Circular Complex-Valued GMDH-Type Neural Network for Real-Valued Classification Problems.

Recently, applications of complex-valued neural networks (CVNNs) to real-valued classification problems have attracted significant attention. However, most existing CVNNs are black-box models with poor explanation performance. This study extends the real-valued group method of data handling (RGMDH)-type neural network to the complex field and constructs a circular complex-valued group method of data handling (C-CGMDH)-type neural network, which is a white-box model. First, a complex least squares method is proposed for parameter estimation. Second, a new complex-valued symmetric regularity criterion is constructed with a logarithmic function to represent explicitly the magnitude and phase of the actual and predicted complex output to evaluate and select the middle candidate models. Furthermore, the property of this new complex-valued external criterion is proven to be similar to that of the real external criterion. Before training this model, a circular transformation is used to transform the real-valued input features to the complex field. Twenty-five real-valued classification data sets from the UCI Machine Learning Repository are used to conduct the experiments. The results show that both RGMDH and C-CGMDH models can select the most important features from the complete feature space through a self-organizing modeling process. Compared with RGMDH, the C-CGMDH model converges faster and selects fewer features. Furthermore, its classification performance is statistically significantly better than the benchmark complex-valued and real-valued models. Regarding time complexity, the C-CGMDH model is comparable with other models in dealing with the data sets that have few features. Finally, we demonstrate that the GMDH-type neural network can be interpretable.