# Improved Reptile Search Optimization Algorithm: Application on Regression and Classification Problems

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## Abstract

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## 1. Introduction

#### 1.1. Contributions and Organization

- An improved reptile search algorithm (IRSA) based on a sine operator and Levy flight was proposed to enhance the performance of the original RSA.
- The proposed IRSA was evaluated using 23 benchmark test functions. Various qualitative, quantitative, comparative, statistical, and complexity analyses were performed to validate the positive effects of the improvisations.
- This research also proposed a hybrid methodology that integrates Multi-Layer Perceptron Neural Network with the improvised RSA for solving various classification problems.
- Finally, the IRSA was applied to train a radial basis function neural network (RBFNN) for short-term wind and solar power predictions.

#### 1.2. Literature Survey

## 2. Proposed Methodology

#### 2.1. Reptile Search Algorithm

#### 2.1.1. Initialization

#### 2.1.2. Encircling Phase (Exploration)

#### 2.1.3. Hunting Phase (Exploitation)

#### 2.2. Proposed Improved Reptile Search Algorithm (IRSA)

Algorithm 1 Pseudocode of IRSA |

Initialize random population x Initialize iteration counter$\tau $= 0, maximum iteration T, alpha, beta while$\tau $< T Evaluate fitness of potential candidates Determines the best solution Update Es,${P}_{\left(j,k\right)}$using Equations (6) and (7) for j = 1: p for k = 1: n If $\tau \le \frac{T}{3}$ Solve using Equation (11) else if $\text{}\tau \le 2\frac{T}{4}\text{}and\text{}\tau \frac{T}{3}$ Solve using Equation (3) else if $\text{}\tau \le 3\frac{T}{4}\text{}and\text{}\tau 2\frac{T}{4}$ Solve using Equation (9) else Solve using Equation (16) end if end for end for t = t + 1 end while Return best solution |

## 3. Experimental Verification Using Benchmark Test Functions

#### 3.1. Qualitative Analysis

#### 3.2. Comparative Analysis

#### 3.3. Time Complexity Analysis

## 4. IRSA for Neural Network Training

#### 4.1. Multi-Layer Perceptron Neural Network (MLPNN)

#### 4.2. Radial Basis Function Neural Network (RBFNN)

#### 4.3. Training of MLPNN and RBFNN Using the Proposed IRSA

## 5. IRSA for Solving Classification Problems

#### Statistical Indicators for Classification

## 6. IRSA for Solving the Regression Problems

#### 6.1. Wind Power Prediction

#### 6.2. Solar Power Prediction

#### 6.3. Statistical Indicators for Regression

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Function Description | Dim | Range | $\text{}{\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|

${f}_{1}\left(x\right)={{\displaystyle \sum}}_{i=1}^{n}{x}_{i}{}^{2}$ | 500, 100, 50, 30 | [−100, 100] | 0 |

${f}_{2}\left(x\right)={{\displaystyle \sum}}_{i=0}^{n}\left|{x}_{i}\right|+{{\displaystyle \prod}}_{i=0}^{n}\left|{x}_{i}\right|$ | 500, 100, 50, 30 | [−10, 10] | 0 |

${f}_{3}\left(x\right)={{\displaystyle \sum}}_{i=1}^{d}{\left({{\displaystyle \sum}}_{j=1}^{i}{x}_{j}\right)}^{2}$ | 500, 100, 50, 30 | [−100, 100] | 0 |

${f}_{4}\left(x\right)=ma{x}_{i}\text{}\left\{\left|{x}_{i}\right|,\text{}1\le i\le n\right\}$ | 500, 100, 50, 30 | [−100, 100] | 0 |

${f}_{5}\left(x\right)={{\displaystyle \sum}}_{i=1}^{n-1}\left[100{\left({x}_{i}{}^{2}-{x}_{i+1}\right)}^{2}+{\left(1-{x}_{i}\right)}^{2}\right]$ | 500, 100, 50, 30 | [−30, 30] | 0 |

${f}_{6}\left(x\right)={{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}+0.5\right)}^{2}$ | 500, 100, 50, 30 | [−100, 100] | 0 |

${f}_{7}\left(x\right)={{\displaystyle \sum}}_{i=0}^{n}i{x}_{i}{}^{4}+rand\left[0,1\right]$ | 500, 100, 50, 30 | [−1.28, 1.28] | 0 |

Function Description | Dim | Range | $\text{}{\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|

${f}_{8}\left(x\right)={{\displaystyle \sum}}_{i=1}^{n}\left(-{x}_{i}\mathit{sin}\left(\sqrt{\lfloor {x}_{i}\rfloor}\right)\right)$ | 500, 100, 50, 30 | [−100, 100] | −418.980 × Dim |

${f}_{9}\left(x\right)={{\displaystyle \sum}}_{i=1}^{n}\left[{x}_{i}{}^{2}-10\mathit{cos}\left(2\pi {x}_{i}\right)+10\right]$ | 500, 100, 50, 30 | [−10, 10] | 0 |

${f}_{10}\left(x\right)=-20{e}^{\left(-0.2\sqrt{\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}{}^{2}})\right)}-{e}^{\left(\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}\mathit{cos}\left(2\pi {x}_{i}\right)\right)}\phantom{\rule{0ex}{0ex}}\text{\hspace{1em}\hspace{1em}\hspace{1em}}+20+e$ | 500, 100, 50, 30 | [−100, 100] | 0 |

${f}_{11}\left(x\right)=1+\frac{1}{4000}{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}{}^{2}-{{\displaystyle \prod}}_{i=1}^{n}\mathit{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)$ | 500, 100, 50, 30 | [−100, 100] | 0 |

${f}_{12}\left(x\right)=\frac{\pi}{n}\left\{10\mathit{sin}\left(\pi {y}_{i}\right)\right\}+\phantom{\rule{0ex}{0ex}}\text{\hspace{1em}\hspace{1em}\hspace{1em}}{{\displaystyle \sum}}_{i=1}^{n-1}{\left({y}_{i}-1\right)}^{2}\left[\begin{array}{c}1+10{\mathit{sin}}^{2}\left(\pi {y}_{i}+1\right)\\ +\\ {{\displaystyle \sum}}_{i=1}^{n-1}u\left({x}_{i},10,100,4\right)\end{array}\right]$ here, ${y}_{i}=1+\frac{{x}_{i}+1}{4}$, $\text{}u\left({x}_{i},a,k,m\right)\left\{\begin{array}{c}K{\left({x}_{i}-a\right)}^{m}\text{}if\text{}{x}_{i}a\\ 0\text{\hspace{1em}\hspace{1em}\hspace{1em}}-a\le {x}_{i}\ge a\\ K{\left(-{x}_{i}-a\right)}^{m}if-a\le {x}_{i}\end{array}\right\}$ | 500, 100, 50, 30 | [−30, 30] | 0 |

${f}_{13}\left(x\right)=0.1\left({\mathit{sin}}^{2}\left(3\pi {x}_{i}\right)\phantom{\rule{0ex}{0ex}}\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{{\displaystyle \sum}}_{i=1}^{n}{({x}_{i}-1)}^{2}\left[1+{\mathit{sin}}^{2}\left(\begin{array}{c}3\pi {x}_{i}\\ +1\end{array}\right)\right]\phantom{\rule{0ex}{0ex}}\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\left({x}_{n}-1\right)}^{2}\left(1+{\mathit{sin}}^{2}\left(2\pi {x}_{n}\right)\right)\right)+{{\displaystyle \sum}}_{i=1}^{n}u({x}_{i},5,100,4)$ | 500, 100, 50, 30 | [−100, 100] | 0 |

Function Description | Dim | Range | $\text{}{\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|

${f}_{14}\left(x\right)={\left(\frac{1}{500}+{{\displaystyle \sum}}_{j=1}^{25}\left(\frac{1}{j+{{\displaystyle \sum}}_{i=1}^{2}\left({x}_{i}-{a}_{ij}\right)}\right)\right)}^{-1}$ | 2 | [−65, 65] | 0.998 |

${f}_{15}\left(x\right)={{\displaystyle \sum}}_{i=1}^{n}{\left[{a}_{i}-\frac{{x}_{1}\left({b}_{i}{}^{2}+{b}_{i}{x}_{2}\right)}{{b}_{i}{}^{2}+{b}_{i}{x}_{3}+{x}_{4}}\right]}^{2}$ | 4 | [−1, 1] | 0 |

${f}_{16}\left(x\right)=4{x}_{1}{}^{2}-2.1{x}_{1}{}^{4}+\frac{1}{3}{x}_{1}{}^{6}+{x}_{1}{x}_{2}-4{x}_{2}{}^{2}\phantom{\rule{0ex}{0ex}}\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+4{x}_{2}{}^{4}$ | 2 | [−5, 5] | −1.0316 |

${f}_{17}\left(x\right)={\left({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}{}^{6}+\frac{5}{\pi}{x}_{1}-6\right)}^{2}+10\left(1-\frac{1}{8\pi}\right)\mathit{cos}{x}_{1}+10$ | 2 | [−4, 4] | 0.398 |

${f}_{18}\left(x\right)=\left[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\left(19-14{x}_{1}+3{x}_{1}{}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}{}^{2}\right)\right]\left[30\phantom{\rule{0ex}{0ex}}\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\left(18-32{x}_{1}+12{x}_{1}{}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}{}^{2}\right)\right]\text{}$ | 2 | [−5, 5] | 3 |

${f}_{19}\left(x\right)=-{{\displaystyle \sum}}_{i=1}^{4}{c}_{i}{e}^{\left(-{{\displaystyle \sum}}_{i=1}^{3}{a}_{ij}{\left({x}_{j}-{p}_{ij}\right)}^{2}\right)}$ | 3 | [−5, 5] | −3.86 |

${f}_{20}(x)=-{{\displaystyle \sum}}_{i=1}^{4}{c}_{i}{e}^{\left(-{{\displaystyle \sum}}_{i=1}^{6}{a}_{ij}{\left({x}_{j}-{p}_{ij}\right)}^{2}\right)}$ | 6 | [−5, 5] | −1.170 |

${f}_{21}\left(x\right)=-{{\displaystyle \sum}}_{i=1}^{5}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}$ | 4 | [−5, 5] | −10.153 |

${f}_{22}\left(x\right)=-{{\displaystyle \sum}}_{i=1}^{7}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}$ | 4 | [−5, 5] | −10.4028 |

${f}_{23}\left(x\right)=-{{\displaystyle \sum}}_{i=1}^{10}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}$ | 4 | [−1, 1] | −10.536 |

Function | Description | $\text{}{\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | Range | Dim |
---|---|---|---|---|

CEC-1 | Storn’s Chebyshev polynomial fitting problem | 1 | [−8192, 8192] | 9 |

CEC-2 | Inverse Hilbert matrix problem | 1 | [−16,384, 16,384] | 16 |

CEC-3 | Lennard–Jones minimum energy cluster | 1 | [−4, 4] | 18 |

CEC-4 | Rastrigin function | 1 | [−100, 100] | 10 |

CEC-5 | Grienwank function | 1 | [−100, 100] | 10 |

CEC-6 | Weierstrass function | 1 | [−100, 100] | 10 |

CEC-7 | Modified Schwefel function | 1 | [−100, 100] | 10 |

CEC-8 | Expanded Schaffer function | 1 | [−100, 100] | 10 |

CEC-9 | Happy CAT function | 1 | [−100, 100] | 10 |

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Fun | Measure | IRSA | RSA | BMO | PSO | GWO | AOA | DGCO |
---|---|---|---|---|---|---|---|---|

F1 | Best | 0 | 0 | 6.030e-129 | 32.4203 | 4.7922e-35 | 6.706e-284 | 1.238e-21 |

Worst | 0 | 0 | 1.961e-121 | 63.6854 | 8.0427e-33 | 1.2838e-114 | 2.1218e-19 | |

Average | 0 | 0 | 2.712e-122 | 43.2608 | 1.4545e-33 | 1.2838e-115 | 4.7446e-20 | |

STD | 0 | 0 | 6.153e-122 | 9.48589 | 2.4625e-33 | 4.0598e-115 | 7.9075e-20 | |

F2 | Best | 0 | 0 | 1.393e-67 | 2.35817 | 1.7406e-20 | 0 | 3.7704e-14 |

Worst | 0 | 0 | 7.383e-65 | 5.32663 | 9.4593e-20 | 0 | 4.68526e-13 | |

Average | 0 | 0 | 1.841e-65 | 3.31147 | 4.5301e-20 | 0 | 1.10333e-13 | |

STD | 0 | 0 | 2.663e-65 | 0.89717 | 2.2708e-20 | 0 | 1.29549e-13 | |

F3 | Best | 0 | 0 | 4.0313e-92 | 5.5826e+03 | 4.5340e-04 | 0.0286 | 0.0025 |

Worst | 0 | 0 | 2.9026e-76 | 9.1971e+03 | 0.0201 | 0.5838 | 25.4054 | |

Average | 0 | 0 | 1.1183e-76 | 7.8891e+03 | 0.0074 | 0.1907 | 5.7823 | |

STD | 0 | 0 | 1.5330e-76 | 1.3842e+03 | 0.0074 | 0.2277 | 10.9998 | |

F4 | Best | 0 | 0 | 7.3742e-58 | 5.43665 | 7.5397e-09 | 4.6289e-88 | 4.97610e-06 |

Worst | 0 | 0 | 2.5295e-52 | 14.1279 | 4.4114e-08 | 0.05287 | 0.000247 | |

Average | 0 | 0 | 4.1512e-53 | 8.305702 | 1.8813e-08 | 0.024560 | 5.9520e-05 | |

STD | 0 | 0 | 8.765e-53 | 2.59027 | 1.274e-08 | 0.02222 | 7.3133e-05 | |

F5 | Best | 45.4705 | 48.9824 | 48.9417 | 5.7675e+03 | 46.1950 | 48.6070 | 46.1974 |

Worst | 47.7889 | 48.9903 | 48.9984 | 1.0132e+04 | 47.8430 | 48.9439 | 47.8660 | |

Average | 46.4656 | 48.9883 | 48.9721 | 8.1573e+03 | 47.0300 | 48.7869 | 47.0569 | |

STD | 0.6506 | 0.0032 | 0.0173 | 2.0349e+03 | 0.8086 | 0.1402 | 0.5917 | |

F6 | Best | 1.58 | 11.97 | 11.39 | 150.00 | 2.25 | 6.43 | 3.48 |

Worst | 2.67 | 12.25 | 12.44 | 284.60 | 2.89 | 7.44 | 5.78 | |

Average | 2.37 | 12.22 | 11.935 | 214.76 | 2.5 | 6.90 | 4.67 | |

STD | 0.25 | 0.0878 | 0.4067 | 47.02 | 0.27 | 0.34 | 0.66 | |

F7 | Best | 7.6084e-06 | 2.6639e-05 | 1.5133e-04 | 0.1347 | 4.8839e-04 | 1.6821e-07 | 0.0014 |

Worst | 2.8963e-05 | 6.7811e-05 | 3.8440e-04 | 0.2985 | 0.0037 | 5.2418e-05 | 0.0154 | |

Average | 8.7740e-05 | 9.0710e-05 | 4.1659e-04 | 0.2155 | 0.0022 | 2.3443e-05 | 0.0071 | |

STD | 9.3116e-05 | 5.2933e-05 | 2.3023e-04 | 0.0592 | 0.0012 | 2.0923e-05 | 0.0055 | |

F8 | Best | −1.0646e+4 | −8.845e+3 | −5.358e+3 | −1.0646e+4 | −9.14e+3 | −8.0208e+03 | −1.0548e+04 |

Worst | −6.8637e+3 | −4.57e+3 | −2.928e+03 | −6.8637e+3 | −6.13e+3 | −6.8298e+03 | −7.3571e+03 | |

Average | −8.7577e+3 | −6.04e+3 | −3.867e+03 | −8.7577e+3 | −7.32e+3 | −7.2473e+03 | −8.5385e+03 | |

STD | 1.0697e+3 | 1.2009e+3 | 863.4437 | 1.0697e+3 | 902.38 | 427.9226 | 849.1905 | |

F9 | Best | 0 | 0 | 0 | 114.0457 | 5.6843e-13 | 0 | 4.4338e-12 |

Worst | 0 | 0 | 0 | 140.5094 | 5.42412 | 0 | 1.4080e-09 | |

Average | 0 | 0 | 0 | 131.6173 | 3.0390 | 0 | 4.7419e-10 | |

STD | 0 | 0 | 0 | 15.2178 | 2.7705 | 0 | 8.0872e-10 | |

F10 | Best | 8.8817e-16 | 8.8817e-16 | 8.8817e-16 | 2.98933 | 3.9968e-14 | 8.8817e-16 | 2.0863e-12 |

Worst | 8.8817e-16 | 8.8817e-16 | 8.8817e-16 | 4.25557 | 5.0626e-14 | 8.8817e-16 | 20.2605 | |

Average | 8.8817e-16 | 8.8817e-16 | 8.881e-16 | 3.62755 | 4.3520e-14 | 8.8817e-16 | 4.04392 | |

STD | 0 | 0 | 0 | 0.38752 | 3.3495e-15 | 0 | 8.52536 | |

F11 | Best | 0 | 0 | 0 | 1.18496 | 0 | 0.009543 | 0 |

Worst | 0 | 0 | 0 | 1.75459 | 0.014698 | 0.428251 | 0.019779 | |

Average | 0 | 0 | 0 | 1.39448 | 0.00368 | 0.159273 | 0.001977 | |

STD | 0 | 0 | 0 | 0.177805 | 0.006091 | 0.138820 | 0.006254 | |

F12 | Best | 0.0868 | 1.1349 | 0.7756 | 0.1567 | 0.0373 | 1.0089 | 0.1162 |

Worst | 0.1513 | 1.4160 | 1.1959 | 0.2635 | 0.0583 | 1.0286 | 0.2057 | |

Average | 0.1534 | 1.3680 | 1.1688 | 0.2182 | 0.0459 | 1.0395 | 0.2604 | |

STD | 0.0373 | 0.1246 | 0.2121 | 0.0552 | 0.0110 | 0.0316 | 0.1348 | |

F13 | Best | 4.5760e-08 | 1.8359 | 0.1800 | 0.1624 | 0.8989 | 4.7901 | 2.8443 |

Worst | 2.9444 | 3.8301 | 0.2521 | 0.3925 | 1.6895 | 4.9737 | 3.4369 | |

Average | 1.9711 | 3.0760 | 0.2071 | 0.2657 | 1.1783 | 4.9183 | 3.0976 | |

STD | 1.6991 | 0.7882 | 0.0293 | 0.0835 | 0.3008 | 0.0811 | 0.2333 |

Fun | Measure | IRSA | RSA | BMO | PSO | GWO | AOA | DGCO |
---|---|---|---|---|---|---|---|---|

F14 | Best | 0.9980 | 2.9821 | 2.0481 | 1.913 | 0.9980 | 7.8740 | 0.9980 |

Worst | 2.9834 | 3.0006 | 11.7187 | 2.981 | 10.7632 | 12.6705 | 2.9821 | |

Average | 2.1313 | 2.9901 | 6.4665 | 2.523 | 5.1025 | 10.7579 | 2.1964 | |

STD | 0.8325 | 0.0139 | 3.0067 | 0.7341 | 4.9092 | 1.7957 | 0.6274 | |

F15 | Best | 0.00030 | 0.00068 | 0.00033 | 0.00038 | 0.00030 | 0.00036 | 0.00035 |

Worst | 0.01550 | 0.00896 | 0.00654 | 0.00142 | 0.02036 | 0.05758 | 0.001224 | |

Average | 0.00406 | 0.00180 | 0.001710 | 0.000813 | 0.004417 | 0.009030 | 0.000709 | |

STD | 0.00548 | 0.00253 | 0.00223 | 0.00043 | 0.008408 | 0.018280 | 0.000237 | |

F16 | Best | −1.03162 | −1.03159 | −1.03162 | −1.03162 | −1.03162 | −1.03162 | −1.03162 |

Worst | −1.03162 | −0.99552 | −0.91896 | −1.03162 | −1.03162 | −1.03162 | −1.03162 | |

Average | −1.03162 | −1.02159 | −1.01681 | −1.03162 | −1.03162 | −1.03162 | −1.03162 | |

STD | 2.8338e-08 | 0.01150 | 0.03493 | 5.4218e-08 | 1.6501e-08 | 4.1202e-09 | 3.1308e-09 | |

F17 | Best | 0.39788 | 0.40742 | 0.39788 | 0.39788 | 0.39788 | 0.39788 | 0.39788 |

Worst | 0.397897 | 1.27099 | 0.42696 | 0.39788 | 0.39788 | 0.39788 | 0.39788 | |

Average | 0.397891 | 0.64371 | 0.40330 | 0.39788 | 0.39788 | 0.39788 | 0.39788 | |

STD | 3.8146e-06 | 0.25782 | 0.01085 | 7.2909e-07 | 4.2567e-07 | 4.3534e-08 | 3.6445e-07 | |

F18 | Best | 3.00000 | 3.00000 | 2.99999 | 3.00000 | 3.00000 | 3.00000 | 3.000000 |

Worst | 3.000001 | 3.39511 | 42.1054 | 3.00011 | 3.00002 | 95.1885 | 3.000000 | |

Average | 3.000000 | 3.04555 | 10.1188 | 3.00002 | 3.00001 | 23.1248 | 3.00000 | |

STD | 5.1797e-07 | 0.12303 | 13.0276 | 3.477e-05 | 7.920e-06 | 28.7189 | 6.458e-10 | |

F19 | Best | −3.86277 | −3.85920 | −3.85436 | −3.86276 | −3.86277 | −3.86277 | −3.86278 |

Worst | −3.85374 | −3.57053 | −1.28898 | −3.86201 | −3.85315 | −1.00081 | −3.86277 | |

Average | −3.86101 | −3.75028 | −3.44530 | −3.86263 | −3.86034 | −1.85933 | −3.86277 | |

STD | 0.00367 | 0.09664 | 0.81662 | 0.00024 | 0.00403 | 1.38235 | 3.6601e-06 | |

F20 | Best | −3.32086 | −2.89690 | −2.45549 | −3.32121 | −3.20304 | −3.32194 | −3.32189 |

Worst | −3.20101 | −0.17365 | −0.54213 | −3.19851 | −3.08259 | −3.15098 | −3.13200 | |

Average | −3.27287 | −1.54187 | −1.24769 | −3.22497 | −3.17957 | −3.27776 | −3.21025 | |

STD | 0.06116 | 1.03850 | 0.75791 | 0.05061 | 0.04241 | 0.071964 | 0.06482 | |

F21 | Best | −10.1206 | −5.0552 | −4.9619 | −10.2512 | −10.0712 | −10.1531 | −10.1494 |

Worst | −5.0552 | −5.0552 | −1.8717 | −5.0552 | −5.0552 | −5.0552 | −5.0549 | |

Average | −9.0840 | −5.0552 | −3.8482 | −6.0943 | −6.0745 | −6.0747 | −7.0926 | |

STD | 2.2573 | 3.8458e-07 | 1.3134 | 2.7922 | 2.2793 | 2.2798 | 2.7900 | |

F22 | Best | −10.3586 | −5.08767 | −5.05414 | −10.4029 | −10.4024 | −10.4029 | −10.3960 |

Worst | −3.72361 | −5.08766 | −2.32919 | −3.72365 | −5.08765 | −3.72429 | −3.72380 | |

Average | −8.44500 | −5.08766 | −3.83632 | −5.87780 | −9.33904 | −7.60893 | −6.66468 | |

STD | 2.83980 | 1.0352e-06 | 0.98429 | 2.449069 | 2.24067 | 2.97300 | 3.25225 | |

F23 | Best | −10.52251 | −5.128480 | −4.99447 | −10.53637 | −10.5361 | −10.5363 | −10.5309 |

Worst | −3.83461 | −5.12847 | −2.49710 | −3.83472 | −5.12845 | −2.29603 | −3.83496 | |

Average | −8.48605 | −5.12847 | −3.77437 | −6.62130 | −8.91314 | −7.34602 | −6.50874 | |

STD | 2.94732 | 2.3686e-06 | 0.76135 | 2.73097 | 2.61167 | 3.32172 | 3.45155 |

**Table 3.**Results for unimodal and multimodal functions for 30, 100, and 500 dimensions, 300 iterations.

Fun | Dim | IRSA | RSA | BMO | PSO | GWO | AOA | DGCO |
---|---|---|---|---|---|---|---|---|

F1 | 30 | 0 | 0 | 2.9729e-75 | 100.429274 | 5.806e-20 | 1.74296e-76 | 5.93134e-10 |

100 | 0 | 0 | 7.502e-46 | 2951.7571 | 0.0012668 | 0.023492033 | 47.4212010 | |

500 | 0 | 0 | 5.948e-41 | 91703.5649 | 109.12381 | 0.5862142 | 113.8580 | |

F2 | 30 | 0 | 0 | 1.0065e-39 | 4.59976602 | 7.467e-11 | 0 | 2.10414e-07 |

100 | 0 | 0 | 4.6351e-25 | 53.7327530 | 0.0134880 | 4.19883e-60 | 0.268447521 | |

500 | 0 | 0 | 3.3724e-23 | 487.76760 | 10.64367 | 0.00161 | 2.519646 | |

F3 | 30 | 0 | 0 | 1.1399e-52 | 1321.79087 | 0.0001101 | 6.9638e-104 | 7.97466788 |

100 | 0 | 0 | 3.164e-28 | 50712.1407 | 14820.47 | 1.531983171 | 80032.5400 | |

500 | 0 | 0 | 6.6866e-19 | 1303817.56 | 513880.05 | 36.035904 | 104.83 | |

F4 | 30 | 0 | 0 | 5.6171e-32 | 8.65232085 | 4.478e-05 | 3.51206e-13 | 0.04110228 |

100 | 0 | 0 | 3.1527e-20 | 31.480699 | 5.7402399 | 0.100756622 | 82.157639 | |

500 | 0 | 0 | 5.7108e-18 | 43.134284 | 65.76152 | 0.17991607 | 0.09866121 | |

F5 | 30 | 25.72424 | 28.99171 | 28.995762 | 1300.12057 | 27.136822 | 28.59270148 | 26.21687570 |

100 | 96.8983 | 98.9901048 | 98.9698060 | 226496.089 | 98.343648 | 98.90012672 | 97.5081886 | |

500 | 498.3830 | 499.6902 | 499.0781 | 27910234.5 | 9445.0271 | 499.1341156 | 4392.7 | |

F6 | 30 | 0.576633 | 7.0346180 | 6.97472880 | 111.668505 | 0.7535595 | 3.03144018 | 1.25053679 |

100 | 11.05309 | 24.7513018 | 24.5099804 | 2069.85282 | 9.4928888 | 18.4072873 | 13.8718994 | |

500 | 111.5654 | 124.751626 | 123.355451 | 64419.3854 | 268.90391 | 117.2773220 | 145.94373 | |

F7 | 30 | 9.50e-05 | 1.9397e-05 | 0.00050276 | 0.05970424 | 0.0019118 | 3.78351e-05 | 0.00390436 |

100 | 3.97e-05 | 2.2064e-05 | 0.0004319 | 1.2096514 | 0.0097439 | 5.96465e-06 | 0.0300833 | |

500 | 8.21e-05 | 0.00046135 | 0.00080821 | 235.665342 | 0.7073979 | 0.000119979 | 0.292147 | |

F8 | 30 | −5214.04 | −3431.1934 | −2834.3518 | −6549.69516 | −6277.543 | −4918.30201 | −5573.32119 |

100 | −64809.8 | −36018.762 | −18000.185 | −56587.461 | −16611.13 | −47550.1316 | −34933.7702 | |

500 | −80242.2 | −44205.61 | −12898.155 | −36269.793 | −58117.96 | −22872.6112 | −24663.3214 | |

F9 | 30 | 0 | 0 | 0 | 103.812346 | 1.9054417 | 0 | 5.86970e-09 |

100 | 0 | 0 | 0 | 587.205492 | 58.233470 | 0 | 20.62413137 | |

500 | 0 | 0 | 0 | 4309.97279 | 902.54104 | 0 | 466.7469966 | |

F10 | 30 | 8.881e-16 | 8.8817e-16 | 8.8817e-16 | 5.2518904 | 1.724e-10 | 8.88178e-16 | 1.63626e-06 |

100 | 8.881e-16 | 8.881e-16 | 8.8817e-16 | 7.8262077 | 5.792e-05 | 8.88178e-16 | 20.523618 | |

500 | 8.881e-16 | 8.881e-16 | 4.440e-15 | 12.927665 | 2.140639 | 0.0083857 | 20.881255 | |

F11 | 30 | 0 | 0 | 0 | 1.8485067 | 0.0298839 | 0.23855046 | 1.26669e-10 |

100 | 0 | 0 | 0 | 28.3884325 | 0.0349668 | 897.847451 | 0.04874511 | |

500 | 0 | 0 | 0 | 867.60304 | 3.1100432 | 13134.6129 | 162.849911 | |

F12 | 30 | 0.048112 | 1.59700365 | 0.93494039 | 0.01698208 | 0.0712204 | 0.81438277 | 0.045133459 |

100 | 0.374642 | 1.28994163 | 1.19529976 | 0.64770498 | 0.1179354 | 1.20962184 | 0.389415151 | |

500 | 0.9743 | 1.2075 | 1.1980 | 2.0190 | 0.5999 | 1.1800 | 1.2411 | |

F13 | 30 | 1.280496 | 1.970635 | 2.6705939 | 0.15916980 | 0.1018868 | 2.9785990 | 1.600784929 |

100 | 7.876407 | 9.878585 | 9.993848 | 4.57433449 | 5.5499672 | 9.963623491 | 8.44409492 | |

500 | 49.99682 | 49.99734 | 49.99725 | 82.318 | 46.351 | 49.998 | 54.177 |

**Table 4.**$\tilde{T}\text{}\left(sec\right)\text{}$ calculation for various metaheuristic algorithms.

Fun | T IRSA | T RSA | T FDO | T BMO | T AOA | T PSO | T GWO | T DGCO | T FPA | T DFA |
---|---|---|---|---|---|---|---|---|---|---|

CEC01 | 440.04 | 498.64 | 9569.9 | 512.80 | 468.96 | 472.38 | 472.49 | 472.884 | 702.73 | 901.60 |

CEC02 | 51.562 | 158.97 | 303.77 | 26.603 | 9.7899 | 10.097 | 11.746 | 14.144 | 420.60 | 858.17 |

CEC03 | 66.843 | 192.89 | 491.69 | 35.672 | 17.950 | 18.158 | 19.011 | 22.222 | 481.66 | 1018.4 |

CEC04 | 33.761 | 99.078 | 514.31 | 26.605 | 9.1060 | 10.075 | 11.035 | 12.389 | 266.56 | 627.08 |

CEC05 | 34.373 | 99.519 | 744.31 | 27.667 | 9.4832 | 10.545 | 11.180 | 12.966 | 267.42 | 491.911 |

CEC06 | 175.14 | 240.27 | 5176.0 | 189.23 | 165.30 | 169.64 | 168.52 | 170.18 | 423.23 | 641.59 |

CEC07 | 34.831 | 99.440 | 384.83 | 24.768 | 9.7759 | 10.604 | 11.566 | 12.993 | 266.66 | 483.13 |

CEC08 | 34.733 | 99.406 | 369.56 | 25.428 | 9.7797 | 10.683 | 11.522 | 12.848 | 267.13 | 481.31 |

CEC09 | 33.670 | 98.69 | 383.27 | 23.551 | 8.3110 | 9.1362 | 10.298 | 11.728 | 266.52 | 492.78 |

Fun | T_{1} | $\tilde{\mathit{T}}$ IRSA | $\tilde{\mathit{T}}$ RSA | $\tilde{\mathit{T}}$ FDO | $\tilde{\mathit{T}}$ BMO | $\tilde{\mathit{T}}$ AOA | $\tilde{\mathit{T}}$ PSO | $\tilde{\mathit{T}}$ GWO | $\tilde{\mathit{T}}$ DGCO | $\tilde{\mathit{T}}$ FPA | $\tilde{\mathit{T}}$ DFA |
---|---|---|---|---|---|---|---|---|---|---|---|

CEC01 | 0.0105 | 43.017 | 48.736 | 934.10 | 50.118 | 45.839 | 46.173 | 46.184 | 46.222 | 68.655 | 88.065 |

CEC02 | 0.0042 | 5.101 | 15.584 | 29.717 | 2.665 | 1.024 | 1.054 | 1.215 | 1.449 | 41.120 | 83.826 |

CEC03 | 0.0044 | 6.5924 | 18.895 | 48.058 | 3.5501 | 1.8205 | 1.84077 | 1.9240 | 2.2374 | 47.079 | 99.466 |

CEC04 | 0.0031 | 3.3636 | 9.7386 | 50.266 | 2.6652 | 0.9572 | 1.05189 | 1.1456 | 1.2777 | 26.085 | 61.272 |

CEC05 | 0.0035 | 3.4234 | 9.7816 | 72.714 | 2.7688 | 0.9940 | 1.09774 | 1.1597 | 1.3340 | 26.169 | 48.079 |

CEC06 | 0.0068 | 17.163 | 23.519 | 505.25 | 18.538 | 16.202 | 16.6257 | 16.517 | 16.679 | 41.376 | 62.688 |

CEC07 | 0.0044 | 3.4681 | 9.7739 | 37.628 | 2.4859 | 1.0226 | 1.10349 | 1.1974 | 1.3367 | 26.095 | 47.222 |

CEC08 | 0.0045 | 3.4585 | 9.7706 | 36.138 | 2.5503 | 1.0230 | 1.1112 | 1.1931 | 1.3225 | 26.141 | 47.044 |

CEC09 | 0.0036 | 3.3547 | 9.7015 | 37.476 | 2.3671 | 0.8796 | 0.9602 | 1.0736 | 1.2132 | 26.081 | 48.164 |

Data Set | # Classes | # Features | No. of training samples | No. of testing samples |
---|---|---|---|---|

Iris | 3 | 4 | 100 | 50 |

Heart | 2 | 13 | 203 | 100 |

Stress-Lysis | 3 | 3 | 1340 | 661 |

Banknote-authentication | 2 | 4 | 919 | 453 |

Blood-transfusion | 2 | 4 | 501 | 247 |

Cryotherapy | 2 | 6 | 60 | 30 |

Diabetes | 2 | 8 | 514 | 254 |

Dataset | Training Accuracy (%) | ||||
---|---|---|---|---|---|

PSONN | BMOANN | AOANN | RSANN | IRSANN | |

Iris | 97 | 94 | 93 | 88.02 | 97 |

Heart | 72.60726 | 66.9967 | 59.73597 | 42.24422 | 88 |

Stress-Lysis | 91.75412 | 95.8021 | 70.16492 | 59.37031 | 99.42171 |

Banknote-authentication | 94.42623 | 97.9235 | 85.68306 | 84.48087 | 99.45355 |

Blood-transfusion | 79.95992 | 78.15631 | 77.35471 | 75.1503 | 77.55511 |

Cryotherapy | 90 | 92 | 71.66667 | 81.66667 | 96.66667 |

Diabetes | 74.80469 | 75 | 72.07031 | 66.79688 | 78.10156 |

Haberman | 75 | 75 | 78.43137 | 79.41176 | 78.43137 |

Dataset | Testing Accuracy (%) | ||||
---|---|---|---|---|---|

PSONN | BMOANN | AOANN | RSANN | IRSANN | |

Iris | 94 | 98.66 | 92 | 91 | 98.23 |

Heart | 70.9571 | 75.90759 | 58.08581 | 46.53465 | 85 |

Stress-Lysis | 91.90405 | 97.0015 | 70.01499 | 57.12144 | 99.7121 |

Banknote-authentication | 93.43545 | 96.49891 | 83.3698 | 85.33917 | 99.56236 |

Blood-transfusion | 77.51004 | 77.91165 | 74.6988 | 79.11647 | 80.72289 |

Cryotherapy | 93.33333 | 86.66667 | 80 | 76.66667 | 93.33333 |

Diabetes | 67.57813 | 75.78125 | 76.1718867 | 69.92188 | 77.26563 |

Haberman | 67.47059 | 76.47059 | 76.64706 | 70.09804 | 80.54902 |

Dataset | Cost Function Comparison | |||||
---|---|---|---|---|---|---|

PSONN | BMOANN | AOANN | RSANN | IRSANN | ||

Iris | Best | 0.078788 | 0.082034 | 0.342085 | 0.143403 | 0.072572 |

Avg | 0.137949 | 0.087161 | 0.742686 | 0.327073 | 0.115797 | |

Std | 0.083667 | 0.00569 | 0.484778 | 0.159949 | 0.108159 | |

Heart | Best | 0.56952 | 0.545925 | 0.995477 | 0.808033 | 0.472722 |

Avg | 0.640322 | 0.581627 | 1.372587 | 0.853887 | 0.517036 | |

Std | 0.062089 | 0.030938 | 0.53151 | 0.076074 | 0.038613 | |

Stress-Lysis | Best | 0.156856 | 0.136333 | 0.321634 | 0.248556 | 0.0911631 |

Avg | 0.159272 | 0.142592 | 0.353229 | 0.627416 | 0.268016 | |

Std | 0.003418 | 0.007051 | 0.027562 | 0.328421 | 0.055127 | |

Banknote-authentication | Best | 0.318751 | 0.154256 | 0.556654 | 0.503623 | 0.085457 |

Avg | 0.331262 | 0.170783 | 0.764328 | 0.836328 | 0.145555 | |

Std | 0.017693 | 0.016899 | 0.207149 | 0.288424 | 0.054529 | |

Blood-transfusion | Best | 0.823567 | 0.882403 | 0.960614 | 0.958203 | 0.887161 |

Avg | 0.830233 | 0.888582 | 1.260426 | 0.976322 | 0.895859 | |

Std | 0.009426 | 0.008437 | 0.259695 | 0.015692 | 0.009459 | |

Cryotherapy | Best | 0.357101 | 0.756437 | 0.234832 | 0.542086 | 0.28827 |

Avg | 0.362329 | 0.888982 | 0.292911 | 0.764291 | 0.305118 | |

Std | 0.007394 | 0.129127 | 0.050999 | 0.192532 | 0.127465 | |

Diabetes | Best | 1.048615 | 0.725924 | 0.730482 | 0.947266 | 0.646058 |

Avg | 1.076385 | 0.743759 | 0.7562 | 0.982109 | 0.724891 | |

Std | 0.042824 | 0.019733 | 0.036372 | 0.040968 | 0.07128 | |

Haberman | Best | 0.892895 | 0.906456 | 0.977251 | 0.994621 | 0.881229 |

Avg | 0.936445 | 0.931774 | 1.269601 | 0.998591 | 0.925252 | |

Std | 0.061589 | 0.00719 | 0.14037 | 0.003469 | 0.038883 |

Data Set | Technique | Training | Testing | ||||
---|---|---|---|---|---|---|---|

Precision | Recall | F1 Score | Precision | Recall | F1 Score | ||

Iris | IRSA | 0.979142 | 0.979923 | 0.979533 | 0.962222 | 0.967178 | 0.969693 |

RSA | 0.891866 | 0.875316 | 0.883513 | 0.832602 | 0.796296 | 0.814045 | |

BMO | 0.944356 | 0.943915 | 0.944136 | 0.977778 | 0.977778 | 0.977778 | |

AOA | 0.680918 | 0.647619 | 0.663851 | 0.695926 | 0.655556 | 0.675138 | |

PSO | 0.972973 | 0.969697 | 0.971332 | 0.964912 | 0.958333 | 0.961612 | |

Heart | IRSA | 0.888428 | 0.884587 | 0.88503 | 0.848726 | 0.851251 | 0.854971 |

RSA | 0.462116 | 0.462022 | 0.462069 | 0.60397 | 0.603725 | 0.603848 | |

BMO | 0.73858 | 0.636917 | 0.683992 | 0.740629 | 0.660205 | 0.698108 | |

AOA | 0.480976 | 0.481498 | 0.481237 | 0.482877 | 0.486242 | 0.484554 | |

PSO | 0.657688 | 0.651564 | 0.654612 | 0.602002 | 0.577957 | 0.589735 | |

Stress-Lysis | IRSA | 0.862953 | 0.869583 | 0.866255 | 0.831595 | 0.842565 | 0.837044 |

RSA | 0.551258 | 0.515412 | 0.514234 | 0.55435 | 0.508844 | 0.61734 | |

BMO | 0.959656 | 0.951436 | 0.955528 | 0.975669 | 0.96373 | 0.969663 | |

AOA | 0.768447 | 0.674813 | 0.718593 | 0.784879 | 0.673437 | 0.7249 | |

PSO | 0.879358 | 0.895391 | 0.887302 | 0.889297 | 0.905763 | 0.897455 | |

Banknote-authentication | IRSA | 0.995792 | 0.994563 | 0.995225 | 0.995902 | 0.995349 | 0.995625 |

RSA | 0.844413 | 0.838669 | 0.841531 | 0.85288 | 0.852147 | 0.852513 | |

BMO | 0.97761 | 0.98115 | 0.979377 | 0.963706 | 0.967257 | 0.965478 | |

AOA | 0.57296 | 0.57642 | 0.694746 | 0.809533 | 0.812795 | 0.811161 | |

PSO | 0.942964 | 0.944103 | 0.943533 | 0.933375 | 0.934613 | 0.933994 | |

Blood-transfusion | IRSA | 0.729753 | 0.595356 | 0.655739 | 0.691121 | 0.570559 | 0.62508 |

RSA | 0.627282 | 0.508097 | 0.561434 | 0.730453 | 0.531909 | 0.615567 | |

BMO | 0.683964 | 0.579004 | 0.627122 | 0.652637 | 0.552494 | 0.598405 | |

AOA | 0.761869 | 0.511741 | 0.612243 | 0.624494 | 0.505248 | 0.558578 | |

PSO | 0.739583 | 0.621758 | 0.675572 | 0.686359 | 0.564815 | 0.619683 | |

Cryotherapy | IRSA | 0.950774 | 0.943611 | 0.957148 | 0.95 | 0.916667 | 0.933036 |

RSA | 0.818449 | 0.806561 | 0.812462 | 0.766667 | 0.767857 | 0.767261 | |

BMO | 0.915882 | 0.928276 | 0.912064 | 0.819444 | 0.830144 | 0.824759 | |

AOA | 0.699177 | 0.690236 | 0.694678 | 0.638889 | 0.633333 | 0.636099 | |

PSO | 0.897321 | 0.902715 | 0.90001 | 0.944444 | 0.928571 | 0.936441 | |

Diabetes | IRSA | 0.772165 | 0.7483 | 0.760045 | 0.683296 | 0.676977 | 0.690122 |

RSA | 0.626005 | 0.601743 | 0.613634 | 0.66266 | 0.610454 | 0.635487 | |

BMO | 0.724919 | 0.726392 | 0.725655 | 0.699808 | 0.722159 | 0.710808 | |

AOA | 0.651455 | 0.614956 | 0.632679 | 0.670297 | 0.643082 | 0.656407 | |

PSO | 0.704908 | 0.676813 | 0.734665 | 0.765362 | 0.699496 | 0.730948 | |

Haberman | IRSA | 0.714555 | 0.561729 | 0.638992 | 0.677083 | 0.54118 | 0.623362 |

RSA | 0.519912 | 0.501389 | 0.510477 | 0.650000 | 0.517637 | 0.576316 | |

BMO | 0.700376 | 0.56504 | 0.625471 | 0.632979 | 0.550607 | 0.588927 | |

AOA | 0.655914 | 0.574148 | 0.612313 | 0.662338 | 0.58316 | 0.620232 | |

PSO | 0.674919 | 0.605186 | 0.638153 | 0.662837 | 0.58249 | 0.620072 |

Data Set | Technique | Training | Testing | Cost | ||||
---|---|---|---|---|---|---|---|---|

RE | RMSE | R^{2} | RE | RMSE | R^{2} | |||

Wind power prediction(winter) | PSO | 0.0157 | 11.8303 | 0.9835 | 0.0886 | 55.6747 | 0.9081 | 0.0121 |

RSA | 0.0170 | 16.6071 | 0.9657 | 0.1065 | 61.2748 | 0.8890 | 0.0326 | |

BMO | 0.0959 | 32.5220 | 0.8278 | 0.2318 | 107.9345 | 0.5259 | 0.0444 | |

AOA | 0.0356 | 26.9534 | 0.8961 | 0.1135 | 59.1465 | 0.8749 | 0.0373 | |

IRSA | 0.0049 | 8.3396 | 0.9918 | 0.0642 | 32.7880 | 0.9665 | 0.0105 | |

Wind power prediction(summer) | PSO | 0.0049 | 2.8918 | 0.9907 | 0.0772 | 9.5937 | 0.9780 | 0.0193 |

RSA | 0.0055 | 4.4855 | 0.9778 | 0.6042 | 38.7233 | 0.6423 | 0.0251 | |

BMO | 0.0808 | 5.8582 | 0.9460 | 0.2983 | 22.3706 | 0.8291 | 0.0428 | |

AOA | 0.0339 | 4.0871 | 0.9779 | 0.4688 | 30.6282 | 0.7565 | 0.0208 | |

IRSA | 0.0015 | 2.8416 | 0.9912 | 0.0632 | 9.3401 | 0.9801 | 0.0184 | |

PV power prediction | PSO | 0.0235 | 94.9000 | 0.9649 | 0.0762 | 298.7930 | 0.9420 | 0.0535 |

RSA | 0.0144 | 119.6847 | 0.9626 | 0.0440 | 223.8701 | 0.9656 | 0.0410 | |

BMO | 0.0359 | 148.5061 | 0.9150 | 0.1148 | 232.2480 | 0.9545 | 0.0827 | |

AOA | 0.0429 | 183.5005 | 0.8987 | 0.1111 | 383.4890 | 0.8913 | 0.0963 | |

IRSA | 0.0146 | 90 | 0.9761 | 0.1285 | 260.8531 | 0.9611 | 0.0234 |

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## Share and Cite

**MDPI and ACS Style**

Khan, M.K.; Zafar, M.H.; Rashid, S.; Mansoor, M.; Moosavi, S.K.R.; Sanfilippo, F.
Improved Reptile Search Optimization Algorithm: Application on Regression and Classification Problems. *Appl. Sci.* **2023**, *13*, 945.
https://doi.org/10.3390/app13020945

**AMA Style**

Khan MK, Zafar MH, Rashid S, Mansoor M, Moosavi SKR, Sanfilippo F.
Improved Reptile Search Optimization Algorithm: Application on Regression and Classification Problems. *Applied Sciences*. 2023; 13(2):945.
https://doi.org/10.3390/app13020945

**Chicago/Turabian Style**

Khan, Muhammad Kamran, Muhammad Hamza Zafar, Saad Rashid, Majad Mansoor, Syed Kumayl Raza Moosavi, and Filippo Sanfilippo.
2023. "Improved Reptile Search Optimization Algorithm: Application on Regression and Classification Problems" *Applied Sciences* 13, no. 2: 945.
https://doi.org/10.3390/app13020945