The VAR model is used to describe the interdependencies among multiple time series. It predicts the current and future values of time series by considering the lagged values of multiple variables, making it suitable for handling multivariate time series data.

The basic equation of the VAR model is as follows:

$y_t=c+A_1{y}_{t-1}+A_2{y}_{t-2}+\cdots +A_p{y}_{t-p}+{ϵ}_t$(1)

where:

To predict the value at a future time $t+h$, the above equation is recursively applied:

$y_{t+h}=c+A_1{y}_{t+h-1}+A_2{y}_{t+h-2}+\cdots +A_p{y}_{t+h-p}$(2)

Random Forest is an ensemble learning method that improves model accuracy and robustness by constructing multiple decision trees and combining their prediction results.

For regression problems, the prediction of Random Forest is expressed as:

$\hat{y}=\underset{b=1}{\overset{B}{{\displaystyle \sum }}}{T}_b\left(x\right)$(3)

where:

The basic idea of the proposed method is as follows: First, the original dataset is divided into a training set and a testing set. A Random Forest model is trained using the training set and is then used to predict the target values for the time steps in the testing set. The training data is combined with the predictions obtained from the Random Forest model to form a new dataset, which is then used to fit and forecast using the VAR model.

The synergy stems from two complementary mechanisms:

${\hat{y}}_{RF}^{k,t}=\frac{1}{B}\underset{b=1}{\overset{B}{{\displaystyle \sum }}}{T}_b^{\left(k\right)}\left(X_t\right)\left(\text{Tree Ensemble}\right)$(4)

where $T_b^{\left(k\right)}$ denotes the $b$-th tree for variable $k$, effectively modeling interactions between lagged terms and external factors.

$\frac{\partial y_{k,t}}{\partial y_{j,t-i}}=A_{k,j,i}\left(\text{Linear Propagation}\right)$(5)

The hybrid model combines these via additive decomposition:

$y_t=\underset{\text{Linear Dynamics}}{\underset{︸}{\underset{i=1}{\overset{p}{{\displaystyle \sum }}}{A}_i{y}_{t-i}}}+\underset{\text{Nonlinear Correction}}{\underset{︸}{\beta \cdot RF\left(X_t\right)}}+{ϵ}_t$(6)

First, use the Random Forest model to predict each variable:

${\hat{y}}_{RF}^{k,t}=R{F}_k\left(X_t\right)$(7)

where:

Feature Engineering Details:

$X_t=\left[y_{1,t-1},y_{1,t-2},\cdots ,y_{1,t-p},\cdots ,y_{K,t-p}\right]\in {ℝ}^{K\times p}$(8)

where $K$ is the number of variables, and $p$ is the maximum lag order (determined via AIC/BIC).

Each RF model $R{F}_k$ is trained to minimize:

$\underset{{\theta }_k}{\text{min}}\underset{t=p+1}{\overset{N_{\text{train}}}{{\displaystyle \sum }}}{\left(y_{k,t}-R{F}_k\left(X_t\right)\right)}^2$(9)

where ${\theta }_k$ represents tree split parameters.

RF predictions may introduce future information, causing endogeneity bias. To address endogeneity issues:

- Training set $Y_{\text{train}}=\left\{y_1,y_2,\cdots ,y_T\right\}$, test set $Y_{\text{test}}=\left\{y_{T+1},\cdots ,y_{T+h}\right\}$.

- Train RF solely on $Y_{\text{train}}$. For prediction, input $X_{T+1}=\left[y_T,y_{T-1},\cdots ,y_{T-p+1}\right]$ to generate ${\hat{y}}_{T+1}^{RF}$.

(a) Dynamically update the training window.

(b) Retrain RF and VAR models to ensure predictions rely only on historical data.

Next, incorporate the Random Forest predictions into the VAR model:

$y_t=c+A_1{y}_{t-1}+A_2{y}_{t-2}+\cdots +A_p{y}_{t-p}+\beta {\hat{y}}_t^{RF}+{ϵ}_t$(10)

where:

The comprehensive model equation becomes:

$y_t=c+A_1{y}_{t-1}+A_2{y}_{t-2}+\cdots +A_p{y}_{t-p}+\beta \cdot RF\left(X_t\right)+{ϵ}_t$(11)

For each variable k, the equation can be expressed as:

$y_{k,t}=c_k+\underset{i=1}{\overset{p}{{\displaystyle \sum }}}\underset{j=1}{\overset{K}{{\displaystyle \sum }}}{A}_{k,j,i}{y}_{j,t-i}+\underset{m=1}{\overset{K}{{\displaystyle \sum }}}{\beta }_{k,m}{\hat{y}}_{m,t}^{RF}+{ϵ}_{k,t},\forall k=1,2,\cdots ,K$(12)

where:

The data used in this study is sourced from the Kaggle dataset, which includes air quality data (AQI dataset) for various continents from 2022 to 2024, as well as country/region mappings, ISO codes, and continent/region data (Country/Region Mapping - ISO, Continent, Region). The two datasets were integrated and matched according to continents, resulting in a new dataset that includes variables such as year, month, and AQI values. By combining the air quality data with the world region classification data, visualization analysis was conducted in Python, and the results are as follows:

This Figure 2 shows the frequency distribution of global Air Quality Index (AQI) statuses in the dataset. The X-axis represents the AQI statuses, including “Good,” “Moderate,” “Unhealthy for Sensitive Groups,” “Unhealthy,” “Very Unhealthy,” and “Hazardous.” The Y-axis represents the frequency of each status globally. The values above each bar indicate the number of occurrences of each AQI status, showing how many countries fall under each category.

This Figure 3 compares the distribution of air quality statuses across different continents, displaying the air quality classification for each region. The X-axis represents the regions, such as Asia, Europe, Africa, etc., while the Y-axis shows the frequency of each air quality status within the region. The values above each bar indicate the specific count of each air quality status in the region, illustrating the distribution of air quality in that area.

This Figure 4 illustrates the distribution of the Air Quality Index (AQI) across different continents, visualized using a boxplot. The X-axis represents regions, such as Asia, Europe, Africa, etc., while the Y-axis shows the AQI values, where higher values indicate more severe air pollution. The median line in the boxplot represents the median AQI for each region (shown as the values on each box). The box represents the inter-quartile range of AQI values in each region, reflecting the central tendency and range of variation in air quality. The points marked as “outliers” represent abnormal values, which are AQI readings that are far from

the majority of the data points.