Linear Twin Quadratic Surface Support Vector Regression

Table 2

Results of Ridge Regression (RR), Linear Regression (LR), nu-SVR, ϵ-SVR, LSSVR, TSVR, LTSVR, and LTQSSVR on quadratic artificial data sets.

Data sets num × fea	Algorithm	RMSE	MAE	SSE/SST	SSR/SST	Time (s)	c	ϵ	σ
AR1; 30 × 10	RR	556.9348 ± 64.6680	472.5440 ± 68.1756	0.5760 ± 0.3548	0.7606 ± 0.3899	0.0037
	LR	399.4814 ± 100.0622	328.3254 ± 88.2998	0.3022 ± 0.1929	1.1167 ± 0.4799	2.0278
	nu-SVR	864.5692 ± 197.0818	673.1263 ± 149.9371	1.1580 ± 0.2129	0.1580 ± 0.2129	0.1191	2^5	0.001	2^0
	ϵ-SVR	865.8422 ± 198.0437	675.8055 ± 151.8056	1.1606 ± 0.2109	0.1606 ± 0.2109	0.1090	2^5	0.001	2^–4
	LSSVR	634.8957 ± 256.3804	520.7602 ± 208.0142	0.5531 ± 0.4678	0.3691 ± 0.1874	1.511	2^2		2^3
	TSVR	509.4136 ± 229.8878	416.2661 ± 194.7960	0.7154 ± 0.3129	0.3126 ± 0.4475	14.91	2^–4	0.1	2^3
	LTSVR	875.5967 ± 178.9845	724.9856 ± 141.0931	0.9595 ± 0.3107	0.2126 ± 0.1810	4.2708	2^–3	0.2	2^3
	LTQSSVR	233.7883 ± 74.5269	184.9046 ± 58.4456	0.0698 ± 0.0375	0.9108 ± 0.1189	3.397	2^–1	0.01
AR2; 50 × 20	RR	772.9401 ± 94.6686	644.1068 ± 80.0471	0.7422 ± 0.2818	1.2858 ± 0.3805	0.0002
	LR	804.7534 ± 205.3123	660.0124 ± 152.2902	0.7986 ± 0.3627	1.4378 ± 0.5091	22.6734
	nu-SVR	1019.9 ± 164.4628	828.6489 ± 138.9097	1.2486 ± 0.4260	0.2486 ± 0.4260	0.1761	2^–4	0.001	2^–2
	ϵ-SVR	1013.4 ± 147.9270	832.0854 ± 131.6954	1.2332 ± 0.4159	0.2332 ± 0.4159	0.1979	2^5	0.001	2^–4
	LSSVR	876.2712 ± 35.8862	656.5610 ± 33.6003	0.3968 ± 0.0538	0.5012 ± 0.0699	0.7977	2^5		2^5
	TSVR	825.4982 ± 192.4043	662.9940 ± 167.3309	0.2658 ± 0.1259	0.7697 ± 0.2515	18.8629	2^–1	0.01	2^5
	LTSVR	2370 ± 767.3638	2016.2 ± 758.3574	2.1851 ± 1.4099	1.3458 ± 1.2179	9.1719	2^–3	0.2	2^4
	LTQSSVR	503.2282 ± 111.1342	391.0349 ± 93.5383	0.1362 ± 0.0736	0.9079 ± 0.0424	20.7707	2^0	0.2
AR3; 100 × 30	RR	951.3805 ± 112.4494	750.8384 ± 88.4491	0.1356 ± 0.0377	0.8898 ± 0.0631	0.0072
	LR	922.5558 ± 132.01183	741.5420 ± 92.6491	0.1268 ± 0.0358	0.9371 ± 0.0548	0.5144
	nu-SVR	2685.5 ± 229.4038	2179.9 ± 162.6985	1.0437 ± 0.0377	0.0437 ± 0.0377	3.7895	2^5	0.001	2^–4
	ϵ-SVR	2701.6 ± 228.9889	2197.2 ± 166.9867	1.0571 ± 0.0578	0.0571 ± 0.0578	0.5144	2^5	0.001	2^–4
	LSSVR	1098 ± 121.5927	889.7450 ± 74.5102	0.0923 ± 0.0197	0.7229 ± 0.0703	2.2872	2^5		2^5
	TSVR	758.5976 ± 149.9849	601.7016 ± 128.2513	0.1026 ± 0.0611	0.8481 ± 0.0975	48.9992	2^5	0.2	2^5
	LTSVR	3841.9 ± 2813.9	3325.2 ± 2771.0	1.7216 ± 2.7258	1.4169 ± 2.4844	64.7086	2^–4	0.2	2^5
	LTQSSVR	863.4386 ± 114.7082	664.7501 ± 96.2545	0.1590 ± 0.0540	0.8574 ± 0.1567	144.6868	2^4	0.001
AR4; 500 × 30	RR	804.0858 ± 29.61234	620.5212 ± 29.0568	0.1167 ± 0.0122	0.8591 ± 0.0527	3.5505
	LR	741.3826 ± 28.4248	589.9683 ± 27.1126	0.0992 ± 0.0108	0.9021 ± 0.0496	6.5738
	nu-SVR	2367.5 ± 67.0977	1864.1 ± 58.4901	1.0067 ± 0.0074	0.0067 ± 0.0074	6.3427	2^5	0.001	2^–4
	ϵ-SVR	2370.9 ± 68.4916	1867.5 ± 60.7537	1.0096 ± 0.0128	0.0096 ± 0.0128	9.1899	2^2	0.001	2^–4
	LSSVR	579.1923 ± 69.7749	460.4036 ± 43.4087	0.0656 ± 0.0110	0.8806 ± 0.0369	46.9875	2^5		2^5
	TSVR	489.2266 ± 23.5683	391.9541 ± 24.6263	0.0612 ± 0.0012	0.9893 ± 0.0012	519.6265	2^5	0.01	2^5
	LTSVR	7321 ± 1491	6689.5 ± 859.5	−	−	2714.643	2^–4	0.2	2^3
	LTQSSVR	529.4326 ± 59.2533	473.1416 ± 2.9623	0.0607 ± 0.0122	1.001 ± 0.0011	442.1526	2^5	0.1
AR5; 1000 × 10	RR	355.4752 ± 13.4051	283.0450 ± 10.6422	0.1897 ± 0.0117	0.6229 ± 0.0495	0.0018
	LR	231.9386 ± 8.2504	181.3980 ± 6.9325	0.0809 ± 0.0064	0.9459 ± 0.0603	1.8952
	nu-SVR	817.6871 ± 22.6287	651.5018 ± 13.5489	1.0028 ± 0.0031	0.0028 ± 0.0031	22.8603	2^5	0.001	2^–4
	ϵ-SVR	817.6872 ± 22.6285	651.5021 ± 13.5487	1.0028 ± 0.0031	0.0028 ± 0.0031	21.8329	2^5	0.001	2^–4
	LSSVR	43.43 ± 1.26	31.09 ± 0.79	0.002 ± 0.0001	0.9815 ± 0.0039	115.985	2^5		2^4
	TSVR	8.25 ± 1.17	5.65 ± 0.83	0.0001 ± 2.33E − 05	0.9967 ± 0.0033	32081.806	2^–4	0.01	2^5
	LTSVR	435.58 ± 431.67	347.57 ± 344.48	0.5135 ± 0.5134	0.5134 ± 0.4865	5105.084	2^–3	0.01	2^5
	LTQSSVR	0.0758 ± 0.0191	0.0731 ± 0.0212	1.34E − 08 ± 6.24E − 09	1.000 ± 1.98E − 05	1546.984	2^2	0.2
AR6; 2000 × 10	RR	348.4587 ± 5.2498	277.5702 ± 4.6563	0.1884 ± 0.0089	0.6374 ± 0.0228	0.0029
	LR	232.5254 ± 5.7657	180.2640 ± 4.9633	0.0840 ± 0.0066	0.9209 ± 0.0215	8.6790
	nu-SVR	804.4539 ± 15.5199	644.4631 ± 13.4341	1.0033 ± 0.0058	0.0033 ± 0.0058	68.0419	2^5	0.001	2^–4
	ϵ-SVR	804.4538 ± 15.5199	644.4633 ± 13.4341	1.0033 ± 0.0058	0.0033 ± 0.0058	95.3783	2^5	0.001	2^–4
	LSSVR	29.2831 ± 0.1239	19.5835 ± 0.5175	0.0016 ± 3.1304e − 05	0.9765 ± 6.0976e − 04	920.0551	2^5		2^4
	TSVR	10.3615 ± 1.5314	7.2374 ± 1.4115	0.0001 ± 1.96E − 05	0.9963 ± 0.0017	10673.192	2^–2	0.01	2^5
	LTSVR	775.3161 ± 133.3652	625.1344 ± 119.2564	1.2355 ± 0.2456	0.3495 ± 0.1316	16739.2	2^–4	0.01	2^–4
	LTQSSVR	0.1074 ± 0.0410	0.1032 ± 0.0416	3.06e − 08 ± 2.23e − 08	1.000 ± 1.3591e − 05	7536.641	2^5	0.01
AR7; 3000 × 5	RR	294.1609 ± 3.7125	237.5383 ± 2.6317	0.1974 ± 0.0039	0.4435 ± 0.0082	0.1503
	LR	143.8636 ± 2.7915	110.1801 ± 1.7634	0.0472 ± 0.0012	0.9656 ± 0.0308	6.5705
	nu-SVR	560.7090 ± 10.0120	442.1160 ± 6.1240	0.7170 ± 0.0127	0.0643 ± 0.0074	1134.1517	2^5	0.001	2^–4
	ϵ-SVR	560.7196 ± 10.0115	442.1315 ± 6.1242	0.7170 ± 0.0127	0.0643 ± 0.0074	170.9971	2^5	0.001	2^–4
	LSSVR	10.4242 ± 0.5163	6.6254 ± 0.2775	0.0002 ± 2.23083e − 05	0.9964 ± 0.0003	2959.8848	2^5		2^4
	TSVR	−	−	−	−	−	2^–4	0.01	2^5
	LTSVR	−	−	−	−	−	2^–3	0.01	2^5
	LTQSSVR	0.0667 ± 0.0049	0.0664 ± 0.0051	1E − 08 ± 1.39E − 09	1.000 ± 1.98E − 05	1546.984	2^2	0.2
AR8; 5000 × 3	RR	121.4335 ± 1.9984	98.4936 ± 1.4719	0.2308 ± 0.0034	0.5152 ± 0.0083	0.0044
	LR	104.7501 ± 2.3667	78.1530 ± 1.0814	0.1717 ± 0.0049	0.8325 ± 0.0107	8.9268
	nu-SVR	7.7250 ± 0.8782	1.3397 ± 0.1542	0.0009 ± 2.0694e − 04	0.9892 ± 0.0018	1219.7795	2^5	0.001	2^–4
	ϵ-SVR	7.7357 ± 0.8790	1.3571 ± 0.1540	0.0009 ± 2.0743e − 04	0.9888 ± 0.0018	438.3024	2^5	0.001	2^–4
	LSSVR	2.0616 ± 0.3076	0.8998 ± 0.3241	6.7308e − 05 ± 1.9711e − 05	0.9967 ± 0.0016	150667.6	2^4		2^4
	TSVR	−	−	−	−	−	2^–4	0.01	2^5
	LTSVR	−	−	−	−	−	2^–3	0.01	2^5
	LTQSSVR	0.0716 ± 0.0240	0.0710 ± 0.0245	8.4658E − 08 ± 6.31E − 09	1.000 ± 1.86311E − 06	22332.86	2^5	0.2