# Multi-Sensor Photoelectric Fire Alarm Device Implementation for Early Fire Detection in Campsites
## 3 Multi-Sensor Photoelectric Fire Detection System
### Relationship Between Weather and Fire Occurrence
#### 3.1.3 Statistical Evaluation  
In this study, we employed various statistical methods to evaluate the accuracy and fit of the multiple linear regression model represented by Equation (1). Validation procedures were conducted to confirm the adherence of the model to its fundamental assumptions. To achieve this, the Ordinary Least Squares (OLS) method, which is a standard approach in multiple linear regression analysis, was employed [28].  
In Figure 7, the R-squared value, which measures the model's goodness of fit, is 0.863, and the adjusted R-squared value is also 0.863. This indicates that the model explains approximately 86.3% of the variance, reflecting a high level of explanatory power. However, assessing a model's reliability based solely on this value is inadequate; therefore, it is necessary to examine other fundamental assumptions of linear regression [29].  
First, the F-statistic is 4.910 \(\times\) 104, with a corresponding \(p\)-value (Prob(F-statistic)) of 0.00%. Given that the \(p\)-value is less than the significance level (0.05), the regression model is considered statistically significant. Furthermore, an examination of the t-statistic values and their associated \(p\)-values (\(p>|t|\)) for each variable confirmed that all variables were statistically significant.  
Figure 6: Multiple linear regression between temperature, humidity and fire occurrence rates.  
Figure 7: OLS linear regression results.  
Next, we assessed whether the basic assumptions of multiple linear regression were satisfied using a residual plot (Figure 8), Q-Q plot (Figure 9), Variance Inflation Factor (VIF), and the Durbin-Watson test, employing the bootstrap method. The VIF, which measures multicollinearity in multiple regression, was calculated based on the R-squared values obtained by regressing each independent variable against the others [30]. The VIF values were 1.00 for the constant term and 1.027525 for both \(\times 1\) and \(\times 2\), indicating that multicollinearity is not a concern because these values are close to 1. Furthermore, the Q-Q plot and residual plot were examined to verify the assumptions of homoscedasticity, normality, linearity, and independence. As shown in Figures 8 and 9, the Q-Q plot shows points clustered along a 45\({}^{\circ}\) line, and the residual plot indicates a random distribution of residuals, confirming that the model adheres to these assumptions.  
However, the Durbin-Watson statistic of 1.165, along with the elevated Omnibus and Jarque-Bera (JB) test values, indicates that the residuals may not fully satisfy the assumptions of normality and independence. Nevertheless, the residual plot indicates a very weak positive autocorrelation, which is acceptable given that the fire occurrence rate relative to temperature and humidity may exhibit some nonlinearity, especially at extreme values.  
In conclusion, the proposed multiple linear regression model, as represented by Equation (1), is statistically significant and satisfies the fundamental assumptions. These results validate the reliability of the proposed model.