Optimization of Circularly Polarized Omnidirectional Base Station Antenna by Machine Learning

Machine learning (ML) is a scientific technology related to data learning that can help machines learn patterns from existing complex data to predict future behavioral outcomes and trends. In this paper, we use 16 different ML algorithms and 5 different hyper-parameters optimal methods to model a circularly polarized omnidirectional base station antenna. The mean absolute error is 8.431, the root mean squared error is 11.100, the coefficient of determination is 0.967, and the adjusted coefficient of determination is 0.944, which means evaluation metrics are excellent.


Introduction
Machine learning (ML) is a multi-field inter-disciplinary that has emerged in recent years, and it involves probability and statistics theory, approximation theory, convex analysis, and algorithm complexity theory.It designs and analyzes algorithms that allow computers to automatically learn and analyze patterns from data, which are used to predict unknown data.ML has been widely applied, such as data mining, securities market analysis, natural language processing, computer vision, speech and handwriting recognition, search engines, strategic games, robot applications, biometric recognition, medical diagnosis, detection of credit card fraud, DNA sequence sequencing, as well as in the field of electromagnetics [1] [2].There are many concern ML algorithms applied in antennas optimization domain, including support vector machine [3], Gaussian process [4], deep Gaussian process [5], student's T process [6], extreme learning machine [7], broad learning system [8], artificial neural network [8], deep neural network [9], convolutional neural network [10], etc. Reference [3] measured electronically steerable parasitic array radiator patterns, and then used the support vector machine training process to handle antenna-based DOA estimation.Reference [4] developed a modeling method named progressive Gaussian process, which test sample with the largest predictive variance was inputted into an electromagnetic solver to compute its results when a Gaussian process was trained, and then added it to the training set to train the Gaussian process progressively.In [5], Gaussian process was used to substitute the fully connected layer of the convolutional neural network to form a kind of deep Gaussian process, and particle swarm optimization was adopted to optimize the network structure parameters simultaneously.In [6], student's T process was used as the surrogate model in Bayesian optimization algorithm, and the estimation strategy function was improved based on the posterior output, realizing the improved Bayesian optimization, finally used for the antennas optimization.In order to design UWB antennas, reference [7] proposed a deep learning model, and the deep belief network structure is determined by particle swarm algorithm, and then combined it with extreme learning machine.Aiming to speed up the antennas design by the easy-to-measure parameter variables achieving more accurate prediction of hard-to-measure quality variables, [8] proposed auto-context for the regression scenario, using the current broad learning system training results as prior knowledge, which was taken as the context information and then combined them with the original inputs becoming the new inputs for further training.Reference [9] presented a new deep neural network approach for the adaptive beamforming of antenna array, where a recurrent neural network based on the gated recurrent unit architecture was used as a beamformer for producing proper complex weights in order to feed the antenna array.In order to make full use of the characteristics of convolutional neural network, [10] converted the 1-dimension input of antenna into the form of an image model, which established a deep surrogate model between the physical parameters of the antenna and its electrical properties, and improved the model's accuracy and generalization ability.In this paper, we apply some ML methods, including Linear Regression, k-Nearest Neighbor, Support Vector Regressor, Multi Layer Perception Neural Network, Gaussian Process Regressor, Decision Tree, Bagging, Random Forest, Gradient Boosting, Extra Trees Regression, AdaBoost, XGBoost, LightGBM, CatBoost, Hist Gradient Boosting, Stacked Generalization, to model a circularly polarized omnidirectional base station antenna.Furthermore, we discuss the hyper-parameters tuning based on the Gird search, Randomized search, Halving grid search, Halving randomized search, and Bayesian optimization.

The Circularly Polarized Omnidirectional Base Station Antenna
The circularly polarized omnidirectional base station antenna, shown in Fig. 1, is from reference [11] and modeled in this paper.It is composed of six-stage cascaded vertical strips and helical loops, and fed by a tapered balun.The axial ration (AR) bandwidth over all the horizontal angles is its main performance, and the initial value is 64 MHz when the four helical loops are in the same orientation.At some certain horizontal angles, the AR bandwidths are very small, limiting the overall bandwidth, because the structure of the antenna is asymmetrical.By arranging the orientations of the four loops, we can achieve an even larger AR bandwidth.Here, when modeling the antenna, the input variables are the rotation angles of the four loops on the xoy plane, i.e., x = [ϕ1, ϕ2, ϕ3, ϕ4], where ϕ=0°is denoted along the x-axis.For each loop, we choose a step width of 120°, resulting in 81 samples simulated by CST Microwave Studio.

Evaluation Metrics
We can evaluate the performance of ML methods using some well-known performance metrics.Here, the mean absolute error (MAE) and the root mean squared error (RMSE) are used, and they are defined below: where n is the total number of test samples, , i o y is the actual value corresponding to the i-th input pattern, and , i p y is the ML predicted output when the i-th input data pattern is as input.In this case, the differences between the predicted values and the actual values are shown by the performance metrics values.Ideally, they are equal to zero.

Numerical Results
We use the simulated samples to model the antenna by ML algorithms, covering Linear Regression, k-Nearest Neighbor, Support Vector Regressor, Multi Layer Perception Neural Network, Gaussian Process Regressor, Decision Tree, Bagging, Random Forest, Gradient Boosting, Extra Trees Regression, AdaBoost, XGBoost, LightGBM, CatBoost, Hist Gradient Boosting, Stacked Generalization, and the evaluation metrics are listed in

The Hyper-parameters Optimization of the Extra Trees Regression
In this section, we will optimize the hyper-parameters optimization of the Extra Trees Regression by Gird search, Randomized search, Halving grid search, Halving randomized search, and Bayesian optimization.For Extra Trees Regression, it has many hyper-parameters.We select n_estimators, min_samples_split, min_samples_leaf, min_impurity_decrease as the optimized objects, where n_estimators is the trees number in the forest, min_samples_split is the samples minimum number required to split an internal node, min_samples_leaf is the samples minimum number required to be at a leaf node, and min_impurity_decrease is a value that a node will be split if this split induces a decrease of the impurity greater than or equal to the value.The optimized results of the Extra Tree Regression hyper-parameters and the corresponding evaluation metrics with ten-fold cross validation are listed in

Conclusion
In recent years, machine learning is developed fast, and it is applied in many fields, including antenna engineering.In this paper, we use 16 different machine learning algorithms to model a circularly polarized omnidirectional base station antenna.Moreover, we optimize the hyper-parameters of the best algorithm by 5 different optimal methods.The evaluation metrics including the mean absolute error, the root mean squared error, the coefficient of determination and the adjusted coefficient of determination are all excellent.

Figure 1 .
Figure 1.The circularly polarized omnidirectional base station antenna This work was supported by the Guangdong Basic and Applied Basic Research Foundation under Grant No. 2023A1515011272, the special project in key fields of Guangdong Universities of China under No. 2022ZDZX1020, and the Tertiary Education Scientific research project of Guangzhou Municipal Education Bureau of China under No. 202234598.

TABLE I .
It should be noted that the hyper-parameters of the ML algorithms in TABLE I are all with default setting.From TABLE I, we can see the Extra Tree Regression has the most small MAE and RMSE with red font, which means it is the best one.

Table 1 .
Evaluation Metrics of The Circularly Polarized Omnidirectional Base Station Antenna TABLE II .From TABLEII, we can conclude that the hyper-parameters (n_estimators, min_samples_split, min_samples_leaf, min_impurity_ decrease) optimized by Halving grid search is (150, 2, 1, 0.1) with the best evaluation metrics, whereas the optimized results given by Bayesian optimization are the worst because its surrogate model is Gaussian process that is not suitable for the circularly polarized omnidirectional base station antenna.In TABLE II , the coefficient of determination (R2) and the adjusted coefficient of determination ( 2R a is one.

Table 2 .
The Optimized Results Of The Hyper-parameters And Corresponding Evaluation Metrics