ORIGINAL ARTICLE On robustness of radial basis function network with input perturbation Prasenjit Dey 1 • Madhumita Gopal 1 • Payal Pradhan 1 • Tandra Pal 1 Received: 30 June 2016 / Accepted: 15 June 2017 Ó The Natural Computing Applications Forum 2017 Abstract In this article, we have proposed a methodology for making a radial basis function network (RBFN) robust with respect to additive and multiplicative input noises. This is achieved by properly selecting the centers and widths for the radial basis function (RBF) units of the hidden layer. For this purpose, firstly, a set of self-orga- nizing map (SOM) networks are trained for center selec- tion. For training a SOM network, random Gaussian noise is injected in the samples of each class of the data set. The number of SOM networks is same as the number of classes present in the data set, and each of the SOM networks is trained separately by the samples belonging to a particular class. The weight vector associated with a unit in the output layer of a particular SOM network corresponding to a class is used as the center of a RBF unit for that class. To determine the widths of the RBF units, p-nearest neighbor algorithm is used class-wise. Proper selection of centers and widths makes the RBFN robust with respect to input perturbation and outliers present in the data set. The weights between the hidden and output layers of RBFN are obtained by pseudo inverse method. To test the robustness of the proposed method in additive and multiplicative noise scenarios, ten standard data sets have been used for classification. Proposed method has been compared with three existing methods, where the centers have been gen- erated in three ways: randomly, using k-means algorithm, and based on SOM network. Simulation results show the superiority of the proposed method compared to those methods. Wilcoxon signed-rank test also shows that the proposed method is statistically better than those methods. Keywords Additive noise  Gaussian noise  Multiplicative noise  Radial basis function network  Robustness  Self-organizing map 1 Introduction Radial basis function network (RBFN), a popular artificial neural network (ANN), has been used in many applications [1, 2]. However, like other neural network models, it is not free from input perturbation. It may encounter input per- turbation when applied to real life applications, e.g., the inputs, which come from electronic sensors, such as microphones, termopars, may be altered [3]. This alteration can be additive [4] or multiplicative [3]. When the per- turbation of an input is proportional to its magnitude, it is called multiplicative perturbation, and when perturbation is done with an additive Gaussian noise, it is called as addi- tive perturbation. Conventional RBFN is sensitive to changes in input [4]. It can be made robust to its input perturbation by proper selection of RBF parameter values [5, 6]. The dimensionality of the hidden layer is an important parameter to make a RBFN efficient. If the number of RBF units of the hidden layer is not sufficient, it may cause underfitting [7]. According to Cover’s theorem [8], the number of units in the hidden layer should be more than the & Prasenjit Dey pd.12cse1107@phd.nitdgp.ac.in Madhumita Gopal madhumitagopal9@gmail.com Payal Pradhan payal.unis.pradhan@gmail.com Tandra Pal tandra.pal@gmail.com 1 Department of Computer Science and Engineering, NIT Durgapur, Durgapur, India 123 Neural Comput & Applic DOI 10.1007/s00521-017-3086-5