Fast learning Circular Complex-valued Extreme Learning Machine (CC-ELM) for real-valued classification problems R. Savitha a , S. Suresh b,⇑ , N. Sundararajan a a School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore b School of Computer Engineering, Nanyang Technological University, Singapore article info Article history: Received 9 November 2010 Received in revised form 14 September 2011 Accepted 6 November 2011 Available online 13 November 2011 Keywords: Complex-valued ELM Orthogonal decision boundaries Circular function Acoustic emission and mammogram classification abstract In this paper, we present a fast learning fully complex-valued extreme learning machine classifier, referred to as ‘Circular Complex-valued Extreme Learning Machine (CC-ELM)’ for handling real-valued classification problems. CC-ELM is a single hidden layer network with non-linear input and hidden layers and a linear output layer. A circular transforma- tion with a translational/rotational bias term that performs a one-to-one transformation of real-valued features to the complex plane is used as an activation function for the input neurons. The neurons in the hidden layer employ a fully complex-valued Gaussian-like (‘sech’) activation function. The input parameters of CC-ELM are chosen randomly and the output weights are computed analytically. This paper also presents an analytical proof to show that the decision boundaries of a single complex-valued neuron at the hidden and output layers of CC-ELM consist of two hyper-surfaces that intersect orthogonally. These orthogonal boundaries and the input circular transformation help CC-ELM to perform real-valued classification tasks efficiently. Performance of CC-ELM is evaluated using a set of benchmark real-valued classification problems from the University of California, Irvine machine learning repository. Finally, the performance of CC-ELM is compared with existing methods on two practical problems, viz., the acoustic emission signal classification problem and a mammogram classification problem. These study results show that CC-ELM performs better than other existing (both) real-valued and complex-valued classifiers, especially when the data sets are highly unbalanced. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction Complex-valued neural networks were originally developed for solving problems involving complex-valued signals. According to Liouville’s theorem, an analytic and bounded function is a constant in the complex plane. To overcome this restriction, non-linear, analytic and almost everywhere bounded fully complex-valued activation functions [14,25–27,42] have been used as activation functions in the literature. Several applications of complex-valued neural networks operating on complex-valued signals have been reported in the literature (e.g. communication channel equalization [26,7] adaptive array signal processing [6,31,28], image reconstruction [32], etc.). Recently, complex-valued neural networks have been shown to have better computational power than real-valued neural networks [16] and also that they are better in performing real-valued classification tasks because of their inherent orthogonal decision boundaries [18,19]. 0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.11.003 ⇑ Corresponding author. Tel.: +65 6790 6185; fax: +65 6792 6559. E-mail address: ssundaram@ntu.edu.sg (S. Suresh). Information Sciences 187 (2012) 277–290 Contents lists available at SciVerse ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins