3.1.1 Vector Auto-Regression (VAR).

One of the most popular statistical models for forecasting multivariate time series is the Vector Auto-Regression (VAR). It is considered an extension of the univariate autoregressive model. The findings of [25] have revealed that the VAR model is particularly valuable in capturing the dynamic characteristics of economic and financial time series, making it a powerful tool for describing their behavior and making forecasts. In [26], VAR was used to forecast daily concentrations of air pollutants (i.e., CO, NO₂, and SO₃) in Tehran city for the next 24h. For such a task, the authors have considered the correlations between air pollutants to get more accurate forecasts. Experimental results have indicated the high efficiency of the proposed method.

3.1.2 Autoregressive Integrated Moving (ARIMA).

Aside from VAR, ARIMA algorithms [27] were applied to forecast air quality. In [28], authors proposed a hybrid method named ARIMAX by combining the advantage of ARIMA and numerical modeling to forecast real-time air pollutants in Hong Kong (i.e., PM2.5, O₃, and NO₂). By employing experimental analysis, the proposed method significantly improves the quality of forecast results in multiple evaluation metrics. Similarly, the findings in [29] have shown the prominent role of ARIMA in forecasting PM10 in Dakar, Senegal. Accordingly, the proposed method combines system observations with multi-agent real-time simulation and evaluates with several simulations.

3.2.1 Support Vector Regression (SVR).

SVR models were used to forecast PM2.5 and PM10 in London [30]. In that paper, the experimental results indicate the SVR’s efficiency in forecasting air quality parameters (i.e., PM2.5 and PM10). A nonlinear dynamic model based on the SVR technique was proposed to forecast AQI in Oviedo, Spain [31]. Accordingly, the proposed model first analyzed the relationship between primary and secondary pollutants. Then, it derived vital factors influencing the air quality and recommended potential enhancements for health and lifestyle. Zhu et al. [32] investigated an application of the SVR algorithm with a quasi-linear kernel for air quality prediction. For such a task, the paper designed a gated linear network to construct the multiple piecewise linear model, and it could be developed through the pre-training of a Winner-Take-All (WTA) autoencoder. This approach could outperform other state-of-the-art methods in the case of complex air quality prediction problems. It is due to the WTA strategy reducing the risk of overfitting and choosing appropriate sparsity parameters.

3.2.2 Random Forest (RF).

Regarding RF [33, 34] proposed a parallel approach combined with Spark to forecast PM2.5 in Beijing. The experimental results revealed the efficiency and scalability of the proposed method in the case of big data. Later, RF was used to select the most important features to improve the quality of real-time air quality prediction [35]. Concretely, the proposed method provides highly accurate predictions of three air pollutants (i.e., PM2.5, NO₂, SO₂) and outperforms other state-of-the-art methods.

3.2.3 Linear regression.

Linear regression is also a state-of-the-art model for air quality prediction. Indeed, many linear regression models have been proposed to predict AQI levels in New Delhi [36]; In Catalonia [37], authors combined factors including the effect of the surface reflectance capacity of urban surfaces with solar radiation and elevation to predict AQI level in Catalonia. The dataset is collected from 75 different air quality monitoring stations. A clustering technique was applied to cluster these stations based on their similarity. Meanwhile, Multiple Linear Regression (MLR) was used to replicate the annual mean values of AQI in Catalonia. Experimental results illustrated that the proposed model provided highly accurate predictions of AQI. Djuric et al. [38] proposed a multiple linear regression to forecast air pollution indices (i.e., SO₂, NO₂, PM10, O₃, and CO) in Belgrade, Serbia. They collected the training and testing sets from the winters of 2011 and 2012/2013, respectively. In addition, the findings show that the proposed model can be scaled up to forecast long-term air quality.

3.3.1 Long short-term memory.

Apart from the traditional machine learning approaches, most deep learning methods, such as Long short-term memory (LSTM), have shown their superiority over many machine learning techniques. Even though in [39], the LSTM model has outperformed MLP and RNN models in predicting PM10 and SO₂ in the Basaksehir district of Istanbul province. In [40], authors have proposed a bidirectional LSTM (Bi-LSTM) model by considering both past and future information to forecast PM2.5 of three cities in Korea and five cities in China. Accordingly, its performance is superior to GRU and LSTM in terms of the air quality forecast for these cities. Concretely, with short-term prediction, these models have similar performances. Meanwhile, with long-term prediction, Bi-LSTM outperformed GRU and LSTM. Wang and colleagues [41] developed a combination of the CT (chi-square test) with the LSTM model to analyze the relationship between air pollution variables. The paper identified the factors influencing air quality by using CT for such a task. Then, the AQI level was predicted by the LSTM model using a dataset collected at Shijiazhuang in the Hebei Province of China. In comparison to other competitive methods (i.e., SVR, MLP, BP neural network, Simple RNN), the proposed method provides an accuracy of 93.7% (the highest one). In addition, the proposed method outperforms the baseline methods in terms of MAE, MSE, and RMSE metrics. In [42], a GRU layer has been added to the LSTM structure to improve the accuracy of the air quality prediction problem. Experimental results with a dataset collected in Delhi show the outperformance of the proposed approach compared to other competitive methods of linear regression, GRU, kNN, and SVM in terms of MAE and R².

3.3.2 Recurrent Neural Networks (RNN).

In [43], the authors have proposed an RNN algorithm to predict PM2.5 in Japan by employing a dynamic method to pre-train the model based on multi-step-ahead time series prediction. [44] apply RNN to predict PM10, O₃, SO₂, CO and NO₂. The dataset is collected from different sensors with intervals of 1 hour. In addition, the authors applied fine-tuning to find the best hyperparameters of neural network structure and optimization function. Moreover, the investigated model can be applied to predict similar pollutants in other neighboring areas.

3.3.3 Gated Recurrent Unit (GRU).

In the current literature, many research results indicate that existing models are best at short-term forecasts. Meanwhile, improving existing approaches to forecasting long-term air quality is necessary. [45] proposed an algorithm that is considered an enhanced version of GRU (named BiAGRU) by combining bidirectional gated recurrent unit integrated with an attention mechanism. By means of experimental analysis, the proposed model is superior to many traditional machine learning models and modern deep learning models.

Referring to [46], a model based on Gated Recurrent Units (GRUs) has been proposed to forecast NO₂ pollutant concentration. The proposed model is assessed and fine-tuned for such a task concerning the number of features, look-backs, neurons, and epochs. Also, in Beijing [47], authors introduced a model based on spatiotemporal CRUs combined with a Geographic Self-Organizing Map (GeoSOM). Concretely, all monitor stations were clustered using time-series features and geographical coordinates. Later, GRU models were proposed for clusters, and Gaussian vector weights were used to weigh different models in predicting the target sequence. Experimental results showed the technique’s efficiency compared to several state-of-the-art ones regarding MAE, MRE, and R2 metrics.

Since existing models do not fully consider the temporal dependencies, spatial correlations, and feature correlations hidden in a given dataset, in [48], authors examined these correlations by introducing a spatiotemporal deep learning model named Conv1D-LSTM based on 1-D convolutional neural network and LSTM for spatial and temporal correlation feature extraction. In addition, a fully connected network exploits these features for the air quality prediction problem. Furthermore, missing data have been imputed to enhance the quality of air quality prediction. The proposed method outperformed other well-known baseline methods through experimental analysis.

3.4.1 Multimodal data.

Air pollution is one of the most worrying issues facing the world today. So, forecasting of particulate matter (PM) is necessary nowadays. Ton et al. [49] pointed out that combining meteorological features and timestamp information in Hanoi air quality datasets improved the results of PM10 and PM2.5 forecasting. The authors extracted two new features, which were weekend and working hour, from the “Date Time” recorded variable. Then, encoding the time into a vector of 0, 1 to include two new variables, weekend and working hour. First, with the variable weekend, the time vector from Monday to Friday was encoded as 0. On the other hand, during Saturday and Sunday weekends, the time vector was 1. Second, with working hour, the time vector was 1 in the range 7 AM to 7 PM, whereas this was 0. According to the authors, the time steps of the two new variables weekend and working hour were synchronized with other weather and air quality variables. It was highly efficient in 68% of the cases compared to other methods by conducting five deep learning models: MLP, 1D-CNN, LSTM, Bi-LSTM, and Stacked LSTM. Besides, in the long-term forecast of PM concentrations, the Vanilla LSTM model with combined features performed better than the other.

Similarly, to predict the PM2.5 air pollution level in the short- and medium-term, Tejima and colleagues [50] also proposed a framework that looks for hidden associations between traffic factors and air pollution. The six steps in their framework can be defined as follows: (1) Use any machine learning algorithm to extract features from the traffic images, (2) Create a new dataset by combining the extracted features dataset and air pollution dataset using time, (3) Use fuzzy rules to convert this new dataset into an uncertain temporal database, and (4) Use uncertain periodic-frequent pattern mining techniques to uncover hidden relationships between various traffic factors and air pollution, (5) estimate air pollution level from a given image using transfer learning on a pre-trained model, and (6) predict air pollution level using estimated air pollution level and mined patterns dataset. Experimental results show that their method can accurately estimate and predict air pollution levels, ranging from 77% to 98%.

3.4.2 Neighbor stations.

Currently, air pollution and urban life influence human health. Therefore, environmental and data science experts always try to find the most accurate way to predict and provide timely warnings to humans. Specifically, Dao et al. [51] use methods of data imputation for the UrbanAir dataset to predict air pollution at a place without a station by using neighbor stations and predict air pollution of Dalat and discover the correlation/association between air pollution and human activities. The authors divided the article into two tasks that need to be performed: Subtask 1 only used environmental data to predict air pollution and only used traffic data in Subtask 2. Subtask 2 accepts training a prediction model using environmental and traffic data, but only traffic data is used to predict air pollution. While Subtask 2 only accepts AQI levels, Subtask 1 requires predicting both the exact value and AQI level of each pollutant concentration. The paper encouraged researchers to develop a generic framework to discover a correlation among different traffic factors, weather, and air pollution in a locality. By using these correlations, the authors improved the accuracy of AQI prediction and understood the mutual impact between urban life and air pollution.

Besides, Nguyen et al. [52] also introduced a dataset containing data about personal life and the surrounding environment, collected periodically along predetermined routes in Ho Chi Minh City, Vietnam. They also introduced self-developed devices and system architectures for data collection, storage, access, and visualization. There were interesting research topics and applications, including understanding the correlation between human health, air pollution, and traffic congestion.

3.4.3 Images.

Human health is mostly impacted by air pollution. Over time, there has been an increase in the number of patients and disease reports related to air pollution. By using lifelog data and urban nature similarity, a method was introduced in [53, 54] that could predict AQI at a local and individual scale with a few images taken from smartphones and open AQI and weather datasets. Various public datasets pertaining to weather, air pollution, and images are used to develop and evaluate image retrieval and prediction model techniques. The outcomes support their hypothesis regarding the strong correlation between the AQI and snapshots of the surrounding area.

3.4.4 Variable selection.

Currently, several statistical and machine learning methods are used to uncover useful information and patterns for enormous datasets. The common model selection (variable selection) methods include Neural Networks (NN) and RF. The statistical methods like the Least Absolute Shrinkage And Selection Operator (LASSO) [55] and principal component analysis (PCA). The authors [56] have proposed combining NN with LASSO or RF for even better results. In addition, they tested these new methods along with classical techniques (ordinary least square and feed-forward NN) using Monte Carlo simulation and real-world air quality data from Italy. The study found that the combined methods achieved lower errors, suggesting they outperform the traditional approaches.

Many methods have been proposed to improve the performance of air quality prediction. However, most investigated methods are based on complete datasets. Therefore, we need to impute missing values to reinforce the prediction models’ performance.

4.1.1 Ignoring.

Ignoring [60, 61] is a method that completely ignores missing values when conducting the analysis process. Although this is a simple method, if the rate of missing data is high enough to influence the analysis outcomes, it is highly dangerous.

4.1.2 Deletion.

An approach of removing/deleting missing observations from raw data is called Deletion [62, 63]. It is also a frequently used method when the missing values of the data are not high, and removing missing values will not affect the analysis results. Nevertheless, when the missing data rate is high, deleting missing values makes the data incomplete and unsuitable for some other analysis applications.

4.1.3 Mean/Median/Mode imputation.

Mean/Median/Mode are simple methods. There, the missing value for a continuous variable is imputed by the mean/median of the observed values. When the missing values for a categorical variable are replaced by the Mode of the observed values, these approaches are quick to compute and simple to implement. Mean/Median/Mode Imputation methods [64] are a solution for better analysis results when they solve the issue of handling missing data values, whereas Ignoring and Deletion methods are thought to provide poor results in the analysis or data mining process when the missing data rate is high. Furthermore, the limitation of these methods is that the bias created by multiple values on the data has the same value, even if the data are MCAR. As a result, it may bias the estimation of skewed distributions.

4.1.4 Regression imputation.

There are two steps in Regression imputation [65, 66]. The first is to estimate a linear regression model using the target variable’s observed values along with the explanatory variables. After that, one can use the model to predict values for the missing cases in the target variable. Missing values of the variable are replaced based on these predictions. There are two types. First is deterministic regression imputation. It means missing values are replaced with the exact prediction of the regression model. The second is stochastic regression imputation, which adds an additional random error term to the predicted value imputed by deterministic regression imputation. Regression imputation is the improvement over Mean/Median/Mode imputation. Besides, it has disadvantages, including the assumptions of error distribution and linear relationship, which are relatively strict and give poor results for heteroscedastic data.

4.1.5 Last Observation Carried Forward (LOCF).

Last Observation Carried Forward [67, 68] fills in missing values by using the last observed value of the given features in each sample; if there is no previous observation, 0 will be filled in. LOCF assumes that the missing data is constant or follows a gradual change. However, if the missing values are not stationary or the sensor readings exhibit abrupt changes, this method may introduce bias and inaccuracies.

4.1.6 Multivariate Imputation by Chained Equations (MICE).

Multiple imputation offers numerous benefits compared to the single imputation methods mentioned above. MICE [69, 70] is one of the most popular multiple imputation techniques. The process uses an iterative set of regression models to impute missing data from a dataset. It imputes missing values in the dataset’s variables by focusing on one variable at a time. Once the focus is placed on one variable, MICE uses all the other variables in the dataset to predict missingness in that variable. The prediction is based on a regression model, with the form of the model depending on the nature of the focus variable. MICE methods perform better and are more reliable for data with a limited sample size.

On the other hand, MICE has several benefits, such as results in unbiased estimates, being easily interpreted in a Bayesian context, and having a large number of workable algorithms built into the MICE framework. It is worth noting that MICE is especially helpful when missing values are associated with the target variable in a way that causes leakage. Users can also state what they believe to be the likely distribution of the missing value using MICE. However, MICE comes at a high computational cost.

4.1.7 First five last three logistic regression imputation (FTLRI).

Chen et al. [71] proposed an interesting approach for data imputation, namely FTLRI, for time-series air quality data. The paper is based on the traditional logistic regression and a presented “first Five & last Three” model. These techniques could explain relationships among disparate attributes and then derive highly relevant data, for both time and attributes, to the missing data, respectively. The results showed that FTLRI has a significant advantage over the compared imputation approaches, particularly in short-term and long-term time-series air quality data. Furthermore, FTLRI can perform better on datasets with relatively high missing rates (about 40%) since it only selects highly relevant data to the missing values instead of relying on all other data like other methods.

4.1.8 Autoregressive Distributed Lag (ARDL).

Selecting criteria is considered an important issue in the Autoregressive Distributed Lag (ARDL) model. El et al. [72] proposed the use of four imputation methods (k-Nearest Neighbors, Expectation-Maximization, Classification, and Regression Tree, and Random Forest) for handling the missing values. Their goal was to improve the accuracy of the model with the optimal order of lags. They compared these methods using real economic data related to foreign direct investment (FDI) in Libya. Their findings suggest that the Expectation-Maximization method performed best compared to the others.

4.1.9 EPK.

Next, Mohamed et al. [73] introduced a new imputation technique called EPK. Using the Monte Carlo simulation, they evaluated the effectiveness of nine different imputation methods, including EPK. The simulations focused on a specific type of statistical model (binary logistic regression) when the missingness mechanism is MAR. Additionally, they tested the methods on real data from social network advertising. The results from both simulations and real-world applications showed that EPK outperformed other imputation methods regardless of where the missing data occurred (independent variables only, dependent variable only, or both).

4.2.1 k-Nearest Neighbor Imputation (kNNI).

kNNI method [14, 69] uses the k-nearest neighbor to identify similar samples with normalized Euclidean distances or some other type of distance and impute the missing values with the average value of its neighbors. The k-nearest neighbor method can impute continuous variables (by using the mean or weighted mean among the k-nearest neighbors) and categorical variables (by using the Mode among the k-nearest neighbors). Both quantitative and qualitative features are handled by kNNI with ease. However, it performs computationally intensively for large data since it searches through all the datasets and requires the specification of hyper-parameters that can greatly affect the results.

4.2.2 MissForest.

The Random Forest (RF) algorithm can also applied for multivariate time series data, employing an average of the corresponding full values. Using proximity data points, this algorithm then iteratively improves the imputation of missing data. Generally, missForest is a technique that was proposed by [74] based on Random Forests. The article showed that RF intrinsically constitutes a multiple imputation scheme by averaging many unpruned classification or regression trees. The imputation error can be estimated without a test set using Random Forest’s built-in out-of-bag error estimates. Furthermore, missForest performs better than K-nearest neighbors and other imputation techniques, giving outstanding results for data containing non-linear relations and/or complex interactions. Additionally, it works well with data containing both qualitative and quantitative features. When using missForest, there is no need to tune parameters, do categorical encoding, or standardize the data. MissForest can be utilized to achieve good imputation results even in high-dimensional datasets with a large number of variables compared to the sample size.

4.3.1 GRU-D.

Chen et al. [75] proposed the GRU-D model, which is a deep learning model based on Gated Recurrent Unit (GRU) that takes two representations of missing patterns, i.e., masking and time interval, and effectively incorporates them into a deep model architecture so that it does not only captures the long-term temporal dependencies in time series but also utilizes the missing patterns to achieve better prediction results.

4.3.2 Deep auto-encoder.

One method that can be used for data imputation is the auto-encoder structure. It extracts features from low-dimensional layers using the encoder and decoder structure, and the decoder recovers missing values. As such, it can function as a methodological feature. [76] presented one technique using deep autoencoders for spatiotemporal challenges involving imputing missing data. The proposed method for capturing temporal and spatial patterns was a convolution bidirectional LSTM. Additionally, the authors analyzed an autoencoder’s latent feature representation in spatiotemporal data and illustrated its performance for missing data imputation. The experimental result illustrated that the convolution recurrent neural network outperforms state-of-the-art methods.

4.3.3 MultiLayer Perceptron (MLP).

Next, [77] estimated the missing values of a variable in multivariate time series data using a MultiLayer Perceptron. To achieve the best prediction performance for the specified time series, an automated technique was employed to identify the optimal MLP model architecture, filling in a long continuous gap instead of relying on isolated, randomly missing observations. The findings demonstrated that using MLP to fill a big gap produces better outcomes, especially when the data behaves nonlinearly.

4.3.4 Raindrop.

Raindrop [78] is a Graph Neural Network-based algorithm embedding irregularly sampled and multivariate time series. It is inspired by how raindrops hit a surface at varying time intervals and create ripple effects propagating throughout the surface. Raindrop helps handle missing data with irregular time series. It represents every sample as a separate sensor graph and models time-varying dependencies between sensors with a novel message-passing operator. It estimates the latent sensor graph structure and leverages the structure together with nearby observations to predict misaligned readouts. This model can be interpreted as a graph neural network that sends messages over graphs that are optimized for capturing time-varying dependencies among sensors. Another typical work comes from Festag et al. [79], where the authors developed a system based on Generative Adversarial Networks that consist of recurrent encoders and decoders with attention mechanisms and can learn the distribution of intervals from multivariate time series conditioned on the periods before and, if available, periods after the values that are to be predicted.

Therefore, it is worthwhile to understand the data types with missing values and propose an effective and robust strategy to fill time-series air quality data with missing values.

6.2.1 Frankfurt dataset.

Table 3 presents the results of the traditional imputation method compared with other imputation methods regarding the experiment accuracy and running time on the Frankfurt air quality dataset. One can see that when the missing rate of the dataset increases from 10% to 80%, Mean and Median methods have an almost constant MAE error (fluctuating around 0.787–0.788); the MRNN model gives the highest error (greater than 0.904). Moreover, the Transformer model has an MAE error from 0.651 to 0.77; the kNNI model alone has the smallest MAE and RMSE errors among the remaining machine learning models, such as MICE, even lower than the currently used neural-network models, such as SAITS and BRITS. In general, in this dataset, the MAE and RMSE errors of the kNNI model both give the lowest and most stable results among the remaining missing data models when the missing rate of the original data is 0%, and the artificial missing data rate gradually increases from 10% to 80%. As depicted in Fig 2c, it is worth noting that the running time of the models used in this dataset mainly increases when the missing rate of the dataset changes from 10% to 80%. On the other hand, compared to traditional models or basic machine learning models, although models based on neural networks have a long calculation time, MRNN gives relatively positive results and is the most effective among the models using neural networks in terms of Mean, Median, and MICE.

On the other hand, when the original dataset is not missing and the data size is larger than one million records (for example, Frankfurt air quality data), kNNI is considered the model that gives the best results with an artificial missing rate of 10%–80% and time execution time gradually increases from 62.684 × 10³ milliseconds to 941.832 × 10³ milliseconds, followed by SAITS with computation time decreasing from 843.803 × 10³ milliseconds to 391.645 × 10³ milliseconds.

6.2.2 Beijing dataset.

Table 4 depicts the results of imputation methods compared with other imputation methods on the Beijing air quality dataset. In this dataset, the SAITS model also gives the lowest MAE measure compared to other machine learning models; the model error varies from 0.142 to 0.349, followed by BRITS. Meanwhile, the traditional data-filling models vary from 0.724 to 0.885 as the missing ratio gradually increases from 10% to 80%. On the other hand, the MAE error of the SAITS model when the artificial missing rate of data changes from 10% to 40% is lower than that of the MICE model; on the contrary, the RMSE error of MICE is lower than that of SAITS, and the lowest in the remaining used models such as BRITS, kNNI, Transformer, Mean, Median, and MRNN. The artificial missing rate of data ranges from 50% to 70%, the MAE and RMSE errors of the SAITS model are stable again, and the experimental results obtained are the smallest among the models. The remaining models give a relatively large error, with MRNN having the largest error. When the missing data rate is at 80%, the kNNI model gives the best MAE and RMSE errors compared to traditional data filling or machine learning models. In addition, Fig 3c shows the computational time of models such as Median, MRNN, MICE, SAITS, and BRITS remains almost constant when the missing rate increases from 10% to 80%. Next, kNNI is a model with large fluctuations in calculation time, gradually increasing as the missing rate increases. Moreover, the calculation time of the Transformer gradually decreases and changes sharply as the missing ratio increases. Compared to traditional machine learning models, MRNN has the most stable and fastest calculation time compared to the remaining models in this dataset.

6.2.3 Taiwan dataset.

Table 5 shows that the MAE error between traditional models and current models using neural networks grows larger as the missing rate of input temporal data increases on the Northern Taiwan air quality dataset. Specifically, SAITS is the model with the lowest error (ranging from 0.121 to 0.284), followed by BRITS (error only from 0.136 to 0.29) in this data. Meanwhile, methods such as Mean, Median, kNNI, or MICE give errors when filling in missing values that deviate greatly from the original value. Besides, when the artificial missing rate of the data changes from 10% to 60%, the MAE and RMSE errors of the SAITS model give the lowest results. However, when the missing data rate reaches 70%, the SAITS model’s MAE error is the smallest, but the RMSE error is higher than that of the kNNI model. When the missing rate is from 80%, the MAE and RMSE error results of kNNI are the lowest among the models, followed by BRITS.

On the one hand, in Fig 4c, we can see the computational time of machine learning models like kNNI is the smallest after traditional missing data filling models like Mean and Median. Besides, MRNN is the model with the least computational time among machine learning models, followed by MICE. The remaining models have fluctuating and irregular calculation times, the highest when the missing data rate is 10%–30%, and the lowest when the missing data rate is 40%–50%. By comparing the performance of the Mean, Median, kNNI, MICE, SAITS, BRITS, MRNN, and Transformer models, we see that SAITS is the best missing data imputation model on Northern Taiwan and Beijing air quality dataset with an artificial missing rate under 70%. One can see that when the original missing rate of the dataset is less than 30% (or 10%). The dataset only has a few hundred thousand records. SAITS seems to be the model with the lowest error, and model execution time also gradually decreased (from 4223.126 × 10³ milliseconds to 1117.453 × 10³ milliseconds with the Northern Taiwan dataset and from 403.223 × 10³ milliseconds to 259.380 × 10³ milliseconds with the Beijing dataset) as the missing rate of the dataset increased.

6.2.4 Dalat dataset.

Table 6 presents similar experimental results on the Dalat air quality dataset [51]. In this table, traditional methods such as Mean and Median give a constant MAE measure (about 0.83–0.86) when the missing rate of data changes from 10% to 30% and almost the result of these measures is the largest compared to the remaining missing data filling models. Meanwhile, the neural network models used in this dataset, such as SAITS and BRITS, give optimal results, which are not much different from traditional machine learning models such as kNNI and MICE (for MAE measurement, the shortest). However, when the artificial missingness ratio of the data is at 10%, MICE gives relatively low MAE and RMSE errors among the models. Furthermore, when increasing the artificial missing rate to 20%, although the MAE error of MICE is the lowest, the RMSE error of the BRITS model is the smallest. Next, when continuing to increase the artificial missing rate of the model to 30%, the experimental results, MICE is the model with the smallest MAE, and kNNI is the model with the lowest RMSE of all. On the other hand, the calculation time of most models increases when the data’s artificial missing rate increases. Accordingly, MRNN is the model with the fastest computation time, followed by SAITS, Transformer, BRITS, and MICE in Fig 5c.

6.2.5 Cau Giay dataset.

The experimental results on the Cau Giay District air quality dataset [49] are presented in Table 7. One can see that the MAE errors of MICE showed the best results among the used models (only from 0.241–0.424) when the artificial missing rate of the data gradually increased from 10% to 80%. The second best is SAITS (from 0.289–0.578), and the third one is BRITS (from 0.337–0.666). However, the RMSE error of SAITS is the lowest with artificial missing rates of 10%–30% and 50%–60%. Meanwhile, when the missing rate is 40% and increases to 70%–80%, MICE almost always gives relatively good results compared to the remaining models. Besides, the MAE and RMSE errors of the models become larger when the missing rate changes from 10% to 80%, especially the MAE and RMSE errors of the two methods Mean and Median are large, only fluctuating around 0.8. In addition, the running time of machine learning and neural network models is significantly slower than Mean imputation (0.379–0.509 milliseconds) and Median imputation (0.933–0.582 milliseconds). Also, the slowest is MICE, with a relatively large running time, ranging from 71.192 × 10³ to 71.527 × 10³ milliseconds.

6.2.6 Minh Khai dataset.

Table 8 presents experimental results on the Minh Khai air quality dataset [49]. Although the experimental time of MICE is the largest, the model gives the most optimal MAE and RMSE error results among the models used, followed by SAITS, BRITS, kNNI, and Transformer. The MAE and RMSE errors of the Mean and Median methods hardly change much when increasing the missing data rate from 10% to 80%; MAE and RMSE errors of the remaining models gradually increase as the missing data rate increases.

Furthermore, one can also see that when the original missing rate of data is less than 5%, MICE is the method that gives the most optimal results among the missing data imputation methods used in this article (specifically with the Cau Giay district and Minh Khai district air quality dataset). Besides, the running time of MICE is high and increases the fastest when the missing rate of data gradually increases with the Minh Khai dataset. On the other hand, the Cau Giay dataset does not change much over time. However, when considering the performance of filling in missing values using neural networks, SAITS is the model with the most optimal performance, followed by BRITS. We knew that Transformer is a deep-learning model designed to solve many problems. However, in this study, we can see that Transformer hardly promotes its strengths, and experimental results on different data all give much larger MAE and RMSE errors than SAITS and BRITS.