## 3 Modeling and Data Sources
### Models
#### 3.1.2 ARIMA Model  
The ARIMA (Autoregressive Integrated Moving Average) model, an extension of the ARMA (Autoregressive Moving Average) model, is esteemed for its robust mathematical foundation and efficient implementation, making it a principal choice in time series analysis [27; 28; 29]. It addresses non-stationary time series data through differencing, converting it into a stationary series. The ARIMA model's utility is particularly evident in forest fire forecasting due to the temporal nature of forest fire occurrences. It assists in predicting the frequency of future forest fires, which is vital for planning and prevention measures.  
The AR model represents future values as a linear combination of past observations and current disturbances. The AR model [30] is presented in Equation (2):  
\[y_{t}=\mu+\sum_{i=1}^{p}\gamma_{i}y_{t-i}+\varepsilon_{t} \tag{2}\]  
where \(y_{t}\) represents the observation, \(\mu\) is a constant term, \(p\) denotes the order of the AR model, \(\gamma_{i}\) is the autocorrelation coefficient, and \(\varepsilon_{t}\) is the error term.  
The MA model [31] accumulates the error terms of the autoregressive model and is given in Equation (3):  
\[y_{t}=\mu+\varepsilon_{t}+\sum_{i=1}^{q}\theta_{i}\varepsilon_{t-i} \tag{3}\]  
where \(y_{t}\) represents the observation, \(\mu\) is a constant term, \(q\) denotes the order of the MA model, \(\theta_{i}\) is the moving average coefficient, and \(\varepsilon_{t}\) is the error term.  
Thus, the ARIMA model [32] is formulated as in Equation (4):  
\[y_{t}=\mu+\varepsilon_{t}+\sum_{i=1}^{q}\theta_{i}\varepsilon_{t-i}+\sum_{i=1 }^{p}\gamma_{i}y_{t-i} \tag{4}\]  
1. ADF Test Prior to conducting time series analysis on the frequency of forest fires in Yunnan Province from 1991 to 2021, it is a prerequisite to employ the ADF (Augmented Dickey-Fuller) test to verify the stationarity of the series [33] ([[https://en.wikipedia.org/wiki/Augmented_[PERSON]%E2%80%93](https://en.wikipedia.org/wiki/Augmented_[PERSON]%E2%80%93) Fuller_test]([https://en.wikipedia.org/wiki/Augmented_[PERSON]%E2%80%93](https://en.wikipedia.org/wiki/Augmented_[PERSON]%E2%80%93) Fuller_test), accessed on 1 November 2023). Establishing an ARIMA model for the time series is contingent upon passing the ADF test. Should the \(p\)-value from the ADF test exceed 0.05, the null hypothesis is accepted, necessitating a first-order differencing of the data.
2. Model Identification In time series analysis, accurate model identification is crucial for the construction of the ARIMA model. This process involves the selection of the order of the autoregressive (AR) terms, the degree of differencing (I), and the order of the moving average (MA) terms. These parameters collectively determine the structure of the model and its predictive accuracy. Appropriate orders allow the model to capture the statistical characteristics of the data, such as trends, seasonality, and cycles, while inappropriate orders may lead to loss of information or overfitting.  
According to the research by [PERSON] [34], the optimal order of the ARIMA model is ascertained by analyzing the plots of the autocorrelation function (ACF) and the partial autocorrelation function (PACF), as demonstrated in Table 1, and by selecting the model with the minimum values of AIC, SC, and HQC. This method effectively identifies the time series correlations and seasonal features, selecting the model and lag order that best align with the data characteristics, thus providing a solid foundation for precise forecasting.