Group screening for ultra-high-dimensional feature under linear model

Figures & data

Figure 1. Boxplots of average model sizes for Example 4.1 under different group number.

Table 1. Simulation results of MEDIAN, TP, FP, CORRECT and ME for Example 4.1.

p	Method	MEDIAN	TP	FP	CORRECT	ME
	ISIS-Group-Lasso	6.00	4.00 (0.00)	2.88 (2.99)	0.12	2.19 (0.32)
p=200	g-ISIS-Group-Lasso	4.00	4.00 (0.00)	0.76 (0.75)	0.51	1.90 (0.38)
	Group Lasso	16.00	4.00 (0.00)	11.78 (2.24)	0.00	2.82 (0.36)
	DC-SIS-Group-Lasso	9.00	3.13 (0.00)	5.76 (2.24)	0.00	4.05 (0.28)
	DC-ISIS-Group-Lasso	13.00	4.00 (0.00)	8.52 (2.24)	0.00	2.73 (0.35)
	ISIS-Group-Lasso	7.00	4.00 (0.00)	3.43 (2.99)	0.09	2.30 (0.39)
p=400	g-ISIS-Group-Lasso	4.00	4.00 (0.00)	0.76 (0.75)	0.51	1.92 (0.46)
	Group Lasso	20.00	4.00 (0.00)	16.11 (2.24)	0.00	2.93 (0.38)
	DC-SIS-Group-Lasso	11.00	3.09 (0.00)	7.48 (2.24)	0.00	4.15 (0.29)
	DC-ISIS-Group-Lasso	14.00	4.00 (0.00)	10.17 (1.49)	0.00	2.77 (0.33)
	ISIS-Group-Lasso	7.00	4.00 (0.00)	3.32 (2.99)	0.14	2.34 (0.49)
p=1000	g-ISIS-Group-Lasso	4.00	3.99 (0.00)	0.83 (0.75)	0.51	1.97 (0.50)
	Group Lasso	28.00	4.00 (0.00)	24.15 (2.99)	0.00	3.02 (0.32)
	DC-SIS-Group-Lasso	13.00	3.04 (0.00)	9.81 (2.24)	0.00	4.17 (0.24)
	DC-ISIS-Group-Lasso	16.00	4.00 (0.00)	11.80 (2.24)	0.00	2.80 (0.32)

Note: Robust standard deviations are given in parentheses. ($\sigma =5$, SNR= 2.92.)

Table 2. Simulation results of MEDIAN, TP, FP, CORRECT and ME for Example 4.2.

p	Method	MEDIAN	TP	FP	CORRECT	ME
	ISIS-Group-Lasso	10.00	8.00 (0.00)	2.89 (2.24)	0.12	1.81 (0.34)
p=200	g-ISIS-Group-Lasso	8.00	8.00 (0.00)	0.73 (0.75)	0.52	1.58 (0.33)
	Group Lasso	34.00	8.00 (0.00)	26.44 (3.73)	0.00	2.74 (0.33)
	DC-SIS-Group-Lasso	19.00	6.22 (0.75)	13.23 (2.24)	0.00	4.40 (1.04)
	DC-ISIS-Group-Lasso	23.00	7.94 (0.00)	14.85 (2.24)	0.00	3.15 (0.44)
	ISIS-Group-Lasso	11.00	8.00 (0.00)	3.08 (2.99)	0.13	1.87 (0.37)
p=400	g-ISIS-Group-Lasso	9.00	8.00 (0.00)	0.87 (0.75)	0.44	1.66 (0.33)
	Group Lasso	45.00	8.00 (0.00)	36.86 (3.73)	0.00	2.95 (0.31)
	DC-SIS-Group-Lasso	21.00	5.47 (0.75)	15.39 (2.24)	0.00	4.99 (0.79)
	DC-ISIS-Group-Lasso	25.00	7.90 (0.00)	17.33 (2.24)	0.00	3.35 (0.43)
	ISIS-Group-Lasso	11.00	8.00 (0.00)	3.37 (2.99)	0.12	1.92 (0.43)
p=1000	g-ISIS-Group-Lasso	8.00	8.00 (0.00)	0.83 (0.75)	0.51	1.66 (0.43)
	Group Lasso	60.00	7.99 (0.00)	52.18 (3.92)	0.00	3.13 (0.33)
	DC-SIS-Group-Lasso	24.00	4.86 (0.75)	19.02 (2.99)	0.00	5.29 (0.62)
	DC-ISIS-Group-Lasso	27.00	7.78 (0.00)	19.43 (2.24)	0.00	3.50 (0.47)

Note: Robust standard deviations are given in parentheses. ($\sigma =4$, SNR= 5.34.)

Table 3. Simulation results of MEDIAN, TP, FP, CORRECT and ME for Example 4.3.

p	Method	MEDIAN	TP	FP	CORRECT	ME
	ISIS-Group-Lasso	7.00	4.00 (0.00)	2.87 (2.24)	0.12	1.64 (0.32)
p=200	g-ISIS-Group-Lasso	4.00	4.00 (0.00)	0.75 (0.75)	0.56	1.37 (0.35)
	Group Lasso	19.00	4.00 (0.00)	15.42 (2.24)	0.00	2.15 (0.28)
	DC-SIS-Group-Lasso	12.00	3.68 (0.75)	8.33 (2.43)	0.00	2.41 (0.67)
	DC-ISIS-Group-Lasso	13.00	4.00 (0.00)	8.73 (1.49)	0.00	2.03 (0.27)
	ISIS-Group-Lasso	6.00	4.00 (0.00)	2.88 (2.24)	0.14	1.64 (0.43)
p=400	g-ISIS-Group-Lasso	4.00	4.00 (0.00)	0.74 (0.75)	0.56	1.37 (0.40)
	Group Lasso	27.00	3.99 (0.00)	22.86 (2.99)	0.00	2.22 (0.26)
	DC-SIS-Group-Lasso	13.00	3.51 (0.75)	9.25 (2.24)	0.00	2.64 (0.71)
	DC-ISIS-Group-Lasso	14.00	4.00 (0.00)	9.88 (1.49)	0.00	2.07 (0.28)
	ISIS-Group-Lasso	7.00	4.00 (0.00)	3.20 (2.99)	0.10	1.77 (0.37)
p=1000	g-ISIS-Group-Lasso	5.00	3.99 (0.00)	0.79 (0.75)	0.44	1.47 (0.41)
	Group Lasso	40.00	4.00 (0.00)	36.01 (3.17)	0.00	2.33 (0.26)
	DCS-Group-Lasso	15.00	3.43 (0.75)	11.71 (2.24)	0.00	2.71 (0.56)
	DC-ISIS-Group-Lasso	15.00	4.00 (0.00)	11.53 (2.24)	0.00	2.15 (0.26)

Note: Robust standard deviations are given in parentheses. ($\sigma =4$, SNR= 2.69.)

Table 4. Simulation results of MEDIAN, TP, FP, CORRECT and ME for Example 4.4.

p	Method	MEDIAN	TP	FP	CORRECT	ME
	ISIS-Group-Lasso	11.00	8.00 (0.00)	3.45 (2.99)	0.13	2.08 (0.43)
p=200	g-ISIS-Group-Lasso	9.00	8.00 (0.00)	0.79 (0.75)	0.46	1.80 (0.32)
	Group Lasso	30.00	8.00 (0.00)	22.17 (2.99)	0.00	3.15 (0.30)
	DC-SIS-Group-Lasso	14.00	5.70 (0.75)	8.08 (2.99)	0.00	5.53 (0.91)
	DC-ISIS-Group-Lasso	18.00	7.52 (0.75)	10.23 (2.24)	0.00	3.52 (1.16)
	ISIS-Group-Lasso	11.00	8.00 (0.00)	3.96 (2.24)	0.07	2.19 (0.38)
p=400	g-ISIS-Group-Lasso	9.00	7.98 (0.00)	0.79 (0.75)	0.49	1.86 (0.34)
	Group Lasso	37.00	7.97 (0.00)	29.55 (3.17)	0.00	3.35 (0.41)
	DC-SIS-Group-Lasso	14.00	5.26 (0.75)	8.99 (2.24)	0.00	5.87 (0.82)
	DC-ISIS-Group-Lasso	18.00	7.44 (0.75)	11.25 (2.99)	0.00	3.69 (1.12)
	ISIS-Group-Lasso	11.00	8.00 (0.00)	3.91 (2.24)	0.07	2.24 (0.39)
p=1000	g-ISIS-Group-Lasso	8.00	7.94 (0.00)	0.75 (0.75)	0.50	1.92 (0.34)
	Group Lasso	49.00	7.91 (0.00)	40.75 (4.48)	0.00	3.60 (0.36)
	DC-SIS-Group-Lasso	16.00	4.88 (0.75)	11.19 (2.99)	0.00	6.07 (0.54)
	DC-ISIS-Group-Lasso	20.00	7.40 (0.75)	12.77 (2.24)	0.00	3.77 (1.11)

Note: Robust standard deviations are given in parentheses. ($\sigma =4$, SNR= 6.78.)

Figure 2. Boxplot of average model sizes for colon data analysis.

Table 5. Results of AMS and CORRECT for colon data.

Method	AMS	PE
ISIS-Group-Lasso	6.58 (2.24)	2.08 (0.34)
g-ISIS-Group-Lasso	3.6 (1.49)	2.10 (0.35)
Group Lasso	18.49 (2.24)	3.09 (0.58)

Note: Robust standard deviations are given in parentheses.