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

BMC Bioinformatics

Table 1 Simulation results of setting 1

Method	RE	EE	Sensitivity		Specificity		cPVE
			1stPC	2ndPC	1stPC	2ndPC	1stPC	2ndPC
P = 500
PCA	31 (9e-1)	1.1 (3e-2)	1.0	1.0	0.0	0.0	4.3e-2 (2e-3)	8.2e-2 (2e-3)
SPCA	34 (3)	1.2 (1e-1)	0.54	0.50	0.95	0.90	2.0e-2 (2e-3)	4.0e-2 (4e-3)
SPC	16 (8)	0.57 (3e-1)	0.57	0.60	0.98	1.0	2.8e-2 (3e-3)	5.5e-2 (6e-3)
Biological information correctly specified
Fused sPCA	25 (6)	0.90 (2e-1)	1.0	1.0	0.73	0.70	2.9e-2 (4e-3)	5.1e-2 (7e-3)
Grouped sPCA	8.0 (6)	0.29 (2e-1)	0.81	0.80	0.97	1.0	3.2e-2 (2e-3)	6.0e-2 (3e-3)
Biological information randomly specified
Fused sPCA	32 (4)	1.1 (2e-1)	0.95	1.0	0.51	0.51	3.0e-2 (4e-3)	5.2e-2 (7e-3)
Grouped sPCA	9.1 (6)	0.33 (2e-1)	0.81	0.80	0.97	1.0	3.2e-2 (2e-3)	5.9e-2 (3e-3)
P = 10,000
PCA	112 (3)	1.3 (2e-2)	1.0	1.0	0.0	0.0	2.6e-2 (1e-3)	5.0e-2 (1e-3)
SPCA	160 (4)	1.9 (3e-2)	0.15	0.15	0.99	0.99	2.3e-3 (5e-4)	4.5e-3 (7e-4)
SPC	172 (4)	2.0 (8e-3)	0.01	0.01	1.0	1.0	1.7e-4 (1e-4)	3.4e-4 (3e-4)
Biological information correctly specified
Fused sPCA	81 (50)	0.94 (0.5)	0.62	0.55	0.99	0.99	1.2e-2 (6e-3)	2.2e-2 (1e-2)
Grouped sPCA	54 (40)	0.62 (0.4)	0.62	0.58	0.99	1.0	1.4e-2 (3e-3)	2.6e-2 (6e-3)
Biological information randomly specified
Fused sPCA	140 (30)	1.6 (0.4)	0.60	0.60	0.68	0.68	8.9e-3 (5e-3)	1.6e-2 (1e-2)
Grouped sPCA	58 (40)	0.67 (0.5)	0.59	0.55	0.99	1.0	1.4e-2 (3e-3)	2.6e-2 (7e-2)

Cumulative proportions of variance explained by true PCs are 0.03 for PC 1 and 0.06 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