2.1. Time-Series Forecasting Methods

Time-series forecasting models can be divided into traditional statistical regression, shallow networks, deep networks, and broad networks. The traditional statistical regression techniques encompass models such as autoregressive model (AR) [19], moving average model (MA) and autoregressive moving average models (ARMA) [20, 21]. The autoregressive integral moving average model (ARIMA) [22–24] is an extension of the ARMA [25]. The AR is more suitable for stationary time series, while the ARIMA [26] is used for nonstationary time-series data. The model combines the AR, MA, and difference method [27]. The traditional statistical regression approach boasts a straightforward structure, underpinned by a rigorous theoretical derivation, endowing it with high interpretability. This characteristic renders it particularly effective for the analysis of online linear data and for the execution of short-term forecasting tasks, often yielding superior outcomes. Concurrently, traditional statistical regression techniques methods exhibit notable limitations. Their efficacy is contingent upon the judicious selection of model parameters and necessitates a high degree of data linearity. This reliance on linear assumptions complicates the accurate fitting of intricate nonlinear datasets. Therefore, for current time-series forecasting problems, the application scenarios of traditional statistical regression methods are relatively limited.

In response to the limitations of traditional statistical regression methods, a machine learning-based forecasting approach has been introduced, mainly including artificial neural networks and support vector machine (SVM), which belong to shallow networks. The artificial neural networks [28] are renowned for their robust learning capabilities. As a quintessential machine learning model, the backpropagation neural network is adept at modeling simple time-series data. SVM is proposed from the best classification surface in the case of linear separability [29]. The machine learning-based prediction model [30] benefits from a well-established theoretical foundation, a straightforward algorithmic structure, and inherent self-learning and adaptability capabilities, making it suitable for modeling nonlinear data. However, the current volume and high volatility of collected time-series data render these machine learning methods inadequate for effectively fitting nonlinear data tasks.

Currently, prediction models based on deep learning methods are widely used in time-series data prediction, mainly including RNNs, long short-term memory networks, and GRUs [31, 32]. The RNN [33] considers the current moment’s input and the earlier moment’s output information when constructed and has a good modeling ability for nonlinear time-series data. Confronted with extensive time-series datasets, RNNs are prone to encountering issues such as vanishing and exploding gradients, thus extending the long short-term memory network [34]. LSTM is equipped with a series of gating mechanisms that effectively integrate short-term and long-term memory, thereby significantly addressing the issue of vanishing gradients. GRU has been introduced to further refine the parameters of the long short-term memory architecture. The architecture of this model incorporates only update and reset gates, which simplifies the computational process and results in a faster computation speed compared to the LSTM network. Deep learning-based prediction models [35] are adept at uncovering the latent features within data, demonstrating a potent capacity for discerning nonlinear relationships. Despite its powerful learning capabilities, deep learning methods require substantial data to achieve optimal performance. Although the deep learning method has a robust learning ability, this learning ability requires a high amount of data. Insufficient data provision typically leads to suboptimal prediction accuracy. Moreover, the presence of noise within the data can impede the learning process of neural networks. The training converges and even leads to overfitting.

In response to challenges encountered in deep learning, the development of broad learning has emerged as an alternative to traditional deep networks. Unlike the conventional approach of extending network depth, broad learning chooses to complete feature transfer by adding neural nodes laterally to solve the problem [36]. Random vector functional link network (RVFL) [37, 38] is a type of planar neural network where the weights between the input data and the hidden layer are randomly initialized to reduce training time and computational demands. RVFL effectively integrates input data with nonlinear features generated during training and establishes a direct connection to the output layer. Compared with the complex deep neural network, the structure of the random vector function link neural network is simple. Compared with the complex neural network, fewer parameters need to be updated, the weight calculation is convenient, and it also has strong generalization ability. However, random vector function linking neural networks face certain limitations. As big data emerge, directly linking input data to the output layer can hinder the system’s maintainability and accessibility.

The existing methods have their limitations in the time-series data prediction field. Traditional RNNs are often plagued by limitations such as susceptibility to local optima and challenges in maintaining stability. Current methodologies typically involve a singular structure with limited generalization capabilities. Concurrently, the field of deep learning is encountering its own set of developmental challenges. The predictive capabilities of models are significantly constrained by dataset-related factors, including data volume and the degree of nonlinearity, which can critically impact both computational efficiency and model accuracy. Furthermore, these models are typically vulnerable to noise interference, which can diminish accuracy and, in severe cases, lead to overfitting.

2.2. Multidimensional Time-Series Forecasting Methods

Multivariate time-series data inherently have strong complexity, nonlinearity, and temporal characteristics. Multivariate time series constantly change, and their data size is generally large. The dimensions of multivariate time series are highly correlated, and it is essential to consider the situation of each dimension comprehensively to obtain more meaningful results. Separating the dimensions can lead to the loss of valuable information. Wang et al. [39] adopted a new template matching method to adaptively extract specific periodic features and reassemble one-dimensional time series into natural structured data of multiple sequences. To reassemble one-dimensional time series into naturally structured multidimensional data, added processing and operations are needed for the original data. This reassembling process may introduce added noise or result in information loss, thereby affecting the accuracy of prediction results. Using a multivariate LSTM network to handle expanded sequences may increase the complexity of the network. Complex network structures can lead to increased time costs for training and inference, as well as higher demands for computational resources.

The structure of multivariate time series is extremely complex. On the one hand, multivariate time series originate from various application domains in real life. The sampling methods and measurement standards differ between different sequences and even among dimensions of the same sequence. Additionally, multivariate time series show frequent fluctuations, noise interference, and nonstationarity. Fan et al. [40] proposed an integrated model combining ARIMA and LSTM. The ARIMA model is used to filter the linear trend in the production time-series data and pass the residual values to the LSTM model. However, this manual approach of handling linear trends may not be flexible or correct enough. For complex or nonlinear trends, the ARIMA model may not effectively capture them, thus affecting the overall predictive ability of the model. Karasu et al. [41] proposed a new prediction model based on support vector regression (SVR) and wrapper-based feature selection method. This method involves multiple parameters such as SMA, EMA, KAMA, etc., which require more computational resources and time to adjust and may increase the risk of overfitting or underfitting. Additionally, multiple-objective optimization techniques are needed to handle the feature selection problem, leading to relatively longer training time. When dealing with large-scale data, slow computation speed may also be met.

Traditional statistical regression methods have complete theoretical derivation processes and procedural modeling steps, making them easy to understand. However, they heavily rely on the judicious selection of parameter models and necessitate a high degree of data linearity, which can render the fitting of intricate nonlinear datasets a formidable task. Shallow networks have a certain degree of self-learning and adaptive capability. However, in the era of big data, shallow networks show limited nonlinear expressive power and cannot capture long-term dependencies in time-series data. Existing time-series networks, to some extent, can capture the correlations between different variables but overlook the temporal nature of multivariate time series. This means that the relationships and trends between variables can change over time. Conventional one-dimensional time-series models have excellent capability to learn temporal dependencies. However, they cannot capture relationships among multiple correlated variables or consider the influence of other potential variables on the current variable. This limitation may result in less correct and complete predictions. But multivariate time-series models simultaneously consider the nonlinearity among multiple variables and the temporal nature of the data. This helps reveal the correlation patterns and influencing factors among different variables. Compared to traditional one-dimensional time-series models that only consider a single variable, multivariate time-series models need to consider the influence of other potential variables on the current variable.

With increased data dimensions, model training and prediction complexity also increase, requiring higher computational capabilities. Therefore, more computational resources and storage space are needed. Although deep neural networks have powerful learning capabilities, they need much data for best performance. Insufficient data can lead to subpar prediction accuracy [42]. Additionally, the presence of noise in the data can affect the learning process of deep neural networks, resulting in training convergence issues or even overfitting. Existing deep neural network time-series models are designed with complex structures, employing sophisticated optimization algorithms and iterative processes to construct and adjust the network [43]. They also involve numerous hyperparameters and require ample training data to achieve high accuracy. Compared to deep neural network models, which often consume a significant amount of time during training, the advantage of incremental learning in broad networks can simplify parameter adjustment and reconstruction processes, significantly saving computational resources and showing promising applications. A single model can only focus on specific features of time-series data, inevitably losing some information when predicting time-series data. Therefore, exploring the combination of models to carry out prediction tasks has become a hot research topic. Wu et al. [44] proposed a broad fuzzy cognitive map system (BFCMS) to address the multidimensional time-series classification problem. However, there was no discussion or solution on how to optimize the BFCMS by removing some redundant nodes to forget some toxic data. Xiong et al. [45] proposed a dynamic adaptive graph convolutional transformer integrated with a broad learning system (DAGCT-BLS) to enhance the prediction accuracy of multidimensional chaotic time series. Due to the integration of multiple deep learning techniques, this model may require substantial computational resources. Su et al. [46] proposed a multihead attentional broad learning system (Multi-Attn BLS) that integrates BLS with a multihead attentional mechanism to effectively extract key spatiotemporal features from chaotic time series, resulting in high prediction performance. However, the effectiveness of this method has not been validated for multistep or long-term predictions. Zhu et al. [47] proposed a multidimensional information fusion and BLS (MIFBLS)-based model for identifying pipeline radiation threat conditions to improve the security of energy pipelines. However, the experimental validation was primarily executed using datasets derived from natural gas pipelines, and the model’s capacity to generalize was not further evaluated across a broader spectrum of pipeline types or under varied environmental conditions. Yuan et al. [48] proposed a joint spatiotemporal feature learning framework for multivariate TSP, consisting of a mixture of multiple sparse autoencoders (SAE) and multiple higher-order fuzzy C-means (HFCM) modules to model the spatiotemporal dynamics of these feature sequences. However, STFCM has many parameters, which makes it difficult to select the appropriate combination of these parameters through grid search. Therefore, more correct predictions can be achieved only by comprehensively considering the nonlinearity and temporal nature of multivariate time series, especially when multiple variables show complex interdependencies and patterns over time.

3.1. Structure Design of Fusion Network

Multidimensional time-series data are inherently complex, exhibiting nonlinearity, temporal dynamics, frequent fluctuations, noise interference, and nonstationarity, which attributes present substantial challenges for forecasting models. Utilizing a BLS framework for multidimensional time-series forecasting can streamline the parameter tuning and model reconstruction process, thereby reducing training duration and enhancing predictive accuracy. Traditional time-series models and deep learning networks can effectively handle multidimensional time-series data’s nonlinear and temporal components. Therefore, fusing deep networks introduces a novel approach for predicting multifaceted time-series data in the broad system framework, simultaneously considering the nonlinear and temporal aspects of time-series feature information. The network structure is illustrated in Figure 1.

The main focus of the proposed fusion network in this article is to address the nonlinear and temporal issues in multidimensional time-series prediction simultaneously. The multidimensional time-series data are classified into temporal and nonlinear features, with solid-colored lines indicating temporal features in the data, that is, data for each variable over a range of time, and miscellaneous-colored lines indicating nonlinear features, that is, data for different variables at the same point in time. The network is structured into an input layer, mapping layer, enhancement layer, and output layer.

3.2. Learning Method of Fusion Network

Based on the design of the fusion network structure, research is conducted on the training and learning methods of the network.

Algorithm 1 details the construction and learning processes of the fusion network. Additionally, the training process of this network is illustrated in Figure 2.

Multidimensional time series continuously change over time, and they typically have a large data size. Each time point in a multidimensional time series corresponds to multiple variables, resulting in a high-dimensional structure. In contrast to deep learning models, which rely on iterative processes for weight updates and are susceptible to challenges such as local optima and gradient-related issues, the broad learning framework circumvents these pitfalls. Additionally, the broad learning framework extends the network in a broad direction by increasing fusion nodes and feature nodes, which helps alleviate the problem of long training time caused by a deep network architecture. For predicting multidimensional time series, the fusion network adopts a BLS framework that incorporates traditional time-series models and deep learning network models to handle nonlinear and temporal data, respectively. This method heightens the model’s capacity for nonlinear representation and its efficacy in capturing prolonged dependencies within time-series data, leveraging the benefits of both broad learning and deep learning methodologies.

4.1. Datasets

The datasets employed in this study are the Fangshan air quality data and wind turbine power data, both of which were sourced from real-world environments.

4.1.1. Fangshan Air Quality Dataset

A city’s air quality directly affects the residents’ well-being and health. Governing atmospheric pollution, improving air quality, strengthening scientific prevention and control of air pollution, and promoting green development are the current trends in social development. This study uses the Fangshan air quality dataset from Beijing, which includes seven variables: CO, NO2, O3, SO2, PM10, AQI, and PM2.5. The Fangshan air quality datasets are collected from real environments, which are fully natural environments and are highly realistic and representative. The data are collected at hourly intervals. PM2.5 concentration is an essential factor influencing the number of days with good air quality, making its prediction highly significant. The proposed model’s predictive performance is evaluated by forecasting the changes in PM2.5 concentration. The original PM2.5 data are shown in Figure 3. The first 18,000 data points are used as the training set, while the last 6000 data points are used as the testing set. The input includes the previous six variables, and the output is the next variable, with a moving step size of three, for example, using multidimensional variable data from time (t − 3) to (t − 1) to predict the PM2.5 concentration data from time t to (t + 3), and so on for further predictions.

4.1.2. Wind Turbine Power Dataset

With the global urgency to reduce carbon emissions and achieve carbon neutrality, wind power plays a significant role as a low-carbon energy source in conducting these goals. Wind power forecasting aids power system operators in accurately understanding future wind power generation, supplying crucial information for power system operation and scheduling. This, in turn, eases better planning and optimization of energy dispatch, avoiding unnecessary energy waste. Wind power forecasting promotes and adopts clean energy, reaching low-carbon environmental objectives and establishing a sustainable future energy system.

This experiment uses the Wind Farm dataset from the EDP Open Data portal, specifically the power dataset of wind turbine T11 in 2016, to validate the proposed model’s performance. Given that the dataset is influenced by human factors, accounting for these elements can enhance the model’s ability to discern underlying patterns and features, thereby improving its predictive accuracy and robustness. And it is publicly available, which increases the reproducibility and comparability of the results, and facilitates the validation and expansion of this research by other researchers. The dataset consists of six variables: average wind speed, average pressure, average humidity, average wind speed within a given time interval, average wind relative direction, and total active power. The measurements are taken at a 30-min interval. Figure 4 presents the raw total active power data, revealing significant fluctuations in wind turbine power. The first 15,000 data points are used as the training set, while the remaining 2475 data points serve as the testing set. The prediction task involves using the first five variables to predict the next variable with a moving step size of three. For instance, predicting the raw total active power data from time (t − 3) to (t − 1) using multidimensional variable data and continuing this pattern for future predictions.

(Data Source: https://opendata.edp.com/explore/dataset/htw-failures-2016/information/?sort=-timestamp%26dataChart=eyJxdWVyaWVzIjpbeyJjaGFydHMiOlt7InR5cG%25E2%80%A6).

4.2. Experimental Results

Symmetric mean absolute percentage error (SMAPE) measures the average absolute percentage deviation between predicted and actual values. A smaller SMAPE indicates higher prediction accuracy. Mean absolute percentage error (MAPE) quantifies the average absolute percentage deviation of individual data points from their arithmetic mean, providing a measure of forecasting accuracy. A lower MAPE indicates better predictive ability of the model. Relative squared error (RSE) is the ratio of the mean-squared error (MSE) to the average variance between predicted and actual values. A smaller RSE indicates a better predictive ability of the model. Relative root-mean-square error (RRMSE) is a statistical metric used to evaluate the predictive accuracy of a model. A smaller RRMSE value signifies higher prediction accuracy of the model. The correlation coefficient, denoted as R, quantifies the linear association between predicted and actual values. An R-value nearing 1 signifies a robust linear relationship.

In this experiment, eight models were evaluated on two datasets separately, using multistep prediction with a moving step size of 3. For the Fangshan dataset, the first six variables were used to predict the last variable. The training set consisted of the first 18,000 data points, while the remaining 6000 data points were gave as the test set. The original data are shown in Figure 3. As for the wind turbine power dataset, the first five variables were used to predict the last variable. The training set included the first 15,000 data points, with the remaining 2475 data points assigned as the test set. The original data are depicted in Figure 4. Each experiment had different levels of data complexity, requiring optimization within different parameter ranges. Different datasets were assigned distinct parameter ranges based on their data fluctuations and magnitudes. The parameter settings for the comparative models used in the experiments are presented in Table 1. The comparative models kept the same parameter configurations.

The predicted PM2.5 concentration results on the Fangshan dataset are shown in Figure 5, while the predicted effects on the wind turbine power dataset are shown in Figure 6. For clarity, the proposed method is represented as Mix-BLS. In the figures, actual data are represented in black, while the red line denotes the predictions made by the Mix-BLS model. Forecasts from other models are distinguished by various colors.

From the prediction result graphs, the Mix-BLS model can fit the changing trends of the real data well on both datasets compared to the comparative models. In Figure 5, during the first 30 time points with relatively mild trend changes, BLS, BESN-2, and Mix-BLS demonstrate more accurate prediction accuracy than other models, indicating the excellent predictive performance of the broad fusion network in multidimensional time-series prediction tasks. From the 30th to the 60th time points, the overall trend of the data shows a complex nonstationary sequence. R-E-BLS and Mix-BLS in the broad fusion network show remarkable learning capabilities, with significantly better prediction results than the comparative models. In the last 40 time points, the data trend displays strong volatility as nonstationary data. At this stage, the Mix-BLS model not only captures the underlying patterns of the data but also achieves smaller prediction errors. In Figure 6, the Mix-BLS model demonstrates its robustness in predicting wind turbine power data across a larger order of magnitude. It accurately fits the real data for the initial 30 time points. Notably, between the 70th and 100th time points, where the data exhibit more pronounced fluctuations, the Mix-BLS model not only maintains a significant accuracy advantage over the comparative model but also outperforms it. Conversely, during the period from the 30th to the 70th time points, where the trend changes are less pronounced, the Mix-BLS model still exhibits higher prediction accuracy than the comparative model. Compared to R-E-BLS, Mix-BLS demonstrates lower prediction errors and can manage complex multidimensional time-series prediction tasks requiring higher accuracy. Overall, from the perspective of prediction results, only Mix-BLS excels in conducting the multifaceted time-series prediction task.

Boxplots can reflect the central tendency and dispersion range of a set or multiple sets of continuous quantitative data distributions. Boxplots of the prediction results for the PM2.5 data and wind turbine power dataset are shown in Figures 7 and 8, respectively.

Boxplots can be used to compare the predictive performance of models from multiple dimensions. In a boxplot, the median of the dataset is denoted by a red line, while the edges of the box represent the upper and lower quartiles, and the whiskers represent the maximum and minimum values. The height of the box reflects the extent of data fluctuation to some degree. From the boxplots, it can be observed that the wind turbine power dataset exhibits more noticeable volatility compared to the PM2.5 dataset, as indicated by the height of the boxes, with a larger range between the maximum and minimum values and a higher median represented by the red line compared to the PM2.5 data. Figure 7 reveals that the BESN-1 model exhibits the smallest box plot, suggesting minimal variability in its predictions, which leads to a more concentrated representation of the information. However, this model’s predictions do not align closely with the data. The Mix-BLS model and the R-E-BLS model have boxes that are close to the real data. However, the Mix-BLS model’s predictions for the PM2.5 dataset exhibit the greatest alignment with the variability observed in the real-world data when contrasted against the other comparative models. In terms of data distribution and median values, the Mix-BLS model performed the best among all comparisons and had the best predictive ability. Figure 8 illustrates that the extreme values of predictions by Mix-BLS, LSTM, and BLS are near the actual data. However, the lower quartile of the LSTM predictions and the upper quartile of the BLS predictions are noticeably more distant from the actual data compared to the Mix-BLS model. Secondly, both R-E-BLS and Mix-BLS closely resemble the appearance of the real data in the boxplot. However, the spread between the highest and lowest values for the R-E-BLS model surpasses that observed in the real data. Accuracy and reliability in wind power prediction are crucial for the wind energy industry’s operation, management, and decision-making. Therefore, overall, only Mix-BLS meets the requirements.

The absolute error stacking plots for the other model predictions on the two datasets are shown in Figures 9 and 10 to compare the prediction results of different models visually. These figures demonstrate that the Mix-BLS model consistently exhibits lower absolute errors compared to the other models at the same time points, reflecting superior prediction accuracy. Specifically, the absolute error of the Mix-BLS model is markedly lower than that of both the ESN and BESN-1 networks. This proves that a single ESN network has insufficient capability when dealing with complex multidimensional time-series prediction tasks and capturing the intricate patterns of real data. Furthermore, in the wind turbine power dataset, which exhibits considerable data fluctuations, the comparative models generally show larger absolute errors owing to the complexity of the multidimensional time series. In contrast, the Mix-BLS model consistently maintains the smallest absolute error. The Mix-BLS model proves stable prediction performance on datasets with more noticeable gradual changes in data trends.

Scatterplots allow for a more visual characterization of changes in the data. The absolute error scatterplots for the prediction results of the two datasets are presented in Figures 11 and 12, respectively. Different colors distinguish the various models; the horizontal axis represents the time steps, while the vertical axis indicates the true values of the absolute prediction errors corresponding to each time step.

In Figure 11, the absolute errors of the prediction results for different models are predominantly concentrated below 40. The four models—LSTM, ESN, BESN-1, and BESN-2—exhibit a higher frequency of error fluctuations at the corresponding time steps compared to Mix-BLS. Additionally, the simple deep networks and broad networks demonstrate instability in multistep prediction tasks. Regarding prediction accuracy, the overall distribution of absolute errors from Mix-BLS is closer to zero, with reduced volatility in the error scatter, thereby offering a significant advantage in both stability and predictive performance. It is clear from Figure 12 that compared to the absolute error scatterplot in Figure 11, the order of magnitude of the error is larger because the order of magnitude of the real data on power loads is approximately in the hundreds of thousands. In the multistep prediction task of the wind turbine power dataset, the absolute error scatter of Mix-BLS is also more concentrated than that of the comparison model, and the overall distribution in the graph is closer to 0, which shows that Mix-BLS also performs stably and has excellent prediction accuracy in the face of more complex wind turbine power data.

In the experiments for predicting two multidimensional time-series datasets, single neural networks (including LSTM, ESN, GRU, and BLS) and broad fusion networks (including BESN-1, BESN-2, and R-E-BLS) were compared to better prove the predictive capabilities of the models. The prediction indicators for each model are shown in Table 2.

Table 2 shows that on the Fangshan dataset, Mix-BLS reduces 0.53% and 0.30% in SMAPE compared to BESN-1 and ESN, respectively. In terms of MAPE, it decreases by 1.01% and 1.21% compared to GRU and LSTM, respectively. Furthermore, it shows 0.61, 0.15, and 0.31 reductions in RRMSE compared to BESN-2, BLS, and R-E-BLS, respectively. The highest value achieved in the R indicator is 0.90. It also requires less training time. On the wind turbine power dataset, Mix-BLS demonstrates reductions of 8.7139%, 4.0641%, and 8.1887% in MAPE compared to BESN-1, BESN-2, and ESN, respectively. It also shows 0.1265, 0.1662, 0.1259, and 0.0112 reductions in RSE compared to GRU, LSTM, BLS, and R-E-BLS, respectively. Mix-BLS exhibits higher prediction accuracy than the comparative models, achieving desirable values in five indicators: SMAPE, MAPE, RRMSE, R, Time etc. The model error is smaller, and the predicted values are more correct. Furthermore, the broad learning architecture facilitates rapid updates of the network’s weight and bias parameters, leading to marked enhancements in the model’s predictive accuracy and a reduction in the duration required for training. Although the training duration of the Mix-BLS model may not be the most expedited, it attains the superior prediction accuracy by trading a modest increment in time for a significant enhancement in precision. Mix-BLS is an improvement over BLS by changing the mapping and enhancement layers. By comparing Mix-BLS with broad fusion networks, it proves better performance, indicating that the Mix-BLS network possesses the advantages of a broad network and exhibits superior predictive capabilities.