Multivariate Joint Probability Function of Earthquake Ground Motion Prediction Equations Based on Vine Copula Approach

Table 4

Estimated copula parameters of vine copula for PSA residuals at different vibration periods.

Copula	Family	Par. 1	Par. 2			Kendall’s
	Tawn1 (180°)	3.4608	0.9346	—	0.7512	0.6745
		(0.0940)	(0.0139)
	T	0.4522	17.1688	0.0173	0.0173	0.2987
		(0.0197)	(8.0943)
	N	0.7799	—	—	—	0.5695
		(0.0079)	—
	BB1z (180°)	0.3726	2.0394	0.4016	0.5952	0.5867
		(0.0547)	(0.0627)
	I	—	—	—	—	—
		—	—
	F	−1.1098	—	—	—	−0.1218
		(0.1497)	—
	BB8 (270°)	−1.4737	−0.7718	—	—	−0.0922
		(0.2539)	(0.1529)
	BB8 (90°)	−1.7090	−0.5839	—	—	−0.0791
		(0.6598)	(0.2916)
	BB8 (90°)	−1.8660	−0.6173	—	—	−0.1055
		(0.6181)	(0.2283)
	I	—	—		—	—
AIC		−5909.0300
BIC		−5834.3600
Log-likelihood		2968.5130

The subscripts 1, 2, 3, 4, and 5 denote PSAs at 0.1, 0.2, 1, 2, and 4 sec, respectively. N, T, F, I, Tawn, BB1, and BB8 the denote normal copula, t copula, Frank copula, independent copula, Tawn copula, BB1 copula, and BB8 copula, respectively. The values in brackets indicate that the fitted pair copula is rotated by original copula, i.e., 90°, 180°, and 270°. The standard errors are in parentheses. Independent copula, normal copula, and Frank copula do not exhibit an upper and lower tail dependence (i.e., and ). Statistical significance at the 5% level; statistical significance at the 1% level.