $r$ is (0.1-0.25) $·$std [16, 17]. The value of FuzzyEn is closely related to $m$, $r$, and the step ($n$). In reference [19], it is mentioned that the embedding dimension $m$ of the FuzzyEn algorithm can be taken as 2 or 3, (0.1-0.25) $·$std. In general, $n$ is set to 2 or 3.Similar to ApEn, SampEn, and FuzzyEn, PermEn is a parameter that quantifies the complexity of time series signals [18]. It also measures their random variation and reflects the regularity of their permutation [19]. ShannonEn of the time series is the change trend information that represents the state parameters in the time dimension [20].

2.2. GRNN

GRNN is a neural network based on mathematical statistics, using a radial basis function as the activation function. Its network structure comprises four layers: input, mode, summation, and output layers. It has a satisfactory prediction effect, especially when dealing with small samples. The main kernel functions of GRNN include the Gaussian kernel (G-kernel), the multiquadratic kernel (IMQ-kernel) and inverse multiple quadratic kernel (IMQ-kernel). The smoothness factor ($\text{Spread}$) or scale parameter ($\sigma$) of the radial basis function is the main parameter affecting the prediction accuracy of the GRNN [9, 10].

5.1. Single entropy feature prediction and computational parameter optimization

To find the optimal parameter combination scheme for calculating time series entropy and effectively capture the characteristics of time series, the scope is expanded to cover the complete interval of “low similarity threshold (capturing fine particle features) - high similarity threshold (capturing overall trend)” by 0.1-2.5 times the standard deviation $\text{std}$ of the coverage signal. Due to the standard deviation $\text{std}$ of the data for each sample in this experiment being 0.05-0.12, $r=0.005+0.02·i$ ($i=$0,…, 14) was selected for this study. The calculation parameters for five types of entropy are set as shown in Table 1.

The above entropy calculation parameters are combined to form 195 different combination schemes. Under different combination schemes, the entropy values of 42 state time series are obtained, and the individual entropy values of the 42 states under each combination scheme are combined with the damping efficiency to form a dataset. Using a 7-fold cross validation method, two different kernel functions, GRNN G-kernel and IMQ-kernel, were selected. The smoothing factor $\text{Spread}$ or scale parameter $\sigma$ interval of the GRNN model kernel function was set for different entropy feature types (the final selected interval is listed in Table 2) to ensure the selection of parameter optimization schemes under good fitting conditions. Calculate three metrics: CV error, F-value, and MI between predicted and true values, to evaluate the prediction stability, robustness, and generalization of the model. A comprehensive score is obtained by weighted combination of these three metrics, which serves as the basis for assessing the superiority of parameter schemes. The fitting curves of cross-validation predictions using single entropy values, the highest comprehensive score, and the degree of fit are plotted in Fig. 2 and 3.

From the data analysis in Table 2, it can be seen that:

When using the GRNN G-kernel function, the highest comprehensive scores for the five entropy feature predictions are as follows: FuzzyEn (4.419) > ApproxEn (3.173) > ShannonEn (0.881) > SampEn (0.430) > PermEn (0.243). Among them, FuzzyEn performs the best and can obtain the highest comprehensive score; ApproxEn comes second, while PermEn has the lowest performance. When using GRNN IMQ-kernel function, the highest comprehensive score ranking of the five entropy feature predictions are as follows: SampEn (3.397) > ApproxEn (3.261) > FuzzyEn (1.206) > ShannonEn (0.707) > PermEn (0.229). Under IMQ-kernel function, the comprehensive score of SampEn is the highest, ApproxEn still ranks second, and PermEn remains the worst performing entropy feature.

By adjusting the parameter range, the ratio of training error to validation error for both kernel functions remains stable within the range of 0.88-0.95 under the optimal parameter combination of entropy features. This result indicates that both kernel function configurations can achieve good model fitting performance and demonstrate excellent generalization ability, effectively avoiding overfitting or underfitting problems.

Comparison of the two kernels’ core metrics: in terms of mean CV error, the G-kernel is significantly lower than the IMQ-kernel; in terms of the overall composite score, the G-kernel likewise outperforms the IMQ-kernel. In summary, the GRNN with a G-kernel shows superior comprehensive performance on the single-entropy feature prediction task and is therefore chosen as the final hyperparameter selection scheme. The optimal parameter combinations for the single-entropy GRNN prediction model are as follows: ApproxEn (2, 0.225, x), SampEn (x, 1, 0.205), PermEn (x, 7), FuzzyEn (x, 3, 0.065, 3), ShannonEn (x, 2).

Fig. 2Learning curves of five entropies under the G-kernel at the optimal parameter combination

Fig. 3Single entropy value cross-validation predicts the optimal comprehensive score and its fitting coefficient

a) The optimal comprehensive score

b) Fitting coefficient

5.2. Prediction results after entropy feature vector reconstruction

To determine the optimal feature combination scheme, based on the parameter optimization results of the five types of entropy values under different states mentioned above, under the optimal parameter conditions corresponding to each entropy value, the five types of entropy features were combined in different numbers – 1 (5 combinations in total), 2 (10 in total), 3 (10 in total), 4 (5 in total), and 5 (1 in total) – resulting in a total of 31 feature vector combinations. Each combination was then paired with the vibration reduction efficiency reconstruction dataset. To evaluate the predictive performance of each feature combination in the system, the last 6 samples of the dataset were fixed as the test set, and the number of training set samples $N$ gradually increased from 7 to 36. When predicting in the GRNN model, G-kernel and IMQ-kernel are also used to set the search range of parameters from 0.0001 to 3, with a step size of 0.001, in order to obtain the optimal model parameters for different feature combinations. Finally, the top 10 feature combination schemes with the best prediction performance were selected, and the optimal prediction results were analyzed. Fig. 4 shows the top 10 combination schemes and their corresponding minimum prediction errors under two kernel functions. Fig. 5 shows the trend of prediction error with the number of training samples $N$ under the optimal prediction scheme for different combinations.

Fig. 4The MAPEmin of the top 10 combinations

a) G-kernel

b) IMQ-kernel

Comparing and analyzing the prediction results of two GRNN kernel functions, it can be seen that the minimum MAPE of the top 10 different entropy feature combinations is less than 3.5 %, indicating that the entropy feature combination can achieve high prediction accuracy. The first two optimal entropy feature combinations with smaller prediction errors selected using different kernel functions are“ApEn+FuzzyEn”and“ApEn”, which are the two entropy combinations with higher comprehensive scores for single features. This indicates that the feature combination scheme can prioritize selecting the entropy with better single entropy prediction for combination, further verifying that the comprehensive scoring method of single entropy prediction is an effective method for evaluating prediction results and can serve as the main basis for selecting optimal entropy feature parameter combinations. However, the optimal results of the entropy feature combination of the two kernel functions are different, and the corresponding MAPE ranking of 3-10 combinations varies. The minimum MAPE of the top 10 combinations predicted by G-kernel is slightly lower than that predicted by IMQ-kernel, while the minimum prediction error of the “ApEn+FuzzyEn” combination (0.571 %) is higher than that predicted by IMQ-kernel (0.264 %). From the perspective of minimum MAPE, using IMQ-kernels can achieve higher prediction accuracy. Therefore, a combination of entropy features and model parameter optimization scheme have been preliminarily selected for the prediction model.

Fig. 5The trend of MAPEmin with different combinations of feature numbers changing with N

a) G-kernel

b) IMQ-kernel

5.3. Evaluation of model performance and industrial application prospects

To evaluate the performance of the entropy feature reconstruction prediction model for the system, firstly, the prediction performance of different GRNN kernel functions was compared under the optimal feature combination conditions to determine the most suitable kernel function; Subsequently, by extracting the prediction MAPE and training MAPE, the generalization and overfitting of the model are judged, and the optimal prediction results and model parameters suitable for the model are selected through the balance criterion; Finally, under the same conditions, the proposed GRNN model was compared with mainstream prediction methods such as LSTM, RF, SVM, and the VMD-box dimension-GRNN model proposed in reference [9], in order to verify the superiority of the entropy feature combination GRNN model in prediction accuracy and stability. The relevant prediction results (Fig. 6) and model parameters are summarized in Table 3.

From Table 3, it can be seen that when using the same GRNN model for prediction, the same feature combination (such as “ApEn+FuzzyEn”) has different minimum MAPE values corresponding to different kernel functions, and the generalization gap also varies greatly. This indicates that there are significant differences in predictive performance under different kernel functions.

The “ApEn+FuzzyEn” GRNN model proposed in this article has a minimum MAPE (0.264 % –0.715 %) lower than the minimum MAPE (1.905 %) of the VMD-box dimension-GRNN combination in reference [9] under three kernel functions, demonstrating its significant advantage in prediction accuracy. However, their corresponding generalization gaps (34 %-45 %) are generally large, indicating that the“ApEn+FuzzyEn”- GRNN model has a certain risk of overfitting and weak generalization ability. Therefore, it cannot be concluded from the perspective of minimum error that the combination of “ApEn+FuzzyEn” is superior to the VMD-box dimension feature combination in predicting vibration reduction efficiency.

Fig. 6GRNN-IMQ model prediction results

a) Curves of generalization gap with kernel parameters under different $N$

b) Curve of testing MAPE -N based on balance criterion

To further evaluate the performance of the model, a balance criterion with a weight of 0.3 is introduced for re evaluation, in order to select the balanced optimal prediction result. The results showed that the prediction errors of each model increased under the balance criterion, but the generalization gap decreased to varying degrees with different kernel functions, reflecting that weakening the prediction accuracy to a certain extent contributes to the overall improvement of the model’s generalization ability. Among them, the GRNN-IMQ kernel function performs particularly well under the balance criterion, with its minimum error (0.95 %) and generalization gap (0.76 %) lower than other kernel functions, demonstrating excellent predictive stability and reliability. In summary, after optimizing the balance criterion, the combination of “ApEn+FuzzyEn” features not only has higher prediction accuracy than VMD-box dimension in most cases, but also has stronger generalization performance, making it more suitable for reliable prediction of vibration reduction efficiency.

By comparing our research model with three mainstream prediction methods, including LSTM, RF, SVM, etc., the minimum error, balanced optimal prediction error, and generalization gap of GRNN-IMQ prediction based on the “ApEn+FuzzyEn” entropy feature combination were found to be optimal. This validates the significant advantages of the model in terms of prediction accuracy and stability, making it suitable for high-precision and high stability prediction tasks and having strong practical application value.

To evaluate the industrial application potential of the proposed model, this study simulates common random interference in industrial environments by superimposing Gaussian white noise with different Signal to Noise Ratio (SNR) levels of 20-45 dB on the original input signal. On this basis, the optimal parameter combination determined by previous optimization is adopted, and the GRNN-IMQ model is used for prediction. The system characterizes its environmental adaptability under different levels of noise interference by quantifying the percentage decrease in model prediction accuracy after adding noise (as shown in Fig. 7). At the same time, in order to further match the core demand for real-time performance in industrial scenarios, this study introduces the model operating efficiency index - by accurately calculating entropy parameters and the actual machine time consumption of GRNN prediction algorithm, analyzing the model’s satisfaction with real-time response requirements in industrial applications from the perspective of computational efficiency, and ultimately providing multi-dimensional (anti-interference, real-time) application references for the feasibility of industrial implementation of the model.

Fig. 7Changes in the prediction performance of balance criteria under different SNR noises

a) Training and testing MAPE performance summary

b) Performance degradation summary

From Fig. 7, it can be seen that as the noise intensity increases, the overall decrease in accuracy shows a decreasing trend, with a decrease of about 5.84 % at SNR = 20 dB, and a decrease of less than 0.5 % at SNR = 30, 35, and 45dB, Higher SNR levels yield prediction performance closer to original signal, the prediction accuracy slightly increases when SNR = 40 dB. At the same time, in ordinary industrial control computers, the model feature extraction takes 0.07 s, the prediction time is 0.08 s, and the average time for a single prediction of the model is less than 0.15 s, which meets the real-time response requirements of industrial scenarios. This indicates that the model has strong anti-interference ability and efficient computing performance, and has good industrial application potential.

To further explore the industrial application feasibility of the algorithm proposed in this paper, we selected the probability screen vibration test dataset [21]. This dataset records vibration signals and corresponding screening efficiency under 30 different states, forming 30 samples. Among them, 25 samples were used as the training set, while the remaining 5 served as the test set. Based on the algorithm, parameters $r$ are chosen from 0 to 50 with an interval of 0.5, as listed in Table 1. The optimal selection results for single entropy values using the GRNN G-kernel as follows: ApproxEn (3, 24.01, x), SampEn (x, 1, 2.51), PermEn (x, 6), FuzzyEn (x, 3, 0.01, 3), and ShannonEn (x, 4). By combining these entropy values in different configurations, the best combination scheme of “FuzzyEn + ShannonEn” was obtained. Under this optimal combination, the balanced criterion was applied to achieve a best prediction accuracy of 98.98 % under the G-kernel, with a generalization error of 2.27 %, demonstrating excellent prediction and generalization performance. This further highlights the industrial application value of the proposed method.