Comparative Analysis of Low Discrepancy Sequence-Based Initialization Approaches Using Population-Based Algorithms for Solving the Global Optimization Problems
Abstract
:1. Introduction
2. Related Work
3. Methodology
3.1. Random Number Generator
3.2. The Sobol Sequence
3.3. The Halton Sequence
Algorithm 1 Halton Sequences |
Halton (): // input: Size = and base = with Dimension = // output: population instances = Fix the interval over For each iteration : do For each particle |
3.4. The Well Sequence
Algorithm 2 WELL Sequences |
WELL (): for Return |
3.5. The Knuth Sequence
3.6. The Torus Sequence
Algorithm 3 Proposed Pseudocode of PSO Using Novel Method of Initialization |
|
Algorithm 4 Proposed Pseudo Code of DE Using Novel Method of Initialization |
Input: 𝑥𝑖 = (𝑥𝑖,1, 𝑥𝑖,2, 𝑥𝑖,3, …, 𝑥𝑖,𝐷), Population size ‘N-P’, Problem Size ‘D’, Mutation Rate ‘F’, Crossover Rate ‘C-R’; Stopping Criteria {Number of Generation, Target}, Upper Bound ‘U’, Lower Bound ‘L’ Output: 𝑥𝑖, = Global fitness vector with minimal fitness value Pop = Initialize of Paraments (N-P, D, U, L); Generate initial population Using WELL,Knuth,Torus While (Stopping Criteria ≠ True) do Best Vector = Evaluate Pop (Pop); vx = Select Rand Vector (Pop); I = Find Index Vector (vx); Select Rand Vector (Pop,v1,v2,v3) where v1 ≠ v2 ≠ v3 ≠ vx vy = v1, + F(v2−v3) For (i = 0; i++; i < D−1) If (randj [0, 1) < C-R) Then U[i] = vx [i]. else U[i] = vy [i] End For loop If (Cost Fun Vector(U) ≤ Cost Fun Vector (vx)) Then Update Pop (U, I, Pop); End IF End While Retune Best Vector |
4. Experimental Setup
5. Simulation Results and Discussion
5.1. Results and Graphs on PSO Approaches
5.2. Friedman and Kruskal–Wallis Test on PSO Approaches
5.3. Discussion on PSO Results
A. Discussion
- Effect of using different initializing PSO approaches
- Effect of using different dimensions for problems
- A comparative analysis
- i.
- Effect of Using Different Initializing PSO Approaches
- ii.
- Effect of Using Different Dimensions for Problems
- iii.
- Comparative Analysis
5.4. Discussion on DE Results
5.5. Results and Graphs on DE Approaches
Functions | DIM × Iter | DE | DE-H | DE-S | DE-TO | DE-WE | DE-KN |
---|---|---|---|---|---|---|---|
F1 | 10 × 1000 | 1.1464 × 10−44 | 2.1338 × 10−44 | 5.8561 × 10−44 | 7.4117 × 10−45 | 7.4827 × 10−39 | 5.7658 × 10−39 |
20 × 2000 | 3.3550 × 10−46 | 7.2338 × 10−46 | 1.3545 × 10−45 | 1.2426 × 10−45 | 9.6318 × 10−45 | 7.1501 × 10−45 | |
30 × 3000 | 8.8946 × 10−47 | 1.2273 × 10−45 | 9.4228 × 10−46 | 1.6213 × 10−46 | 6.2007 × 10−46 | 5.7425 × 10−46 | |
F2 | 10 × 1000 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 |
20 × 2000 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | |
30 × 3000 | 1.8392 × 10+01 | 1.1846 × 10+01 | 1.8871 × 10+01 | 3.7132 × 10−01 | 5.0821 × 10+00 | 6.6313 × 10+00 | |
F3 | 10 × 1000 | 5.00325 × 10−44 | 1.5019 × 10−38 | 9.3956 × 10−44 | 4.7807 × 10−44 | 1.6251 × 10−38 | 1.3411 × 10−38 |
20 × 2000 | 2.56987 × 10−45 | 4.1485 × 10−44 | 1.5339 × 10−44 | 3.0262 × 10−45 | 9.5984 × 10−44 | 1.3606 × 10−43 | |
30 × 3000 | 1.01692 × 10−45 | 2.7349 × 10−45 | 4.0581 × 10−45 | 4.5726 × 10−45 | 4.5686 × 10−45 | 5.4659 × 10−45 | |
F4 | 10 × 1000 | 5.81825 × 10−42 | 3.0950 × 10−36 | 2.2300 × 10−41 | 1.6903 × 10−41 | 1.1331 × 10−36 | 3.8869 × 10−36 |
20 × 2000 | 2.70747 × 10−43 | 1.0658 × 10−41 | 1.6730 × 10−42 | 1.3490 × 10−42 | 1.3094 × 10−41 | 6.0053 × 10−42 | |
30 × 3000 | 2.99887 × 10−43 | 1.4032 × 10−42 | 4.4442 × 10−42 | 5.9186 × 10−43 | 4.6922 × 10−43 | 1.4829 × 10−42 | |
F5 | 10 × 1000 | 1.65318 × 10−43 | 4.7939 × 10−38 | 7.0329 × 10−43 | 4.8106 × 10−43 | 4.3219 × 10−38 | 3.5770 × 10−38 |
20 × 2000 | 1.39082 × 10−44 | 3.6325 × 10−43 | 4.2191 × 10−44 | 2.7448 × 10−44 | 5.8557 × 10−43 | 1.4008 × 10−43 | |
30 × 3000 | 6.07162 × 10−45 | 1.7557 × 10−44 | 1.6295 × 10−44 | 2.0582 × 10−44 | 8.6773 × 10−45 | 4.2285 × 10−44 | |
F6 | 10 × 1000 | 7.8201 × 10−96 | 3.8819 × 10−96 | 9.7956 × 10−96 | 2.3292 × 10−95 | 8.4774 × 10−94 | 2.8037 × 10−95 |
20 × 2000 | 1.6847 × 10−125 | 8.6880 × 10−124 | 5.9005 × 10−122 | 8.7800 × 10−123 | 3.7438 × 10−124 | 1.3947 × 10−124 | |
30 × 3000 | 2.4533 × 10−140 | 1.5487 × 10−139 | 5.7211 × 10−138 | 4.4492 × 10−137 | 6.5749 × 10−140 | 3.4442 × 10−137 | |
F7 | 10 × 1000 | 8.0217 × 10−75 | 7.3243 × 10−67 | 5.7807 × 10−66 | 1.0243 × 10−73 | 1.9035 × 10−67 | 1.4359 × 10−65 |
20 × 2000 | 4.0682 × 10−71 | 1.5037 × 10−70 | 1.5747 × 10−69 | 1.0623 × 10−70 | 5.5546 × 10−70 | 2.3507 × 10−70 | |
30 × 3000 | 8.5895 × 10−68 | 6.6009 × 10−68 | 3.3919 × 10−67 | 2.6036 × 10−67 | 1.1587 × 10−67 | 2.1901 × 10−67 | |
F8 | 10 × 1000 | 7.0221 × 10−120 | 3.4271 × 10−108 | 2.7718 × 10−108 | 6.3092 × 10−118 | 3.9423 × 10−106 | 9.9394 × 10−108 |
20 × 2000 | 5.2096 × 10−108 | 7.7158 × 10−89 | 1.4732 × 10−106 | 8.8720 × 10−107 | 3.4490 × 10−107 | 2.2539 × 10−106 | |
30 × 3000 | 1.2538 × 10−98 | 1.8071 × 10−98 | 1.1085 × 10−95 | 7.2462 × 10−98 | 2.5375 × 10−99 | 5.8040 × 10−98 | |
F9 | 10 × 1000 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 |
20 × 2000 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | |
30 × 3000 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | 0.0000 × 10+00 | |
F10 | 10 × 1000 | 1.3459 × 10−75 | 2.6493 × 10−66 | 2.6884 × 10−66 | 3.6168 × 10−67 | 3.8397 × 10−67 | 1.8408 × 10−66 |
20 × 2000 | 3.0478 × 10−71 | 1.6106 × 10−69 | 5.5253 × 10−69 | 2.7746 × 10−70 | 5.3662 × 10−70 | 5.0931 × 10−70 | |
30 × 3000 | 8.2514 × 10−68 | 1.0937 × 10−66 | 4.1120 × 10−67 | 1.3055 × 10−67 | 3.5397 × 10−68 | 6.0736 × 10−67 | |
F11 | 10 × 1000 | 2.3417 × 10−42 | 1.2483 × 10−41 | 1.3726 × 10−41 | 6.3337 × 10−42 | 4.8161 × 10−42 | 7.3464 × 10−42 |
20 × 2000 | 8.4769 × 10−44 | 3.5140 × 10−43 | 3.3777 × 10−43 | 2.4721 × 10−43 | 1.9553 × 10−43 | 3.6961 × 10−43 | |
30 × 3000 | 3.6888 × 10−44 | 6.9938 × 10−44 | 2.5123 × 10−43 | 1.4710 × 10−43 | 4.0019 × 10−44 | 3.9503 × 10−43 | |
F12 | 10 × 1000 | 2.3304 × 10+00 | 4.4354 × 10+00 | 3.4520 × 10+00 | 5.1229 × 10+00 | 3.8782 × 10+00 | 2.7840 × 10+00 |
20 × 2000 | 3.1768 × 10+04 | 3.9596 × 10+04 | 3.8814 × 10+04 | 2.9488 × 10+04 | 4.1181 × 10+04 | 4.0914 × 10+04 | |
30 × 3000 | 1.1760 × 10+06 | 1.0300 × 10+06 | 1.3402 × 10+06 | 1.2008 × 10+06 | 1.0916 × 10+06 | 1.0160 × 10+06 | |
F13 | 10 × 1000 | 1.3940 × 10−65 | 1.3756 × 10−64 | 3.1956 × 10−66 | 9.3609 × 10−64 | 5.4864 × 10−63 | 9.2695 × 10−63 |
20 × 2000 | 2.0163 × 10−111 | 8.5333 × 10−110 | 8.5260 × 10−111 | 3.9836 × 10−109 | 5.0102 × 10−115 | 4.4624 × 10−110 | |
30 × 3000 | 1.4146 × 10−156 | 4.3434 × 10−156 | 4.4702 × 10−154 | 4.3862 × 10−151 | 1.0781 × 10−153 | 1.0142 × 10−149 | |
F14 | 10 × 1000 | 9.1259 × 10−24 | 2.1900 × 10−23 | 2.5559 × 10−23 | 2.9039 × 10−23 | 1.9174 × 10−23 | 3.3427 × 10−23 |
20 × 2000 | 2.6867 × 10−25 | 3.8631 × 10−25 | 1.5177 × 10−24 | 5.5714 × 10−25 | 4.5049 × 10−25 | 5.6503 × 10−25 | |
30 × 3000 | 5.9241 × 10−26 | 8.6401 × 10−26 | 8.4348 × 10−26 | 1.4630 × 10−25 | 9.7932 × 10−26 | 1.4921 × 10−25 | |
F15 | 10 × 1000 | 1.0493 × 10−185 | 4.0276 × 10−181 | 5.0331 × 10−182 | 3.1770 × 10−183 | 1.1698 × 10−180 | 2.6563 × 10−182 |
20 × 2000 | 2.9407 × 10−159 | 9.9152 × 10−159 | 2.1401 × 10−158 | 9.0345 × 10−156 | 3.8871 × 10−158 | 8.0144 × 10−160 | |
30 × 3000 | 4.6769 × 10−138 | 1.0737 × 10−137 | 7.0544 × 10−138 | 8.0376 × 10−138 | 4.9091 × 10−139 | 1.1054 × 10−137 | |
F16 | 10 × 1000 | 1.8635 × 10−04 | 1.8109 × 10−02 | 4.9798 × 10−02 | 5.8605 × 10−04 | 1.4858 × 10−02 | 3.7220 × 10−02 |
20 × 2000 | 1.1032 × 10+00 | 1.6605 × 10+00 | 1.7157 × 10+00 | 1.4875 × 10+00 | 1.5697 × 10+00 | 1.2008 × 10+00 | |
30 × 3000 | 2.8283 × 10+01 | 2.2049 × 10+01 | 2.9388 × 10+01 | 2.8205 × 10+01 | 2.5794 × 10+01 | 2.9526 × 10+01 |
5.6. Friedman and Kruskal–Wallis Test on DE Approaches
Friedman Value | Kruskal-Wallis | |
---|---|---|
DE | 63.74 | 65.11 |
DE-H | 59.31 | 60.41 |
DE-S | 64.01 | 65.05 |
DE-TO | 63.76 | 65.35 |
DE-WE | 63.35 | 63.93 |
DE-KN | 63.33 | 64.06 |
6. Comparison of PSO and DE Regarding Data Classification
6.1. NN Classifications with PSO-Based Initialization Approaches
Discussion
6.2. NN Classifications with DE-Based Initialization Approaches
Discussion
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Parameter | Value | ||
---|---|---|---|
Search Space | [−100, 100] | ||
Dimensions | 10 | 20 | 30 |
Iterations | 1000 | 2000 | 3000 |
Population size | 50 | ||
Number of Runs | 10 |
Algorithm | Parameters |
---|---|
PSO | c1 = c2 = 1.49, w = linearly decreasing |
DE | 𝐹 ∈ [0.4, 1], 𝐶𝑅 ∈ 0.6 |
Sr.# | Function Name | Objective Function | Search Space | Optimal Value |
---|---|---|---|---|
01 | Sphere | 0 | ||
02 | Rastrigin | 0 | ||
03 | Axis parallel hyper-ellipsoid | 0 | ||
04 | Rotated hyper ellipsoid | 0 | ||
05 | Moved Axis | 0 | ||
06 | Sum of different power | 0 | ||
07 | ChungReynolds | 0 | ||
08 | Csendes | 0 | ||
09 | Schaffer | 0 | ||
10 | Schumer_Steiglitz | 0 | ||
11 | Schwefel | 0 | ||
12 | Schwefel1.2 | 0 | ||
13 | Schwefel 2.21 | 0 | ||
14 | Schwefel 2.22 | 0 | ||
15 | Schwefel 2.23 | 0 | ||
16 | Zakharov | 0 |
Functions | DIM × Itr | PSO | SO-PSO | H-PSO | TO-PSO | WE-PSO | KN-PSO |
---|---|---|---|---|---|---|---|
Mean | Mean | Mean | Mean | Mean | Mean | ||
F1 | 10 × 1000 | 2.33 × 10−74 | 2.74 × 10−76 | 3.10 × 10−77 | 5.57 × 10−78 | 5.91 × 10−78 | 0.0000 × 10+00 |
20 × 2000 | 1.02 × 10−84 | 8.20 × 10−88 | 1.76 × 10−90 | 1.30 × 10−90 | 4.95 × 10−90 | 3.14001 × 10−217 | |
30 × 3000 | 1.77 × 10−26 | 7.67 × 10−20 | 4.13 × 10−32 | 1.25 × 10−51 | 1.30 × 10−42 | 8.91595 × 10−88 | |
F2 | 10 × 1000 | 4.97 × 10−01 | 4.97 × 10−01 | 7.96 × 10−01 | 3.98 × 10−01 | 2.98 × 10−01 | −8602.02 |
20 × 2000 | 8.17 × 10+00 | 6.47 × 10+00 | 3.58 × 10+00 | 2.89 × 10+00 | 3.11 × 10+00 | −31,433.3 | |
30 × 3000 | 1.01 × 10+01 | 9.86 × 10+00 | 9.45 × 10+00 | 8.16 × 10+00 | 7.76 × 10+00 | −60,711.8 | |
F3 | 10 × 1000 | 8.70 × 10−80 | 1.79 × 10−79 | 4.87 × 10−79 | 3.91 × 10−82 | 4.40 × 10−81 | 0.0000 × 10+00 |
20 × 2000 | 2.62144 | 7.86432 | 2.62144 | 7.07 × 10−90 | 1.78 × 10−89 | 4.78718 × 10−237 | |
30 × 3000 | 2.62 × 10+01 | 1.57 × 10+01 | 1.05 × 10+01 | 7.70 × 10−35 | 3.87 × 10−57 | 1.57084 × 10−97 | |
F4 | 10 × 1000 | 4.46 × 10−147 | 3.86 × 10−147 | 9.78 × 10−145 | 7.29 × 10−148 | 1.24 × 10−150 | 0.0000 × 10+00 |
20 × 2000 | 3.14 × 10−155 | 9.27 × 10−154 | 2.75 × 10−159 | 5.14 × 10−158 | 4.96 × 10−159 | 0.0000 × 10+00 | |
30 × 3000 | 1.82 × 10−133 | 2.36 × 10−135 | 8.53 × 10−130 | 3.13 × 10−138 | 2.54 × 10−136 | 1.6439 × 10−228 | |
F5 | 10 × 1000 | 4.35 × 10−79 | 8.95 × 10−79 | 2.43 × 10−78 | 2.04 × 10−80 | 2.20 × 10−80 | 0.0000 × 10+00 |
20 × 2000 | 1.31 × 10+01 | 3.93 × 10+01 | 1.31 × 10+01 | 3.54 × 10−89 | 3.12 × 10−89 | 2.39359 × 10−236 | |
30 × 3000 | 1.31 × 10+02 | 7.86 × 10+01 | 5.24 × 10+01 | 3.85 × 10−34 | 1.94 × 10−56 | 2.9093 × 10−87 | |
F6 | 10 × 1000 | 1.70 × 10−61 | 4.45 × 10−64 | 7.29 × 10−66 | 2.46 × 10−66 | 4.62 × 10−66 | 3.04226 × 10−318 |
20 × 2000 | 3.25 × 10−112 | 4.39 × 10−112 | 5.01 × 10−109 | 2.56 × 10−115 | 4.45 × 10−113 | 8.59557 × 10−277 | |
30 × 3000 | 7.21 × 10−135 | 4.10 × 10−124 | 1.51 × 10−134 | 6.22 × 10−137 | 6.96 × 10−135 | 2.33033 × 10−223 | |
F7 | 10 × 1000 | 2.96 × 10−157 | 2.39 × 10−157 | 1.28 × 10−157 | 4.89 × 10−159 | 2.47 × 10−163 | 0.0000 × 10+00 |
20 × 2000 | 8.79 × 10−177 | 1.77 × 10−184 | 3.49 × 10−183 | 3.09 × 10−187 | 3.41 × 10−186 | 0.0000 × 10+00 | |
30 × 3000 | 1.23 × 10−82 | 1.25 × 10−116 | 5.99 × 10−130 | 5.01 × 10−135 | 4.60 × 10−134 | 8.03288 × 10−175 | |
F8 | 10 × 1000 | 4.39 × 10−200 | 1.98 × 10−194 | 4.51 × 10−197 | 1.26 × 10−202 | 8.99 × 10−201 | 4.9228 × 10−67 |
20 × 2000 | 1.57 × 10−20 | 1.04 × 10−93 | 1.10 × 10−148 | 2.84 × 10−157 | 4.09 × 10−151 | 4.5887 × 10−16 | |
30 × 3000 | 1.89 × 10−09 | 4.54 × 10−10 | 1.14 × 10−08 | 1.40 × 10−10 | 1.34 × 10−09 | 2.2334 × 10−08 | |
F9 | 10 × 1000 | 5.49 × 10−01 | 1.30 × 10−01 | 2.02 × 10−01 | 1.26 × 10−01 | 1.42 × 10−01 | 0.824968 |
20 × 2000 | 2.05 × 10+00 | 7.83 × 10−01 | 6.83 × 10−01 | 5.84 × 10−01 | 4.32 × 10−01 | 4.56265 | |
30 × 3000 | 1.12 × 10+00 | 9.99 × 10−01 | 9.56 × 10−01 | 9.06 × 10−01 | 9.12 × 10−01 | 7.25675 | |
F10 | 10 × 1000 | 2.23 × 10−138 | 2.23 × 10−138 | 4.35 × 10−137 | 1.02 × 10−140 | 1.10 × 10−139 | 0.0000 × 10+00 |
20 × 2000 | 3.79 × 10−148 | 7.87 × 10−149 | 4.19 × 10−147 | 3.78 × 10−151 | 8.73 × 10−153 | 0.0000 × 10+00 | |
30 × 3000 | 4.43 × 10−126 | 7.52 × 10−133 | 1.57 × 10−128 | 2.03 × 10−134 | 1.38 × 10−133 | 2.26229 × 10−221 | |
F11 | 10 × 1000 | 3.75 × 10−187 | 1.57 × 10−192 | 2.15 × 10−191 | 5.57 × 10−198 | 8.99 × 10−198 | 0.0000 × 10+00 |
20 × 2000 | 5.29 × 10−193 | 2.53 × 10−195 | 8.45 × 10−195 | 8.45 × 10−195 | 9.83 × 10−197 | 0.0000 × 10+00 | |
30 × 3000 | 4.82 × 10−154 | 8.84 × 10−159 | 5.49 × 10−168 | 2.04 × 10−170 | 5.75 × 10−173 | 9.00586 × 10−278 | |
F12 | 10 × 1000 | 1.13 × 10−01 | 1.67 × 10−02 | 2.28 × 10−02 | 4.78 × 10−03 | 2.89 × 10−03 | 2.739 × 10−12 |
20 × 2000 | 1.39 × 10+01 | 5.03 × 10+00 | 2.95 × 10+00 | 1.28 × 10+00 | 1.67 × 10+00 | 7.819 × 10+00 | |
30 × 3000 | 7.45 × 10+00 | 1.22 × 10+01 | 8.74 × 10+00 | 2.94 × 10+00 | 4.94 × 10+00 | 2.239 × 10+01 | |
F13 | 10 × 1000 | 8.04 × 10−26 | 8.01 × 10−27 | 3.59 × 10−27 | 1.24 × 10−27 | 1.41 × 10−27 | 0.0000 × 10+00 |
20 × 2000 | 1.42 × 10−08 | 2.64 × 10−11 | 3.29 × 10−10 | 2.99 × 10−10 | 2.14 × 10−12 | 0.0000 × 10+00 | |
30 × 3000 | 6.20 × 10−03 | 1.41 × 10−03 | 9.36 × 10−03 | 1.12 × 10−03 | 1.41 × 10−03 | 0.0000 × 10+00 | |
F14 | 10 × 1000 | 3.62 × 10−38 | 3.62 × 10−38 | 5.92 × 10−36 | 6.92 × 10−39 | 1.95 × 10−38 | 7.78286 × 10−197 |
20 × 2000 | 6.27 × 10−10 | 1.38 × 10−09 | 7.91 × 10−13 | 2.49 × 10−12 | 1.17 × 10−13 | 6.6163 × 10−12 | |
30 × 3000 | 2.56 × 10−06 | 4.80 × 10+01 | 1.34 × 10−06 | 5.40 × 10−11 | 4.88 × 10−09 | 9.3032 × 10−06 | |
F15 | 10 × 1000 | 1.10 × 10−294 | 3.19 × 10−301 | 2.78 × 10−307 | 1.94 × 10−307 | 3.21 × 10−308 | 6.26612 × 10−138 |
20 × 2000 | 6.16 × 10−271 | 5.09 × 10−276 | 3.74 × 10−270 | 1.60 × 10−276 | 4.85 × 10−268 | 1.29033 × 10−25 | |
30 × 3000 | 3.08 × 10−207 | 1.04 × 10−200 | 8.12 × 10−209 | 2.34 × 10−215 | 3.06 × 10−212 | 2.27 × 10−06 | |
F16 | 10 × 1000 | 5.4835385 | 8.5299 × 10−17 | 3.3074 × 10−16 | 1.224803 | 8.3354 × 10−07 | 2.26476 × 10−27 |
20 × 2000 | 83.467 | 1.6344 | 0.18037 | 49.16841 | 5.1322 | 7.17014 × 10−72 | |
30 × 3000 | 265.90708 | 282.1864 | 45.0408 | 133.9679 | 67.0301 | 5.45179 × 10−251 |
Friedman Value | Kruskal–Wallis | |
---|---|---|
PSO | 39.09 | 39.33 |
SO-PSO | 37.47 | 38.39 |
H-PSO | 38.50 | 38.91 |
TO-PSO | 41.79 | 42.67 |
WE-PSO | 41.88 | 42.50 |
KN-PSO | 18.24 | 23.31 |
Features | |||||
---|---|---|---|---|---|
Sr. No | Data Set | Continuous | Nature | No. of Inputs | No. of Classes |
1 | Diabetes | 8 | Real | 8 | 2 |
2 | Heart | 13 | Real | 13 | 2 |
3 | Wine | 13 | Real | 13 | 3 |
4 | Seed | 7 | Real | 7 | 3 |
5 | Vertebral | 6 | Real | 6 | 2 |
6 | Blood Tissue | 5 | Real | 5 | 2 |
7 | Mammography | 6 | Real | 6 | 2 |
Sr. No | Data Sets | Type | BPA | PSONN | SO-PSONN | H-PSONN | TO-PSONN | WE-PSONN | KN-PSONN |
---|---|---|---|---|---|---|---|---|---|
Ts. Acc | Ts. Acc | Ts. Acc | Ts. Acc | Ts. Acc | Ts. Acc | Ts. Acc | |||
1 | Diabetes | 2-Class | 65.3% | 69.1% | 69.1% | 71.6% | 73.3% | 74.1% | 78.5% |
2 | Heart | 2-Class | 68.3% | 72.5% | 67.5% | 72.5% | 77.5 | 77.5% | 79% |
3 | Wine | 3-Class | 62.17% | 61.11% | 66.66% | 67.44% | 69.44% | 69.6% | 72% |
4 | Seed | 3-Class | 70.56% | 77.77% | 84.44% | 77.77% | 88.88% | 91.11% | 93% |
5 | Vertebral | 2-Class | 84.95% | 92.85% | 92.85% | 92.85% | 94.64% | 94.64% | 96% |
6 | Blood Tissue | 2-Class | 73.47% | 78.6% | 78.66% | 70% | 82.66% | 84% | 87% |
7 | Memo Graphy | 2-Class | 71.26% | 76.66% | 63% | 85% | 88.88% | 96.66% | 98% |
Parameter | Relation | Sum of Squares | df | Mean Square | F | Significance |
---|---|---|---|---|---|---|
Testing Accuracy | Among groups | 1318.2 | 6 | 219.697 | 2.3676 | 0.04639 |
Sr. No | Data Sets | Type | BPA | DE | DE-S | DE-WE | DE-TO | DE-KN | DE-H |
---|---|---|---|---|---|---|---|---|---|
Ts. Acc | Ts. Acc | Ts. Acc | Ts. Acc | Ts. Acc | Ts. Acc | Ts. Acc | |||
1 | Diabetes | 2-Class | 65.3% | 66.1% | 68.16% | 69.6% | 71.30% | 67.17% | 75.50% |
2 | Heart | 2-Class | 68.3% | 70.5% | 72.5% | 71.5% | 74.50% | 72.56% | 76.34% |
3 | Wine | 3-Class | 62.17% | 64.7% | 65.19% | 66.20% | 66.59% | 68.25% | 70.51% |
4 | Seed | 3-Class | 70.56% | 75.16% | 75.29% | 75.77% | 82.13% | 86.76% | 91.54% |
5 | Vertebral | 2-Class | 79.95% | 82.13% | 84.26% | 86.15% | 87.64% | 90.17% | 96.25% |
6 | Blood Tissue | 2-Class | 73.47% | 76.23% | 74.16% | 72..21% | 84.76% | 81.34% | 86.45% |
7 | Mammography | 2-Class | 71.26% | 74.39% | 68.37% | 82.45% | 86.17% | 96.66% | 99.21% |
Parameter | Relation | Sum of Squares | df | Mean Square | F | Significance |
---|---|---|---|---|---|---|
Testing Accuracy | Among groups | 1180.0 | 6 | 196.672 | 2.8453 | 0.02043 |
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Bangyal, W.H.; Nisar, K.; Ag. Ibrahim, A.A.B.; Haque, M.R.; Rodrigues, J.J.P.C.; Rawat, D.B. Comparative Analysis of Low Discrepancy Sequence-Based Initialization Approaches Using Population-Based Algorithms for Solving the Global Optimization Problems. Appl. Sci. 2021, 11, 7591. https://doi.org/10.3390/app11167591
Bangyal WH, Nisar K, Ag. Ibrahim AAB, Haque MR, Rodrigues JJPC, Rawat DB. Comparative Analysis of Low Discrepancy Sequence-Based Initialization Approaches Using Population-Based Algorithms for Solving the Global Optimization Problems. Applied Sciences. 2021; 11(16):7591. https://doi.org/10.3390/app11167591
Chicago/Turabian StyleBangyal, Waqas Haider, Kashif Nisar, Ag. Asri Bin Ag. Ibrahim, Muhammad Reazul Haque, Joel J. P. C. Rodrigues, and Danda B. Rawat. 2021. "Comparative Analysis of Low Discrepancy Sequence-Based Initialization Approaches Using Population-Based Algorithms for Solving the Global Optimization Problems" Applied Sciences 11, no. 16: 7591. https://doi.org/10.3390/app11167591
APA StyleBangyal, W. H., Nisar, K., Ag. Ibrahim, A. A. B., Haque, M. R., Rodrigues, J. J. P. C., & Rawat, D. B. (2021). Comparative Analysis of Low Discrepancy Sequence-Based Initialization Approaches Using Population-Based Algorithms for Solving the Global Optimization Problems. Applied Sciences, 11(16), 7591. https://doi.org/10.3390/app11167591