## 4 Applicability of TR and Its Approximate Combination Rules
### Applicability of TR1 and TR2 Rule  
In the range of \(\rho\geq\) 0.4, \(\sigma_{2}\)/\(\sigma_{1}\)\(\leq\) 0.4, and \(\sigma_{2}\)/\(\sigma_{1}\)\(\geq\) 2.2, the relative errors of the combination extremes between TR1, TR2, and CQC rules were compared and analyzed to determine the application range of TR1 and TR2 rules. Figure 2 demonstrates how the relative error between the TR1 rule- and CQC rule-based combination extremes vary in terms of correlation coefficient and standard deviation ratio.  
Figure 1: The relative error between the Turkstra (TR) rule- and complete quadratic combination (CQC) rule-based combination extreme.  
As shown in Figure 2a, when the effect components are partially positively correlated, the relative error of the TR1 rule-and CQC rule-based combination extreme first increases and then decreases as \(\sigma_{2}\)/\(\sigma_{1}\) increases, and reaches the maximum when \(\sigma_{2}\)/\(\sigma_{1}=1\), and the largest relative error is \(-8.2\%\); when the effect components are completely positively correlated, the relative error of the TR1 rule- and CQC rule-based combination extreme remains unaffected by \(\sigma_{2}\)/\(\sigma_{1}\), and the relative error is \(-0.7\%\). As shown in Figure 2b, the relative error of the TR1 rule-and CQC rule-based combination extreme first increases and then decreases with the increase of \(\rho\) when \(\sigma_{2}\)/\(\sigma_{1}\) remains constant, and reaches the maximum when \(\rho=-0.4\) and the maximum relative error is \(-11.5\%\). The relative error of the TR1 rule- and CQC rule-based combination extreme is always negative, that is, the TR1 rule-based combination extreme always underestimates the combined effect. It can be seen from Figure 2 that the relative error of the TR1 rule- and CQC rule-based combination extreme is less than \(10\%\) in the range of \(\rho\geq 0.7\), \(\sigma_{2}\)/\(\sigma_{1}\leq 0.4\), \(\sigma_{2}\)/\(\sigma_{1}\geq 2.4\).  
As can be seen from Figure 3a, when \(0.4\leq\rho\leq 0.7\), the relative error of the TR2 rule- and CQC rule-based combination extreme increases first and then decreases as \(\sigma_{2}\)/\(\sigma_{1}\) increases, and reaches the maximum when \(\sigma_{2}\)/\(\sigma_{1}=1\). The maximum relative error is \(-6.6\%\); when \(\rho\geq 0.8\), the relative error of the TR2 rule-and CQC rule-based combination extreme changes very slightly as \(\sigma_{2}\)/\(\sigma_{1}\) goes up, which is almost a constant value. As shown in Figure 3b, the relative error of the TR2 rule- and CQC rule-based combination extreme, first increases and then decreases as the correlation coefficient increases when \(\sigma_{2}\)/\(\sigma_{1}\) remains constant, and reaches the maximum when \(\rho=-0.4\) and the largest relative error is \(-11.4\%\); the relative error of the TR2 rule- and CQC rule-based combination extreme is positive or negative. When \(\rho\leq 0.7\), the TR2 rule-based combination extreme is smaller than the CQC rule-based combination extreme, and when \(\rho>0.7\), the TR2 rule-based combination extreme is greater than the CQC rule-based combination extreme. It can be seen from Figure 3 that the relative error of the TR2 rule- and CQC rule-based combination extreme is less than \(10\%\) in the range of \(\rho\geq 0.4\), \(\sigma_{2}\)/\(\sigma_{1}\leq 0.4\), and \(\sigma_{2}\)/\(\sigma_{1}\geq 2.4\).