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Boosting ensembles have deserved much attention because their high performance. But they are also sensitive to adverse conditions, such as noisy environments or the presence of outliers. A way to fight against their degradation is to modify the forms of the emphasis weighting which is applied to train each new learner. In this paper, we propose to use a general form for that emphasis function, which not only includes an error dependent and a proximity to the classification boundary dependent term, but also a constant value which serves to control how much emphasis is applied. Two convex combinations are used to consider these terms, and this makes possible to control their relative influence. Experimental results support the effectiveness of this general form of boosting emphasis. (C) 2017 Elsevier B.V. All rights reserved.