Forecasting nonlinear time series of energy consumption using a hybrid dynamic model
Introduction
Energy-related issues are a priority owing to the major role that energy sources, such as coal, oil, gas, and wind, play in daily life and the global economy. Energy consumption has greatly increased owing to a burgeoning population growth and elevated living standards [1], [2]. For instance, the Energy Information Administration (EIA) of the United States has forecast that global energy consumption will increase by 49% from 2007 to 2035 [3]. Also, the energy consumption of public buildings is increasing in proportionate to overall national use [4]. In economics, energy consumption has significantly and positively affected Asian economic growth [5]. Accordingly, a highly precise model for forecasting energy consumption must be developed. Based on such a model, energy policy makers can either implement an energy conservation policy or allocate a certain amount of energy to public buildings.
The autoregressive integrated moving average (ARIMA) model is extensively used to forecast time-series data [6]. However, the forecasting accuracy of the ARIMA model is poor when data are few or nonlinear [7]. Forecasting models (such as the ARIMA model) that are based on conventional statistical methods are limited because real-world data are commonly few or fail to satisfy statistical assumptions.
Forecasting models can also be developed using data-mining approaches such as artificial neural networks (ANNs), evolutionary algorithms (EAs), and mixed-integer programming [8], [9]. However, the hidden layers in ANNs are difficult to explain, and the relationship between input and output variables in ANNs is difficult to express as a clear forecasting equation. To solve this problem and compare the forecasting accuracy with ANNs, some studies [7] have applied genetic programming (GP) to construct a clear forecasting equation and compared the forecasting accuracy with other models. GP is more accurate than ANNs in forecasting or classification problems [7], [10], [11]. In forecasting energy consumption, Togun and Baysec [12] found that GP performs as well as ANNs. In contrast to the ANNs model, GP uses symbolic regression to derive a clear forecasting equation [7], [10], [12], [13], [14].
The grey model (GM) of grey system theory has been adopted in many forecasting studies [15], [16], [17], [18] with only four or more observations. Real-word data sets are often difficult to collect and data sets include a few observations. Although linear regression (such as the ARIMA model) is often utilized to forecast time-series data, it is inaccurate when observations are few or do not satisfy statistical assumptions. GM(1,1), the first-order one-variable GM, has been widely applied in various fields [15], [16], [17], [18]. Although capable of forecasting using small time-series data accurately, GM(1,1) may fail to do so for nonlinear time-series data.
Many researchers have been developed GM models to increase their forecasting accuracy. For instance, Hsu and Wang [17] estimated the parameters of a grey differential function using the Bayesian method to increase the accuracy of GM(1,1). Wang and Hsu [18] estimated the parameters of grey differential function using genetic algorithms (GAs) to increase the forecasting accuracy of GM(1,1). To improve further the performance of GM(1,1) models, some studies have developed innovative approaches to forecast the residual series of GM(1,1). For instance, Hsu and Chen [15] combined residual modification with residual ANN sign estimation to forecast the residual series of GM(1,1). Hsu [16] combined residual modification with residual Markov-chain sign estimation to forecast residual series of GM(1,1). To increase the predictive accuracy of the method of Hsu and Chen [15], Lee and Tong [10] combined residual modification with residual GP sign estimation to increase the effectiveness of ANN in estimating the residual signs of GM(1,1). When the time-series data are nonlinear, the forecasting accuracy of GM(1,1) or an improved GM(1,1) may be poor. Hence, Zhou and Hu [19] developed a hybrid GM(1,1) model that combines GM(1,1) modeling in original time-series data with ARIMA modeling in residual series to increase forecasting accuracy of GM(1,1). However, their approach adopts a linear model (ARIMA) to forecast the residual series. Small or nonlinear residual-series data may obtain inaccurate outcomes using ARIMA model.
GM(1,1) is normally constructed using a entire data set. Akay and Atak [20] developed a grey predictive model with a rolling mechanism (GPRM), in which only a minimal amount of recent data are used, to increase forecasting accuracy. Based on the structure of GM(1,1), GPRM can be used efficiently to increase the forecasting accuracy of GM(1,1) in each rolling process when applied to exponential or chaotic data sets. Although capable of increasing the forecasting accuracy of GM(1,1), GPRM does not model the residual series in each rolling process to increase forecasting accuracy. Furthermore, improved forecasting models [10], [15], [16], [19] fail to enhance significantly the accuracy of GM(1,1) modeling [10], [15], [16] or ARIMA modeling [19] in forecasting the residual series. To enhance the accuracy of the residual series, heuristic methods, such as symbolic regression, must be utilized since they perform well in forecasting [13]. Lee and Tong [7] claimed that the conventional linear time-series model (ARIMA model) cannot easily be used to fit nonlinear time-series data and therefore developed a heuristic approach to improve the accuracy of residual series.
To increase the accuracy of GM(1,1) applied to original time-series data and to prevent inaccurate forecasting using conventional linear time-series models when residual series are complex patterns (such as nonlinear patterns), this work develops a novel hybrid dynamic forecasting model in which dynamic grey prediction is applied to the time-series data and GP prediction is applied to the residual-series data of the dynamic grey prediction, to ensure high forecasting accuracy.
The rest of this paper is organized as follows. Section 2 reviews available models for forecasting energy consumption. Section 3 then describes the proposed novel hybrid dynamic GM for forecasting energy consumption. Next, based on real-world examples, Section 4 evaluates the forecasting accuracy of the proposed model, and compares it to other energy consumption models. Section 5 draws conclusions.
Section snippets
Energy consumption models
This section describes three models that are used in forecasting energy consumption. The first one, the GM(1,1) model, is commonly adopted when only a few time-series data are available. The second one, the dynamic GM(1,1) model, is known for its robustness in forecasting each rolling time-series data. The third one, the GP model, is often used either to forecast nonlinear time-series data [12], [14] or to elucidate a complex data-structure.
Hybrid dynamic grey forecasting
This section describes a novel nonlinear hybrid dynamic forecasting model that combines the dynamic grey model with GP. The proposed model is derived as follows.
Step 1: Assume that original time-series of energy consumption data is yt (n data points), and that is predicted using a novel DGM(1,1) model (NDGM(1,1)). Because GM(1,1) requires at least four data points to construct the forecasting model, the NDGM(1,1) model utilizes the most recent four data points to predict the next data
Computational results
To demonstrate the effectiveness of the proposed hybrid dynamic GM, two energy consumption data sets from the United States [28] and China [29] are used to evaluate the accuracy of the proposed model. Energy consumption data from the United States from 1970 to 2008 provide a total of 39 observations. Annual energy consumption data from the US from 1974 to 1998 form a training set (25 observations), and data from 1999 to 2008 form a testing set (10 observations). The China energy consumption
Conclusions
Developing a high-precision energy consumption model is rather complex owing to various uncertain factors that affect it. Methodologies in the literature often use whole data sets to construct an energy consumption model. However, many uncontrolled factors affect annual energy consumption. The use of available data to construct a forecasting model may be unreliable when historical observations of energy consumption vary significantly. This work develops a novel hybrid dynamic GM which combines
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