Ordinal regression problems are those machine learning problems where the objective is to classify patterns using a categorical scale which shows a natural order between the labels. Many real-world applications present this labelling structure and that has increased the number of methods and algorithms developed over the last years in this field. Although ordinal regression can be faced using standard nominal classification techniques, there are several algorithms which can specifically benefit from the ordering information. Therefore, this paper is aimed at reviewing the state of the art on these techniques and proposing a taxonomy based on how the models are constructed to take the order into account. Furthermore, a thorough experimental study is proposed to check if the use of the order information improves the performance of the models obtained, considering some of the approaches within the taxonomy. The results confirm that ordering information benefits ordinal models improving their accuracy and the closeness of the predictions to actual targets in the ordinal scale.

In this paper, the well known stagewise additive modeling using a multiclass exponential (SAMME) boosting algorithm is extended to address problems where there exists a natural order in the targets using a cost-sensitive approach. The proposed ensemble model uses an extreme learning machine (ELM) model as a base classifier (with the Gaussian kernel and the additional regularization parameter). The closed form of the derived weighted least squares problem is provided, and it is employed to estimate analytically the parameters connecting the hidden layer to the output layer at each iteration of the boosting algorithm. Compared to the state-of-the-art boosting algorithms, in particular those using ELM as base classifier, the suggested technique does not require the generation of a new training dataset at each iteration. The adoption of the weighted least squares formulation of the problem has been presented as an unbiased and alternative approach to the already existing ELM boosting techniques. Moreover, the addition of a cost model for weighting the patterns, according to the order of the targets, enables the classifier to tackle ordinal regression problems further. The proposed method has been validated by an experimental study by comparing it with already existing ensemble methods and ELM techniques for ordinal regression, showing competitive results.

Artificial neural networks (ANNs) have traditionally been seen as black-box models, because, although they are able to find “hidden” relations between inputs and outputs with a high approximation capacity, their structure seldom provides any insights on the structure of the functions being approximated. Several research papers have tried to debunk the black-box nature of ANNs, since it limits the potential use of ANNs in many research areas. This paper is framed in this context and proposes a methodology to determine the individual and collective effects of the input variables on the outputs for classification problems based on the ANOVA-functional decomposition. The method is applied after the training phase of the ANN and allows researchers to rank the input variables according to their importance in the variance of the ANN output. The computation of the sensitivity indices for product unit neural networks is straightforward as those indices can be calculated analytically by evaluating the integrals in the ANOVA decomposition. Unfortunately, the sensitivity indices associated with ANNs based on sigmoidal basis functions or radial basis functions cannot be calculated analytically. In this paper, the indices for those kinds of ANNs are proposed to be estimated by the (quasi-) Monte Carlo method.