# Tuning Machine Learning Models Using a Group Search Firefly Algorithm for Credit Card Fraud Detection

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## Abstract

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## 1. Introduction

- The development of the novel improved version of the well-known FA metaheuristic that addresses the known drawbacks of the original implementation.
- The application of the devised algorithm for tuning three machine learning classifiers for the particular task of fraud detection, with a goal to enhance the classifiers’ accuracy, as well as other performance metrics.
- The comprehensive comparative analysis of different swarm intelligence metaheuristics for ML tuning against practical credit card fraud challenge.

## 2. Literature Review and Background

#### 2.1. Support Vector Machine

#### 2.2. Extreme Learning Machine

#### 2.3. The XGBoost Algorithm

#### 2.4. Swarm Intelligence

#### 2.5. Machine Learning Model Tuning by Swarm Intelligence Metaheuristics

#### 2.6. Credit Card Fraud Detection Overview

## 3. Proposed Method

#### 3.1. Original Firefly Algorithm

#### 3.2. Motivation and Proposed Improved Group Search Firefly Algorithm

Algorithm 1 The GSFA pseudo-code. |

Define global parameters N and T Generate the initial population of solutions ${x}_{i}$, ($i=1,2,3,\dots N$) Define basic FA control parameters Define specific GSFA control parameters Set initial values of dynamic parameters while$t<T$dofor $i=1$ to N dofor $j=1$ to i doif ${I}_{j}<{I}_{i}$ then Move the firefly j in the direction of the firefly i in D dimension Attractiveness changes with distance r as exp[$-\gamma r$] Evaluate the new solution, replace the worst solution with better one and update intensity of light end ifend forend for Sort population according to fitness in descending order and determine solution with index -1 (${x}_{worst}$) if $t>gss$ thenif $t<cmt$ then Generate new solution ${x}_{new}$ by group search mode 1 operator else Generate new solution ${x}_{new}$ by group search mode 2 operator end if Perform greedy selection between ${x}_{new}$ and ${x}_{worst}$ end if All solution are ranked in order to find the current best solution end whileOutput the global best solution ${x}^{*}$ Post-process results and perform visualization |

## 4. Experimental Findings, Comparative Analysis, and Discussion

#### 4.1. Datasets Used in Experiments

#### 4.2. Experimental Setup, Proposed Encoding Scheme, and Flow-Chart Diagram

- C, boundaries: $[{2}^{-5},{2}^{15}]$, type: continuous,
- $\gamma $, boundaries: $[{2}^{-15},{2}^{3}]$, type: continuous, and
- kernel type, boundaries: $[0,3]$, type: integer, where value 0 denotes polynomial (poly), 1 marks radial basis function (rbf), 2 represents sigmoid, and finally 3 represents linear kernel type.

- learning rate ($\eta $), boundaries: $[0.1,0.9]$, type: continuous,
- $min\_child\_weight$, boundaries: $[0,10]$, type: continuous,
- subsample, boundaries: $[0.01,1]$,type: continuous,
- collsample_bytree, boundaries: $[0.01,1]$, type: continuous,
- max_depth, boundaries: $[3,10]$, type: integer and
- $gamma$, boundaries: $[0,0.5]$, type: continuous.

#### 4.3. Comparative Analysis and Discussion

^{®}Core™ i9-11900K Processor with 64 GB of RAM and Windows 11 O.S. was used as a simulation platform. All employed datasets were relatively large, and the cache argument for SVM and XGBoost models in scikit-learn environment was set to 32 GB to improve the computation speed. On the other hand, the ELM is implemented by using the cupy instead of numpy library for operations with matrices, because the cupy supports execution on GPU, and, in this case, NVIDIA Geforce GTX 1080 GPU with 8 GB of memory is employed for such computations.

#### 4.4. Statistical Tests

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 3.**Flow-chart diagram of proposed methodology in this study. (

**a**) The SVM/ELM/XGBoost GSFA flow chart. (

**b**) Fitness calculation.

**Figure 4.**Convergence speed graphs of swarm algorithms for SVM, ELM, and XGBoost models for original and synthetic (SMOTE) credit card fraud datasets.

**Figure 5.**Confusion matrices for SVM, ELM, and XGBoost models tuned by GSFA, FA, and SNS for original credit card fraud dataset.

**Figure 6.**Precision–recall curves for SVM, ELM, and XGBoost models tuned by GSFA and arbitrary chosen approaches for synthetic credit card fraud dataset.

Parameter | Expression | Description |
---|---|---|

$gsp$ | $gsp=gsp-\frac{t}{T}$ | dynamic group search parameter, starting value 2 |

$gss$ | $gss=\frac{T}{2}$ | group search start |

$cmt$ | $cmt=gss+\frac{T}{3}$ | change mode trigger |

Metrics | SVM-GSFA | SVM-FA | SVM-BA | SVM-ABC | SVM-SCA | SVM-MBO | SVM-HHO | SVM-EHO | SVM-WOA | SVM-SNS |
---|---|---|---|---|---|---|---|---|---|---|

best (%) | 99.9552 | 99.9064 | 99.8830 | 99.9532 | 99.9064 | 99.8361 | 99.9064 | 99.9064 | 99.9064 | 99.9064 |

worst (%) | 99.9064 | 99.8596 | 99.8596 | 99.9064 | 99.9064 | 99.8127 | 99.9064 | 99.8127 | 99.9064 | 99.8596 |

mean (%) | 99.9220 | 99.8752 | 99.8674 | 99.9298 | 99.9064 | 99.8283 | 99.9064 | 99.8752 | 99.9064 | 99.8908 |

median (%) | 99.9064 | 99.8596 | 99.8596 | 99.9298 | 99.9064 | 99.8361 | 99.9064 | 99.9064 | 99.9064 | 99.9064 |

std | 0.000270 | 0.000270 | 0.000135 | 0.000234 | 0.000000 | 0.000135 | 0.000000 | 0.000541 | 0.000000 | 0.000270 |

Metrics | ELM-GSFA | ELM-FA | ELM-BA | ELM-ABC | ELM-SCA | ELM-MBO | ELM-HHO | ELM-EHO | ELM-WOA | ELM-SNS |
---|---|---|---|---|---|---|---|---|---|---|

best (%) | 99.9462 | 99.9111 | 99.9345 | 99.9345 | 99.9169 | 99.9333 | 99.9181 | 99.9263 | 99.9192 | 99.9111 |

worst (%) | 99.9427 | 99.8841 | 99.9134 | 99.8947 | 99.9029 | 99.8982 | 99.8947 | 99.9075 | 99.9075 | 99.8947 |

mean (%) | 99.9442 | 99.8947 | 99.9207 | 99.9058 | 99.9105 | 99.9099 | 99.9081 | 99.9160 | 99.9160 | 99.9029 |

median (%) | 99.9438 | 99.8917 | 99.9175 | 99.8970 | 99.9111 | 99.9040 | 99.9099 | 99.9151 | 99.9157 | 99.9029 |

std | 0.000018 | 0.000123 | 0.000095 | 0.000192 | 0.000059 | 0.000163 | 0.000109 | 0.000077 | 0.000136 | 0.000070 |

Metrics | XGBoost-GSFA | XGBoost-FA | XGBoost-BA | XGBoost-ABC | XGBoost-SCA | XGBoost-MBO | XGBoost-HHO | XGBoost-EHO | XGBoost-WOA | XGBoost-SNS |
---|---|---|---|---|---|---|---|---|---|---|

best (%) | 99.9707 | 99.9684 | 99.9672 | 99.9684 | 99.9661 | 99.9672 | 99.9661 | 99.9672 | 99.9661 | 99.9661 |

worst (%) | 99.9696 | 99.9649 | 99.9625 | 99.9661 | 99.9649 | 99.9649 | 99.9649 | 99.9649 | 99.9661 | 99.9637 |

mean (%) | 99.9704 | 99.9668 | 99.9649 | 99.9668 | 99.9653 | 99.9664 | 99.9657 | 99.9657 | 99.9661 | 99.9649 |

median (%) | 99.9707 | 99.9672 | 99.9649 | 99.9661 | 99.9649 | 99.9672 | 99.9661 | 99.9649 | 99.9661 | 99.9649 |

std | 0.000007 | 0.000018 | 0.000023 | 0.000014 | 0.000007 | 0.000014 | 0.000007 | 0.000014 | 0.000000 | 0.000012 |

Metrics | SVM-GSFA | SVM-FA | SVM-BA | SVM-ABC | SVM-SCA | SVM-MBO | SVM-HHO | SVM-EHO | SVM-WOA | SVM-SNS |
---|---|---|---|---|---|---|---|---|---|---|

best (%) | 96.9519 | 96.5064 | 95.1700 | 96.2954 | 96.2954 | 95.1465 | 95.1700 | 96.9285 | 96.9519 | 95.1700 |

worst (%) | 96.8039 | 96.2704 | 94.0674 | 96.0005 | 95.9408 | 94.6151 | 94.1505 | 96.2706 | 96.4701 | 94.3300 |

mean (%) | 96.8643 | 96.3557 | 94.4115 | 96.0875 | 96.0431 | 94.8514 | 94.4080 | 96.6505 | 96.6271 | 94.9307 |

median (%) | 96.8504 | 96.3691 | 94.5049 | 96.0651 | 96.0251 | 94.8352 | 94.3261 | 96.5141 | 96.7155 | 94.9480 |

std | 0.001050 | 0.007410 | 0.056500 | 0.002150 | 0.008980 | 0.010500 | 0.045500 | 0.007450 | 0.006660 | 0.074500 |

Metrics | ELM-GSFA | ELM-FA | ELM-BA | ELM-ABC | ELM-SCA | ELM-MBO | ELM-HHO | ELM-EHO | ELM-WOA | ELM-SNS |
---|---|---|---|---|---|---|---|---|---|---|

best (%) | 97.1716 | 97.3140 | 96.6270 | 97.0186 | 96.5133 | 96.4165 | 96.6229 | 96.6364 | 96.1867 | 96.5695 |

worst (%) | 97.1046 | 97.2455 | 96.4971 | 97.0046 | 96.3211 | 96.3961 | 96.4708 | 96.5071 | 95.8648 | 96.3695 |

mean (%) | 97.1440 | 97.2669 | 96.5881 | 97.0126 | 96.3971 | 96.4055 | 96.5207 | 96.6044 | 96.0175 | 96.4710 |

median (%) | 97.1595 | 97.261 | 96.5794 | 97.0069 | 96.4671 | 96.4087 | 96.4981 | 96.5951 | 96.0266 | 96.4898 |

std | 0.000321 | 0.000355 | 0.004440 | 0.000456 | 0.000059 | 0.000429 | 0.084200 | 0.025200 | 0.023600 | 0.003450 |

Metrics | XGBoost-GSFA | XGBoost-FA | XGBoost-BA | XGBoost-ABC | XGBoost-SCA | XGBoost-MBO | XGBoost-HHO | XGBoost-EHO | XGBoost-WOA | XGBoost-SNS |
---|---|---|---|---|---|---|---|---|---|---|

best (%) | 99.9842 | 99.9766 | 99.9818 | 99.9842 | 99.9830 | 99.9683 | 99.9818 | 99.9812 | 99.9766 | 99.9818 |

worst (%) | 99.9841 | 99.9740 | 99.9786 | 99.9803 | 99.9810 | 99.9642 | 99.9793 | 99.9762 | 99.9743 | 99.9756 |

mean (%) | 99.9841 | 99.9750 | 99.9800 | 99.9836 | 99.9823 | 99.9662 | 99.9800 | 99.9795 | 99.9757 | 99.9794 |

median (%) | 99.9841 | 99.9751 | 99.9801 | 99.9841 | 99.9824 | 99.9669 | 99.9801 | 99.9786 | 99.9756 | 99.9786 |

std | 0.000001 | 0.000020 | 0.000095 | 0.000034 | 0.000034 | 0.000072 | 0.000084 | 0.000085 | 0.000067 | 0.000105 |

Metrics | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Metaheuristic | Accuracy (%) | Precision 0 | Precision 1 |
M.Avg.
Precision | Recall 0 | Recall 1 |
M.Avg.
Recall | F1 Score 0 | F1 Score 1 |
M.Avg.
F1 Score |
M.Avg.
ROC AUC |
M.Avg.
PR AUC |

SVM-GSFA | 99.9532 | 0.999765 | 0.875000 | 0.999532 | 0.999765 | 0.875000 | 0.999532 | 0.999765 | 0.875000 | 0.999532 | 1.00 | 1.00 |

SVM-FA | 99.9064 | 0.999531 | 0.750000 | 0.999064 | 0.999531 | 0.750000 | 0.999064 | 0.999531 | 0.750000 | 0.999064 | 1.00 | 1.00 |

SVM-BA | 99.8830 | 0.999063 | 0.800000 | 0.998690 | 0.999765 | 0.500000 | 0.998830 | 0.999414 | 0.615385 | 0.998695 | 1.00 | 1.00 |

SVM-ABC | 99.9532 | 0.998251 | 0.002015 | 0.996385 | 0.535413 | 0.500000 | 0.535346 | 0.696993 | 0.004014 | 0.695695 | 0.52 | 0.56 |

SVM-SCA | 99.9064 | 0.999531 | 0.750000 | 0.999064 | 0.999531 | 0.750000 | 0.999064 | 0.999531 | 0.750000 | 0.999064 | 1.00 | 1.00 |

SVM-MBO | 99.8361 | 0.998127 | 0.000000 | 0.996258 | 0.999765 | 0.000000 | 0.997893 | 0.998946 | 0.000000 | 0.997075 | 0.50 | 0.50 |

SVM-HHO | 99.9064 | 0.999531 | 0.750000 | 0.999064 | 0.999531 | 0.750000 | 0.999064 | 0.999531 | 0.750000 | 0.999064 | 1.00 | 1.00 |

SVM-EHO | 99.9064 | 0.999297 | 0.833333 | 0.998986 | 0.999765 | 0.625000 | 0.999064 | 0.999531 | 0.714286 | 0.998997 | 1.00 | 1.00 |

SVM-WOA | 99.9064 | 0.999531 | 0.750000 | 0.999064 | 0.999531 | 0.750000 | 0.999064 | 0.999531 | 0.750000 | 0.999064 | 1.00 | 1.00 |

SVM-SNS | 99.9064 | 0.999531 | 0.750000 | 0.999064 | 0.999531 | 0.750000 | 0.999064 | 0.999531 | 0.750000 | 0.999064 | 1.00 | 1.00 |

Metrics | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Metaheuristic | Accuracy (%) | Precision 0 | Precision 1 |
M.Avg.
Precision | Recall 0 | Recall 1 |
M.Avg.
Recall | F1 Score 0 | F1 Score 1 |
M.Avg.
F1 Score |
M.Avg.
ROC AUC |
M.Avg.
PR AUC |

ELM-GSFA | 99.9462 | 0.999648 | 0.875000 | 0.999441 | 0.999812 | 0.788732 | 0.999462 | 0.999730 | 0.829630 | 0.999448 | 1.00 | 1.00 |

ELM-FA | 99.9111 | 0.999274 | 0.851064 | 0.999027 | 0.999836 | 0.563380 | 0.999111 | 0.999555 | 0.677966 | 0.999020 | 1.00 | 1.00 |

ELM-BA | 99.9345 | 0.999637 | 0.816176 | 0.999332 | 0.999707 | 0.781690 | 0.999345 | 0.999672 | 0.798561 | 0.999338 | 1.00 | 1.00 |

ELM-ABC | 99.9345 | 0.999555 | 0.852459 | 0.999310 | 0.999789 | 0.732394 | 0.999345 | 0.999672 | 0.787879 | 0.999320 | 1.00 | 1.00 |

ELM-SCA | 99.9169 | 0.999332 | 0.858586 | 0.999098 | 0.999836 | 0.598592 | 0.999169 | 0.999584 | 0.705394 | 0.999095 | 1.00 | 1.00 |

ELM-MBO | 99.9333 | 0.999578 | 0.834646 | 0.999304 | 0.999754 | 0.746479 | 0.999333 | 0.999666 | 0.788104 | 0.999314 | 1.00 | 1.00 |

ELM-HHO | 99.9181 | 0.999402 | 0.827273 | 0.999116 | 0.999777 | 0.640845 | 0.999181 | 0.999590 | 0.722222 | 0.999129 | 1.00 | 1.00 |

ELM-EHO | 99.9263 | 0.998338 | 0.000000 | 0.996679 | 1.000000 | 0.000000 | 0.998338 | 0.999168 | 0.000000 | 0.997508 | 0.50 | 0.50 |

ELM-WOA | 99.9192 | 0.999391 | 0.841121 | 0.999128 | 0.999801 | 0.633803 | 0.999192 | 0.999596 | 0.722892 | 0.999136 | 1.00 | 1.00 |

ELM-SNS | 99.9111 | 0.999274 | 0.851064 | 0.999027 | 0.999836 | 0.563380 | 0.999111 | 0.999555 | 0.677966 | 0.999020 | 1.00 | 1.00 |

Metrics | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Metaheuristic | Accuracy (%) | Precision 0 | Precision 1 |
M.Avg.
Precision | Recall 0 | Recall 1 |
M.Avg.
Recall | F1 Score 0 | F1 Score 1 |
M.Avg.
F1 Score |
M.Avg.
ROC AUC |
M.Avg.
PR AUC |

XGBoost-GSFA | 99.9707 | 0.999754 | 0.968000 | 0.999701 | 0.999953 | 0.852113 | 0.999707 | 0.999853 | 0.906367 | 0.999698 | 1.00 | 1.00 |

XGBoost-FA | 99.9684 | 0.999742 | 0.960000 | 0.999676 | 0.999941 | 0.845070 | 0.999684 | 0.999842 | 0.898876 | 0.999674 | 1.00 | 1.00 |

XGBoost-BA | 99.9672 | 0.999730 | 0.959677 | 0.999664 | 0.999941 | 0.838028 | 0.999672 | 0.999836 | 0.894737 | 0.999661 | 1.00 | 1.00 |

XGBoost-ABC | 99.9684 | 0.999742 | 0.960000 | 0.999676 | 0.999941 | 0.845070 | 0.999684 | 0.999842 | 0.898876 | 0.999674 | 1.00 | 1.00 |

XGBoost-SCA | 99.9661 | 0.999719 | 0.959350 | 0.999652 | 0.999941 | 0.830986 | 0.999661 | 0.999830 | 0.890566 | 0.999648 | 1.00 | 1.00 |

XGBoost-MBO | 99.9672 | 0.999730 | 0.959677 | 0.999664 | 0.999941 | 0.838028 | 0.999672 | 0.999836 | 0.894737 | 0.999661 | 1.00 | 1.00 |

XGBoost-HHO | 99.9661 | 0.999719 | 0.959350 | 0.999652 | 0.999941 | 0.830986 | 0.999661 | 0.999830 | 0.890566 | 0.999648 | 1.00 | 1.00 |

XGBoost-EHO | 99.9672 | 0.999742 | 0.952381 | 0.999663 | 0.999930 | 0.845070 | 0.999672 | 0.999836 | 0.895522 | 0.999663 | 1.00 | 1.00 |

XGBoost-WOA | 99.9661 | 0.999730 | 0.952000 | 0.999651 | 0.999930 | 0.838028 | 0.999661 | 0.999830 | 0.891386 | 0.999650 | 1.00 | 1.00 |

XGBoost-SNS | 99.9661 | 0.999719 | 0.959350 | 0.999652 | 0.999941 | 0.830986 | 0.999661 | 0.999830 | 0.890566 | 0.999648 | 1.00 | 1.00 |

Metrics | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Metaheuristic | Accuracy (%) | Precision 0 | Precision 1 |
M.Avg.
Precision | Recall 0 | Recall 1 |
M.Avg.
Recall | F1 Score 0 | F1 Score 1 |
M.Avg.
F1 Score |
M.Avg.
ROC AUC |
M.Avg.
PR AUC |

SVM-GSFA | 96.9519 | 0.945285 | 0.996530 | 0.970913 | 0.996717 | 0.942335 | 0.969520 | 0.970320 | 0.968675 | 0.969497 | 0.99 | 0.99 |

SVM-FA | 96.5064 | 0.943228 | 0.989151 | 0.966195 | 0.989681 | 0.940459 | 0.965064 | 0.965896 | 0.964191 | 0.965043 | 0.99 | 0.99 |

SVM-BA | 95.1700 | 0.911891 | 1.000000 | 0.955956 | 1.000000 | 0.903422 | 0.951700 | 0.953915 | 0.949261 | 0.951587 | 0.99 | 0.98 |

SVM-ABC | 96.2954 | 0.931381 | 0.999494 | 0.965446 | 0.999531 | 0.926394 | 0.962954 | 0.964253 | 0.961557 | 0.962905 | 1.00 | 1.00 |

SVM-SCA | 96.2954 | 0.931381 | 0.999494 | 0.965446 | 0.999531 | 0.926395 | 0.962954 | 0.964253 | 0.961557 | 0.962905 | 1.00 | 1.00 |

SVM-MBO | 95.1465 | 0.536780 | 0.533067 | 0.534923 | 0.506567 | 0.563057 | 0.534818 | 0.521236 | 0.547652 | 0.534447 | 0.54 | 0.53 |

SVM-HHO | 95.1700 | 0.911891 | 1.000000 | 0.955956 | 1.000000 | 0.903422 | 0.951700 | 0.953915 | 0.949261 | 0.951587 | 0.99 | 0.98 |

SVM-EHO | 96.9285 | 0.946054 | 0.995054 | 0.970560 | 0.995310 | 0.943272 | 0.969285 | 0.970057 | 0.968472 | 0.969264 | 0.99 | 0.99 |

SVM-WOA | 96.9519 | 0.946875 | 0.994568 | 0.970727 | 0.994841 | 0.944210 | 0.969519 | 0.970265 | 0.968735 | 0.969500 | 0.99 | 0.99 |

SVM-SNS | 95.1700 | 0.911891 | 1.000000 | 0.955956 | 1.000000 | 0.903422 | 0.951700 | 0.953915 | 0.949261 | 0.951587 | 0.99 | 0.98 |

Metrics | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Metaheuristic | Accuracy (%) | Precision 0 | Precision 1 |
M.Avg.
Precision | Recall 0 | Recall 1 |
M.Avg.
Recall | F1 Score 0 | F1 Score 1 |
M.Avg.
F1 Score |
M.Avg.
ROC AUC |
M.Avg.
PR AUC |

ELM-GSFA | 97.1716 | 0.961894 | 0.982012 | 0.971931 | 0.982475 | 0.960909 | 0.971716 | 0.972076 | 0.971346 | 0.971712 | 1.00 | 1.00 |

ELM-FA | 97.3140 | 0.967921 | 0.978501 | 0.973200 | 0.978837 | 0.967418 | 0.973140 | 0.973349 | 0.972928 | 0.973139 | 1.00 | 1.00 |

ELM-BA | 96.6270 | 0.966703 | 0.965835 | 0.966270 | 0.965957 | 0.966584 | 0.966270 | 0.966330 | 0.966209 | 0.966270 | 0.99 | 0.99 |

ELM-ABC | 97.0186 | 0.958294 | 0.982770 | 0.970506 | 0.983294 | 0.957020 | 0.970186 | 0.970633 | 0.969724 | 0.970180 | 0.99 | 0.99 |

ELM-SCA | 96.5133 | 0.948796 | 0.982790 | 0.965755 | 0.983493 | 0.946692 | 0.965132 | 0.965833 | 0.964403 | 0.965119 | 0.99 | 0.99 |

ELM-MBO | 96.4165 | 0.949218 | 0.980219 | 0.964685 | 0.980966 | 0.947291 | 0.964165 | 0.964831 | 0.963474 | 0.964154 | 0.99 | 0.99 |

ELM-HHO | 96.6229 | 0.952841 | 0.980501 | 0.966641 | 0.981165 | 0.951227 | 0.966229 | 0.966796 | 0.965642 | 0.966220 | 0.99 | 0.99 |

ELM-EHO | 96.6364 | 0.957048 | 0.976114 | 0.966560 | 0.976708 | 0.955974 | 0.966364 | 0.966778 | 0.965939 | 0.966359 | 0.99 | 0.99 |

ELM-WOA | 96.1867 | 0.958182 | 0.965630 | 0.961898 | 0.966062 | 0.957654 | 0.961867 | 0.962106 | 0.961626 | 0.961866 | 0.99 | 0.99 |

ELM-SNS | 96.5695 | 0.960938 | 0.970575 | 0.965746 | 0.971011 | 0.960357 | 0.965695 | 0.965948 | 0.965439 | 0.965694 | 0.99 | 0.99 |

Metrics | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Metaheuristic | Accuracy (%) | Precision 0 | Precision 1 |
M.Avg.
Precision | Recall 0 | Recall 1 |
M.Avg.
Recall | F1 Score 0 | F1 Score 1 |
M.Avg.
F1 Score |
M.Avg.
ROC AUC |
M.Avg.
PR AUC |

XGBoost-GSFA | 99.9842 | 0.999988 | 0.999695 | 0.999842 | 0.999696 | 0.999988 | 0.999842 | 0.999842 | 0.999841 | 0.999842 | 1.00 | 1.00 |

XGBoost-FA | 99.9766 | 0.999965 | 0.999565 | 0.999766 | 0.999567 | 0.999965 | 0.999766 | 0.999766 | 0.999765 | 0.999766 | 1.00 | 1.00 |

XGBoost-BA | 99.9818 | 0.999988 | 0.999648 | 0.999818 | 0.999649 | 0.999988 | 0.999818 | 0.999819 | 0.999818 | 0.999818 | 1.00 | 1.00 |

XGBoost-ABC | 99.9842 | 0.999965 | 0.999718 | 0.999842 | 0.999719 | 0.999965 | 0.999842 | 0.999842 | 0.999841 | 0.999842 | 1.00 | 1.00 |

XGBoost-SCA | 99.9830 | 0.999965 | 0.999695 | 0.999830 | 0.999696 | 0.999965 | 0.999830 | 0.999830 | 0.999830 | 0.999830 | 1.00 | 1.00 |

XGBoost-MBO | 99.9683 | 0.999941 | 0.999425 | 0.999684 | 0.999427 | 0.999941 | 0.999683 | 0.999684 | 0.999683 | 0.999683 | 1.00 | 1.00 |

XGBoost-HHO | 99.9818 | 0.999965 | 0.999671 | 0.999818 | 0.999672 | 0.999965 | 0.999818 | 0.999819 | 0.999818 | 0.999818 | 1.00 | 1.00 |

XGBoost-EHO | 99.9812 | 0.999965 | 0.999660 | 0.999812 | 0.999661 | 0.999965 | 0.999812 | 0.999813 | 0.999812 | 0.999812 | 1.00 | 1.00 |

XGBoost-WOA | 99.9766 | 0.999930 | 0.999601 | 0.999766 | 0.999602 | 0.999930 | 0.999766 | 0.999766 | 0.999765 | 0.999766 | 1.00 | 1.00 |

XGBoost-SNS | 99.9818 | 0.999977 | 0.999659 | 0.999818 | 0.999661 | 0.999976 | 0.999818 | 0.999819 | 0.999818 | 0.999818 | 1.00 | 1.00 |

Method/Parameters | No SMOTE | With SMOTE | ||||
---|---|---|---|---|---|---|

C | $\mathit{\gamma}$ | Kernel Type | C | $\mathit{\gamma}$ | Kernel Type | |

SVM-GSFA | 1816.0411 | 0.0492 | 1 | 0.031 | 0.1015 | 0 |

SVM-FA | 16316.8042 | $3\times {10}^{-5}$ | 0 | 0.031 | 1.6601 | 1 |

SVM-BA | 32768 | 0.1172 | 2 | 2106.3912 | 7.6593 | 0 |

SVM-ABC | 6064.1918 | 0.0142 | 1 | 8512.3559 | $3\times {10}^{-5}$ | 2 |

SVM-SCA | 15,430.8553 | $3\times {10}^{-5}$ | 0 | 8591.6538 | $3\times {10}^{-5}$ | 2 |

SVM-MBO | 14,167.7370 | 1.9988 | 2 | 631.3854 | 5.1329 | 0 |

SVM-HHO | 22,160.9077 | $3\times {10}^{-5}$ | 0 | 500.4590 | 2.3178 | 0 |

SVM-EHO | 0.031 | 2.3360 | 0 | 2425.9645 | 0.0234 | 0 |

SVM-WOA | 22,320.8262 | $3\times {10}^{-5}$ | 0 | 0.0353 | 0.0912 | 0 |

SVM-SNS | 32,768 | $3\times {10}^{-5}$ | 0 | 8277.7914 | 0.6790 | 0 |

Method/ | No SMOTE | With SMOTE |
---|---|---|

Parameters | Number of Neurons | Number of Neurons |

ELM-GSFA | 88 | 67 |

ELM-FA | 60 | 74 |

ELM-BA | 30 | 85 |

ELM-ABC | 85 | 86 |

ELM-SCA | 53 | 150 |

ELM-MBO | 97 | 135 |

ELM-HHO | 50 | 48 |

ELM-EHO | 56 | 79 |

ELM-WOA | 64 | 133 |

ELM-SNS | 64 | 90 |

Method/ | No SMOTE | |||||
---|---|---|---|---|---|---|

Parameters | eta | min_child_weight | Subsample | colsample_bytree | max_depth | Gamma |

XGBoost-GSFA | 0.6109 | 5.3438 | 0.6276 | 0.7093 | 8.0268 | 0.4437 |

XGBoost-FA | 0.8330 | 7.1837 | 0.9261 | 0.7303 | 3.7943 | 0.1307 |

XGBoost-BA | 0.7028 | 6.7516 | 0.6247 | 0.6904 | 5.7910 | 0.3833 |

XGBoost-ABC | 0.8340 | 7.7698 | 0.5957 | 0.7957 | 6.2033 | 0.0021 |

XGBoost-SCA | 0.5143 | 1.5846 | 1 | 0.7242 | 6.6324 | 0.4547 |

XGBoost-MBO | 0.6572 | 3.9720 | 0.5423 | 0.7891 | 5.3727 | 0.1232 |

XGBoost-HHO | 0.6329 | 2.2029 | 0.9299 | 0.9118 | 7.8752 | 0.5 |

XGBoost-EHO | 0.6274 | 10 | 1 | 0.6247 | 5.1698 | 0.5 |

XGBoost-WOA | 0.5940 | 5.5582 | 0.5548 | 0.6715 | 6.0848 | 0.4435 |

XGBoost-SNS | 0.6841 | 1 | 0.9168 | 1 | 5.8061 | 0.1113 |

With SMOTE | ||||||

Parameters | eta | min_child_weight | Subsample | colsample_bytree | max_depth | Gamma |

XGBoost-GSFA | 0.8753 | 5.5543 | 0.9998 | 0.7219 | 9.5889 | 0.3941 |

XGBoost-FA | 0.7744 | 5.7581 | 0.9646 | 0.4266 | 8.9161 | 0.0730 |

XGBoost-BA | 0.8889 | 1 | 1 | 0.4705 | 9.9235 | 0.3647 |

XGBoost-ABC | 0.8581 | 3.9509 | 0.7588 | 0.3775 | 10 | 0.0874 |

XGBoost-SCA | 0.9 | 1 | 0.9368 | 0.5866 | 10 | 0 |

XGBoost-MBO | 0.7940 | 2.1121 | 0.4077 | 0.7334 | 8.7939 | 0.2791 |

XGBoost-HHO | 0.8227 | 5.0387 | 0.9566 | 0.4330 | 9.6767 | 0.4883 |

XGBoost-EHO | 0.9 | 3.9113 | 0.8527 | 0.7699 | 10 | 0.5 |

XGBoost-WOA | 0.8606 | 2.5538 | 0.9666 | 0.9613 | 9.5649 | 0.1993 |

XGBoost-SNS | 0.9 | 1 | 0.8993 | 0.5963 | 10 | 0.0858 |

Methods | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Problem | GSFA | FA | BA | ABC | SCA | MBO | HHO | EHO | WOA | SNS |

SVM | $6.15\times {10}^{-1}$ | $4.24\times {10}^{-1}$ | $7.05\times {10}^{-1}$ | $4.37\times {10}^{-1}$ | $8.29\times {10}^{-1}$ | $6.02\times {10}^{-1}$ | $7.40\times {10}^{-1}$ | $4.40\times {10}^{-1}$ | $1.23\times {10}^{-1}$ | $4.23\times {10}^{-1}$ |

SVM Smote | $8.06\times {10}^{-1}$ | $7.49\times {10}^{-1}$ | $5.82\times {10}^{-1}$ | $4.92\times {10}^{-1}$ | $3.05\times {10}^{-1}$ | $5.93\times {10}^{-1}$ | $8.08\times {10}^{-1}$ | $3.63\times {10}^{-1}$ | $3.33\times {10}^{-1}$ | $4.51\times {10}^{-1}$ |

ELM | $6.37\times {10}^{-1}$ | $5.32\times {10}^{-1}$ | $6.52\times {10}^{-1}$ | $5.30\times {10}^{-1}$ | $8.42\times {10}^{-2}$ | $6.05\times {10}^{-1}$ | $5.55\times {10}^{-1}$ | $2.84\times {10}^{-1}$ | $2.85\times {10}^{-1}$ | $8.52\times {10}^{-2}$ |

ELM Smote | $1.56\times {10}^{-1}$ | $9.46\times {10}^{-2}$ | $4.52\times {10}^{-1}$ | $6.95\times {10}^{-1}$ | $1.34\times {10}^{-1}$ | $1.12\times {10}^{-1}$ | $8.58\times {10}^{-2}$ | $3.50\times {10}^{-1}$ | $7.92\times {10}^{-1}$ | $3.10\times {10}^{-1}$ |

XGB | $7.98\times {10}^{-1}$ | $4.24\times {10}^{-1}$ | $9.52\times {10}^{-2}$ | $7.35\times {10}^{-1}$ | $7.29\times {10}^{-1}$ | $7.49\times {10}^{-1}$ | $5.32\times {10}^{-1}$ | $6.93\times {10}^{-1}$ | $6.07\times {10}^{-1}$ | $7.43\times {10}^{-1}$ |

XGB Smote | $3.94\times {10}^{-1}$ | $3.84\times {10}^{-1}$ | $1.75\times {10}^{-1}$ | $5.23\times {10}^{-1}$ | $4.01\times {10}^{-1}$ | $5.69\times {10}^{-1}$ | $6.66\times {10}^{-1}$ | $7.24\times {10}^{-1}$ | $5.21\times {10}^{-1}$ | $5.62\times {10}^{-1}$ |

Methods | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

GSFA | FA | BA | ABC | SCA | MBO | HHO | EHO | WOA | SNS | |

p-value | $3.20\times {10}^{-3}$ | $8.34\times {10}^{-3}$ | $9.71\times {10}^{-3}$ | $6.81\times {10}^{-3}$ | $3.28\times {10}^{-3}$ | $9.19\times {10}^{-3}$ | $8.90\times {10}^{-3}$ | $2.32\times {10}^{-3}$ | $4.58\times {10}^{-3}$ | $8.34\times {10}^{-3}$ |

Functions | GSFA | FA | BA | ABC | SCA | MBO | HHO | EHO | WOA | SNS |
---|---|---|---|---|---|---|---|---|---|---|

SVM | 1 | 7.5 | 9 | 2 | 4 | 10 | 4 | 7.5 | 4 | 6 |

SVM Smote | 1 | 4 | 9 | 5 | 6 | 8 | 10 | 2 | 3 | 7 |

ELM | 1 | 10 | 2 | 8 | 4 | 6 | 7 | 3 | 5 | 9 |

ELM Smote | 2 | 1 | 4 | 6 | 9 | 8 | 5 | 3 | 10 | 7 |

XGB | 1 | 2.5 | 9.5 | 2.5 | 8 | 4 | 6.5 | 6.5 | 5 | 9.5 |

XGB Smote | 1 | 7 | 4 | 2 | 3 | 10 | 9 | 8 | 6 | 5 |

Average Ranking | 1.17 | 5.33 | 6.25 | 4.25 | 5.67 | 7.67 | 6.92 | 5.00 | 5.50 | 7.25 |

Rank | 1 | 4 | 7 | 2 | 6 | 10 | 8 | 3 | 5 | 9 |

Functions | GSFA | FA | BA | ABC | SCA | MBO | HHO | EHO | WOA | SNS |
---|---|---|---|---|---|---|---|---|---|---|

SVM | 9 | 46.5 | 49 | 11 | 13 | 50 | 13 | 46.5 | 13 | 24 |

SVM Smote | 1 | 5 | 59 | 7 | 8 | 58 | 60 | 2 | 3 | 57 |

ELM | 10 | 48 | 16 | 43 | 35 | 38 | 40 | 21 | 36 | 44 |

ELM Smote | 6 | 4 | 42 | 52 | 55 | 54 | 51 | 15 | 56 | 53 |

XGB | 20 | 25.5 | 32.5 | 25.5 | 31 | 27 | 29.5 | 29.5 | 28 | 32.5 |

XGB Smote | 17 | 37 | 22 | 18 | 19 | 45 | 41 | 39 | 34 | 23 |

Average Ranking | 10.50 | 27.67 | 36.75 | 26.08 | 26.83 | 45.33 | 39.08 | 25.50 | 28.33 | 38.92 |

Rank | 1 | 5 | 7 | 3 | 4 | 10 | 9 | 2 | 6 | 8 |

Comparison | p-Value | Rank | 0.05/($\mathit{k}-\mathit{i}$) | 0.1/($\mathit{k}-\mathit{i}$) |
---|---|---|---|---|

GSFA vs. MBO | $1.00\times {10}^{-4}$ | 0 | 0.005556 | 0.011111 |

GSFA vs. SNS | $2.51\times {10}^{-4}$ | 1 | 0.006250 | 0.012500 |

GSFA vs. HHO | $5.02\times {10}^{-4}$ | 2 | 0.007143 | 0.014286 |

GSFA vs. BA | $1.82\times {10}^{-3}$ | 3 | 0.008333 | 0.016667 |

GSFA vs. SCA | $5.02\times {10}^{-3}$ | 4 | 0.010000 | 0.020000 |

GSFA vs. WOA | $6.59\times {10}^{-3}$ | 5 | 0.012500 | 0.025000 |

GSFA vs. FA | $8.57\times {10}^{-3}$ | 6 | 0.016667 | 0.033333 |

GSFA vs. EHO | $1.42\times {10}^{-2}$ | 7 | 0.025000 | 0.050000 |

GSFA vs. ABC | $3.89\times {10}^{-2}$ | 8 | 0.050000 | 0.100000 |

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## Share and Cite

**MDPI and ACS Style**

Jovanovic, D.; Antonijevic, M.; Stankovic, M.; Zivkovic, M.; Tanaskovic, M.; Bacanin, N.
Tuning Machine Learning Models Using a Group Search Firefly Algorithm for Credit Card Fraud Detection. *Mathematics* **2022**, *10*, 2272.
https://doi.org/10.3390/math10132272

**AMA Style**

Jovanovic D, Antonijevic M, Stankovic M, Zivkovic M, Tanaskovic M, Bacanin N.
Tuning Machine Learning Models Using a Group Search Firefly Algorithm for Credit Card Fraud Detection. *Mathematics*. 2022; 10(13):2272.
https://doi.org/10.3390/math10132272

**Chicago/Turabian Style**

Jovanovic, Dijana, Milos Antonijevic, Milos Stankovic, Miodrag Zivkovic, Marko Tanaskovic, and Nebojsa Bacanin.
2022. "Tuning Machine Learning Models Using a Group Search Firefly Algorithm for Credit Card Fraud Detection" *Mathematics* 10, no. 13: 2272.
https://doi.org/10.3390/math10132272