Metamaterial Antennas are a type of antenna that uses metamaterial to enhance performance. The bandwidth restriction associated with small antennas can be solved using metamaterial antennas. Machine learning is gaining popularity as a way to improve solutions in a range of fields. Machine learning approaches are currently a big part of current research, and they’re likely to be huge in the future. The model utilized determines the accuracy of the prediction in large part. The goal of this paper is to develop an optimized ensemble model for forecasting the metamaterial antenna’s bandwidth and gain. The basic models employed in the developed ensemble are Support Vector Regression (SVR), K-Nearest Regression (KNR), Multi-Layer Perceptron (MLP), Decision Trees (DT), and Random Forest (RF). The percentages of contribution of these models in the ensemble model are weighted and optimized using the dipper throated optimization (DTO) algorithm. To choose the best features from the dataset, the binary (bDTO) algorithm is exploited. The proposed ensemble model is compared to the base models and results are recorded and analyzed statistically. In addition, two other ensembles are incorporated in the conducted experiments for comparison. These ensembles are average ensemble and K-nearest neighbors (KNN)-based ensemble. The comparison is performed in terms of eleven evaluation criteria. The evaluation results confirmed the superiority of the proposed model when compared with the basic models and the other ensemble models.

In all sectors of science and engineering, machine learning (ML) has been widely used to automate everyday tasks and provide breakthrough insights. Practitioners of machine learning have changed the foundations of various industries and fields of study. One of the newest fields is the design and optimization of metamaterial antennas. Given the current state of the world’s huge data, machine learning (ML) has received a lot of attention. In the design and prediction of antenna behavior, machine learning has a lot of promise since it allows for a lot of speed while maintaining high accuracy [

Closed-form solutions are uncommon in metamaterial antennas due to their complex shapes. The function of electromagnetic fields in the construction of antennas is described using Maxwell’s equations in computational electromagnetics (CEM). To get a physical understanding of the antenna’s design, a series of approximate solutions is usually used. Integral equations, for example, may be used to solve linear antennas using sophisticated numerical methods. Maxwell’s equations were later solved using differential and integral equation solvers as computer technology evolved [

Due to the inherent nonlinearities of antenna designs, machine learning (ML) has been extensively investigated as a supplement to CEM in enhancing and creating a wide range of antenna designs. Because statistics and data science are frequently referenced, ML is a subset of artificial intelligence (AI) that focuses on extracting useful information from data. Researchers have been able to create systems using machine learning’s data-driven methodology, bringing us closer to fully autonomous systems that can match, compete with, and occasionally surpass human abilities and intuition. Machine learning approaches, on the other hand, rely on data quality, quantity, and accessibility, which might be difficult to come by in some cases [

For metamaterial antennas, such as those used in computer vision, there is no standardized dataset available. From the aspect of antenna design, this dataset must be collected if it isn’t already accessible. This may be done by simulating the intended antenna over a wide range of values using CEM simulation software. Training, testing, and cross-validation may all be done with the same dataset. These components are used to train and test the capacity of the machine learning model to generalize to new inputs. At this point, it is up to the designer’s vision and talent to find out how to validate the model and improve its generality. In this case, normal processes include plotting learning curves and evaluating bias and variance values. In most cases, the designer’s intuition plays a big influence in improving a model’s performance [

The application of machine learning to antenna parameter optimization considerably accelerates the design process. Traditional methods of getting ideal parameters for a particular antenna design, as shown in

To apply machine learning into the antenna design challenge, follow the methods below in general. A series of simulations is used to estimate an antenna’s electromagnetic characteristics. These characteristics are subsequently kept in a database and fed into a machine learning algorithm. Finally, according on the designer’s specifications, the algorithm selects the Antenna that gives the best results.

Machine learning (ML) is a technology that uses algorithms to learn from data without having to pre-program them. There are three forms of reinforcement learning: supervised, unsupervised, and reinforcement. Extensive interconnections of neurons; which are fundamental processing cells, are employed to achieve excellent performance in Artificial Neural Networks (ANN). When complex functions with numerous features are identified, neural networks may be used to do machine learning. An input layer, an output layer, and hidden layers between the input and output layers are all layers in a neural network [

Machine learning algorithms have been employed in smart grid networks to predict dangerous occurrences, wireless networks to forecast wireless users’ mobility patterns and content demands, and voice recognition. Training a learning algorithm on data from previous simulations to enhance antenna parameters is one way to use machine learning in antenna design.

Because they are intelligent and have past knowledge of random search, metaheuristic algorithms tackle unanticipated problems. These algorithms are either versatile, straightforward, or capable of avoiding local perfection. The aspects of population-based heuristic algorithms include exploration and exploitation. Exploration and exploitation are chosen by the metaheuristic algorithm. The approach extensively inspects the search space while exploring. Local search in the region is currently being used. In recent decades, several natural-inspired global optimization algorithms have been created. A number of scenarios can benefit from population-based metaheuristics, sometimes known as general-purpose algorithms. Metaheuristics can be metaphor-based or non-metaphor-based. Metaphors, on the other hand, use algorithms to reflect natural events or human behavior in today’s society [

The process of feature selection and extraction are referred to as feature engineering. This process is essential to all machine learning operations. Although extraction and selection of features are similar in certain aspects, they are frequently used interchangeably. The feature selection approach aims to find the most consistent, relevant, and nonredundant qualities. The search area for feature selection is limited to two binary values: 0 and 1. Consequently, the binary version of the optimization algorithm should be employed to fit the feature selection task. The main idea of the binary version is to employ the sigmoid function to get the binary values from the continuous results of the optimizer.

When it comes to artificial intelligence problems, ensemble strategies are becoming more popular. The average ensemble is one of the most fundamental ensemble algorithms for integrating and computing the mean of base regressor outputs. This approach computes the mean value by combining the results of several regressors. This type of ensembles is used in conjunction with KNN-based ensemble to prove the effectiveness of the proposed weighted ensemble model. The proposed weighted ensemble for bandwidth and gain prediction is based on three phases namely, preprocessing, selection of relevant features, and optimization of the weighted outputs of five regression models, as illustrated in

Eleven Metamaterial Antenna properties are included in the dataset used in this investigation. The collection of antenna designs is available on the Kaggle dataset which is employed in this research [

The preprocessing of the dataset is performed in terms of three steps. Firstly, data cleaning, in which the null values are replaces with the average between the surrounding values for each feature. Secondly, scaling the features values using the min-max scaler. Thirdly, the split of the dataset into training and testing based on the 80% and 20% recommendation rule.

This algorithm is proven to be an effective metaheuristic optimization algorithm based on the hunting dipper throated bird’s quick bending motions [

Because the search space is confined to two binary values, 0 and 1, picking features presents a unique problem. As a result, we employed the sigmoid function to transform the output of the conventional optimizer into binary values. The following equation is used to convert the continuous answer to binary in order to fit the feature selection task.

These are the explanations behind the outcomes in this section. The findings are described using support vector regression (SVR), k-nearest regressor (KNR), random forest (RF), decision tree (DT), and multi-layer perceptron (MLP) regressors, as well as the suggested weighted average ensemble model. After that, the outcomes of feature selection are used to offer the suggested model’s performance.

The evaluation metrics employed in this research are presented in ^{th} estimated and observed PV power values, and

Metrics | Equation | |
---|---|---|

Average error | = | |

Average fitness | = | |

Average fitness size | = | |

Best fitness | = | |

Worst fitness | = | |

STD (Standard Deviation) fitness | = | |

RRMSE | = | |

RMSE | = | |

MAPE | = | |

r | = | |

WI | = |

The first set of experiments was conducted to measure the performance of the feature selection methods.

Avg. error | Avg. select size | Avg. fitness | Best fitness | Worst fitness | Std. fitness | |
---|---|---|---|---|---|---|

bGWO | 0.46768 | 0.60328 | 0.52988 | 0.45018 | 0.51708 | 0.34068 |

bGWO_PSO | 0.50698 | 0.73658 | 0.53818 | 0.49168 | 0.60168 | 0.35888 |

bPSO | 0.50148 | 0.60328 | 0.52828 | 0.50858 | 0.57628 | 0.34008 |

bBA | 0.51108 | 0.74268 | 0.55118 | 0.44088 | 0.54248 | 0.34998 |

bWAO | 0.50128 | 0.76668 | 0.53608 | 0.50018 | 0.57628 | 0.34228 |

bBBO | 0.46968 | 0.76708 | 0.53398 | 0.52368 | 0.61018 | 0.38498 |

bMVO | 0.47818 | 0.69978 | 0.55798 | 0.48318 | 0.60118 | 0.39078 |

bSBO | 0.50978 | 0.77358 | 0.56798 | 0.51108 | 0.59078 | 0.40098 |

bGWO_GA | 0.48778 | 0.52608 | 0.53598 | 0.51378 | 0.58998 | 0.34128 |

bFA | 0.49988 | 0.63778 | 0.58018 | 0.49888 | 0.59648 | 0.37688 |

bGA | 0.48128 | 0.54568 | 0.54128 | 0.44458 | 0.55968 | 0.34228 |

Once the significant features are selected, the optimized weighted ensemble model is employed to predict the gain values of metamaterial antenna. The prediction results are analyzed and presented in

GWO | PSO | GA | WOA | ||
---|---|---|---|---|---|

Num. values | 14 | 14 | 14 | 14 | |

Range | 0.002 | 0.002 | 0.002 | 0.00027 | |

Minimum | 0.004547 | 0.005678 | 0.006785 | 0.009662 | |

Median | 0.005547 | 0.006678 | 0.007846 | 0.009932 | |

Maximum | 0.006547 | 0.007678 | 0.008785 | 0.009932 | |

Mean | 0.005554 | 0.006685 | 0.007881 | 0.009891 | |

25% Percentile | 0.005547 | 0.006678 | 0.007846 | 0.009907 | |

75% Percentile | 0.005547 | 0.006678 | 0.007846 | 0.009932 | |

Std. Error of Mean | 0.000105 | 0.000105 | 0.000114 | 2.34E-05 | |

Std. Deviation | 0.000393 | 0.000393 | 0.000426 | 8.76E-05 | |

Sum | 0.07776 | 0.09359 | 0.1103 | 0.1385 |

The null and alternative hypotheses are analyzed using a one-way analysis of variance (ANOVA) test. For the null hypothesis H0 (i.e., DTO = GWO = PSO = GA = WOA), the algorithm’s mean values are set equal. Under the alternative hypothesis, H1, the means of the algorithms are not similar. The results of the ANOVA test are presented in

Criteria | SS | DF | MS | F (DFn, DFd) | |
---|---|---|---|---|---|

Treatment | 0.000454 | 4 | 0.000113 | F (4, 65) = 1136 | |

Residual | 6.49E-06 | 65 | 9.99E-08 | ||

Total | 0.00046 | 69 |

The statistical difference between each two algorithms is used to compute the p-values between the optimization of the weighted ensemble using DTO and four other optimization techniques. This study used Wilcoxon’s rank-sum test. The two basic hypotheses in this test are the null and alternative hypotheses. For the null hypothesis given by H0, DTO = GWO, DTO = PSO, DTO = GA, DTO = WOA Under the alternative hypothesis, H1, the algorithms’ means aren’t similar. The Wilcoxon rank-sum test’s findings are shown in

GWO | PSO | GA | WOA | ||
---|---|---|---|---|---|

Number of values | 14 | 14 | 14 | 14 | |

Actual median | 0.005547 | 0.006678 | 0.007846 | 0.009932 | |

Theoretical median | 0 | 0 | 0 | 0 | |

Sum of positive ranks | 105 | 105 | 105 | 105 | |

Sum of signed ranks (W) | 105 | 105 | 105 | 105 | |

Exact or estimate? | Exact | Exact | Exact | Exact | |

Significant (alpha=0.05)? | Yes | Yes | Yes | Yes | |

Sum of negative ranks | 0 | 0 | 0 | 0 | |

Discrepancy | 0.005547 | 0.006678 | 0.007846 | 0.009932 | |

P value (two tailed) | 0.0001 | 0.0001 | 0.0001 | 0.0001 |

On the other hand, the prediction results of the metamaterial gain are recorded using five separate machine learning regressors and two ensemble models in addition to the proposed weighted ensemble model. These results are analysis using eight evaluation criteria and the results are presented in

RMSE | MSE | MBE | r | R2 | RRNSE | NSE | WI | |
---|---|---|---|---|---|---|---|---|

MLP | 0.102 | 0.016 | −0.010 | 0.378 | 0.143 | 10.754 | 0.053 | 0.692 |

KNR | 0.103 | 0.016 | −0.009 | 0.316 | 0.100 | 10.842 | 0.038 | 0.689 |

DT | 0.100 | 0.016 | −0.009 | 0.481 | 0.232 | 10.540 | 0.091 | 0.704 |

SVR | 0.106 | 0.042 | 0.018 | 0.000 | 0.000 | 11.217 | −0.030 | 0.202 |

RF | 0.099 | 0.017 | −0.009 | 0.564 | 0.318 | 10.461 | 0.104 | 0.681 |

AVG Ensemble | 0.089 | 0.021 | −0.002 | 0.834 | 0.695 | 10.461 | 0.270 | 0.612 |

KNR Ensemble | 0.060 | 0.036 | 0.002 | 0.823 | 0.677 | 6.339 | 0.671 | 0.316 |

The histogram of the gain values using the proposed weighted ensemble model that is optimized by DTO algorithm and four other optimizers and the RMSE of the predicted gain values using the proposed weighted ensemble model that is optimized by DTO algorithm and four other optimizers are presented in

To prove the generalization of the proposed approach, the bandwidth of the metamaterial antenna is predicted using the proposed weighted ensemble model. The first step is to select the significant features from the given dataset. The feature selection is performed using bDTO, and the evaluation of the performance of features selection for this task is presented in

Avg. Error | Avg. select size | Avg. fitness | Best fitness | Worst fitness | Std. fitness | |
---|---|---|---|---|---|---|

bGWO | 0.54428 | 0.67988 | 0.60648 | 0.52678 | 0.59368 | 0.41728 |

bGWO_PSO | 0.58358 | 0.81318 | 0.61478 | 0.56828 | 0.67828 | 0.43548 |

bPSO | 0.57808 | 0.67988 | 0.60488 | 0.58518 | 0.65288 | 0.41668 |

bBA | 0.58768 | 0.81928 | 0.62778 | 0.51748 | 0.61908 | 0.42658 |

bWAO | 0.57788 | 0.84328 | 0.61268 | 0.57678 | 0.65288 | 0.41888 |

bBBO | 0.54628 | 0.84368 | 0.61058 | 0.60028 | 0.68678 | 0.46158 |

bMVO | 0.55478 | 0.77638 | 0.63458 | 0.55978 | 0.67778 | 0.46738 |

bSBO | 0.58638 | 0.85018 | 0.64458 | 0.58768 | 0.66738 | 0.47758 |

bGWO_GA | 0.56438 | 0.60268 | 0.61258 | 0.59038 | 0.66658 | 0.41788 |

bFA | 0.57648 | 0.71438 | 0.65678 | 0.57548 | 0.67308 | 0.45348 |

bGA | 0.55788 | 0.62228 | 0.61788 | 0.52118 | 0.63628 | 0.41888 |

GWO | PSO | GA | WOA | ||
---|---|---|---|---|---|

Num. values | 14 | 14 | 14 | 14 | |

Range | 0.001896 | 0.003 | 0.00202 | 0.00198 | |

Minimum | 0.005544 | 0.005789 | 0.006679 | 0.007998 | |

Median | 0.005744 | 0.006789 | 0.00787 | 0.009978 | |

Maximum | 0.00744 | 0.008789 | 0.008699 | 0.009978 | |

Mean | 0.005902 | 0.006853 | 0.007767 | 0.009737 | |

25% Percentile | 0.005744 | 0.006789 | 0.00787 | 0.009753 | |

75% Percentile | 0.005744 | 0.006789 | 0.00787 | 0.009978 | |

Std. Error of Mean | 0.00013 | 0.000165 | 0.000131 | 0.000148 | |

Std. Deviation | 0.000486 | 0.000617 | 0.000491 | 0.000552 | |

Sum | 0.08263 | 0.09594 | 0.1087 | 0.1363 |

The ANOVA test and Wilcoxon test results are presented in

Criteria | SS | DF | MS | F (DFn, DFd) | |
---|---|---|---|---|---|

Treatment | 0.00042 | 4 | 0.000105 | F (4, 65) = 451.2 | |

Residual | 1.51E-05 | 65 | 2.33E-07 | ||

Total | 0.000435 | 69 |

DTO | GWO | PSO | GA | WOA | |
---|---|---|---|---|---|

Number of values | 14 | 14 | 14 | 14 | 14 |

Actual median | 0.002324 | 0.005744 | 0.006789 | 0.00787 | 0.009978 |

Theoretical median | 0 | 0 | 0 | 0 | 0 |

Sum of positive ranks | 105 | 105 | 105 | 105 | 105 |

Sum of signed ranks (W) | 105 | 105 | 105 | 105 | 105 |

Significant (alpha=0.05)? | Yes | Yes | Yes | Yes | Yes |

Exact or estimate? | Exact | Exact | Exact | Exact | Exact |

Sum of negative ranks | 0 | 0 | 0 | 0 | 0 |

Discrepancy | 0.002324 | 0.005744 | 0.006789 | 0.00787 | 0.009978 |

P value (two tailed) | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 |

On the other hand, the prediction results of the metamaterial bandwidth are recorded using five separate machine learning regressors and two ensemble models in addition to the proposed weighted ensemble model. These results are analysis using eight evaluation criteria and the results are presented in

RMSE | MSE | MBE | r | R2 | RRNSE | NSE | WI | |
---|---|---|---|---|---|---|---|---|

MLP | 0.101 | 0.067 | −0.022 | 0.758 | 0.575 | 11.009 | 0.542 | 0.616 |

KNR | 0.105 | 0.029 | −0.016 | 0.734 | 0.539 | 11.415 | 0.508 | 0.833 |

DT | 0.060 | 0.017 | −0.004 | 0.917 | 0.840 | 6.576 | 0.837 | 0.899 |

SVR | 0.096 | 0.052 | −0.007 | 0.863 | 0.746 | 10.436 | 0.589 | 0.698 |

RF | 0.055 | 0.019 | −0.007 | 0.953 | 0.908 | 5.925 | 0.867 | 0.889 |

AVG Ensemble | 0.074 | 0.031 | −0.011 | 0.927 | 0.860 | 5.925 | 0.757 | 0.820 |

KNR Ensemble | 0.054 | 0.015 | −0.006 | 0.946 | 0.894 | 5.879 | 0.869 | 0.913 |

The histogram of the gain values using the proposed weighted ensemble model that is optimized by DTO algorithm and four other optimizers and the RMSE of the predicted bandwidth values using the proposed weighted ensemble model that is optimized by DTO algorithm and four other optimizers are presented in

Machine learning approaches are currently a big part of current study, and they’re likely to be huge in the future. The model utilized determines the accuracy of the forecast in large part. To choose the best characteristics from the metamaterial antenna dataset, this research use the DTO method. Metamaterial antennas are able to overcome the gain and bandwidth limitations of small antennas. Machine learning is attracting a lot of attention for its potential to improve solutions in a range of fields. For estimating the bandwidth and gain of the metamaterial antenna, the optimum ensemble model produced satisfactory results. SVR, RF, KNR, DT, and MLP are the fundamental models that have been examined. The best characteristics from the datasets were chosen using the DTO method. Five regression models were tested against the suggested technique. According to the data, the proposed method is better to others in terms of properly predicting antenna bandwidth and gain.

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R300), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.