International Journal of Forecasting 27 (2011) 689–699 www.elsevier.com/locate/ijforecast Conditionally dependent strategies for multiple-step-ahead prediction in local learning Gianluca Bontempi ∗ , Souhaib Ben Taieb Machine Learning Group, D´ epartement d’Informatique, Facult´ e des Sciences, ULB, Universit´ e Libre de Bruxelles, 1050 Bruxelles, Belgium Available online 8 January 2011 Abstract Computational intelligence approaches to multiple-step-ahead forecasting rely on either iterated one-step-ahead predictors or direct predictors. In both cases the predictions are obtained by means of multi-input single-output modeling techniques. This paper discusses the limitations of single-output approaches when the predictor is expected to return a long series of future values, and presents a multi-output approach to long term prediction. The motivation for this work is that, when predicting multiple steps ahead, the forecasted sequence should preserve the stochastic properties of the training series. However, this may not be the case, for instance in direct approaches where predictions for different horizons are produced independently. We discuss here a multi-output extension of conventional local modeling approaches, and present and compare three distinct criteria for performing conditionally dependent model selection. In order to assess the effectiveness of the different selection strategies, we carry out an extensive experimental session based on the 111 series in the NN5 competition. c ⃝ 2010 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. Keywords: Multiple-step-ahead; Local learning 1. Introduction The multiple-step-ahead prediction (also known as trace forecasting) of a time series ϕ t in a nonlinear setting is considered a difficult task in both the parametric (Guo, Bai, & An, 1999) and nonparametric (Chen, Yang, & Hafner, 2004) time series literature. While the parametric form of the multiple-step- ahead predictor depends only on the one-step- ahead parametric model under linear assumptions, in nonlinear settings the analytical derivation of the multiple-step predictor requires a knowledge ∗ Corresponding author. Tel.: +32 2 6505591. E-mail addresses: gbonte@ulb.ac.be (G. Bontempi), sbentaie@ulb.ac.be (S. Ben Taieb). of the innovation distribution. Various numerical and Monte Carlo methods have been proposed in the literature for computing the multiple-step-ahead predictors (see Pemberton, 1987; Tong, 1983) when both the nonlinear one-step-ahead predictor and the innovation distribution are known. Additional results for unspecified innovation distributions have been given by Guo et al. (1999). However, these approaches are based on parametric assumptions, and their resulting accuracy depends heavily on the adequacy of the model. Over the last two decades, computational intelligence methods (such as artificial neural networks, see Lapedes & Farber, 1987, and Zhang, Patuwo, & Hu, 1998; and nearest-neighbors techniques, see Farmer & Sidorowich, 1987) have drawn the attention of the forecasting community. 0169-2070/$ - see front matter c ⃝ 2010 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ijforecast.2010.09.004