## 2 Methodology
### Seasonal Autoregressive Integrated Moving Average (SARIMA)  
The seasonal autoregressive integrated moving average (SARIMA) model is a widely recognized statistical approach for forecasting time series data that exhibit seasonality and trends. Its structure is defined as SARIMA (\(p\),\(d\),\(q\))(\(P\),\(D\),\(Q\))\(S\), where \(p\), \(d\), and \(q\) represent the non-seasonal autoregressive order, degree of differencing, and moving average order, respectively, while \(P\), \(D\), and \(Q\) correspond to the seasonal components of the model, and \(S\) denotes the seasonal period length ([PERSON] et al., 2015). The SARIMA model is particularly advantageous in fields such as epidemiology, economics, and environmental science, where seasonal patterns are prevalent. \[\varphi_{p}(B)\Phi_{p}(B^{s})W_{t}=\theta_{q}(B)\Theta_{Q}(B^{s})\omega_{t} \tag{9}\]  
The notation for Equation (17) is described as follows: The parameters \(p\), \(d\), and \(q\) represent the order of the non-seasonal components of the model. Specifically,  
* \(P\) denotes the order of the seasonal autoregressive (AR) model. * \(D\) refers to the number of seasonal differencing required to achieve stationarity. * \(Q\) indicates the order of the seasonal moving average (MA) component. * \(S\) represents the length of the seasonal cycle or periodicity. In addition, \(\omega_{t}\) refers to the white noise value at time t, while B denotes the backward shift operator. The SARIMA (\(p\),\(d\),\(q\)) \(\times\) (\(P\),\(D\),\(Q\))\(S\) model is particularly effective for analyzing a variety of time series due to its relatively compact order, making it computationally efficient. The periodicity (\(s\)) of the time series is determined based on the seasonal characteristics of the dataset. One of the key strengths of the SARIMA model is its ability to capture linear relationships within time series data effectively. For instance, studies have demonstrated that SARIMA outperforms nonlinear models like back propagation neural networks (BPNNs) in forecasting disease incidence, as seen in the analysis of rubella incidence in Chongqing, China (2015). This finding is echoed in the context of mumps forecasting in Zibo City, where SARIMA exhibited superior performance compared to standard regression models, which often fail to account for autocorrelations and trends adequately ([PERSON] et al., 2015). Furthermore, the SARIMA model has been successfully applied to predict various infectious diseases, including tuberculosis and dengue fever, highlighting its versatility and reliability in public health forecasting [67]. However, the application of the SARIMA model is not without limitations. It is essential that the data used for modeling are relatively smooth, as the model may struggle with complex nonlinear data, potentially leading to a loss of information during the transformation process [68]. Additionally, while SARIMA is effective for many types of seasonal data, it may not always outperform hybrid models that combine SARIMA with other techniques, such as ANN or GA [69; 70]. For example, a hybrid SARIMA-ANN model has shown improved forecasting accuracy in various studies, suggesting that combining different modeling approaches can enhance predictive performance [71; 72]. In conclusion, SARIMA remains a robust method for forecasting seasonal time series data, particularly in fields requiring accurate predictions of trends and cyclic patterns. Its ability to model linear relationships effectively makes it a preferred choice in many applications, although researchers should remain cognizant of its limitations and consider hybrid approaches for more complex datasets. Continued exploration of SARIMA's hyperparameters and its integration with other forecasting methods will likely yield further improvements in predictive accuracy across diverse domains [73; 74]. The development of an ARIMA model involves three key stages: identification, parameter estimation, and diagnostic testing [75]. 1.