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The Kohonen self-organising map as an alternative to cluster analysis: an application to direct marketing.

Article Excerpt
This paper examines the potential of the Kohonen self-organising map (SOM) in a marketing context. It deals specifically with consumer attitudes towards direct marketing. The SOM belongs to the general class of neural network (NN) models, but differs from the now orthodox way in which NNs are implemented. The major difference is that network learning is 'unsupervised', in which case the SOM is related to clustering methods. The result of an SOM is a two-dimensional grid of related 'prototypes' rather than non-overlapping clusters. The method involves iterative adjustment of the prototypes in such a way as to capture and preserve the properties of the data. We show how the resulting maps offer useful new perspectives.

Introduction

This paper describes the use of a technique relatively new in the Marketing field, namely the Kohonen self-organising map (SOM), which analyses consumer data regarding attitudes to direct marketing. The SOM (see for example Kohonen 1995) is generally regarded as a form of neural network (NN). In common with other better known NN models its applications are as a statistical technique. For marketing purposes, the SOM provides a method of segmentation which differs from traditional statistical techniques such as cluster analysis. Traditional clustering methods (Everitt 1993) may involve a variety of algorithms but almost invariably build distinct self-contained clusters; in contrast, an SOM market map shows neighbouring segments which are linked by having many similar characteristics but differ perhaps on one or two.

The main technical difference between the SOM and more 'conventional' neural networks is that it involves 'unsupervised' learning (i.e. without targets or outputs) and is more closely related to techniques for data transformation or data reduction than it is to predictive methods such as regression analysis.

The aim of this paper is to provide a further examination of the value of the SOM in a marketing context. As our literature review shows, there has been a relatively limited coverage of the technique. It could perhaps be said that the paper by Mazanec (1995) is the only one devoted to marketing issues per se. Our own paper is an attempt to provide an additional case study of the technique by applying it to the study of consumer attitudes towards direct marketing. We are interested in the value of the SOM vis-a-vis more traditional statistical techniques such as regression, factor analysis and cluster analysis. As well as technical comparisons, we are interested in the 'value added' offered by the feature maps produced by SOMs. We seek to derive a taxonomy of consumers and to discover previously hidden structure in the data, as represented in the 'market maps' which are obtained. Such added value can be seen to depend significantly on the skills of the analyst and will inevitably vary according to the circumstances in which it is employed.

Basic properties of the SOM

It is well known that the most important property of feedforward NNs such as the backpropagation network, is that they can provide approximations to almost any 'reasonable' underlying function: see for example Curry et al. (2001). SOMs can also be considered in a similar way, although little emphasis has been placed on this aspect in the literature on their applications. Kohonen (1995) views the SOM as a device for data transformation or reduction: the resulting map can be regarded as an approximation to the probability distribution generating the data. In the terminology of the SOM, the transformation to a two-dimensional grid of network nodes is 'topology preserving': i.e. it preserves the underlying properties of the data. In this sense the SOM can also be seen as a non-linear version of principal components analysis which is based on linear transformations of the data.

Along similar lines, the SOM has been shown (Mulier & Cherkassy 1995) as being an implied kernel smoothing process. Kernel methods (Hardle 1992) are another interesting statistical technique for non-linear modelling without assumed functional forms, and Mulier and Cherkassy have pointed out that there are important formal similarities between kernel estimates and the implied non-linear transformation carried out in the SOM.

The main idea behind the SOM is that a set of input data (input vectors) is subject to a topology-preserving transformation such that the data are effectively described by a collection of 'prototypes' (the SOM equivalent of clusters). Each node in the grid is a prototype in the sense that it possesses a set of weights which are values for the set of inputs. The position of each node in the grid vis-a-vis its neighbouring nodes is of major importance, particularly during the training process.

The network model can be considered as having two main groupings of nodes. In the first place we have input nodes, which are essentially the same as inputs in more standard networks. Each node represents a measurable attribute relating to data points. An input vector is a collection of attribute measures for each data unit, e.g. a firm or consumer. What gives the SOM its primary distinctive feature is the two-dimensional grid of Kohonen nodes. The grid serves to relate the nodes together, rather than them being taken as separate clusters. Each node in the grid is a 'prototype' rather than a cluster in the conventional sense. It represents a particular set of attribute values, these being comprised of its weights. (In our attitude analysis later in the paper we use the more usual marketing term 'segments'.) For each Kohonen node the number of weights is the same as the number of inputs to the network. The structure of the network is illustrated in Figure 1.

[FIGURE 1 OMITTED]

Once the weights have been established, the network operates simply by finding the Kohonen node which is the nearest match to a given input vector, measured in terms of the Euclidean distance between the input vector and the weights of the node. This classifies the input data by linking each data point to a single prototype or segment.

In the standard approach, referred to as Kohonen's training rule, actually establishing the weights ('training' in NN parlance) involves a similar theme, giving rise to 'competitive' or 'winner takes all' learning. Input vectors are presented repeatedly to the network, as with more conventional models, and at each presentation the 'winning' Kohonen node is identified. This being the prototype for which the weights are the best representation of a particular input vector, the weights are then adjusted to move nearer towards that vector. Where the SOM becomes more interesting however is through the fact that it is not only the winning node which is adjusted. Other nodes within a defined 'neighbourhood' of the winner are also subject to adjustment, thus exploiting the fact that the nodes are positioned within a grid. The shape of the neighbourhood may for example be characterised by a square or a diamond: alternatively, a Gaussian or other decay function may be used to model the spatial aspect (see e.g. Ritter et al. 1992).

More formally, we denote the input data by an m by n matrix X, each row of which contains a data point (vector) of observed values. Each node k in the SOM grid is characterised by a one by n vector [w.sup.(k)] of weights. The Euclidean distance between the kth node and the jth input vector is then given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the observed values of the attributes of each data vector are indexed by i.

During training, the winning node is that with the smallest distance from the current data vector. The distance is in fact modified to allow for the frequency with which nodes have...

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