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45 Self-Organizing Feature Maps
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Shortcut winner search does not perform a search for the BMN over the entire map, but just within the neighborhood of the previous BMN, thereby substantially reducing computational complexity
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Clustering and Visualization
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The e ect of the SOM training process is to cluster together similar patterns, while preserving the topology of input space After training, all that is given is the set of trained weights with no explicit cluster boundaries An additional step is required to nd these cluster boundaries One way to determine and visualize these cluster boundaries is to calculate the uni ed distance matrix (U-matrix) [403], which contains a geometrical approximation of the codebook vector distribution in the map The U-matrix expresses for each neuron, the distance to the neighboring codebook vectors Large values within the U-matrix indicate the position of cluster boundaries Using a gray-scale scheme, Figure 44(a) visualizes the U-matrix for the iris classi cation problem For the same problem, Figure 44(b) visualizes the clusters on the actual map Boundaries are usually found by using Ward clustering [23] of the codebook vectors Ward clustering follows a bottom-up approach where each neuron initially forms its own cluster At consecutive iterations, two clusters that are closest to one another are merged, until the optimal or speci ed number of clusters has been constructed The end result of Ward clustering is a set of clusters with a small variance over its members, and a large variance between separate clusters The Ward distance measure is used to decide which clusters should be merged The distance measure is de ned as drs = nr ns ||wr ws ||2 2 nr + ns (441)
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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 (ie 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 = and nq = nr + ns (443) 1 (nr wr + ns ws ) nr + ns (442)
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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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4 Unsupervised Learning Neural Networks
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(a) U-Matrix
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(b) Map Illustration
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(c) Component Map for Third Input (d) Component Map for Fourth Input
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Figure 44 Visualization of SOM Clusters for Iris Classi cation
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46 Assignments
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Using SOM
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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 44(b), the relative component values in the codebook vectors can be visualized as illustrated in the same gure 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 nd the values of the input parameters for that region A trained SOM can also be used as a classi er 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 (eg take the average of the weight values of all neurons in the neighborhood of the BMN)
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