# Improved Salp Swarm Optimization Algorithm: Application in Feature Weighting for Blind Modulation Identification

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Improvements on Salp Swarm Algorithm

#### 3.1. SSA, the Basic Algorithm

#### 3.2. Motivation and Improvements

## 4. Comparison Methodology and MATERIALS

#### 4.1. Comparison Methodology

#### 4.2. Materials

## 5. Benchmarking of SSA and ISSA

#### 5.1. Comparison Based on Solution Accuracy

#### 5.2. Comparison Based on Convergence

#### 5.3. Statistical Analysis

## 6. ISSA for Feature Weighting in DMI

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 2.**Average best solution (i.e., the achieved minimum, described by the y-axis) in terms of the number of objective function evaluations (x-axis) for the benchmark function ${f}_{3}$ using (

**a**) SSA and (

**c**) ISSA; corresponding convergence visualizations (

**b**,

**d**) in accordance with (

**a**,

**c**). (

**b**,

**d**) are the contour plots of the function ${f}_{3}$, the x and y-axis represent the dimensions of the function, and the markers represent the evaluated points by running the optimization algorithms.

**Figure 3.**Average best solution (i.e., the achieved minimum, described by the y-axis) in terms of the number of objective function evaluations (x-axis) for the benchmark function ${f}_{4}$ using (

**a**) SSA and (

**c**) ISSA; corresponding convergence visualizations (

**b**,

**d**) in accordance with (

**a**,

**c**). (

**b**,

**d**) are the contour plots of the function ${f}_{4}$, the x and y-axis represent the dimensions of the function, and the markers represent the evaluated points by running the optimization algorithms.

**Figure 5.**Average best solution (i.e., the achieved minimum, described by the y-axis) in terms of FES (the number of objective function evaluations, described by the x-axis) for the benchmark functions ${f}_{1}-{f}_{2}$, ${f}_{3}-{f}_{4}$, ${f}_{5}-{f}_{6}$, ${f}_{7}-{f}_{8}$, ${f}_{9}-{f}_{10}$, ${f}_{11}-{f}_{12}$, ${f}_{13}-{f}_{14}$, ${f}_{15}-{f}_{16}$ in row-major order. Optimization algorithms are SSA [□], STSSA [▽], IWSSA [✸], and ISSA [○].

**Figure 7.**Average best solution (i.e., the achieved minimum, described by the y-axis) in terms of FES (the number of objective function evaluations, described by the x-axis). SSA [□], STSSA [▽], IWSSA [✸], and ISSA [○].

**Figure 8.**Confusion matrices given by (

**a**) SSA, (

**b**) STSSA, (

**c**) IWSSA, (

**d**) ISSA, (

**e**) w/o FW, and (

**f**) z-score.

**Table 1.**Mean, SEM, and SD for functions ${f}_{1}-{f}_{16}$ obtained by SSA, STSSA, IWSSA, and ISSA.

SSA | STSSA | IWSSA | ISSA | ||
---|---|---|---|---|---|

${f}_{1}$ | Mean | 4.03 × 10${}^{-9}$ | 7.12 × 10${}^{-10}$ | 1.01 × 10${}^{-23}$ | 0.00 × 10${}^{\mathbf{0}}$ |

SD | 9.52 × 10${}^{-10}$ | 2.78 × 10${}^{-10}$ | 3.48 × 10${}^{-23}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

SEM | 1.73 × 10${}^{-10}$ | 5.08 × 10${}^{-11}$ | 6.35 × 10${}^{-24}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

${f}_{2}$ | Mean | 1.90 × 10${}^{-1}$ | 8.90 × 10${}^{-6}$ | 9.00 × 10${}^{-13}$ | 1.20 × 10${}^{-\mathbf{180}}$ |

SD | 7.10 × 10${}^{-1}$ | 1.54 × 10${}^{-6}$ | 1.17 × 10${}^{-12}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

SEM | 1.20 × 10${}^{-1}$ | 2.82 × 10${}^{-7}$ | 2.13 × 10${}^{-13}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

${f}_{3}$ | Mean | 1.93 × 10${}^{-9}$ | 2.20 × 10${}^{-10}$ | 1.78 × 10${}^{-23}$ | 0.00 × 10${}^{\mathbf{0}}$ |

SD | 1.01 × 10${}^{-9}$ | 1.24 × 10${}^{-10}$ | 3.61 × 10${}^{-23}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

SEM | 1.84 × 10${}^{-10}$ | 2.28 × 10${}^{-11}$ | 6.60 × 10${}^{-24}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

${f}_{4}$ | Mean | 1.50 × 10${}^{-5}$ | 7.73 × 10${}^{-6}$ | 9.02 × 10${}^{-13}$ | 2.27 × 10${}^{-\mathbf{172}}$ |

SD | 4.17 × 10${}^{-6}$ | 1.78 × 10${}^{-6}$ | 9.66 × 10${}^{-13}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

SEM | 7.63 × 10${}^{-7}$ | 7.63 × 10${}^{-7}$ | 1.76 × 10${}^{-13}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

${f}_{5}$ | Mean | 1.40 × 10${}^{2}$ | 89.4 × 10${}^{-1}$ | 89.2 × 10${}^{-\mathbf{1}}$ | 89.2 × 10${}^{-\mathbf{1}}$ |

SD | 3.59 × 10${}^{2}$ | 1.60 × 10${}^{-2}$ | 1.40 × 10${}^{-\mathbf{2}}$ | 1.20 × 10${}^{-\mathbf{2}}$ | |

SEM | 6.56 × 10${}^{1}$ | 0.3 × 10${}^{-2}$ | 0.2 × 10${}^{-\mathbf{2}}$ | 0.2 × 10${}^{-\mathbf{2}}$ | |

${f}_{6}$ | Mean | 6.31 × 10${}^{-\mathbf{10}}$ | 0.13 × 10${}^{1}$ | 6.00 × 10${}^{-1}$ | 6.30 × 10${}^{-1}$ |

SD | 2.72 × 10${}^{-\mathbf{10}}$ | 3.30 × 10${}^{-1}$ | 1.40 × 10${}^{-1}$ | 1.70 × 10${}^{-1}$ | |

SEM | 4.98 × 10${}^{-\mathbf{11}}$ | 0.60 × 10${}^{-1}$ | 0.20 × 10${}^{-1}$ | 0.30 × 10${}^{-1}$ | |

${f}_{7}$ | Mean | 0.60 × 10${}^{-2}$ | 8.19 × 10${}^{-5}$ | 4.44 × 10${}^{-\mathbf{5}}$ | 4.69 × 10${}^{-5}$ |

SD | 0.40 × 10${}^{-2}$ | 8.38 × 10${}^{-5}$ | 4.16 × 10${}^{-\mathbf{5}}$ | 4.58 × 10${}^{-5}$ | |

SEM | 0.40 × 10${}^{-3}$ | 1.53 × 10${}^{-5}$ | 7.60 × 10${}^{-\mathbf{6}}$ | 8.37 × 10${}^{-6}$ | |

${f}_{8}$ | Mean | −2.83 × 10${}^{3}$ | −2.23 × 10${}^{3}$ | −2.093 × 10${}^{\mathbf{3}}$ | −2.096 × 10${}^{3}$ |

SD | 2.54 × 10${}^{2}$ | 1.91 × 10${}^{2}$ | 1.54 × 10${}^{\mathbf{2}}$ | 1.65 × 10${}^{2}$ | |

SEM | 4.65 × 10${}^{1}$ | 3.50 × 10${}^{1}$ | 2.82 × 10${}^{\mathbf{1}}$ | 3.01 × 10${}^{1}$ | |

${f}_{9}$ | Mean | 1.94 × 10${}^{1}$ | 1.01 × 10${}^{-10}$ | 0.00 × 10${}^{\mathbf{0}}$ | 0.00 × 10${}^{\mathbf{0}}$ |

SD | 0.75 × 10${}^{1}$ | 3.62 × 10${}^{-11}$ | 0.00 × 10${}^{\mathbf{0}}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

SEM | 0.13 × 10${}^{1}$ | 6.61 × 10${}^{-12}$ | 0.00 × 10${}^{\mathbf{0}}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

${f}_{10}$ | Mean | 7.50 × 10${}^{-1}$ | 6.17 × 10${}^{-6}$ | 6.11 × 10${}^{-13}$ | 8.88 × 10${}^{-\mathbf{16}}$ |

SD | 8.10 × 10${}^{-1}$ | 1.57 × 10${}^{-6}$ | 8.03 × 10${}^{-13}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

SEM | 1.40 × 10${}^{-1}$ | 2.87 × 10${}^{-7}$ | 1.46 × 10${}^{-13}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

${f}_{11}$ | Mean | 2.60 × 10${}^{-1}$ | 9.36 × 10${}^{-10}$ | 0.00 × 10${}^{\mathbf{0}}$ | 0.00 × 10${}^{\mathbf{0}}$ |

SD | 1.40 × 10${}^{-1}$ | 4.36 × 10${}^{-10}$ | 0.00 × 10${}^{\mathbf{0}}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

SEM | 0.20 × 10${}^{-1}$ | 7.97 × 10${}^{-11}$ | 0.00 × 10${}^{\mathbf{0}}$ | 0.00 × 10${}^{\mathbf{0}}$ | |

${f}_{12}$ | Mean | 2.70 × 10${}^{-1}$ | 3.60 × 10${}^{-1}$ | 1.40 × 10${}^{-1}$ | 1.30 × 10${}^{-\mathbf{1}}$ |

SD | 4.60 × 10${}^{-1}$ | 1.20 × 10${}^{-1}$ | 0.50 × 10${}^{-1}$ | 0.50 × 10${}^{-\mathbf{1}}$ | |

SEM | 0.80 × 10${}^{-1}$ | 0.20 × 10${}^{-1}$ | 0.10 × 10${}^{-1}$ | 0.10 × 10${}^{-\mathbf{1}}$ | |

${f}_{13}$ | Mean | 0.10 × 10${}^{-\mathbf{2}}$ | 7.00 × 10${}^{-1}$ | 3.40 × 10${}^{-1}$ | 3.50 × 10${}^{-1}$ |

SD | 0.30 × 10${}^{-\mathbf{2}}$ | 1.70 × 10${}^{-1}$ | 0.70 × 10${}^{-1}$ | 0.90 × 10${}^{-1}$ | |

SEM | 0.60 × 10${}^{-\mathbf{3}}$ | 0.30 × 10${}^{-1}$ | 0.10 × 10${}^{-1}$ | 0.10 × 10${}^{-1}$ | |

${f}_{14}$ | Mean | 0.10 × 10${}^{\mathbf{1}}$ | 0.20 × 10${}^{1}$ | 0.14 × 10${}^{1}$ | 0.14 × 10${}^{1}$ |

SD | 1.80 × 10${}^{-\mathbf{1}}$ | 0.11 × 10${}^{1}$ | 6.10 × 10${}^{-1}$ | 6.70 × 10${}^{-1}$ | |

SEM | 0.30 × 10${}^{-\mathbf{1}}$ | 2.10 × 10${}^{-1}$ | 1.10 × 10${}^{-1}$ | 1.20 × 10${}^{-1}$ | |

${f}_{15}$ | Mean | 0.10 × 10${}^{-2}$ | 0.90 × 10${}^{-3}$ | 0.60 × 10${}^{-\mathbf{3}}$ | 0.60 × 10${}^{-\mathbf{3}}$ |

SD | 0.30 × 10${}^{-2}$ | 0.40 × 10${}^{-3}$ | 0.10 × 10${}^{-\mathbf{3}}$ | 0.20 × 10${}^{-\mathbf{3}}$ | |

SEM | 0.60 × 10${}^{-3}$ | 8.92 × 10${}^{-5}$ | 3.03 × 10${}^{-\mathbf{5}}$ | 4.32 × 10${}^{-\mathbf{5}}$ | |

${f}_{16}$ | Mean | −10.3 × 10${}^{-1}$ | −10.3 × 10${}^{-1}$ | −10.2 × 10${}^{-\mathbf{1}}$ | −10.2 × 10${}^{-\mathbf{1}}$ |

SD | 9.59 × 10${}^{-15}$ | 3.01 × 10${}^{-14}$ | 0.10 × 10${}^{-\mathbf{2}}$ | 0.20 × 10${}^{-\mathbf{2}}$ | |

SEM | 1.75 × 10${}^{-15}$ | 5.49 × 10${}^{-15}$5 | 0.2 × 10${}^{-\mathbf{3}}$ | 0.3 × 10${}^{-\mathbf{3}}$ | |

Best for | 3/16 | 0/16 | 7/16 | 11/16 |

SSA | STSSA | IWSSA | ISSA | ||
---|---|---|---|---|---|

${f}_{1}$ | MeanFES | 24,699.3 | 23,743.1 | 961.8 | 427.1 |

SR (%) | 100 | 100 | 100 | 100 | |

${f}_{2}$ | MeanFES | 29,272.6 | 27,581.9 | 3262.9 | 538.5 |

SR (%) | 20 | 100 | 100 | 100 | |

${f}_{3}$ | MeanFES | 24,189.5 | 23,052.8 | 975.5 | 314.3 |

SR (%) | 100 | 100 | 100 | 100 | |

${f}_{4}$ | MeanFES | 28,211.3 | 27,320.7 | 2709.9 | 595.5 |

SR (%) | 100 | 100 | 100 | 100 | |

${f}_{5}$ | MeanFES | NaN | NaN | NaN | NaN |

SR (%) | 0 | 0 | 0 | 0 | |

${f}_{6}$ | MeanFES | 23,569.4 | NaN | NaN | NaN |

SR (%) | 100 | 0 | 0 | 0 | |

${f}_{7}$ | MeanFES | NaN | 22,196.0 | 10,977.3 | 16,115.1 |

SR (%) | 0 | 70 | 90 | 90 | |

${f}_{8}$ | MeanFES | 6670.3 | 8318.1 | 7208.3 | 7298.2 |

SR (%) | 100 | 70 | 40 | 46.7 | |

${f}_{9}$ | MeanFES | NaN | 22,598.3 | 765.4 | 342.9 |

SR (%) | 0 | 100 | 100 | 100 | |

${f}_{10}$ | MeanFES | 27,726.4 | 27,119.7 | 2840.9 | 489.7 |

SR (%) | 50 | 100 | 100 | 100 | |

${f}_{11}$ | MeanFES | NaN | 23,761.7 | 1082.5 | 339.9 |

SR (%) | 0 | 100 | 100 | 100 | |

${f}_{12}$ | MeanFES | 20,934.8 | NaN | NaN | NaN |

SR (%) | 63.3 | 0 | 0 | 0 | |

${f}_{13}$ | MeanFES | 21,824.07 | NaN | NaN | NaN |

SR (%) | 90 | 0 | 0 | 0 | |

${f}_{14}$ | MeanFES | 6371.1 | 2980.2 | 9078 | 6266.7 |

SR (%) | 97 | 16.67 | 20 | 24 | |

${f}_{15}$ | MeanFES | 10,698.4 | 9842.8 | 9280.5 | 11,614.1 |

SR (%) | 66.67 | 66.67 | 96.67 | 90 | |

${f}_{16}$ | MeanFES | 10,090.9 | 11,945.7 | 13,526 | 13,367.5 |

SR (%) | 100 | 100 | 20 | 33.3 | |

Best for | MeanFES | 5/16 | 1/16 | 2/16 | 7/16 |

SR (%) | 9/16 | 8/16 | 9/16 | 8/16 |

SSA | STSSA | IWSSA | ISSA | |
---|---|---|---|---|

Mean | 2.289 × 10${}^{-1}$ | 2.290 × 10${}^{-1}$ | 2.29 × 10${}^{-2}$ | 1.06 × 10${}^{-\mathbf{2}}$ |

SD | 3.24 × 10${}^{-2}$ | 2.57 × 10${}^{-2}$ | 5.5 × 10${}^{-3}$ | 1.5 × 10${}^{-\mathbf{3}}$ |

SEM | 5.9 × 10${}^{-3}$ | 4.7 × 10${}^{-3}$ | 9.954 × 10${}^{-4}$ | 2.801 × 10${}^{-\mathbf{4}}$ |

SSA | STSSA | IWSSA | ISSA | |
---|---|---|---|---|

MeanFES | NaN | 10,134 | 810.3 | 2773.7 |

SR (%) | 0 | 86.67 | 100 | 100 |

w/o FW | z-Score | SSA | STSSA | IWSSA | ISSA |
---|---|---|---|---|---|

0.1888 | 0.0384 | 0.2418 | 0.0213 | 0.0137 | 0.0089 |

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**MDPI and ACS Style**

Ben Chaabane, S.; Belazi, A.; Kharbech, S.; Bouallegue, A.; Clavier, L. Improved Salp Swarm Optimization Algorithm: Application in Feature Weighting for Blind Modulation Identification. *Electronics* **2021**, *10*, 2002.
https://doi.org/10.3390/electronics10162002

**AMA Style**

Ben Chaabane S, Belazi A, Kharbech S, Bouallegue A, Clavier L. Improved Salp Swarm Optimization Algorithm: Application in Feature Weighting for Blind Modulation Identification. *Electronics*. 2021; 10(16):2002.
https://doi.org/10.3390/electronics10162002

**Chicago/Turabian Style**

Ben Chaabane, Sarra, Akram Belazi, Sofiane Kharbech, Ammar Bouallegue, and Laurent Clavier. 2021. "Improved Salp Swarm Optimization Algorithm: Application in Feature Weighting for Blind Modulation Identification" *Electronics* 10, no. 16: 2002.
https://doi.org/10.3390/electronics10162002