. 2021 Jan 3;53(1):459–515. doi: 10.1007/s11063-020-10380-y

Table 3.

Performance comparison of RHN-TSVR with SVR, TSVR, $ε$ -AHSVR, $ε$ -SVQR and HN-TSVR using Gaussian kernel on synthetic data sets with Uniform and Gaussian noise

Dataset (Train size, Test size)	SVR RMSE $(C, ε, μ)$ Time	TSVR RMSE $(C_{1} = C_{2},$ $ε, μ)$ Time	$ε$ -AHSVR RMSE $(C, ε_{α} = ε_{β}, ζ_{α},$ $ζ_{β}, μ)$ Time	$ε$ -SVQR RMSE $(C, ε,$ $θ, μ)$ Time	HN-TSVR RMSE $(C_{1} = C_{2}, ε_{1} = ε_{2},$ $ε_{1}^{} = ε_{2}^{}, μ)$ Time	RHN-TSVR RMSE $(C_{1} = C_{2}, ε_{1} = ε_{2},$ $ε_{1}^{} = ε_{2}^{}, μ)$ Time
Function1 (150X2,500X2)	0.060254 (10¹,0.1,2⁰) 0.24846	0.07662 (10⁰,10⁻³,2⁰) 0.07161	0.102363 (10³,10⁻³, 1.345, 1.345, 2¹) 0.00237	0.08825 (10¹,0.1,0.3,2¹) 0.11429	0.07716 (10¹, 0.1, 0.1, 2⁰) 0.13004	0.12935 (10⁻⁵,10⁻⁵, 0.9, 0.1, 2⁰) 0.0068301
Function2 (150X2,500X2)	0.084763 (10⁵,0.1,2⁻¹) 0.27253	0.02778 (10⁰,10⁻¹,2⁰) 0.09476	0.025112 (10⁵,10⁻³, 1, 1.345, 2¹) 0.00214	0.04269 (10³,0.1,0.3,2⁻¹) 0.13124	0.02775 (10¹, 0.9, 0.3, 2⁰) 0.10073	0.02199 (10¹,10⁻⁵, 0.9, 0.9, 2¹) 0.00916
Function3 (150X6,500X6)	0.01362 (10⁵,0.1,2⁻³) 0.23213	0.01921 (10³,10⁻³,2⁻²) 0.17381	0.01156 (10⁵,10⁻¹, 1, 1.345, 2⁻²) 0.00603	0.01356 (10⁵,0.3,0.5,2⁻³) 0.06577	0.02379 (10⁵, 0.5, 0.3, 2⁰) 0.19114	0.00988 (10⁵,10⁻⁵, 0.1, 0.3, 2⁻³) 0.0118904
Function4 (150X6,500X6)	0.061408 (10⁴,0.1,2⁻⁵) 0.23359	0.01346 (10⁴,10⁻¹,2⁰) 0.10612	0.00632 (10⁵,10⁻³, 0.1, 1.345, 2³) 0.00569	0.02464 (10⁵,0.1,0.5,2⁻³) 0.06941	0.01333 (10⁵, 0.9, 0.5, 2⁰) 0.19064	0.00602 (10¹,10⁻⁵, 0.9, 0.5, 2³) 0.0154294
Function5 (150X3,500X3)	0.051805 (10⁴,0.1,2⁵) 0.23124	0.03731 (10³,10⁻¹,2⁵) 0.10908	0.031527 (10³,10⁻¹, 1, 1.345, 2⁵) 0.01027	0.04023 (10⁵,0.3,0.5,2⁵) 0.07013	0.03731 (10³, 0.9, 0.5, 2⁵) 0.16043	0.03919 (10⁵,10⁻⁵, 0.9, 0.5, 2⁵) 0.0139697
Function6 (150X3,500X3)	0.068543 (10⁴,0.1,2²) 0.24664	0.00725 (10²,10⁻¹,2⁵) 0.10067	0.004275 (10⁵,10⁻³, 0.1, 1.345, 2⁵) 0.00794	0.03068 (10⁵,0.1,0.2,2³) 0.07966	0.00727 (10³, 0.3, 0.9, 2⁵) 0.12189	0.00406 (10⁰,10⁻⁵, 0.1, 0.5, 2⁵) 0.015654
Function7 (150X6,500X6)	0.017707 (10⁴,0.1,2⁻⁴) 0.21608	0.008 (10³,10⁻³,2¹) 0.10053	0.010915 (10⁵,10⁻¹, 1, 1.345, 2⁻⁴) 0.01112	0.02645 (10⁵,0.1,0.6,2⁻⁵) 0.07513	0.00799 (10⁵, 0.1, 0.1, 2¹) 0.12071	0.00795 (10⁵,10⁻⁵, 0.1, 0.7, 2⁰) 0.0161369
Function8 (150X6,500X6)	0.062695 (10³,0.1,2¹) 0.23092	0.00418 (10¹,10⁻¹,2³) 0.10265	0.002216 (10⁵,10⁻³, 0.1, 1.345, 2²) 0.00669	0.02145 (10⁵,0.1,0.1,2⁻¹) 0.07909	0.00418 (10⁵, 0.5, 0.3, 2³) 0.10959	0.00169 (10¹,10⁻⁵, 0.1, 0.1, 2⁰) 0.0169987
Function9 (150X2,500X2)	0.088101 (10²,0.1,2⁰) 0.27421	0.08105 (10⁰,10⁻¹,2⁰) 0.07858	0.083806 (10²,10⁻³, 1.345, 1.345, 2¹) 0.00213	0.09462 (10³,0.1,0.5,2⁻¹) 0.12864	0.08129 (10¹, 0.9, 0.1, 2⁰) 0.07415	0.08129 (10¹,10⁻³, 0.9, 0.1, 2⁰) 0.0151367
Function10 (150X2,500X2)	0.078086 (10²,0.1,2⁰) 0.48633	0.0158 (10²,10⁻³,2¹) 0.07912	0.011333 (10⁵,10⁻³, 0.1, 1.345, 2¹) 0.0013	0.04075 (10⁵,0.1,0.2,2⁻¹) 0.14998	0.01579 (10⁵, 0.1, 0.9, 2¹) 0.06906	0.0112 (10⁻⁵,10⁻⁵, 0.1, 0.1, 2¹) 0.0126496
Function11 (150X3,500X3)	0.027556 (10⁵,0.1,2²) 0.25697	0.01022 (10⁵,10⁻¹,2⁴) 0.10644	0.016189 (10⁵,10⁻¹, 1, 1.345, 2³) 0.01082	0.04191 (10⁵,0.3,0.1,2⁵) 0.07923	0.0138 (10⁵, 0.1, 0.7, 2³) 0.15925	0.01293 (10⁵,10⁻⁵, 0.9, 0.3, 2³) 0.017398
Function12 (150X3,500X3)	0.061853 (10⁴,0.1,2³) 0.28748	0.00696 (10⁰,10⁻¹,2⁵) 0.08644	0.004961 (10⁵,10⁻³, 0.1, 1.345, 2⁴) 0.00726	0.03326 (10³,0.1,0.1,2³) 0.07364	0.00689 (10³, 0.9, 0.1, 2⁵) 0.09879	0.00531 (10⁰,10⁻⁵, 0.9, 0.3, 2⁵) 0.0173283
Function13 (200X2,450X2)	0.136885 (10⁰,0.1,2¹) 0.26544	0.11783 (10¹,10⁻³,2⁻¹) 0.12948	0.08155 (10¹,10⁻¹, 1, 1.345, 2¹) 0.00388	0.10239 (10¹,0.9,0.2,2¹) 0.12211	0.11656 (10⁵, 0.1, 0.5, 2⁻¹) 0.10624	0.08749 (10⁵,10¹, 0.3, 0.5, 2¹) 0.0225593
Function14 (200X2,500X2)	0.186935 (10⁰,0.1,2⁴) 0.25514	0.12023 (10⁵,10⁻³,2⁰) 0.10377	0.184837 (10¹,10⁻¹, 1, 1.345, 2⁵) 0.00275	0.15779 (10¹,0.5,0.1,2⁵) 0.10492	0.12034 (10³, 0.1, 0.7, 2⁰) 0.09102	0.10566 (10⁵,10¹, 0.1, 0.9, 2³) 0.0235153
Function15 (200X2,500X2)	0.034488 (10²,0.1,2⁻⁵) 0.46716	0.01879 (10³,10⁻¹,2⁻⁵) 0.16623	0.019379 (10¹,10⁻¹, 1, 1.345, 2⁻⁴) 0.00225	0.02682 (10³,0.3,0.5,2⁻⁵) 0.22456	0.01878 (10⁵, 0.9, 0.7, 2⁻⁵) 0.30141	0.01878 (10⁵,10⁻³, 0.9, 0.7, 2⁻⁵) 0.024471
Function16 (200X2,500X2)	0.036077 (10²,0.1,2⁻³) 0.47924	0.02116 (10⁻⁵,10⁻³,2 ⁻³) 0.15487	0.0222 (10³,10⁻², 1, 1.345, 2⁻³) 0.0046	0.02166 (10³,0.1,0.5,2⁻⁵) 0.22893	0.0219 (10⁰, 0.9, 0.1, 2⁻³) 0.236	0.01955 (10⁵,10¹, 0.1, 0.5, 2⁻³) 0.028278
Function17 (200X2,500X2)	0.147868 (10⁰,0.1,2³) 0.46149	0.15262 (10⁻⁵,10⁻¹,2¹) 0.143	0.100892 (10¹,10⁻¹, 1.345, 1.345, 2⁴) 0.00233	0.18809 (10¹,0.7,0.8,2⁵) 0.24851	0.15262 (10⁻⁵, 0.1, 0.1, 2¹) 0.14581	0.07032 (10¹,10⁻¹, 0.9, 0.5, 2³) 0.0183895
Function18 (200X2,500X2)	0.052586 (10¹,0.1,2³) 0.44357	0.05026 (10⁰,10⁻³,2³) 0.14172	0.044265 (10²,10⁻², 1, 1.345, 2³) 0.0035	0.03809 (10¹,0.1,0.5,2³) 0.21918	0.05027 (10¹, 0.9, 0.1, 2³) 0.2795	0.05027 (10¹,10⁻³, 0.9, 0.1, 2³) 0.0184345

Dataset
(Train size, Test size)

SVR
RMSE

(C, ε, μ)

Time

TSVR
RMSE

(C_{1} = C_{2},

ε, μ)

Time

ε

-AHSVR
RMSE

(C, ε_{α} = ε_{β}, ζ_{α},

ζ_{β}, μ)

Time

ε

-SVQR
RMSE

(C, ε,

θ, μ)

Time

HN-TSVR
RMSE

(C_{1} = C_{2}, ε_{1} = ε_{2},

ε_{1}^{*} = ε_{2}^{*}, μ)

Time

RHN-TSVR
RMSE

(C_{1} = C_{2}, ε_{1} = ε_{2},

ε_{1}^{*} = ε_{2}^{*}, μ)

Time

Function1

(150X2,500X2)

0.060254

(10¹,0.1,2⁰)

0.24846

0.07662

(10⁰,10⁻³,2⁰)

0.07161

0.102363

(10³,10⁻³, 1.345, 1.345, 2¹)

0.00237

0.08825

(10¹,0.1,0.3,2¹)

0.11429

0.07716

(10¹, 0.1, 0.1, 2⁰)

0.13004

0.12935

(10⁻⁵,10⁻⁵, 0.9, 0.1, 2⁰)

0.0068301

Function2

(150X2,500X2)

0.084763

(10⁵,0.1,2⁻¹)

0.27253

0.02778

(10⁰,10⁻¹,2⁰)

0.09476

0.025112

(10⁵,10⁻³, 1, 1.345, 2¹)

0.00214

0.04269

(10³,0.1,0.3,2⁻¹)

0.13124

0.02775

(10¹, 0.9, 0.3, 2⁰)

0.10073

0.02199

(10¹,10⁻⁵, 0.9, 0.9, 2¹)

0.00916

Function3

(150X6,500X6)

0.01362

(10⁵,0.1,2⁻³)

0.23213

0.01921

(10³,10⁻³,2⁻²)

0.17381

0.01156

(10⁵,10⁻¹, 1, 1.345, 2⁻²)

0.00603

0.01356

(10⁵,0.3,0.5,2⁻³)

0.06577

0.02379

(10⁵, 0.5, 0.3, 2⁰)

0.19114

0.00988

(10⁵,10⁻⁵, 0.1, 0.3, 2⁻³)

0.0118904

Function4

(150X6,500X6)

0.061408

(10⁴,0.1,2⁻⁵)

0.23359

0.01346

(10⁴,10⁻¹,2⁰)

0.10612

0.00632

(10⁵,10⁻³, 0.1, 1.345, 2³)

0.00569

0.02464

(10⁵,0.1,0.5,2⁻³)

0.06941

0.01333

(10⁵, 0.9, 0.5, 2⁰)

0.19064

0.00602

(10¹,10⁻⁵, 0.9, 0.5, 2³)

0.0154294

Function5

(150X3,500X3)

0.051805

(10⁴,0.1,2⁵)

0.23124

0.03731

(10³,10⁻¹,2⁵)

0.10908

0.031527

(10³,10⁻¹, 1, 1.345, 2⁵)

0.01027

0.04023

(10⁵,0.3,0.5,2⁵)

0.07013

0.03731

(10³, 0.9, 0.5, 2⁵)

0.16043

0.03919

(10⁵,10⁻⁵, 0.9, 0.5, 2⁵)

0.0139697

Function6

(150X3,500X3)

0.068543

(10⁴,0.1,2²)

0.24664

0.00725

(10²,10⁻¹,2⁵)

0.10067

0.004275

(10⁵,10⁻³, 0.1, 1.345, 2⁵)

0.00794

0.03068

(10⁵,0.1,0.2,2³)

0.07966

0.00727

(10³, 0.3, 0.9, 2⁵)

0.12189

0.00406

(10⁰,10⁻⁵, 0.1, 0.5, 2⁵)

0.015654

Function7

(150X6,500X6)

0.017707

(10⁴,0.1,2⁻⁴)

0.21608

0.008

(10³,10⁻³,2¹)

0.10053

0.010915

(10⁵,10⁻¹, 1, 1.345, 2⁻⁴)

0.01112

0.02645

(10⁵,0.1,0.6,2⁻⁵)

0.07513

0.00799

(10⁵, 0.1, 0.1, 2¹)

0.12071

0.00795

(10⁵,10⁻⁵, 0.1, 0.7, 2⁰)

0.0161369

Function8

(150X6,500X6)

0.062695

(10³,0.1,2¹)

0.23092

0.00418

(10¹,10⁻¹,2³)

0.10265

0.002216

(10⁵,10⁻³, 0.1, 1.345, 2²)

0.00669

0.02145

(10⁵,0.1,0.1,2⁻¹)

0.07909

0.00418

(10⁵, 0.5, 0.3, 2³)

0.10959

0.00169

(10¹,10⁻⁵, 0.1, 0.1, 2⁰)

0.0169987

Function9

(150X2,500X2)

0.088101

(10²,0.1,2⁰)

0.27421

0.08105

(10⁰,10⁻¹,2⁰)

0.07858

0.083806

(10²,10⁻³, 1.345, 1.345, 2¹)

0.00213

0.09462

(10³,0.1,0.5,2⁻¹)

0.12864

0.08129

(10¹, 0.9, 0.1, 2⁰)

0.07415

0.08129

(10¹,10⁻³, 0.9, 0.1, 2⁰)

0.0151367

Function10

(150X2,500X2)

0.078086

(10²,0.1,2⁰)

0.48633

0.0158

(10²,10⁻³,2¹)

0.07912

0.011333

(10⁵,10⁻³, 0.1, 1.345, 2¹)

0.0013

0.04075

(10⁵,0.1,0.2,2⁻¹)

0.14998

0.01579

(10⁵, 0.1, 0.9, 2¹)

0.06906

0.0112

(10⁻⁵,10⁻⁵, 0.1, 0.1, 2¹)

0.0126496

Function11

(150X3,500X3)

0.027556

(10⁵,0.1,2²)

0.25697

0.01022

(10⁵,10⁻¹,2⁴)

0.10644

0.016189

(10⁵,10⁻¹, 1, 1.345, 2³)

0.01082

0.04191

(10⁵,0.3,0.1,2⁵)

0.07923

0.0138

(10⁵, 0.1, 0.7, 2³)

0.15925

0.01293

(10⁵,10⁻⁵, 0.9, 0.3, 2³)

0.017398

Function12

(150X3,500X3)

0.061853

(10⁴,0.1,2³)

0.28748

0.00696

(10⁰,10⁻¹,2⁵)

0.08644

0.004961

(10⁵,10⁻³, 0.1, 1.345, 2⁴)

0.00726

0.03326

(10³,0.1,0.1,2³)

0.07364

0.00689

(10³, 0.9, 0.1, 2⁵)

0.09879

0.00531

(10⁰,10⁻⁵, 0.9, 0.3, 2⁵)

0.0173283

Function13

(200X2,450X2)

0.136885

(10⁰,0.1,2¹)

0.26544

0.11783

(10¹,10⁻³,2⁻¹)

0.12948

0.08155

(10¹,10⁻¹, 1, 1.345, 2¹)

0.00388

0.10239

(10¹,0.9,0.2,2¹)

0.12211

0.11656

(10⁵, 0.1, 0.5, 2⁻¹)

0.10624

0.08749

(10⁵,10¹, 0.3, 0.5, 2¹)

0.0225593

Function14

(200X2,500X2)

0.186935

(10⁰,0.1,2⁴)

0.25514

0.12023

(10⁵,10⁻³,2⁰)

0.10377

0.184837

(10¹,10⁻¹, 1, 1.345, 2⁵)

0.00275

0.15779

(10¹,0.5,0.1,2⁵)

0.10492

0.12034

(10³, 0.1, 0.7, 2⁰)

0.09102

0.10566

(10⁵,10¹, 0.1, 0.9, 2³)

0.0235153

Function15

(200X2,500X2)

0.034488

(10²,0.1,2⁻⁵)

0.46716

0.01879

(10³,10⁻¹,2⁻⁵)

0.16623

0.019379

(10¹,10⁻¹, 1, 1.345, 2⁻⁴)

0.00225

0.02682

(10³,0.3,0.5,2⁻⁵)

0.22456

0.01878

(10⁵, 0.9, 0.7, 2⁻⁵)

0.30141

0.01878

(10⁵,10⁻³, 0.9, 0.7, 2⁻⁵)

0.024471

Function16

(200X2,500X2)

0.036077

(10²,0.1,2⁻³)

0.47924

0.02116

(10⁻⁵,10⁻³,2 ⁻³)

0.15487

0.0222

(10³,10⁻², 1, 1.345, 2⁻³)

0.0046

0.02166

(10³,0.1,0.5,2⁻⁵)

0.22893

0.0219

(10⁰, 0.9, 0.1, 2⁻³)

0.236

0.01955

(10⁵,10¹, 0.1, 0.5, 2⁻³)

0.028278

Function17

(200X2,500X2)

0.147868

(10⁰,0.1,2³)

0.46149

0.15262

(10⁻⁵,10⁻¹,2¹)

0.143

0.100892

(10¹,10⁻¹, 1.345, 1.345, 2⁴)

0.00233

0.18809

(10¹,0.7,0.8,2⁵)

0.24851

0.15262

(10⁻⁵, 0.1, 0.1, 2¹)

0.14581

0.07032

(10¹,10⁻¹, 0.9, 0.5, 2³)

0.0183895

Function18

(200X2,500X2)

0.052586

(10¹,0.1,2³)

0.44357

0.05026

(10⁰,10⁻³,2³)

0.14172

0.044265

(10²,10⁻², 1, 1.345, 2³)

0.0035

0.03809

(10¹,0.1,0.5,2³)

0.21918

0.05027

(10¹, 0.9, 0.1, 2³)

0.2795

0.05027

(10¹,10⁻³, 0.9, 0.1, 2³)

0.0184345

The best result is shown as boldface