Development of Integrated Choice and Latent Variable (ICLV) Models Using Matrix-Based Analytic Approximation and Automatic Differentiation Methods on TensorFlow Platform

Table 1

BVN and TVN evaluation results for the different methods.

Dim	Measure	Numerical integration methods		Analytic approximation methods (TensorFlow)
		Gauss	TF	ME	BME	TVBS
K = 2 (BVN)	MAE	4.8^∗e − 10	1.9^∗e − 15	0.0037	—	—
	%MAE > 0.005	0.0	0.0	18.8	—	—
	MAPE	8.6^∗e − 8	1.3^∗e − 12	1.28	—	—
	%MAPE > 2	0.0	0.0	15.2	—	—
	Time (s)	0.001	0.002	0.003	—	—
K = 3 (TVN)	MAE	8.8^∗e − 10	6.1^∗e − 11	0.0027	0.0018	—
	%MAE > 0.005	0.0	0.0	18.1	12.4	—
	MAPE	1.9^∗e − 7	2.9^∗e − 8	1.34	1.00	—
	%MAPE > 2	0.0	0.0	16.7	11.0	—
	Time (s)	0.005	0.005	0.006	0.003	—

Note. “—“ indicates that the corresponding analytic approximation methods cannot calculate the values of BVN or TVN.