۱- Introduction

۲- Time series forecasting models

۳- Proposed linear combination method

۴- Empirical results and discussions

۵- Conclusions and future work

References

Abstract

Research indicates that forecast combination is one of the most important and effective approaches for time series forecasting. The success of forecast combination depends on how well component models are selected and combination weights are determined. A forecast combination model resulting from a new neural networks-based linear ensemble framework (NNsLEF) is proposed in this study. The principle of the proposed framework adheres to three primary aspects. (a) Four kinds of neural network models, namely, back-propagation neural network, dynamic architecture for artificial neural network, Elman artificial neural network, and echo state network, are selected as component forecasting models. (b) An input-hidden selection heuristic (IHSH) is designed to determine the input-hidden neuron combination for each component neural network. (c) An in-sample training–validation pair-based neural network weighting (ITVPNNW) mechanism is studied to generate the associated combination weights. In particular, the four neural network models are applied to impart their superior performance to the combination approach while maintaining their diversity. Meanwhile, IHSH is investigated to improve the performance of each component neural network model by attempting to solve the familiar overfitting problem of networks. Lastly, the ITVPNNW mechanism is studied to search for a set of appropriate combination weights that will primarily affect the accuracy of the linear ensemble framework.

Introduction

In the past few decades, time series analysis has become a popular research topic and has attracted a great deal of attention. Time series analysis has established itself as a powerful tool for characterizing complex systems from observed data [1–۴]. Researchers have applied time series analysis in many fields, such as clustering [5], pattern recognition [6], classification [7] and prediction [8,9]. As a branch of time series analysis, time series forecasting plays an important role in practice applications. Examples include diverse forecasting applications in energy [10], finance [11], tourism [12,13], and electricity load [14,15]. However, improving the performance of forecasting is an important yet frequently difficult task. A substantial number of studies have been conducted and several approaches, such as autoregressive integrated moving average, support vector machine, and artificial neural networks (ANNs), have been proposed to address this issue. For situations where no dominant approach has been identified, forecast combination has been one of the most important, effective, and popular research perspectives applied since its introduction by Bates and Granger [16]in the 1960s. A theoretical justification of forecast combination can be established by viewing the problem from the perspective of Bayesian model averaging [17]. That is, several forecasting models can be tested and their forecasts are averaged according to the probabilities of the component models, in which knowledge of the precise data to generate a time series process is lacking. Forecast combination is motivated by an impressive result, which shows that this approach can generally yield more accurate and reliable results than single forecasting methods, as evidenced in the literature review [18–۲۰].