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features into a small number of intervals, where each interval is mapped to a discrete symbol. The benefits of these techniques are simplified data description and easy-to-understand data and final data-mining results. Also, more data-mining techniques are applicable with discrete feature values. An “old-fashioned” discretization is made manually, based on our a priori knowledge about the feature. For example, using common sense or consensus, a person’s age, given at the beginning of a data-mining process as a continuous value (between 0 and 150 years), may be classified into categorical segments: child, adolescent, adult, middle age, and elderly. Cutoff points are subjectively defined (Fig. 3.5). Two main questions exist about this reduction process:

1. What are the cutoff points?

2. How does one select representatives of intervals?

Figure 3.5. Discretization of the age feature.

Without any knowledge about a feature, a discretization is much more difficult and, in many cases, arbitrary. A reduction in feature values usually is not harmful for real-world data-mining applications, and it leads to a major decrease in computational complexity. Therefore, we will introduce, in the next two sections, several automated discretization techniques.

Within a column of a data set (set of feature values), the number of distinct values can be counted. If this number can be reduced, many data-mining methods, especially the logic-based methods explained in Chapter 6, will increase the quality of a data analysis. Reducing the number of values by smoothing feature values does not require a complex algorithm because each feature is smoothed independently of other features and the process is performed only once, without iterations.

Suppose that a feature has a range of numeric values, and that these values can be ordered from the smallest to the largest using standard greater-than and less-than operators. This leads naturally to the concept of placing the values in bins—partitioning into groups with close values. Typically, these bins have a close number of elements. All values in a bin will be merged into a single concept represented by a single value—usually either the mean or median of the bin’s values. The mean or the mode is effective for a moderate or large number of bins. When the number of bins is small, the closest boundaries of each bin can be candidates for representatives in a given bin.

For example, if a set of values for a given feature f is {3, 2, 1, 5, 4, 3, 1, 7, 5, 3}, then, after sorting, these values will be organized into an ordered set:

Now, it is possible to split the total set of values into three bins with a close number of elements in each bin:

In the next step, different representatives can be selected for each bin. If the smoothing is performed based on bin modes, the new set of values for each bin will be

If the smoothing is performed based on mean values, then the new distribution for reduced set of values will be

and finally, if all the values in a bin are replaced by the closest of the boundary values, the new set will be

One of the main problems of this method is to find the best cutoffs for bins. In theory, a decision about cutoffs cannot be made independently of other features. Still, heuristic decisions for every feature independently give good results in many data-mining applications. The value-reduction problem can be stated as an optimization problem in the selection of k bins. Given the number of k bins, distribute the values in the bins to minimize the average distance of a value from its bin mean or median. The distance is usually measured as the squared distance for a bin mean and as the absolute distance for a bin median. This algorithm can be computationally very complex, and a modified heuristic procedure is used to produce a near-optimum solution. The procedure consists of the following steps:

1. Sort all values for a given feature.

2. Assign approximately equal numbers of sorted adjacent values (vi) to each bin, where the number of bins is given in advance.