For a given set of nrandom variables, the total correlation is defined as the Kullback–Leibler divergence from the joint distribution to the independent distribution of, This divergence reduces to the simpler difference of entropies, where is the information entropy of variable, and is the joint entropy of the variable set. In terms of the discrete probability distributions on variables, the total correlation is given by The total correlation is the amount of information shared among the variables in the set. The sum represents the amount of information in bits that the variables would possess if they were totally independent of one another, or, equivalently, the average code length to transmit the values of all variables if each variable was coded independently. The term is the actual amount of information that the variable set contains, or equivalently, the average code length to transmit the values of all variables if the set of variables was coded together. The difference between these terms therefore represents the absolute redundancy present in the given set of variables, and thus provides a general quantitative measure of the structure or organization embodied in the set of variables . The total correlation is also the Kullback-Leibler divergence between the actual distribution and its maximum entropy product approximation. Total correlation quantifies the amount of dependence among a group of variables. A near-zero total correlation indicates that the variables in the group are essentially statistically independent; they are completely unrelated, in the sense that knowing the value of one variable does not provide any clue as to the values of the other variables. On the other hand, the maximum total correlation is given by and occurs when one of the variables determines all of the other variables. The variables are then maximally related in the sense that knowing the value of one variable provides complete information about the values of all the other variables, and the variables can be figuratively regarded as cogs, in which the position of one cog determines the positions of all the others. It is important to note that the total correlation counts up all the redundancies among a set of variables, but that these redundancies may be distributed throughout the variable set in a variety of complicated ways. For example, some variables in the set may be totally inter-redundant while others in the set are completely independent. Perhaps more significantly, redundancy may be carried in interactions of various degrees: A group of variables may not possess any pairwise redundancies, but may possess higher-order interaction redundancies of the kind exemplified by the parity function. The decomposition of total correlation into its constituent redundancies is explored in a number sources.
Conditional total correlation
Conditional total correlation is defined analogously to the total correlation, but adding a condition to each term. Conditional total correlation is similarly defined as a Kullback-Leibler divergence between two conditional probabilitydistributions, Analogous to the above, conditional total correlation reduces to a difference of conditional entropies,
Uses of total correlation
and feature selection algorithms based on total correlation have been explored by Watanabe. Alfonso et al. applied the concept of total correlation to the optimisation of water monitoring networks.
