Incorporating biological information in sparse principal component analysis with application to genomic data

BMC Bioinformatics

Table 2 Simulation results of setting 2

Method	RE	EE	Sensitivity		Specificity		cPVE
			1stPC	2ndPC	1stPC	2ndPC	1stPC	2ndPC
P = 500
PCA	31 (0.9)	1.1 (3e-2)	1.0	1.0	0.0	0.0	4.3e-2 (2e-3)	8.2e-2 (2e-3)
SPCA	35 (2)	1.3 (9e-2)	0.49	0.50	0.95	1.0	1.9e-2 (3e-3)	3.9e-2 (4e-3)
SPC	15 (7)	0.54 (3e-1)	0.57	0.60	0.98	1.0	2.8e-2 (3e-3)	5.6e-2 (5e-3)
Biological information correctly specified
Fused sPCA	27 (4)	0.93 (2e-1)	1.0	1.0	0.70	0.70	3.0e-2 (3e-3)	5.3e-2 (5e-3)
Grouped sPCA	7.9 (5)	0.29 (2e-1)	0.80	0.80	0.97	1.0	3.2e-2(2e-3)	6.0e-2 (3e-3)
Biological information randomly specified
Fused sPCA	32 (5)	1.1 (2e-1)	0.96	1.0	0.52	0.50	2.9e-2 (5e-3)	5.1e-2 (8e-3)
Grouped sPCA	9.2 (6)	0.33 (0.2)	0.79	0.8	0.97	1.0	3.2e-2 (2e-3)	5.9e-2 (4e-3)
P = 10,000
PCA	112 (3)	1.3 (2e-2)	1.0	1.0	0.0	0.0	2.7e-2 (1e-3)	5.0e-2 (1e-3)
SPCA	162 (4)	1.9 (3e-2)	0.16	0.16	1.0	1.0	2.0e-3 (5e-4)	4.0e-3 (8e-4)
SPC	173 (4)	2.0 (5e-3)	5.0e-3	5.0e-3	1.0	1.0	1.6e-4 (1e-4)	3.2e-4 (2e-4)
Biological information correctly specified
Fused sPCA	77 (40)	0.89 (0.5)	0.65	0.57	0.99	1.0	1.3e-2 (5e-3)	2.3e-2 (9e-3)
Grouped sPCA	46 (30)	0.53 (0.4)	0.65	0.62	0.99	1.0	1.5e-2 (2e-3)	2.8e-2 (5e-3)
Biological information randomly specified
Fused sPCA	140 (30)	1.6 (0.4)	0.59	0.60	0.68	0.70	9.0e-3 (5e-3)	1.7e-2 (1e-2)
Grouped sPCA	53 (40)	0.61 (0.4)	0.63	0.60	0.99	1.0	1.5e-2 (3e-3)	2.7e-2 (6e-3)

Cumulative proportions of variance explained by true PCs are 0.15 for PC 1 and 0.30 for PC 1 and 2. P, number of variables. RE, reconstruction error, defined as \(||\mathbf {X}_{test}\mathbf {A}\mathbf {A}^{T} -\mathbf {X}_{test}\hat {\mathbf {A}}\hat {\mathbf {A}}^{\text {T}}||^{2}_{F}\), where A=(α ₁ α ₂). EE, estimation error, defined as \(||\mathbf {A}\mathbf {A}^{T} -\hat {\mathbf {A}}\hat {\mathbf {A}}^{\text {T}}||^{2}_{F}\). cPVE, proportions of cumulative variation explained. ·(·), mean(std)

ISSN: 1471-2105