An algebraic approach to data mining: some examples

We introduce an algebraic approach to the foundations of data mining. Our approach is based upon two algebras of functions defined over a common state space X and a pairing between them. One algebra is an algebra of state space observations, and the other is an algebra of labeled sets of states. We interpret H as the algebraic encoding of the data and the pairing as the misclassification rate when the classifier f is applied to the set of states X. We give a realization theorem giving conditions on formal series of data sets built from D that imply there is a realization involving a state space X, a classifier f ∈ R and a set of labeled states χ ∈ R₀ that yield this series.