Theory of dependence values

Abstract

A new model to evaluate dependencies in data mining problems is presented and discussed. The well-known concept of the association rule is replaced by the new definition of dependence value, which is a single real number uniquely associated with a given itemset. Knowledge of dependence values is sufficient to describe all the dependencies characterizing a given data mining problem. The dependence value of an itemset is the difference between the occurrence probability of the itemset and a corresponding "maximum independence estimate." This can be determined as a function of joint probabilities of the subsets of the itemset being considered by maximizing a suitable entropy function. So it is possible to separate in an itemset of cardinality k the dependence inherited from its subsets of cardinality (k - 1) and the specific inherent dependence of that itemset. The absolute value of the difference between the probability P (i) of the event i that indicates the presence of the itemset {a, b, . . . }and its maximum independence estimate is constant for any combination of values of (a, b, . . . ). In addition, the Boolean function specifying the combinations of values for which the dependence is positive is a parity function. So the determination of such combinations is immediate. The model appears to be simple and powerful.