# Experimental Study on the Effect of Wind Speed on Hail Impacts
## 4 Multivariate Regression Analysis of Hail Impact Forces  
In engineering practice, multiple linear regression is commonly employed to explore the relationships between dependent variables and multiple independent variables [39]. This is because when actual phenomena occur, the observed results are usually influenced by multiple interacting determinants. Multivariate regression analysis, which incorporates an optimal combination of independent variables to predict or estimate dependent variables, demonstrates superior predictive accuracy and empirical validity compared to univariate approaches that rely on a single predictor. This multivariate approach not only mitigates confounding effects but also aligns more closely with the complex causal architectures inherent in nature. In this study, to numerically calculate and predict the impact forces of hailstones on structures for engineering design reference, hail impact simulations under varying wind speeds were conducted in a wind tunnel laboratory using a self-developed integrated hail impact simulation device. A systematic comparative analysis was performed to investigate the interrelationships among wind speed, hailstone diameter, hail velocity, and hail impact force. Based on the experimental results, a peak hail impact load prediction  
Figure 9: Peak impact force against kinetic energy of hail. model applicable to diverse conditions was proposed, offering significant implications for structural design to combat wind-hail disasters. To perform the multivariate regression analysis, the experimental data were processed using SPSS 27.0 software. Through iterative trial and error, the regression model was hypothesized as follows:  
\[F_{p}=k_{1}V_{w}+k_{2}d+k_{3}V_{hail}+k_{4} \tag{4}\]  
In the equation, \(F_{p}\): peak load of hail impact on the structure (unit: N); \(V_{w}\): wind speed (unit: m/s); \(d\): diameter of the hailstone (unit: m); and \(V_{hail}\): velocity of the hailstone (unit: m/s). The results of the multivariate regression analysis model are presented in Table 5:  
As shown in Table 5, hail diameter, hail velocity, and wind speed all had statistically significant effects on the peak hail impact force. Based on the magnitudes of the standardized coefficient beta, velocity exerted the strongest influence on peak displacement, followed by hailstone diameter, with wind speed demonstrating the weakest effect. The estimated coefficients of the model were \(k_{1}=2.321\), \(k_{2}=6375.502\), \(k_{3}=9.487\), and \(k_{4}=-262.402\). Based on the significance levels of each coefficient, all coefficients exhibited \(p\leq 0.001<0.05\), indicating statistically significant effects on hail impact forces. Furthermore, the overall model validity was confirmed through an F-test, leading to the establishment of the linear regression equation as follows:  
\[F_{p}=2.321V_{w}+6375.502d+9.487V_{hail}-262.402 \tag{5}\]  
In the equation, \(F_{p}\): peak load of hail impact on the structure (unit: N); \(V_{w}\): wind speed (unit: m/s); \(d\): diameter of the hailstone (unit: m); and \(V_{hail}\): velocity of the hailstone (unit: m/s). The following F-test [40] was conducted to evaluate the regression equation. Original hypothesis (H\({}_{0}\)): \(k_{1}=k_{2}=k_{3}=0\) (F-test original hypothesis: all parameters in the model are zero). Alternative hypothesis (H\({}_{1}\)): At least one of \(k_{1}\), \(k_{2}\), and \(k_{3}\) is non-zero (F-test alternative hypothesis: not all parameters in the model are zero). If the original hypothesis holds, the independent variables have no significant effect on the dependent variable. Conversely, if the original hypothesis is rejected, the independent variables significantly influence the dependent variable, indicating a statistically significant linear relationship between the variables.