Efficient selection of feature sets possessing high coefficients of determination based on incremental determinations

Feature selection is problematic when the number of potential features is very large. Absent distribution knowledge, to select a best feature set of a certain size requires that all feature sets of that size be examined. This paper considers the question in the context of variable selection for prediction based on the coefficient of determination (CoD). The CoD varies between 0 and 1, and measures the degree to which prediction is improved by using the features relative to prediction in the absence of the features. It examines the following heuristic: if we wish to find feature sets of size m with CoD exceeding δ, what is the effect of only considering a feature set if it contains a subset with CoD exceeding λ < δ? This means that if the subsets do not possess sufficiently high CoD, then it is assumed that the feature set itself cannot possess the required CoD. As it stands, the heuristic cannot be applied since one would have to know the CoDs beforehand. It is meaningfully posed by assuming a prior distribution on the CoDs. Then one can pose the question in a Bayesian framework by considering the probability P(θ > δ|max{θ₁,θ₂,⋯,θ_v } < λ), where θ is the CoD of the feature set and θ₁,θ₂,⋯,θ_v are the CoDs of the subsets. Such probabilities allow a rigorous analysis of the following decision procedure: the feature set is examined if max{θ₁,θ₂,⋯,θ_v} ≥ λ. Computational saving increases as λ increases, but the probability of missing desirable feature sets increases as the increment δ-λ decreases; conversely, computational saving goes down as λ decreases, but the probability of missing desirable feature sets decreases as δ-λ increases. The paper considers various loss measures pertaining to omitting feature sets based on the criteria. After specializing the matter to binary features, it considers a simulation model, and then applies the theory in the context of microarray-based genomic CoD analysis. It also provides optimal computational algorithms.