A great deal of interesting real-world data is encountered through the analysis of continuous variables, however many of the robust tools for rule discovery and data characterization depend upon the underlying data existing in an ordinal, enumerable or discrete data domain. Tools that fall into this category include much of the current work in fuzzy logic and rough sets, as well as all forms of event-based pattern discovery tools based on probabilistic inference. Through the application of discretization techniques, continuous data is made accessible to the analysis provided by the strong tools of discrete-valued data mining. The most common approach for discretization is quantization, in which the range of observed continuous valued data are assigned to a fixed number of quanta, each of which covers a particular portion of the range within the bounds provided by the most extreme points observed within the continuous domain. This chapter explores the effects such quantization may have, and the techniques that are available to ameliorate the negative effects of these efforts, notably fuzzy systems and rough sets.
Real-world data sets are only infrequently composed of discrete data, and any reasonable knowledge discovery approach must take into account the fact that the underlying data will be based on continuous-valued or mixed mode data. If one examines the data at the UCI Machine-Learning Repository (Newman, Hettich, Blake & Merz, 1998) one will see that many of the data sets within this group are continuous-valued; the majority of the remainder are based on measurements of continuous valued random variables that have been pre-quantized before being placed in the database.
The tools of the data mining community may be considered to fall into the following three groups:
minimum-error-fit and other gradient descent models, such as: support vector machines (Cristianini & Shawe-Taylor, 2000; Duda, Hart & Stork, 2001; Camps-Valls, Martínez-Ramón, Rojo-Álvarez & Soria-Olivas, 2004); neural networks (Rumelhart, Hinton & Williams, 1986); and other kernel or radial-basis networks (Duda, Hart & Stork, 2001; Pham, 2006)
Bayesian-based learning tools (Duda, Hart & Stork, 2001), including related random-variable methods such as Parzen window estimation
statistically based pattern and knowledge discovery algorithms based on an event-based model. Into this category falls much of the work in rough sets (Grzymala-Busse, & Ziarko, 1999; Pawlak, 1982,1992; Singh & Minz, 2007; Slezak & Wroblewski, 2006), fuzzy knowledge representation (Boyen & Wehenkel, 1999; Gabrys 2004; Hathaway & Bezdek 2002; Höppner, Klawonn, Kruse & Runkler, 1999), as well as true statistical methods such as “pattern discovery” (Wang & Wong, 2003; Wong & Wang, 2003; Hamilton-Wright & Stashuk, 2005, 2006).
The methods in the last category are most affected by quantization and as such will be specifically discussed in this chapter. These algorithms function by constructing rules based on the observed association of data values among different quanta. The occurrence of a feature value within particular quanta may be considered an “event” and thereby all of the tools of information theory may be brought to bear. Without the aggregation of data into quanta, it is not possible to generate an accurate count of event occurrence or estimate of inter-event relationships.