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where r and s are cluster indices, nr and ns are the number of patterns within the clusters, and wr and ws are the centroid vectors of these clusters (i.e. the average of all the codebook vectors within the cluster). The two clusters are merged if their distance, drs, is the smallest. For the newly formed cluster, q, wq = 1
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Note that, in order to preserve topological structure, two clusters can only be merged if they are adjacent. Furthermore, only clusters that have a nonzero number of patterns associated with them are merged.
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The SOM has been applied to a variety of real-world problems, including image analysis, speech recognition, music pattern analysis, signal processing, robotics, telecommunications, electronic-circuit design, knowledge discovery and time series analysis. The main advantage of SOMs comes from the easy visualization and interpretation of clusters formed by the map. In addition to visualizing the complete map as illustrated in Figure 4.4(b), the relative component values in the codebook vectors can be visualized as illustrated in the same figure. Here a component refers to an input attribute. That is, a component plane can be constructed for each input parameter (component) to visualize the distribution of the corresponding weight (using some color scale representation). The map and component planes can be used for exploratory data analysis. For example, a marked region on the visualized map can be projected onto the component planes to find the values of the input parameters for that region. A trained SOM can also be used as a classifier. However, since no target information is available during training, the clusters formed by the map should be manually inspected and labeled. A data vector is then presented to the map, and the winning neuron determined. The corresponding cluster label is then used as the class. Used in recall mode, the SOM can be used to interpolate missing values within a pattern. Given such a pattern, the BMN is determined, ignoring the inputs with missing values. A value is then found by either replacing the missing value with the corresponding weight of the BMN, or through interpolation among a neighborhood of neurons (e.g. take the average of the weight values of all neurons in the neighborhood of the BMN).
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This chapter gave a short introduction to unsupervised learning algorithms, with emphasis on LVQ-I and SOMs. These algorithms are very useful in performing clustering, with applications in analysis of mammograms and landsat images, customer profiling, stock prediction, and many more. The next chapter presents learning
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1. Implement and test a LVQ-I network to distinguish between different alphabetical characters of different fonts. 2. Explain why it is necessary to retrain a supervised NN on all the training data, including any new data that becomes available at a later stage. Why is this not an issue with unsupervised NNs 3. Discuss an approach to optimize the LVQ-I network architecture. 4. How can PSO be used for unsupervised learning 5. What is the main difference between the LVQ-I and SOM as an approach to cluster multi-dimensional data 6. For a SOM, if the training set contains PT patterns, what is the upper bound on the number of neurons necessary to fit the data Justify your answer. 7. Explain the purpose of the neighborhood function of SOMs. 8. Assuming a Gaussian neighborhood function for SOMs, what can be done to reduce the number of weight updates in a sensible way 9. Explain how a SOM can be used to distinguish among different hand gestures. 10. Discuss a number of ways in which the SOM can be adapted to reduce its computational complexity.
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