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k,y := N,l ; { Invariant: O ^ f c ^ N A y = X5 where 5x2 k = ( Bound function: k } do fc>0 k,y := k-l,y2 ; if a[fc] = 0 skip

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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In order to illustrate the technique, attention will be focused initially on the former of the two cases. First, consider evaluation of the entity F(a) as defined in equation (6.18); using this particular choice ofwr and moving into plane polar coordinates gives:

{ y = XM } .

Using equation (6.47) to substitute r with r', noting that r'dr' rT_drT_, and accounting for the change in limits, i.e. when:

You should make use of Exercise 10.21 in order to develop the loop body. (In fact, the reason Exercise 10.21 was included was in order to prepare the way for the current exercise.) You will need to make the substitutions XM for C and z for

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

then obtain:

Exercise 13.16 (All Zeros). Given an array of integers, specify formally and develop an algorithm that will determine whether all values stored in the array are zero. Specify formally and develop an algorithm that will determine whether at least one of the values stored in the array is zero. D Exercise 13.17 (Binary Split). Given an array of booleans, specify formally and develop an algorithm that will determine whether there is an index k such that all array elements with index less than k are true and all elements with index at least k are false. What answer should be returned if (a) the length of the array is zero, (b) all elements of the array are true, and (c) all elements of the array are false

and by integrating by parts:

Exercise 13.18 (Array Equality). Given are two arrays a and b both of length M. Given also is a binary relation R. (Think of the equality relation, or the less-than relation.) Specify formally and develop a program to determine whether corresponding elements of a and b are all related by relation R. D

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