Data Mining - Mehmed Kantardzic [43]
Practical experiments show that noise reduction and consequently better model estimation may be obtained by adding features that are presumably redundant. Therefore, we have to be very careful in the preprocessing analysis. Yes, perfectly correlated variables are truly redundant in the sense that no additional information is gained by adding them. But, even variables with relatively high correlation (or anti-correlation) do not guarantee absence of variables’ complementarity. We can find cases where a feature looks completely useless by itself, and it is ranked very low, but it can provide significant information to the model and performance improvement when taken with others. These features by themselves may have little correlation with the output, target concept, but when combined with some other features, they can be strongly correlated with the target feature. Unintentional removal of these features can result in poor mining performance.
The previous simple methods test features separately. Several features may be useful when considered separately, but they may be redundant in their predictive ability. If the features are examined collectively, instead of independently, additional information can be obtained about their characteristics and mutual relations. Assuming normal distributions of values, it is possible to describe an efficient technique for selecting subsets of features. Two descriptors characterize a multivariate normal distribution:
1. M, a vector of the m feature means, and
2. C, an m × m covariance matrix of the means, where Ci,i are simply the variance of feature i, and Ci,j terms are correlations between each pair of features
where
v(k,i) and v(k,j) are the values of features indexed with i and j,
m(i) and m(j) are feature means, and
n is the number of dimensions.
These two descriptors, M and C, provide a basis for detecting redundancies in a set of features. If two classes exist in a data set, then the heuristic measure, DM, for filtering features that separate the two classes is defined as
where M1 and C1 are descriptors of samples for the first class, and M2 and C2 for the second class. Given the target of k best features, all subsets of k from m features must be evaluated to find the subset with the largest DM. With large data sets that have large numbers of features, this can be a huge search space, and alternative heuristic methods should be considered. One of these methods selects and ranks features based on an entropy measure. Detailed explanations are given in Section 3.4. The other heuristic approach, explained in the following, is based on a combined correlation and covariance analyses and ranking of all features.
Existing efficient feature-selection algorithms usually rank features under the assumption of feature independence. Under this framework, features are ranked as relevant mainly based on their individual high correlations with the output feature. Because of the irreducible nature of feature interactions, these algorithms cannot select interacting features. In principle, it can be shown that a feature is relevant due to two reasons: (1) It is strongly correlated with the target feature, or (2) it forms a feature subset and the subset is strongly correlated with the target. A heuristic approach is developed to analyze features of type (2) in the selection process.
In the first part of a selection process, the features are ranked in descending order based on their correlation values with output using a previously defined technique. We may assume that a set of features S can be divided into subset S1 including relevant features, and subset S2 containing irrelevant ones. Heuristically, critical for removal are features in S2 first, while features in S1 are more likely