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 | max{θ1,θ2,...,θv} < λ), where θ is the CoD of the feature set and θ1,θ2,...,θv 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,...,θv} ≥ 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.