Published in the Proceedings of the 4th TICSP Workshop on Computational Systems Biology (WCSB 2006), Tampere University of Technology, Finland, June 12-13, 2006, pp. 27-30. MULTILAYER NEURAL NETWORK BASED ON MULTI-VALUED NEURONS (MLMVN) APPLIED TO CLASSIFICATION OF MICRORARRAY GENE EXPRESSION DATA Igor Aizenberg 1 , Pekka Ruusuvuori 2 , Olli Yli-Harja 2 and Jaakko T. Astola 2 1 Texas A&M University-Texarkana Department of Computer and Information Sciences P.O. Box 5518, 2600 N. Robison Rd. Texarkana, Texas 75505 USA, 2 Institute of Signal Processing, Tampere University of Technology, P.O. Box 553, FI-33101 Tampere, Finland, igor.aizenberg@gmail.com, pekka.ruusuvuori@tut.fi, olli.yli-harja@tut.fi, jaakko.astola@tut.fi ABSTRACT Classification of microarray gene expression data is a common problem in bioinformatics. Classification problems with more than two output classes require more attention than the normal binary classification. Here we apply a multilayer neural network based on multi-valued neurons (MLMVN) to the multiclass classification of microarray gene expression data. Two four-class test cases are considered. The results show that MLMVN can be used for classifying microarray data accurately. 1. INTRODUCTION A multilayer neural network based on multi-valued neurons (MLMVN) has been introduced in [1] and then it has been developed in [2]. This network and its backpropagation learning is comprehensively observed and developed further in [3]. The MLMVN consists of multi-valued neurons (MVN). That is a neuron with complex-valued weights and an activation function, defined as a function of the argument of a weighted sum. MVN is based on the principles of multiple-valued threshold logic over the field of complex numbers. A comprehensive observation of the discrete-valued MVN, its properties and learning is presented in [4]. A continuous-valued MVN and its learning are considered in [1]-[3]. The most important properties of MVN are: the complex-valued weights, inputs and output coded by the k th roots of unity (a discrete-valued MVN) or lying on the unit circle (a continuous-valued MVN), and an activation function, which maps the complex plane into the unit circle. Both MVN and MLMVN learning are reduced to the movement along the unit circle. The most important property and advantage of their learning is that it does not require differentiability of the activation function. The MVN learning algorithm [3], [4] is based on a simple linear error correction rule. This learning rule is generalized for the MLMVN as a backpropagation learning algorithm [3], which is simpler and more efficient than traditional backpropagation learning. MLMVN outperforms a classical multilayer feedforward network (usually referred to as a multilayer perceptron - MLP) and different kernel-based networks in the terms of learning speed, network complexity, and classification/prediction rate tested for such popular benchmark problems as the parity n, the two spirals, the sonar, and the Mackey-Glass time series prediction [1]- [3]. These properties of MLMVN show that it is more flexible and adapts faster in comparison with other solutions based on neural networks. It is important to note that since MLMVN (as well as a single MVN) implements such mappings that are described by multiple-valued (up to infinite-valued) functions, it can be an efficient mean for solving the multiclass classification problems. In this paper we apply MLMVN to the multiclass classification of microarray gene expression data. After presenting the basic properties of MLMVN and its backpropagation learning algorithm we will consider two four-class test cases of microarray gene expression data classification. The classification results of MLMVN classifier are compared to those given by nearest neighbor classifiers with different number of neighbors. 2. MULTILAYER NEURAL NETWORK BASED ON MULTI-VALUED NEURONS 2.1. Multi-valued neuron (MVN) MVN [4] is a neural element based on the principles of multiple-valued threshold logic over the field of complex numbers. A single MVN performs a mapping between n inputs and a single output. For the discrete-valued MVN this mapping is described by a multiple-valued (k- valued) function of n variables 1 ( ) n f x , ..., x with n+1 complex-valued weights as parameters: 1 0 1 1 ( ) ( ) n n n f x , ..., x Pw wx ... wx = + + + , (1) where 1 ( ) n X x ,...,x = is a vector of inputs (a pattern vector) and 0 1 ( ) n W w ,w , ...,w = is a weighting vector. The inputs and output of the discrete-valued MVN are the k th roots of unity: exp( 2 ) j i j/K ε π = , 0 ..., 1 j , k- = ,