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Kohonen developed the self-organizing feature map (SOM) [474, 475, 476], as motivated by the self-organization characteristics of the human cerebral cortex Studies of the cerebral cortex showed that the motor cortex, somatosensory cortex, visual cortex and auditory cortex are represented by topologically ordered maps These topological maps form to represent the structures sensed in the sensory input signals The self-organizing feature map is a multidimensional scaling method to project an I-dimensional input space to a discrete output space, e ectively performing a compression of input space onto a set of codebook vectors The output space is usually a two-dimensional grid The SOM uses the grid to approximate the probability density function of the input space, while still maintaining the topological structure of input space That is, if two vectors are close to one another in input space, so is the case for the map representation The SOM closely resembles the learning vector quantizer discussed in the previous section The di erence between the two unsupervised algorithms is that neurons are usually organized on a rectangular grid for SOM, and neighbors are updated to also perform an ordering of the neurons In the process, SOMs e ectively cluster the input vectors through a competitive learning process, while maintaining the topological structure of the input space Section 451 explains the standard stochastic SOM training rule, while a batch version is discussed in Section 452 A growing approach to SOM is given in Section 453 Di erent approaches to speed up the training of SOMs are overviewed in Section 454 Section 455 explains the formation of clusters for visualization purposes Section 456 discusses in brief di erent ways how the SOM can be used after training

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SOM training is based on a competitive learning strategy Assume I-dimensional input vectors zp , where the subscript p denotes a single training pattern The rst step of the training process is to de ne a map structure, usually a two-dimensional grid (refer to Figure 43) The map is usually square, but can be of any rectangular shape The number of elements (neurons) in the map is less than the number of training patterns Ideally, the number of neurons should be equal to the number of independent training patterns

Figure 43 Self-organizing Map Each neuron on the map is associated with and I-dimensional weight vector that forms the centroid of one cluster Larger cluster groupings are formed by grouping together similar neighboring neurons Initialization of the codebook vectors can occur in various ways: Assign random values to each weight wkj = (wkj1 , wkj2 , , wKJI ), with K the number of rows and J the number of columns of the map The initial values are bounded by the range of the corresponding input parameter While random initialization of weight vectors is simple to implement, this form of initialization introduces large variance components into the map which increases training time Assign to the codebook vectors randomly selected input patterns That is, wkj = zp with p U (1, PT ) This approach may lead to premature convergence, unless weights are perturbed with small random values Find the principal components of the input space, and initialize the codebook vectors to re ect these principal components A di erent technique of weight initialization is due to Su et al [818], where the objective is to de ne a large enough hyper cube to cover all the training patterns [818] The algorithm starts by nding the four extreme points of the map by determining the four extreme training patterns Firstly, two patterns are found with the largest inter-pattern Euclidean distance A third pattern is (425)

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