In recent years, a plethora of approaches have been proposed to deal with the increasingly challenging task of multi-output regression. This paper provides a survey on state-of-the-art multi-output regression methods, that are categorized as problem transformation and algorithm adaptation methods. In addition, we present the mostly used performance evaluation measures, publicly available data sets for multi-output regression real-world problems, as well as open-source software frameworks.

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Mixtures of polynomials (MoPs) are a nonparametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one-and multidimensional (marginal) MoPs from data have recently been proposed. In this paper, we introduce two methods for learning MoP approximations of conditional densities from data. Both approaches are based on learning MoP approximations of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities is found. We illustrate and study the methods using data sampled from known parametric distributions, and demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method. C 2014 Wiley Periodicals, Inc.

Multi-label classification problems require each instance to be assigned a subset of a defined set of labels. This problem is equivalent to finding a multi-valued decisión function that predicts a vector of binary classes. In this paper we study the decisión boundaries of two widely used approaches for building multi-label classifiers, when Bayesian network-augmented naive Bayes classifiers are used as base models: Binary relevance method and chain classifiers. In particular extending previous single-label results to multi-label chain classifiers, we find polynomial expressions for the multi-valued decisión functions associated with these methods. We prove upper boundings on the expressive power of both methods and we prove that chain classifiers provide a more expressive model than the binary relevance method.