Extending Hybrid Random Fields: Continuous-Valued Variables

In Section 1.2 we introduced the concept of feature, or attribute. The idea is that, in order to apply statistics or learning machines to phenomena occurring in the real world, it is necessary to describe them in a proper feature space, such that each phenomenon can be thought of as a random variable (if a single attribute is used), or a random vector (if multiple features are extracted). The feature extraction process is crucial for the success of an application, since there is no good model that can compensate for wrong, or poor, features. Although several types of attributes are sometimes referred to in the literature (e.g., categorical, nominal, ordinals, etc.), two main families of features can be pointed out, namely discrete and continuous-valued (or, simply, continuous). Attributes are discrete when they belong to a domain which is a countable set, such that each feature can be seen as a symbol from a given alphabet. Examples of discrete features spaces are any subsets of the integer numbers, or arbitrary collections of alphanumeric characters. Continuous-valued feature spaces are compact subsets of ℝ—in which case the attribute is thought of as the outcome of a real-valued random variable—or ℝ

d

for a certain integer

d

 > 1—in which case the

d

-dimensional feature vector is assumed to be the outcome of a real-valued random vector.