HRLR regression

Figures & data

Table 1. Median TRMSE and the SE values for Example 1 for the five methods considered with four error distributions

Error Dist.		HRLR	relaxed	lasso	ridge	enet
1	TRMSE	3.381257	4.015348	3.396064	3.393395	3.383454
	SE	0.537501	0.722440	0.503587	0.459631	0.503179
2	TRMSE	1.927973	2.457771	2.126671	2.231816	2.116753
	SE	0.440730	0.849136	0.541952	0.577668	0.491403
3	TRMSE	1.253082	2.142927	1.327364	1.486484	1.352891
	SE	0.357699	1.045325	0.577568	0.595865	0.558552
4	TRMSE	0.792765	1.707721	1.046893	1.187845	1.037665
	SE	0.389845	0.925814	0.953209	0.996269	0.936685

Figure 1. TRMSE distributions of the five methods considered with four error distributions for Example 1

Figure 2. TRMSE distributions of the five methods considered with four error distributions for Example 2

Figure 3. TRMSE distributions of the five methods considered with four error distributions for Example 3

Table 2. Median TRMSE and the SE values for Example 2 for the five methods considered with four error distributions

Error Dist.		HRLR	relaxed	lasso	ridge	enet
1	TRMSE	3.625042	3.775470	3.507422	3.262799	3.414795
	SE	0.395082	0.513873	0.405024	0.346396	0.386352
2	TRMSE	2.270207	2.311954	2.161612	1.972279	2.118149
	SE	0.480255	0.544875	0.512514	0.380335	0.464744
3	TRMSE	1.540670	1.786979	1.529741	1.263564	1.458128
	SE	0.286257	0.374682	0.361605	0.354150	0.369028
4	TRMSE	1.208735	1.288259	1.273977	1.001277	1.121465
	SE	0.254703	0.315132	0.736371	0.584408	0.680291

Table 3. Median TRMSE and the SE values for Example 3 for the five methods considered with four error distributions

Error Dist.		HRLR	relaxed	lasso	ridge	enet
1	TRMSE	16.63881	17.73929	16.45876	16.05251	16.35268
	SE	1.049380	1.694161	0.750628	0.755907	0.823232
2	TRMSE	3.230074	7.631660	3.771314	3.589136	3.650914
	SE	1.335897	1.319291	1.626837	1.585390	1.581940
3	TRMSE	1.584614	6.978842	2.144613	2.168660	2.093443
	SE	0.2784299	0.627340	1.390488	1.224328	1.302390
4	TRMSE	1.217255	6.957786	1.785902	1.820204	1.723091
	SE	0.291042	0.783199	1.427069	1.251697	1.335950

Table 4. Diabetes data

Patient	Age	Sex	BMI	BP	Serum measurements						$y$
1	59	2	32.1	101	157	93.2	38	4	4.9	87	151
2	48	1	21.6	87	183	103.2	70	3	3.9	69	75
3	72	2	30.5	93	156	93.6	41	4	4.7	85	141
4	24	1	25.3	84	198	131.4	40	5	4.9	89	206
5	50	1	23.0	101	192	125.4	52	4	4.3	80	135
.	.	.	.	.	.	.	.	.	.	.	.
.	.	.	.	.	.	.	.	.	.	.	.
.	.	.	.	.	.	.	.	.	.	.	.
441	36	1	30.0	95	201	125.2	42	5	5.1	85	220
442	36	1	19.6	71	250	133.2	97	3	4.6	92	57

Table 5. Coefficient estimates for diabetes data for the five methods considered

	Coefficient estimates
Varable	HRLR	relaxed	lasso	ridge	enet
age	0.00000000	0.00000000	0.00000000	0.01742117	0.00000000
sex	−0.02908175	0.00000000	−0.01539929	−0.03656159	0.00000000
bmi	0.33552927	0.35071797	0.34203828	0.30647396	0.33093482
map	0.13958292	0.13658879	0.13781028	0.14539792	0.13229055
tc	−0.13690801	−0.14827839	−0.12538283	−0.03965807	−0.01895922
ldl	0.00000000	0.00000000	0.00000000	−0.07896509	−0.06693646
hdl	−0.04423102	−0.03158335	−0.04174263	−0.08563,881	−0.07253518
tch	0.00000000	0.00000000	0.00000000	0.02245709	0.00000000
ltg	0.33620499	0.35037166	0.33560931	0.26525703	0.28081150
glu	0.07397086	0.07484722	0.06815829	0.08763129	0.06180458
TRMSE	0.6782023	0.6853457	0.6799639	0.6796829	0.6860057

Table 6. Coefficient estimates for bodyFat data for the five methods considered

	Coefficient estimates
Varable	HRLR	relaxed	lasso	ridge	enet
age	0.07910159	0.046044482	0.04391435	0.090061076	0.05667671
weight_lbs	0.00000000	0.000000000	0.00000000	0.005741097	0.00000000
height_in	−0.07738729	−0.059388509	−0.04079588	−0.030997615	−0.03472945
neck_cm	0.00000000	−0.057102588	−0.11440429	−0.111206362	−0.11674649
chest_cm	0.00000000	0.000000000	−0.11915296	−0.050638714	−0.11758926
abdomen_cm	0.00000000	0.000000000	0.00000000	0.349843231	0.38413309
hip_cm	0.00000000	0.000000000	0.00000000	0.029781385	0.00000000
thigh_cm	0.00000000	0.022723755	0.05315623	0.075930628	0.05893404
knee_cm	0.00000000	0.000000000	−0.03966939	−0.034524515	−0.04357893
ankle_cm	0.00000000	0.000000000	0.05916103	0.042586820	0.06092822
biceps_cm	0.00000000	0.000000000	0.03821674	0.031027182	0.03830752
forearm_cm	0.00000000	0.002414099	0.04565175	0.032540975	0.04583286
wrist_cm	−0.12437328	−0.108517197	−0.16465279	−0.197619699	−0.20312188
bmi_C	0.00000000	0.000000000	0.00000000	0.079378228	0.00000000
age_sq	0.00000000	0.045627887	0.10885672	0.093930394	0.10213594
abdomen_wrist	0.83846021	0.846912806	0.91846915	0.384097332	0.53899245
am	0.00000000	0.034282390	0.07859840	0.080403867	0.09020931
TRMSE	0.5210758	0.5182996	0.5236091	0.534444	0.5253579