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by first finding one or a few points (called split points or cut points) to split the entire attribute range, and then repeats this recursively on the resulting intervals, it is called top-down discretization or splitting. This contrasts with bottom-up discretization or merging, which starts by considering all of the continuous values as potential split-points, removes some by merging neighborhood values to form intervals, and then recursively applies this process to the resulting intervals.

Data discretization and concept hierarchy generation are also forms of data reduction. The raw data are replaced by a smaller number of interval or concept labels. This simplifies the original data and makes the mining more efficient. The resulting patterns mined are typically easier to understand. Concept hierarchies are also useful for mining at multiple abstraction levels.

The rest of this section is organized as follows. First, normalization techniques are presented in Section 3.5.2. We then describe several techniques for data discretization, each of which can be used to generate concept hierarchies for numeric attributes. The techniques include binning (Section 3.5.3) and histogram analysis (Section 3.5.4), as well as cluster analysis, decision tree analysis, and correlation analysis (Section 3.5.5). Finally, Section 3.5.6 describes the automatic generation of concept hierarchies for nominal data.

3.5.2. Data Transformation by Normalization

The measurement unit used can affect the data analysis. For example, changing measurement units from meters to inches for height, or from kilograms to pounds for weight, may lead to very different results. In general, expressing an attribute in smaller units will lead to a larger range for that attribute, and thus tend to give such an attribute greater effect or “weight.” To help avoid dependence on the choice of measurement units, the data should be normalized or standardized. This involves transforming the data to fall within a smaller or common range such as [−1, 1] or [0.0, 1.0]. (The terms standardize and normalize are used interchangeably in data preprocessing, although in statistics, the latter term also has other connotations.)

Normalizing the data attempts to give all attributes an equal weight. Normalization is particularly useful for classification algorithms involving neural networks or distance measurements such as nearest-neighbor classification and clustering. If using the neural network backpropagation algorithm for classification mining (Chapter 9), normalizing the input values for each attribute measured in the training tuples will help speed up the learning phase. For distance-based methods, normalization helps prevent attributes with initially large ranges (e.g., income) from outweighing attributes with initially smaller ranges (e.g., binary attributes). It is also useful when given no prior knowledge of the data.

There are many methods for data normalization. We study min-max normalization, z-score normalization, and normalization by decimal scaling. For our discussion, let A be a numeric attribute with n observed values, v1, v2, …, vn.

Min-max normalization performs a linear transformation on the original data. Suppose that minA and maxA are the minimum and maximum values of an attribute, A. Min-max normalization maps a value, vi, of A to v′i in the range [new_minA, new_maxA] by computing

(3.8)

Min-max normalization preserves the relationships among the original data values. It will encounter an “out-of-bounds” error if a future input case for normalization falls outside of the original data range for A.

Min-max normalization

Suppose that the minimum and maximum values for the attribute income are $12,000 and $98,000, respectively. We would like to map income to the range [0.0, 1.0]. By min-max normalization, a value of $73,600 for income is transformed to .

In z -score normalization (or zero-mean normalization), the values for an attribute, A, are normalized based on the mean (i.e., average) and standard deviation of A. A value, vi, of A is normalized