Self-Organizing Maps for imprecise data

Abstract

Self-Organizing Maps (SOMs) consist of a set of neurons arranged in such a way that there are neighbourhood relationships among neurons. Following an unsupervised learning procedure, the input space is divided into regions with common nearest neuron (vector quantization), allowing clustering of the input vectors. In this paper, we propose an extension of the SOMs for data imprecisely observed (Self-Organizing Maps for imprecise data, SOMs-ID). The learning algorithm is based on two distances for imprecise data. In order to illustrate the main features and to compare the performances of the proposed method, we provide a simulation study and different substantive applications.

Introduction

Self-Organizing Maps (SOMs) (Kohonen [1], [2]) consist of a set of neurons arranged in a linear (one-dimensional) or rectangular (two-dimensional) configuration, such that there are neighbourhood relations among the neurons. Each neuron is attached to a weight vector of the same dimension as the input space (the multi-dimensional space of the units or input vectors). After completion of training, by assigning each input vector to the neuron with the nearest weight vector (reference vector), the SOMs are able to divide the input space into regions with common nearest weight vector (vector quantization), allowing clustering of the input vectors. Moreover, under appropriate training, because of the neighbourhood relation contributed by the interconnection among neurons, the SOMs exhibit the important property of topology preservation. In other words, if two input vectors are close in the input space, the corresponding reference vectors (closest neurons) will also be close in the neural network. Therefore, at least in two-dimensional neural networks, visualization is also possible. The density of the weight vectors of an organized map reflects the density of the input: in clustered areas the weight vectors are close to each other, and in the empty space between the clusters they are more sparse.

In the literature, a great deal of attention has been paid to the SOMs for traditional (numeric) data (“precise” observations). However, there are various real situations in which the data are not precisely observed (imprecise observations). In the literature on the SOMs, there are only few works. Bock [3], [4] proposed to visualize symbolic data (i.e. interval-valued data) by using SOMs. Chen et al. [5] suggested a batch version of the SOMs for symbolic data. DʼUrso and De Giovanni [6] proposed Midpoint Radius-Based Self-Organizing Maps (MR-SOMs) for interval-valued data showing a suggestive telecommunications application. Hajjar and Hamdan [7] suggested an algorithm to train the SOMs for interval data based on city-block distance. Yang et al. [8] suggested SOMs for symbolic data. According to our knowledge, in the literature, there are no works on SOMs for fuzzy data.

In this paper, by considering two distance measures for fuzzy data [9], [10], we proposed SOMs allowing clustering and vector quantization of imprecise data (i.e. fuzzy data).

In particular, in Section 2 we show the fuzzy management of data imprecisely observed. In Section 3, following a fuzzy formalization of the data, two distance measures for imprecise observations are illustrated [9], [10]; in particular, we prove that the measure introduced by Coppi et al. [9] is a metric. These metrics are utilized in the Self-Organizing Maps for imprecise data (SOMs-ID) proposed in Section 4. In order to point out the performances of the suggested SOMs-ID a simulation study is illustrated in Section 5. Several suggestive applications are shown in Section 6. Final remarks are made in Section 7.