Prediction of PM_2.5 concentration based on the CEEMDAN-RLMD-BiLSTM-LEC model

CEEMDAN

Due to the “modal overlap” caused by the EMD algorithm and the noise residual caused by EEMD, this article introduces the CEEMDAN algorithm (Torres et al., 2011), which overcomes the defects of EEMD decomposition in terms of loss of completeness and modal aliasing by adaptively adding white noise.

RLMD

LMD is an improved algorithm based on EMD, which is a decomposition of the signal into a series of product function components (PF) (Smith, 2005). When each PF component is separated from the original signal, the residual component of the signal u_k(t) is obtained, which indicates that the original signal can be considered the sum of the residual component and all PF components, i.e., as Eq. (1).

(1) $$x(t)=\sum _{p=1}^k{\mathrm{P}\mathrm{F}}_{\mathrm{p}}(t)+u_k(t)$$

The main influencing factors of LMD are mainly three points, which are boundary conditions, envelope estimation, and stopping criterion of screening. The specific optimization steps of the robust mean algorithm (RLMD) are as follows: boundary conditions: determine the symmetry points of the left and right ends of the signal using the mirror expansion algorithm; envelope estimation: obtain the optimal subset according to statistical theory; screening stopping principle: minimize the error function, and then RLMD can automatically determine the fixed subset size of the moving average algorithm and the optimal number of screening iterations in the screening process, so it becomes a time-frequency effective tool for analysis (Liu et al., 2017).

BiLSTM

Compared with the traditional unidirectional LSTM neural network, adding a reverse LSTM layer on top of the original one can make it have a two-way propagation loop structure (Zhao et al., 2017). The BiLSTM allows the past and future states of the implicit layer to be passed and fed back through a bidirectional network.

Dynamic weighting method

CEEMDAN effectively solves the modal blending problem and endpoint effect of the EMD algorithm and extracts the feature information of the original signal more fully, but there are problems of noise redundancy and spurious components, while RLMD can effectively solve the problem of spurious components among them and achieve optimization for CEEMDAN. Therefore, this article organically combines the prediction results of the two by the dynamic weighting method. pred(t) of the CEEMDAN-RLMD-BiLSTM prediction results are as Eq. (2).

(2) $$pred(t)=w_{1}pre{d}_1(t)+w_{2}pre{d}_2(t)$$where pred₁(t) is the CEEMDAN prediction result, pred₂(t) is the RLMD prediction result, and w₁ and w₂ are the weights. The weights are calculated by taking the initial w₁ = 1 and w₂ = 0, varying them in steps of 0.01 to find the corresponding pred(t), and using the root mean square error (RMSE) as the criterion. The weight corresponding to the smallest RMSE is taken as the result.

Prediction accuracy analysis

The parameter settings of each model and the prediction results of each model are shown in Table 1 and Table S1.

In all experiments conducted, the proposed CEEMDAN-RLMD-BiLSTM-LEC model showed the smallest MAE, RMSE, and SMAPE, outperforming all other models in these evaluation metrics. These results underscore the superior predictive accuracy of the CEEMDAN-RLMD-BiLSTM-LEC model in comparison to other comparable models. Among the four input step lengths, the smallest RMSE and SMAPE were both at a step length of 9, while the smallest MAE was at a step length of 12.

For instance, at a 12-step length, the CEEMDAN-RLMD-BiLSTM-LEC model’s MAE of 3.9147 µg/m³ was significantly lower than the MAEs of the SVR, BPNN, RNN, LSTM, GRU, BiLSTM, and CNN models, which were 24.1578, 20.1145, 18.3365, 16.4255, 23.1458, 9.8019, and 10.0212 µg/m³, respectively. The MAE and SMAPE metrics also followed a similar trend. Interestingly, the GRU model performed noticeably poorer than the LSTM model, which in turn was outperformed by the BiLSTM model. This is because the GRU model is a simplified version of the LSTM model and might perform weaker with sufficient sample data. The BiLSTM model, with an additional backward LSTM layer, capitalizes on the advantage of bidirectional transfer. However, the CNN model did not perform as well as the BiLSTM model in prediction, and therefore, future decomposition prediction models will no longer involve the BPNN, RNN, LSTM, GRU and CNN models. Additionally, the performance of SVR in time series prediction was considerably weaker than the above neural network models; hence, it will not be discussed further.

Second, when compared to individual decomposition models, the combined decomposition models demonstrated a significant performance boost, showcasing smaller prediction errors and enhanced stability. For instance, at a nine-step length, the SMAPE of the CEEMDAN-RLMD-BiLSTM model was reduced by 0.1374 and 0.1354 compared to the CEEMDAN-BiLSTM and RLMD-BiLSTM models, respectively. Despite RLMD achieving complete decomposition, it lacks comprehensive feature extraction, leading to larger prediction errors. Conversely, although CEEMDAN may have false components, its feature extraction is complete, resulting in smaller prediction errors. This further indicates that the combined prediction models can complement the algorithms, grasp the essence of the original sequence features, and significantly enhance prediction accuracy.

The experimental results indicate that the smallest RMSE for all predictions in the LEC process steps is 9. This suggests that when the difference in PM_2.5 concentration at adjacent time points exceeds 9, the predictive accuracy of CEEMDAN-RLMD-BiLSTM could be affected, necessitating LEC treatment. For instance, at a 12-step length, the RMSE was reduced by 5.6378% following LEC treatment, with the MAE and SMAPE reduced by 20.5040% and 54.5537%, respectively. This shows that while LEC did not significantly enhance the overall stability of the model’s prediction (RMSE), it did markedly improve the model’s predictive stability at points of drastic data change (MAE, SMAPE) and reduced errors caused by random fluctuations in PM_2.5 concentration. Compared to the Transformer, it also outperforms in all metrics, particularly the MAE, highlighting the effectiveness of LEC.

Taking the prediction test set curves of the nine-step and 12-step lengths as representative examples, illustrated in Fig. S2, the performance of each model can be compared against the original sequence. Notably, the CEEMDAN-RLMD-BiLSTM-LEC model demonstrates superior adaptability, maintaining a high degree of fit even in the face of significant local data oscillations. Furthermore, when looking at the overall fit with the entire sequence, this model clearly surpasses the others, providing compelling empirical evidence of its superior predictive capabilities. The robust performance of the CEEMDAN-RLMD-BiLSTM-LEC model underscores its potential as a powerful tool for time-series prediction tasks, particularly in dealing with complex and volatile datasets such as PM_2.5 concentrations.

Prediction error analysis

This section delves further into the analysis of prediction errors. Figures 4A and 5A illustrate the actual prediction errors of various comparative models and the proposed model at different time points, using the nine-step and 12-step lengths as examples. The results indicate that the prediction error curve of the CEEMDAN-RLMD-BiLSTM-LEC model is closely aligned with the x-axis, suggesting that at each time point, the prediction errors are approximately distributed around zero, significantly smaller than the prediction errors of other models at corresponding time points. Figures 4B, 4C, 5B and 5C display the histograms of error frequency distributions and kernel density diagrams for all models considered in this study. The distribution of error values for the target models is concentrated within the 5–10 µg/m³ range, with a relatively small variance. Moreover, in both step-length experiments, the frequency of the error probability distribution within the 5–10 µg/m³ range exceeds 700, significantly presenting the highest kurtosis. These observations provide more profound validation of the accuracy and robustness of the CEEMDAN-RLMD-BiLSTM-LEC model.

Decomposition results

As depicted in Fig. S3, the RLMD decomposition results in five PF components and one residual component, whereas the CEEMDAN decomposition yields 14 IMF components. Figure 6 further presents the CE correlation coefficients between the original data and the outcomes produced by these two decomposition methods. The decomposition effects adhere to the same principles as those discussed in “Decomposition Results”. Upon closer inspection, it becomes evident that, despite the discrepancy in the quantity of decomposed components, both methods exhibit considerable effectiveness in extracting significant features and patterns embedded within the data. This observation provides additional support for our analysis articulated in “Decomposition Results”, affirming the high efficiency and precision of both decomposition methods in their designated operations.

Following experimental trials of PE feature selection, it has been discerned that the prediction performance is optimal when RLMD does not remove any component, while for CEEMDAN, the best prediction results are obtained when IMF₁ and IMF₂ are excluded. Therefore, a PE threshold of 0.95 was chosen (Fig. 7).

Analysis of prediction results

Building on the conclusions derived from “Prediction Accuracy Analysis”, this section will concentrate exclusively on the Transformer, RLMD-BiLSTM, RLMD-BiLSTM, CEEMDAN-RLMD-BiLSTM, and CEEMDAN-RLMD-BiLSTM-LEC models, consequently excluding consideration of the SVR, BPNN, RNN, GRU, LSTM, and CNN models. These selected models have emerged as top performers in our preceding evaluations. The components they predict comprise IMF₃ to IMF₁₄, PF₁ to PF₅, along with the residual component. Our comparative analysis will adhere to the parameter settings detailed in Table 1, thereby ensuring a consistent basis for evaluation. The insights gleaned from this comparative endeavor are presented in Table S2.

The experimental analysis of PM_2.5 concentration data from Beijing Station 1001A yielded results that aligned with the conclusions presented in “Prediction Accuracy Analysis”. For instance, in a nine-step forecast, the RLMD decomposition prediction method produced an MAE of 4.6689 µg/m³, an RMSE of 7.2767 µg/m³, and an SMAPE of 0.1072. These were the least favorable outcomes among the four models scrutinized, reflecting the limitations of RLMD in fully capturing and extracting data features. However, upon integration with CEEMDAN, the values of MAE, RMSE, and SMAPE were reduced to _2.5320 µg/m³, 3.6558 µg/m³, and 0.0482, respectively. This decrement signifies reductions of 45.7688%, 49.7602%, and 55.0373% compared to the RLMD decomposition prediction, respectively. These results surpassed the performance of the CEEMDAN decomposition prediction by 5.4448%, 3.3981%, and 12.9983%, respectively. These figures illustrate the improved capability of the CEEMDAN-RLMD decomposition prediction in extracting and preserving data features compared to a standalone decomposition prediction. This statement maintains an objective tone devoid of a first-person perspective. Upon the application of LEC, the model findings indicated an LEC threshold of 8, implying that when the absolute difference between two adjacent data points exceeds 8, the incorporation of LEC could reduce the MAE, RMSE, and SMAPE of the model by 27.7291%, 15.2214%, and 19.2946%, respectively. This decrease underscores an enhanced sensitivity of the model to local variations, suggesting that LEC can effectively mitigate random disturbances in PM_2.5, allowing the model to adapt to significant data shifts and thereby delivering robust predictive performance for data exhibiting high numerical elasticity. The same trend was evident in the 12-step prediction, with the proposed model consistently outperforming the Transformer model.

To illustrate the predictive fitting efficiency of the proposed model, Fig. S4 presents the time series curves of the 12 models juxtaposed against the original sequence. The results disclose that CEEMDAN-RLMD-BiLSTM-LEC tightly overlays the original sequence curve. In the face of larger local data fluctuations, it showcases a stronger fit with the original data compared to the other predictive curves, implying that the model is proficient in effectively capturing all the features of the original sequence.

Prediction error analysis

The analysis in this section further corroborates the outstanding performance of our proposed CEEMDAN-RLMD-BiLSTM-LEC model in the context of generalized experiments. Specifically, Figs. 8 and 9 present the prediction errors at each time point for nine-step and 12-step predictions respectively, along with the corresponding histograms of frequency distributions and kernel density curves. Notably, from Figs. 8A and 9A, it is evident that the prediction errors of the CEEMDAN-RLMD-BiLSTM-LEC model are more concentrated around zero, significantly surpassing other models.

This observation is powerfully reinforced by the frequency distribution curves and kernel density curves depicted in Figs. 8B, 8C, 9B and 9C. Compared to other models, our model exhibits a marked peak in the range of small errors, indicating that its errors conform to a normal distribution with a lower variance. This representation not only highlights its superiority statistically but also empirically demonstrates that the absolute values of prediction errors for the CEEMDAN-RLMD-BiLSTM-LEC model are relatively small.

Taken together, in combination with the results from “Decomposition Results” and “Analysis of Prediction Results”, it can be concluded that the CEEMDAN-RLMD-BiLSTM-LEC model possesses exceptional precision and stability. When dealing with data incorporating noise and disturbances, it demonstrates commendable robustness and generalization ability. Therefore, we can confidently infer that this model holds considerable application value and broad prospects in tackling various complex and dynamic real-world problems.