Compositional random forest

원문 초록

Compositional data have been analysed in the literature from three main points of view: directly on the simplex, in log-ratio coordinates, and as directional data on the unit sphere. In this paper, we construct a random forest framework for compositional responses that accommodates all three. Random forests are ensembles of decision trees, and the split criterion determines how each tree grows. We use the log-likelihood gain as the split criterion and consider distributions defined on the different spaces in which compositions can be represented. The Dirichlet likelihood has been studied in the literature, but its covariance is fixed by its mean up to a single scalar, and its maximum likelihood estimator has no closed form. To obtain more flexible criteria, we use two well-known transformations. First, the square-root transformation maps compositions onto the unit sphere, where existing directional distributions require numerical estimation. We introduce a tangent-normal (TN) distribution, which shares its parameters with the Kent distribution, and construct a split criterion whose gain is computed from additive sufficient statistics: the shape parameters are maximised out in closed form, up to an exponentially small truncation constant, so that no parameter is estimated iteratively. Second, the log-ratio transformation maps compositions to Euclidean space, where the multivariate logistic-normal distribution has a flexible covariance and a closed-form maximum log-likelihood, also based on additive sufficient statistics. Both criteria respond to a shift of the mean, measured relative to the covariance, and to a change in the covariance itself. In simulations, the most accurate forest depended on the covariance structure of the composition. The TN forest was the most accurate in two of the three designs.