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Traditional model-free prediction approaches, such as neural networks or fuzzy models use all training data without preference in building their prediction models. Alternately, one may make predictions based only on a set of the most recent data without using other data. Usually, such local prediction schemes may have better performance in predicting time series than global prediction schemes do. However, local prediction schemes only use the most recent information and ignore information bearing on far away data. As a result, the accuracy of local prediction schemes may be limited. In this paper a novel prediction approach, termed the Markov-Fourier gray model (MFGM), is proposed. The approach builds a gray model from a set of the most recent data and a Fourier series is used to fit the residuals produced by this gray model. Then, the Markov matrices are employed to encode possible global information generated also by the residuals. It is evident that MFGM can provide the best performance among existing prediction schemes. Besides, we also implemented a short-term MFGM approach, in which the Markov matrices only recorded information for a period of time instead of all data. The predictions using MFGM again are more accurate than those using short-term MFGM. Thus, it is concluded that the global information encoded in the Markov matrices indeed can provide useful information for predictions.