# Efficient Feature Selection Using Weighted Superposition Attraction Optimization Algorithm

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Wrapper Method for Feature Selection

#### 2.2. Fitness Function

#### 2.3. Metaheuristics

#### 2.3.1. Differential Evolution

Algorithm 1: Differential evolution | ||

Objective function $f\left(x\right),x=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right)$ Initialize population G = 0; Initialize all NP individuals WHILE $Iterations<MaximumIterations$ | ||

FOR $i\leftarrow 1$ to NP | ||

GENERATE three individuals randomly based on the condition that ${r}_{1}\ne {r}_{2}\ne {r}_{3}\ne i$ MUTATION Form the donor vector using the formula: ${v}_{i}={x}_{{r}_{1}}+F\left({x}_{{r}_{2}}-{x}_{{r}_{3}}\right)$ CROSSOVER The trial vector u _{i} is developed either from the elements of the target vector x_{i} or the elements of the donor vector v_{i} as follows: | ||

${u}_{i,j}=\left(\right)open="\{">\begin{array}{c}{v}_{i,j}\mathrm{if}{r}_{i,j}\le CR\mathrm{or}j={j}_{rand}\\ {x}_{i,j}\mathit{otherwise}\end{array}$ | ||

where $i=\left\{1,\dots ,NP\right\}$,$j=\left\{1,\dots ,D\right\}$, CR is the crossover rate, ${r}_{i,j}~\cup \left(0,1\right)$ is random number generated for each $j$ and ${j}_{rand}\in \left\{1,\dots ,D\right\}$ is random integer to ensure that ${u}_{i}\ne {x}_{i}$ in all cases EVALUATE if $f\left({u}_{i}\right)\le f\left({x}_{i}\right)$ replace the individual ${x}_{i}$ with the trial vector ${u}_{i}$ | ||

$Iterations=Iterations+NP$ | ||

END | ||

$G=G+1$ | ||

END |

#### 2.3.2. Genetic Algorithm

_{k}be the fitness value for each chromosome ${C}_{k}$. The probability ${p}_{k}$ is calculated using the following expression:

_{i}is the fitness value of ith individual and n is the total number of individuals.

Algorithm 2: Genetic algorithm (roulette wheel) | |||

Objective function $f\left(x\right),x=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right)$ Initialize population G = 0; Initialize all NP individuals WHILE Iterations < Maximum Iterations | |||

FOR i ← 1 to NP | |||

sum += fitness of this individual FOR all members of population probability = sum of probabilities + (fitness/sum) | |||

sum of probabilities += probability | |||

END number = random between 0 and 1 for all members of population if number > probability but less than next probability Iterations = Iterations + NP | |||

END G = G + 1 | |||

END |

Algorithm 3: Genetic algorithm (tournament selection) | ||||

Objective function $f\left(x\right),x=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right)$ Initialize population $G=0$; Initialize all $NP$ individuals WHILE $Iterations<MaximumIterations$ | ||||

FOR $i\leftarrow 1toNP$ | ||||

WHILE need to generate more offspring | ||||

IF $populationsizek$ then | ||||

Refill: move all individuals from the temporary population $T$ to population $S$ | ||||

END IF | ||||

sampling $k$ individuals without replacement from population $S$ select the winner from the tournament move the $k$ sampled individuals into temporary population $T$ return the winner | ||||

END $Iterations=Iterations+NP$ | ||||

END $G=G+1$ | ||||

END |

#### 2.3.3. Particle Swarm Optimization

Algorithm 4: Particle swarm optimization | |||

Objective function $f\left(x\right),x=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right)$ Initialize population FOR $t=1:maxgeneration$ | |||

FOR $i=1:N$ | |||

If $f\left({x}_{i,d}\left(t\right)\right)<f\left({p}_{i}\left(t\right)\right)$ then ${p}_{i}\left(t\right)={x}_{i,d}\left(t\right),f({p}_{g}\left(t\right))=mi{n}_{t}f\left({p}_{i}\left(t\right)\right)$ | |||

END | |||

FOR $d=1:d$ | |||

${v}_{i,d}\left(t+1\right)=w{v}_{i,d}\left(t\right)+{c}_{1}{r}_{1}\left({p}_{i}-{x}_{i,d}\left(t\right)\right)+{c}_{2}{r}_{2}\left({p}_{g}-{x}_{i,d}\left(t\right)\right)$ ${x}_{i,d}\left(t+1\right)={x}_{i,d}\left(t\right)+{v}_{i,d}\left(t+1\right)$ | |||

IF ${v}_{i,d}\left(t+1\right)>{v}_{max}$ THEN | |||

${v}_{i,d}\left(t+1\right)={v}_{max}$ | |||

ELSE IF ${v}_{i,d}\left(t+1\right)<{v}_{min}$ THEN | |||

${v}_{i,d}\left(t+1\right)={v}_{min}$ | |||

IF ${x}_{i,d}\left(t+1\right)>{x}_{max}$ THEN | |||

${x}_{i,d}\left(t+1\right)={x}_{max}$ | |||

IF ${x}_{i,d}\left(t+1\right)<{x}_{min}$ THEN | |||

${x}_{i,d}\left(t+1\right)={x}_{min}$ | |||

END | |||

END |

#### 2.3.4. Flower Pollination Algorithm

- (a)
- A biotic process is global pollination and obeys Levy flights.
- (b)
- On the other hand, an abiotic process is local pollination.
- (c)
- Pollinators are the probabilities of reproduction.
- (d)
- Probability switches between local and global pollination.

Algorithm 5: Flower pollination algorithm | |||

Objective function $f\left(x\right),x=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right)$ Initialize population Find the best solution ${g}_{*}$ in the initial population Define a switch probability $p\in \left[0,1\right]$ FOR $t=1:maxgeneration$ | |||

FOR $i=1:n$ | |||

IF $rand<p$ | |||

Draw d-dimensional step vector L Global phase Equation (6) | |||

ELSE | |||

Draw $\rho $ in Equation (8) from a uniform distribution [0, 1] Local phase Equation (8) | |||

END | |||

Assess new solutions If new solutions > old solutions, update population | |||

END current best solution ${g}_{*}$ | |||

END |

#### 2.3.5. Symbiotic Organism’s Search

Algorithm 6: Symbiotic organisms search | |

Objective function $f\left(x\right),x=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right)$ Initialize population FOR $t=1:maxgeneration$ | |

Mutual interaction phase Commensalism interaction phase Parasitic interaction phase current best solution | |

END |

#### 2.3.6. Marine Predators’ Algorithm

_{i,j}represents jth dimension of ith prey.

Algorithm 7: Marine predators algorithm | ||

Objective function $f\left(x\right),x=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right)$ Initialize population Compute the fitness values, elite matrix and memory saving FOR t = 1: max generation | ||

IF $Iter<\frac{1}{3}MaxIter$ | ||

${\overrightarrow{stepsize}}_{i}={\overrightarrow{R}}_{B}\otimes \left({\overrightarrow{Elite}}_{i}-{\overrightarrow{R}}_{B}\otimes {\overrightarrow{\mathit{Pr}ey}}_{i}\right)\left(i=1,2,\dots ,n\right)$ ${\overrightarrow{\mathit{Pr}ey}}_{i}={\overrightarrow{\mathit{Pr}ey}}_{i}+P\xb7\overrightarrow{R}\otimes \u2942{\overrightarrow{stepsize}}_{i}$ | ||

ELSE IF $\frac{1}{3}MaxIter<Iter<\frac{2}{3}MaxIter$ | ||

${\overrightarrow{stepsize}}_{i}={\overrightarrow{R}}_{L}\otimes \left({\overrightarrow{Elite}}_{i}-{\overrightarrow{R}}_{L}\otimes {\overrightarrow{\mathit{Pr}ey}}_{i}\right)\left(i=1,2,\dots ,n/2\right)$ The first half of the population is updated by ${\overrightarrow{\mathit{Pr}ey}}_{i}={\overrightarrow{\mathit{Pr}ey}}_{i}+P\xb7CF\otimes \u2942{\overrightarrow{stepsize}}_{i}$ The second half of the population is updated by ${\overrightarrow{\mathit{Pr}ey}}_{i}={\overrightarrow{Elite}}_{i}+P\xb7CF\otimes \u2942{\overrightarrow{stepsize}}_{i}$ | ||

ELSE IF $Iter>\frac{2}{3}MaxIter$ | ||

${\overrightarrow{stepsize}}_{i}={\overrightarrow{R}}_{L}\otimes \left(\overrightarrow{{R}_{L}}\otimes {\overrightarrow{Elite}}_{i}-{\overrightarrow{\mathit{Pr}ey}}_{i}\right)\left(i=1,2,\dots ,n\right)$ ${\overrightarrow{\mathit{Pr}ey}}_{i}={\overrightarrow{Elite}}_{i}+P\xb7CF\otimes \u2942{\overrightarrow{stepsize}}_{i}$ | ||

END IF | ||

Accomplish elite update and memory saving based on ${\overrightarrow{\mathrm{Pr}ey}}_{i}=\left(\right)open="\{">\begin{array}{c}\mathrm{Pr}e{y}_{i}+CF\left[{X}_{\mathrm{min}}+R\otimes \left({X}_{\mathrm{max}}-{X}_{\mathrm{min}}\right)\right]\otimes U\mathrm{if}r\le FADs\\ \mathrm{Pr}e{y}_{i}+\left[FADs\left(1-r\right)+r\right]\left({\overrightarrow{\mathrm{Pr}ey}}_{{r}_{1}}-{\overrightarrow{\mathrm{Pr}ey}}_{{r}_{2}}\right)\mathrm{if}rFADs\end{array}$ (where $FADs=0.2$) current best solution | ||

END |

#### 2.3.7. Manta Ray Foraging Optimization

Algorithm 8: Manta ray foraging | |||

Objective function $f\left(x\right),x=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right)$ Initialize population and maximum iterations Compute the fitness of each individual and obtain the best solutions FOR $t=1:maxgeneration$ | |||

IF $rand<0.5$ THEN use cyclone foraging | |||

IF $t/{T}_{max}<rand$ THEN | |||

${x}_{rand}={x}_{l}+rand\xb7\left({x}_{u}-{x}_{l}\right)$ ${x}_{i}\left(t+1\right)=\left(\right)open="\{">\begin{array}{c}{x}_{rand}+r\xb7\left({x}_{rand}-{x}_{i}\left(t\right)\right)+\beta \xb7\left({x}_{rand}-{x}_{i}\left(t\right)\right)\\ {x}_{rand}+r\xb7\left({x}_{i-1}-{x}_{i}\left(t\right)\right)+\beta \xb7\left({x}_{rand}-{x}_{i}\left(t\right)\right)\end{array}$ | |||

ELSE ${x}_{i}\left(t+1\right)=\left(\right)open="\{">\begin{array}{c}{x}_{best}+r\xb7\left({x}_{best}-{x}_{i}\left(t\right)\right)+\beta \xb7\left({x}_{best}-{x}_{i}\left(t\right)\right)\\ {x}_{best}+r\xb7\left({x}_{i-1}-{x}_{i}\left(t\right)\right)+\beta \xb7\left({x}_{best}-{x}_{i}\left(t\right)\right)\end{array}$ | |||

END IF | |||

ELSE use chain foraging | |||

${x}_{i}\left(t+1\right)=\left(\right)open="\{">\begin{array}{c}{x}_{i}\left(t\right)+r\xb7\left({x}_{best}-{x}_{i}\left(t\right)\right)+\alpha \xb7\left({x}_{best}-{x}_{i}\left(t\right)\right)\\ {x}_{i}\left(t\right)+r\xb7\left({x}_{i-1}-{x}_{i}\left(t\right)\right)+\alpha \xb7\left({x}_{best}-{x}_{i}\left(t\right)\right)\end{array}$ | |||

END IF | |||

Calculate the fitness of the individuals using $f\left({x}_{i}\left(t+1\right)\right)$ | |||

IF $f\left({x}_{i}\left(t+1\right)\right)<f\left({x}_{best}\right)$ THEN ${x}_{best}={x}_{i}\left(t+1\right)$ | |||

For somersault foraging ${x}_{i}\left(t+1\right)={x}_{i}\left(t\right)+S\xb7\left({r}_{2}\xb7{x}_{best}-{r}_{3}\xb7{x}_{i}\left(t\right)\right)$ | |||

Calculate the fitness of the individuals using $f\left({x}_{i}\left(t+1\right)\right)$ | |||

IF $f\left({x}_{i}\left(t+1\right)\right)<f\left({x}_{best}\right)$ THEN | |||

${x}_{best}={x}_{i}\left(t+1\right)$ | |||

END current best solution |

#### 2.3.8. Weighted Superposition Attraction Algorithm

Algorithm 9: Weighted superposition attraction algorithm | |

Objective function $f\left(x\right),x=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right)$) Initialize population and maximum iterations Compute the fitness of each individual and obtain the best solutions FOR $t=1:maxgeneration$ | |

Ranking solutions by the fitness Determining the target point to move the simulated iteration towards it Evaluating the fitness value of the target Determination of the search direction for the solutions Each solution is moved toward its determined direction Update the fitness solutions for $t=t+1$ | |

END current best solution |

## 3. Results and Discussion

#### 3.1. Dataset Description

#### 3.2. Metaheuristic Parameters

#### 3.3. Performance of the Metaheuristic on the Real-World Datasets

#### 3.4. Comparison with the State of the Art

## 4. Conclusions

- WSA > MPA > GA (T) > MRFO > GA (R) > FPA > PSO > DE > SOS is the order of the algorithms under consideration that provide the best fitness value. In contrast, while comparing these algorithms based on mean best fitness and standard deviation, WSA > MPA > MRFO > FPA > GA > GA > GA > GA > GA > DE > PSO > SOS is the order of their performance.
- The convergence of WSA and MPA is found to be superior to other algorithms.
- WSA > GA (T) > GA (R) > DE > MPA > MRFO > PSO > FPA > SOS is the ranking order of the algorithms with respect to the highest classification accuracy. On the other hand, in terms of mean best classification accuracy and standard deviation, WSA > MPA > MRFO > GA (T) > GA (R) > FPA > DE > PSO > SOS is the order of the algorithms.
- FPA and DE are noticed to be computationally faster than the other algorithms, and thus, based on the best computation time, the algorithms can be ranked as FPA > DE > WSA > PSO > MRFO > MPA > GA (T) > GA (R) > SOS. In terms of mean computational time and standard deviation, the ranking of these algorithms is FPA > DE > PSO > WSA > MRFO > MPA > GA (T) > GA (R) > SOS.
- With respect to the lowest number of features selected, the ranking is MPA > WSA > SOS > GA (T) > GA (R) > PSO > FPA > MRFO > DE, whereas for the mean and standard deviation of the number of features selected, the ranking is derived as SOS > MPA > GA (T) > GA (R) > WSA > PSO > FPA > DE > MRFO.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Convergence of each algorithm for different datasets (

**a**) Heart (

**b**) Ionosphere (

**c**) German (

**d**) Breast (

**e**) Sonar (

**f**) Ovarian cancer (

**g**) Australian (

**h**) Colon (

**i**) Diabetes.

Dataset | Instances | Features | Source |
---|---|---|---|

Heart | 303 | 13 | https://archive.ics.uci.edu/ml/datasets/heart+Disease (accessed on 1 February 2023) |

Ionosphere | 351 | 34 | https://archive.ics.uci.edu/ml/datasets/ionosphere (accessed on 1 February 2023) |

German | 1000 | 24 | http://www.liacc.up.pt/ML/old/statlog/datasets.html (accessed on 1 February 2023) |

Breast | 699 | 9 | https://archive.ics.uci.edu/ml/datasets/breast+cancer+wisconsin+(diagnostic) (accessed on 1 February 2023) |

Sonar | 208 | 60 | https://archive.ics.uci.edu/ml/datasets/Connectionist+Bench+(Sonar,+Mines+vs.+Rocks) (accessed on 1 February 2023) |

Ovarian | 216 | 4000 | Conrads et al. [57] |

Australian | 690 | 14 | https://archive.ics.uci.edu/ml/datasets/statlog+(australian+credit+approval) (accessed on 1 February 2023) |

Colon | 62 | 2000 | Alon et al. [58] |

Diabetes | 768 | 8 | https://archive.ics.uci.edu/ml/datasets/diabetes (accessed on 1 February 2023) |

Algorithm | Parameter | Value |
---|---|---|

Common parameters | K | 5 |

Iteration limit | 200 | |

Search agents | 30 | |

Independent runs | 20 | |

Validation data | 20% | |

DE | Crossover rate | 0.9 |

Constant factor | 0.5 | |

GA (R) | Crossover rate | 0.8 |

Mutation rate | 0.01 | |

GA (T) | Crossover rate | 0.8 |

Mutation rate | 0.01 | |

Tournament size | 3 | |

PSO | ${c}_{1}$ | 2 |

${c}_{2}$ | 2 | |

$w$ | 0.9 | |

FPA | Levy component | 1.5 |

Switch probability | 0.8 | |

MPA | Levy component | 1.5 |

Constant | 0.5 | |

Fish aggregating devices effect | 0.2 | |

MRFO | Somersault factor | 2 |

WSA | $\tau $ | 0.8 |

$\phi $ | 0.001 | |

$\lambda $ | 0.75 | |

Step length | 0.035 |

Dataset | DE | GA (R) | GA (T) | PSO | FPA | SOS | MPA | MRFO | WSA |
---|---|---|---|---|---|---|---|---|---|

Heart | 0.06908 | 0.08635 | 0.08481 | 0.05258 | 0.08635 | 0.15312 | 0.10131 | 0.08558 | 0.08481 |

Ionosphere | 0.06245 | 0.04478 | 0.05951 | 0.03123 | 0.01591 | 0.10194 | 0.02946 | 0.02976 | 0.01503 |

German | 0.20920 | 0.18320 | 0.18153 | 0.18940 | 0.19847 | 0.23187 | 0.19268 | 0.20217 | 0.16793 |

Breast | 0.01157 | 0.01046 | 0.00667 | 0.01046 | 0.01869 | 0.01046 | 0.01157 | 0.01647 | 0.01157 |

Sonar | 0.00433 | 0.00183 | 0.00267 | 0.00250 | 0.00283 | 0.00500 | 0.00100 | 0.00183 | 0.00100 |

Ovarian | 0.00566 | 0.00324 | 0.00455 | 0.00451 | 0.00346 | 0.00486 | 0.00001 | 0.00011 | 0.00001 |

Australian | 0.08966 | 0.09612 | 0.08106 | 0.10975 | 0.08823 | 0.16354 | 0.10975 | 0.08894 | 0.09683 |

Colon | 0.00489 | 0.00263 | 0.00242 | 0.08677 | 0.00456 | 0.08721 | 0.00001 | 0.00001 | 0.00001 |

Diabetes | 0.20559 | 0.20684 | 0.20559 | 0.22250 | 0.19515 | 0.21331 | 0.19265 | 0.19912 | 0.18618 |

Average rank | 6.39 | 4.78 | 4.11 | 5.39 | 5.17 | 8.11 | 4.06 | 4.39 | 2.61 |

**Table 4.**Mean (and standard deviation) of best fitness value achieved by each algorithm for each dataset on repeated trials.

Dataset | DE | GA (R) | GA (T) | PSO | FPA | SOS | MPA | MRFO | WSA |
---|---|---|---|---|---|---|---|---|---|

Heart | 0.11 (0.04) | 0.14 (0.04) | 0.13 (0.03) | 0.13 (0.05) | 0.14 (0.06) | 0.21 (0.05) | 0.13 (0.03) | 0.11 (0.04) | 0.1 (0.01) |

Ionosphere | 0.11 (0.04) | 0.06 (0.02) | 0.07 (0.02) | 0.08 (0.03) | 0.06 (0.04) | 0.11 (0.01) | 0.04 (0.01) | 0.06 (0.03) | 0.03 (0.01) |

German | 0.22 (0.01) | 0.21 (0.02) | 0.22 (0.03) | 0.21 (0.02) | 0.21 (0.01) | 0.24 (0.01) | 0.2 (0.01) | 0.21 (0.01) | 0.2 (0.02) |

Breast | 0.02 (0.01) | 0.03 (0.01) | 0.02 (0.01) | 0.02 (0) | 0.03 (0.01) | 0.02 (0.01) | 0.02 (0.01) | 0.02 (0.01) | 0.02 (0.01) |

Sonar | 0.02 (0.02) | 0.04 (0.03) | 0.04 (0.04) | 0.04 (0.02) | 0.02 (0.01) | 0.11 (0.09) | 0.02 (0.02) | 0.05 (0.04) | 0.01 (0.01) |

Ovarian | 0.02 (0.01) | 0.02 (0.01) | 0.03 (0.02) | 0.02 (0.02) | 0.02 (0.01) | 0.05 (0.03) | 0 (0) | 0 (0) | 0 (0) |

Australian | 0.12 (0.02) | 0.11 (0.01) | 0.12 (0.03) | 0.13 (0.02) | 0.11 (0.02) | 0.2 (0.05) | 0.12 (0.01) | 0.11 (0.01) | 0.11 (0.01) |

Colon | 0.14 (0.11) | 0.05 (0.05) | 0.05 (0.05) | 0.17 (0.06) | 0.07 (0.07) | 0.12 (0.05) | 0 (0) | 0.02 (0.04) | 0 (0) |

Diabetes | 0.23 (0.02) | 0.23 (0.02) | 0.22 (0.01) | 0.24 (0.01) | 0.21 (0.02) | 0.25 (0.04) | 0.22 (0.02) | 0.23 (0.02) | 0.22 (0.02) |

Av. Rank (mean) | 5.61 | 5.00 | 5.44 | 6.33 | 5.11 | 8.56 | 3.00 | 4.33 | 1.61 |

Av. Rank (SD) | 5.89 | 5.44 | 5.56 | 5.44 | 5.00 | 6.44 | 2.67 | 5.22 | 3.33 |

Combined av. rank | 5.75 | 5.22 | 5.50 | 5.89 | 5.06 | 7.50 | 2.83 | 4.78 | 2.47 |

Dataset | DE | GA (R) | GA (T) | PSO | FPA | SOS | MPA | MRFO | WSA |
---|---|---|---|---|---|---|---|---|---|

Heart | 0.9333 | 0.9167 | 0.9167 | 0.9500 | 0.9167 | 0.8500 | 0.9000 | 0.9167 | 0.9167 |

Ionosphere | 0.9429 | 0.9857 | 0.9571 | 0.9714 | 0.9429 | 0.9000 | 0.9714 | 0.9714 | 0.9857 |

German | 0.7950 | 0.8200 | 0.8200 | 0.8150 | 0.8050 | 0.7700 | 0.8100 | 0.8000 | 0.8350 |

Breast | 0.9928 | 0.9928 | 1.0000 | 0.9928 | 0.9856 | 0.9928 | 0.9928 | 0.9856 | 0.9928 |

Sonar | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9512 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Ovarian | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Australian | 0.9130 | 0.9058 | 0.9203 | 0.8913 | 0.9130 | 0.8406 | 0.8913 | 0.9130 | 0.9058 |

Colon | 1.0000 | 1.0000 | 1.0000 | 0.9167 | 1.0000 | 0.9167 | 1.0000 | 1.0000 | 1.0000 |

Diabetes | 0.7974 | 0.7974 | 0.7974 | 0.7778 | 0.8105 | 0.7909 | 0.8105 | 0.8039 | 0.8170 |

Average rank | 4.94 | 4.28 | 3.89 | 5.33 | 5.61 | 7.39 | 5.00 | 5.00 | 3.56 |

**Table 6.**Mean (and standard deviation) of best classification accuracy achieved by each algorithm for each dataset in repeated trials.

Dataset | DE | GA (R) | GA (T) | PSO | FPA | SOS | MPA | MRFO | WSA |
---|---|---|---|---|---|---|---|---|---|

Heart | 0.89 (0.04) | 0.87 (0.04) | 0.87 (0.03) | 0.87 (0.05) | 0.86 (0.06) | 0.79 (0.05) | 0.88 (0.03) | 0.89 (0.04) | 0.9 (0.01) |

Ionosphere | 0.9 (0.04) | 0.94 (0.04) | 0.95 (0.02) | 0.92 (0.03) | 0.91 (0.03) | 0.9 (0.01) | 0.96 (0.01) | 0.94 (0.03) | 0.97 (0.01) |

German | 0.79 (0.01) | 0.8 (0.02) | 0.79 (0.03) | 0.79 (0.02) | 0.79 (0.01) | 0.76 (0.01) | 0.8 (0.01) | 0.79 (0.01) | 0.81 (0.02) |

Breast | 0.99 (0.01) | 0.98 (0.01) | 0.99 (0.01) | 0.99 (0) | 0.98 (0.01) | 0.98 (0.01) | 0.99 (0.01) | 0.98 (0.01) | 0.99 (0.01) |

Sonar | 0.98 (0.02) | 0.98 (0.01) | 0.97 (0.03) | 0.97 (0.02) | 0.94 (0.02) | 0.89 (0.09) | 0.99 (0.02) | 0.96 (0.04) | 0.99 (0.01) |

Ovarian | 0.99 (0.01) | 0.98 (0.01) | 0.99 (0.01) | 0.99 (0.02) | 0.99 (0.01) | 0.95 (0.03) | 1 (0) | 1 (0) | 1 (0) |

Australian | 0.89 (0.02) | 0.89 (0.01) | 0.88 (0.03) | 0.87 (0.02) | 0.89 (0.02) | 0.8 (0.05) | 0.88 (0.01) | 0.89 (0.01) | 0.89 (0.01) |

Colon | 0.87 (0.11) | 0.95 (0.05) | 0.95 (0.05) | 0.83 (0.06) | 0.93 (0.07) | 0.88 (0.05) | 1 (0) | 0.98 (0.04) | 1 (0) |

Diabetes | 0.77 (0.02) | 0.77 (0.02) | 0.78 (0.01) | 0.76 (0.01) | 0.79 (0.02) | 0.75 (0.04) | 0.79 (0.02) | 0.77 (0.02) | 0.79 (0.02) |

Av. Rank (mean) | 5.33 | 5.44 | 5.00 | 6.17 | 5.61 | 8.44 | 3.11 | 4.28 | 1.61 |

Av. Rank (SD) | 5.94 | 5.17 | 5.06 | 5.61 | 5.56 | 6.67 | 2.83 | 4.89 | 3.28 |

Combined av. rank | 5.64 | 5.31 | 5.03 | 5.89 | 5.58 | 7.56 | 2.97 | 4.58 | 2.44 |

Dataset | DE | GA (R) | GA (T) | PSO | FPA | SOS | MPA | MRFO | WSA |
---|---|---|---|---|---|---|---|---|---|

Heart | 37.3069 | 58.4510 | 59.3564 | 36.9822 | 37.2579 | 125.6186 | 71.1589 | 66.0726 | 35.9040 |

Ionosphere | 4.8874 | 14.3396 | 14.8998 | 9.2014 | 4.9101 | 18.4415 | 9.8246 | 8.2109 | 5.4902 |

German | 41.4371 | 126.0874 | 150.9288 | 94.9779 | 46.2631 | 171.6129 | 87.2630 | 75.1673 | 45.2955 |

Breast | 42.1620 | 64.1089 | 65.3039 | 42.1937 | 37.8479 | 137.6404 | 74.1079 | 62.2471 | 45.1631 |

Sonar | 32.8366 | 56.9434 | 54.5112 | 32.6148 | 32.5553 | 129.3449 | 65.7633 | 56.0746 | 34.6354 |

Ovarian | 126.6548 | 137.8998 | 137.5729 | 122.3213 | 103.2913 | 155.3433 | 85.1104 | 83.8790 | 169.3444 |

Australian | 38.4591 | 61.5668 | 62.1789 | 38.4547 | 38.7062 | 139.0013 | 77.3190 | 53.8778 | 37.9527 |

Colon | 38.3924 | 56.6966 | 56.6441 | 47.9606 | 37.6771 | 109.1697 | 67.4664 | 53.6178 | 42.1641 |

Diabetes | 86.3755 | 133.5861 | 130.1496 | 81.9149 | 78.9590 | 324.7326 | 73.3953 | 62.7816 | 39.9376 |

Average rank | 3.00 | 6.67 | 6.67 | 3.67 | 2.44 | 8.89 | 6.22 | 4.33 | 3.11 |

**Table 8.**Mean (and standard deviation) of computational time achieved by each algorithm for each dataset in repeated trials.

Dataset | DE | GA (R) | GA (T) | PSO | FPA | SOS | MPA | MRFO | WSA |
---|---|---|---|---|---|---|---|---|---|

Heart | 37.67 (0.62) | 60.41 (2.35) | 62.8 (5.57) | 37.34 (0.23) | 41.35 (5.03) | 130.76 (4.1) | 74.69 (2.17) | 68.33 (1.93) | 38.59 (2.44) |

Ionosphere | 4.97 (0.11) | 15.37 (0.9) | 15.85 (0.63) | 9.4 (0.2) | 5.05 (0.1) | 19.48 (0.81) | 10.29 (0.36) | 8.71 (0.32) | 5.62 (0.1) |

German | 63.37 (29.87) | 138.45 (10.19) | 154.66 (5.76) | 96.22 (1.04) | 46.88 (0.71) | 183.47 (11.72) | 89.79 (2.52) | 78.52 (2.36) | 46.01 (0.92) |

Breast | 43.33 (0.96) | 69.46 (6.36) | 67.98 (3.03) | 43.26 (0.93) | 38.24 (0.27) | 142.41 (4.31) | 76.91 (1.78) | 64.25 (2.11) | 48 (2.63) |

Sonar | 32.96 (0.1) | 58.05 (1.22) | 55.76 (1.05) | 33.3 (1.2) | 32.63 (0.11) | 132.15 (1.64) | 66.85 (0.87) | 56.81 (0.58) | 34.78 (0.25) |

Ovarian | 131.01 (3.26) | 142.65 (3.87) | 138.86 (1.46) | 123.76 (1.4) | 103.8 (0.42) | 168.66 (9.57) | 90.99 (7.2) | 91.38 (7.58) | 217.57 (82.28) |

Australian | 39.88 (1.03) | 62.9 (1.76) | 62.87 (0.6) | 39.64 (1.79) | 40.54 (2.29) | 147.99 (7.3) | 78.32 (0.81) | 68.75 (9.5) | 39.71 (2.51) |

Colon | 39.07 (0.59) | 57.78 (1.11) | 57.14 (0.43) | 48.27 (0.18) | 37.8 (0.09) | 121.52 (8.99) | 67.78 (0.21) | 55.4 (1.89) | 42.31 (0.22) |

Diabetes | 88.07 (2.02) | 139.5 (4.17) | 138.37 (7.08) | 86.84 (3.4) | 83.91 (3.36) | 334.15 (7.5) | 75.12 (1.27) | 64.34 (1.14) | 40.15 (0.34) |

Av. Rank (mean) | 3.00 | 6.78 | 6.33 | 3.44 | 2.44 | 8.89 | 6.11 | 4.67 | 3.33 |

Av. Rank (SD) | 3.89 | 6.78 | 5.78 | 3.56 | 3.00 | 8.22 | 4.22 | 5.22 | 4.33 |

Combined av. rank | 3.44 | 6.78 | 6.06 | 3.50 | 2.72 | 8.56 | 5.17 | 4.94 | 3.83 |

Dataset | DE | GA (R) | GA (T) | PSO | FPA | SOS | MPA | MRFO | WSA |
---|---|---|---|---|---|---|---|---|---|

Heart | 4 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 3 |

Ionosphere | 12 | 6 | 7 | 10 | 11 | 2 | 3 | 10 | 3 |

German | 12 | 9 | 8 | 11 | 9 | 7 | 6 | 10 | 7 |

Breast | 2 | 3 | 3 | 3 | 3 | 3 | 2 | 3 | 2 |

Sonar | 26 | 10 | 10 | 12 | 19 | 6 | 6 | 23 | 9 |

Ovarian | 2014 | 1326 | 1294 | 1805 | 1888 | 2 | 5 | 1944 | 44 |

Australian | 3 | 1 | 1 | 3 | 3 | 4 | 2 | 7 | 1 |

Colon | 951 | 526 | 484 | 791 | 912 | 1 | 1 | 942 | 2 |

Diabetes | 4 | 3 | 3 | 2 | 3 | 4 | 3 | 4 | 2 |

Average rank | 7.78 | 4.67 | 4.39 | 5.72 | 6.28 | 3.89 | 2.67 | 6.89 | 2.72 |

**Table 10.**Mean (and standard deviation) of features selected by each algorithm for each dataset on repeated trials.

Dataset | DE | GA (R) | GA (T) | PSO | FPA | SOS | MPA | MRFO | WSA |
---|---|---|---|---|---|---|---|---|---|

Heart | 4.4 (0.55) | 4 (1.41) | 4.6 (1.14) | 4.6 (1.34) | 5 (1.58) | 3.2 (0.45) | 4 (1.22) | 4 (1.58) | 4.6 (1.14) |

Ionosphere | 15.4 (3.58) | 9.8 (2.39) | 9.4 (1.95) | 10.4 (0.55) | 14.4 (3.97) | 3.6 (1.14) | 4.2 (1.3) | 17.6 (5.03) | 5.2 (1.79) |

German | 14.6 (2.3) | 11.8 (2.68) | 9 (0.71) | 12.8 (1.64) | 11 (2.35) | 10.8 (2.59) | 9 (2.35) | 14.6 (3.21) | 11.4 (3.05) |

Breast | 4.4 (1.52) | 3.6 (1.34) | 4.8 (1.3) | 5 (1.41) | 4.4 (1.14) | 4.2 (0.84) | 4.4 (1.67) | 4.4 (1.52) | 3.4 (1.52) |

Sonar | 33.2 (4.21) | 14.8 (3.11) | 14.4 (6.23) | 16.6 (3.36) | 21.8 (2.77) | 7 (1.41) | 10.8 (4.32) | 31 (5.79) | 16 (10.42) |

Ovarian | 2443.2 (324.56) | 1374.2 (39.86) | 1325 (24.9) | 1846.4 (30.06) | 1924.4 (23) | 5.4 (3.58) | 6.4 (1.67) | 2010.8 (91.38) | 97.6 (56.42) |

Australian | 4.8 (1.48) | 3.4 (1.52) | 3.4 (1.52) | 3.6 (0.89) | 4.2 (0.84) | 5 (1.22) | 3.8 (1.3) | 9.4 (1.82) | 3.8 (1.64) |

Colon | 1061 (134.75) | 543 (18.32) | 499.6 (12.1) | 836.8 (28.25) | 936.8 (17.66) | 2.2 (1.3) | 1.6 (0.55) | 970.4 (21.38) | 3.8 (3.03) |

Diabetes | 4.4 (0.89) | 4.2 (0.84) | 4.2 (0.84) | 3.6 (1.52) | 4.4 (1.14) | 4.2 (0.45) | 4.4 (1.14) | 4.4 (0.55) | 3.2 (1.1) |

Av. Rank (mean) | 7.61 | 3.94 | 4.11 | 5.78 | 6.67 | 2.67 | 3.22 | 7.39 | 3.61 |

Av. Rank (SD) | 5.78 | 5.44 | 4.28 | 4.67 | 4.61 | 2.11 | 4.56 | 7.39 | 6.17 |

Combined av. rank | 6.69 | 4.69 | 4.19 | 5.22 | 5.64 | 2.39 | 3.89 | 7.39 | 4.89 |

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**MDPI and ACS Style**

Ganesh, N.; Shankar, R.; Čep, R.; Chakraborty, S.; Kalita, K.
Efficient Feature Selection Using Weighted Superposition Attraction Optimization Algorithm. *Appl. Sci.* **2023**, *13*, 3223.
https://doi.org/10.3390/app13053223

**AMA Style**

Ganesh N, Shankar R, Čep R, Chakraborty S, Kalita K.
Efficient Feature Selection Using Weighted Superposition Attraction Optimization Algorithm. *Applied Sciences*. 2023; 13(5):3223.
https://doi.org/10.3390/app13053223

**Chicago/Turabian Style**

Ganesh, Narayanan, Rajendran Shankar, Robert Čep, Shankar Chakraborty, and Kanak Kalita.
2023. "Efficient Feature Selection Using Weighted Superposition Attraction Optimization Algorithm" *Applied Sciences* 13, no. 5: 3223.
https://doi.org/10.3390/app13053223