# Cluster Analysis for Data Mining

What Cluster Analysis Is Cluster analysis groups objects (observations, events) based on the information found in the data describing the objects or their relationships. The goal is that the objects in a group will be similar (or related) to one other and different from (or unrelated to) the objects in other groups. The greater the similarity (or homogeneity) within a group, and the greater the difference between groups, the better or more distinct the clustering. The definition of what constitutes a cluster is not well defined, and, in many applications clusters are not well separated from one another. Nonetheless, most cluster analysis seeks as a result, a crisp classification of the data into non-overlapping groups. Fuzzy clustering, described in section 7.5, is an exception to this, and allows an object to partially belong to several groups. To better understand the difficulty of deciding what constitutes a cluster, consider figures 1a through 1d, which show twenty points and three different ways that they can be divided into clusters. If we allow clusters to be nested, then the most reasonable interpretation of the structure of these points is that there are two clusters, each of which has three subclusters. However, the apparent division of the two larger clusters into three subclusters may simply be an artifact of the human visual system. Finally, it may not be unreasonable to say that the points form four clusters. Thus, we stress once again that the definition of what constitutes a cluster is imprecise, and the best definition depends on the type of data and the desired results. The Data Matrix Objects (samples, measurements, patterns, events) are usually represented as points (vectors) in a multi-dimensional space, where each dimension represents a distinct attribute (variable, measurement) describing the object. For simplicity, it is normally assumed that values are present for all attributes. (Techniques for dealing with missing values are described in section 9.1.) Thus, a set of objects is represented (at least conceptually) as an m by n matrix, where there are m rows, one for each object, and n columns, one for each attribute. This matrix has different names, e.g., pattern matrix or data matrix, depending on the particular field.Figure 2, below, provides a concrete example of some points and their corresponding data matrix. The data is sometimes transformed before being used. One reason for this is that different attributes may be measured on different scales, e.g., centimeters and kilograms. In cases where the range of values differs widely from attribute to attribute, these differing attribute scales can dominate the results of the cluster analysis and it is common to standardize the data so that all attributes are on the same scale.

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