Another way of classifying a variable, based on its values, is to look at it as a continuous variable or a discrete variable.
Continuous variables are also known as quantitative or metric variables. They are measured using either an interval scale or a ratio scale. Both scales allow the underlying variable to be defined or measured theoretically with infinite precision. The difference between these two scales lies in how the 0 point is defined in the scale. The 0 point in the interval scale is placed arbitrarily, and thus it does not indicate the complete absence of whatever is being measured. The best example of the interval scale is the temperature scale, where 0 degrees Fahrenheit does not mean a total absence of temperature. Because of the arbitrary placement of the 0 point, the ratio relation does not hold true for variables measured using interval scales. For example, 80 degrees Fahrenheit does not imply twice as much heat as 40 degrees Fahrenheit. In contrast, a ratio scale has an absolute 0 point and, consequently, the ratio relation holds true for variables measured using this scale. Quantities such as height, length, and salary use this type of scale. Continuous variables are represented in large data sets with values that are numbers—real or integers.
Discrete variables are also called qualitative variables. Such variables are measured, or their values defined, using one of two kinds of nonmetric scales—nominal or ordinal. A nominal scale is an orderless scale, which uses different symbols, characters, and numbers to represent the different states (values) of the variable being measured. An example of a nominal variable, a utility, customer-type identifier with possible values is residential, commercial, and industrial. These values can be coded alphabetically as A, B, and C, or numerically as 1, 2, or 3, but they do not have metric characteristics as the other numeric data have. Another example of a nominal attribute is the zip code field available in many data sets. In both examples, the numbers used to designate different attribute values have no particular order and no necessary relation to one another.
An ordinal scale consists of ordered, discrete gradations, for example, rankings. An ordinal variable is a categorical variable for which an order relation is defined but not a distance relation. Some examples of an ordinal attribute are the rank of a student in a class and the gold, silver, and bronze medal positions in a sports competition. The ordered scale need not be necessarily linear; for example, the difference between the students ranked fourth and fifth need not be identical to the difference between the students ranked 15th and 16th. All that can be established from an ordered scale for ordinal attributes with greater-than, equal-to, or less-than relations. Typically, ordinal variables encode a numeric variable onto a small set of overlapping intervals corresponding to the values of an ordinal variable. These ordinal variables are closely related to the linguistic or fuzzy variables commonly used in spoken English, for example, AGE (with values young, middle aged, and old) and INCOME (with values low-middle class, upper middle class, and rich). More examples are given in Figure 2.1, and the formalization and use of fuzzy values in a data-mining process are given in Chapter 14.
Figure 2.1. Variable types with examples.
A special class of discrete variables is periodic variables. A periodic variable is a feature for which the distance relation exists, but there is no order relation. Examples are days of the week, days of the month, or days of the year. Monday and Tuesday, as the values of a feature, are closer than Monday and Thursday, but Monday can come before or after Friday.
Finally, one additional dimension of classification of data is based on its behavior with respect to time. Some data do not change with time, and we consider them static data. On the other hand, there are attribute values that change with time, and this type of data we call dynamic or temporal data. The majority of data-mining methods are more suitable for static data, and special consideration and some preprocessing are often required to mine dynamic data.
Most data-mining problems arise because there are large amounts of samples with different types of features. Additionally, these samples are very often high dimensional, which means they have extremely large number of measurable features. This additional dimension of large data sets causes the problem known in data-mining terminology as “the curse of dimensionality.” The “curse of dimensionality” is produced because of the geometry of high-dimensional spaces, and these kinds of data spaces are typical for data-mining problems. The properties of high-dimensional spaces often appear counterintuitive because our experience with the physical world is in a low-dimensional space, such as a space with two or three dimensions. Conceptually, objects in high-dimensional spaces have a larger surface area for a given volume than objects in low-dimensional spaces. For example, a high-dimensional hypercube, if it could be visualized, would look like a porcupine, as in Figure 2.2. As the dimensionality grows larger, the edges grow longer relative to the size of the central part of the hypercube. Four important properties of high-dimensional data affect the interpretation of input data and data-mining results.
1. The size of a data set yielding the same density of data points in an
