Evaluation of Moving Average Model and Autoregressive Moving Average Model (ARMA) for Prediction of Industrial Electricity Consumption in Nigeria

In this paper, evaluation of moving average model and autoregressive moving average model (ARMA) for prediction of industrial electricity consumption in Nigeria is presented. Industrial electricity consumption data obtained from Central Bank of Nigeria (CBN) Statistical Bulletin for the year 1979-2014 is used to determine the model parameters and prediction performance in terms of Root Mean Square Error (RMSE) and Coefficient of determination r 2 values. The results show that the Autoregressive Moving Average (ARMA) model with coefficient of determination value of 66.0% and RMSE value of 68.628 gives better prediction performance than the Moving Average with coefficient of determination value of 42.6% and value of 84.749. However, coefficient of determination value of 66% is not particularly adequate for acceptable prediction accuracy. In that case, for better prediction accuracy for the industrial electricity consumption in Nigeria, other models may need to be examined apart from the two models considered in this paper.


Introduction
Globally, reliable and adequate electricity supply has been identified as paramount to socio-economic and technological development of every nation [1][2][3][4][5][6][7][8]. Particularly, the industrial sector with their job creation and poverty alleviation potentials is heavily reliant on electricity. In this wise, electricity is central to industrialization of any nation [9][10][11][12][13]. However, in Nigeria, there has been perennial acute shortage of power supply [14][15][16]. Some of the challenges include insufficient power generation, excessive power losses, poor maintenance of power plants, lack of political will to tackle the problems associated with the power sector.
In any case, since the last decades, Nigerian government has continued to introduce reforms in the power sector ranging from privation to expansion in the power generation and distribution capacities [17][18][19][20][21][22][23][24][25]. In all, there has been noticeable marginal improvement in the power supply across the nation. However, the industrial sectors are still overwhelmed by grossly inadequate power supply. The effect has been on high cost of production in Nigeria and disproportional dependence on imported goods. In order to plan for adequate power supply to the industrial sector, the existing and projected industrial energy demand profile is needed. In this wise, models need to be developed and evaluated for prediction and possibly forecasting of the industrial electricity consumption in Nigeria.
Consequently, in this paper, two time series models are considered, namely, Moving Average Model and Autoregressive Moving Average Model (ARMA) [26][27][28][29][30][31][32][33][34]. The two models are used to predict industrial electricity consumption in Nigeria. The prediction performance of the two models are then compared in terms of RMSE and values. The study is meant to evaluate the adequacy of the prediction performance of the selected models for industrial electricity consumption prediction. Particularly, the prediction Model (ARMA) for Prediction of Industrial Electricity Consumption in Nigeria performance is expressed in terms of Root Mean Square Error (RMSE) and Coefficient of determination .

Theoretical Background
In this paper, two time series models are considered, namely, Moving Average Model and Autoregressive Moving Average Model (ARMA). The two models are used to predict industrial electricity consumption in Nigeria. The prediction performance of the two models are then compared in terms of RMSE and r 2 values.

Moving Average Model (MA) (of Order 1)
The moving average model of order 1 is given as; 0 θ and 1 θ are the model parameters and t U is the error term with mean 0 and constant variance.
Making t U the subject in equation (1) gives Taking partial derivatives of equation (3) with respect to 0 θ and 1 θ gives Arranging equations 10 to 13 in matrix form gives Let Then equation 14 becomes The solution of equation (18) gives the parameters of the Moving Average model.

Autoregressive Moving Average Model, (ARMA)
The ARMA can be expressed as: Where, 0 1 0 1 , , and β β θ θ are the parameters of the model and t a is the error term.
Let D denote sum of square errors Therefore; To obtain the parameter of equation (22), D have to be minimized. Then, Divide through by -2 gives Arranging the sets of equations 29 to-34 in matrix form give    The solution of equation (37) gives the parameters of the ARMA model.

Coefficient of Determination
The coefficient of determination is used to determine the effectiveness of using the models in predicting the predict industrial electricity consumption in Nigeria.
gives the coefficient of the total variance in the department variable explained by the model. 2

RSS r TSS
RSS for model is = Generally, the RSS is: TSS for model Where, J is the matrix, Sum of square Error Model (ARMA) for Prediction of Industrial Electricity Consumption in Nigeria (SSE) = SST-SSR (41)

Root Mean Square Error (RMSE)
The Means Square Error (MSE) is computed using the formula: The Root Means Square Error (RMSE) is given as Where, t Y is the actual industrial electricity consumption and ˆt Y is the predicted value from the model.  Table 1 gives results showing the model parameters and the model performance parameters namely r 2 and RMSE for the moving average model. Based on the results, the Moving Average (MA) model of order 1 for predicting the industrial electricity consumption in Nigeria is given as:

Moving Average Model
The results also show that the Moving Average model accounted for 42.6 percent of the variation in industrial electricity consumption (r 2 = 0.426 = 42.6%). The actual and the Moving Average model predicted values of industrial electricity consumption are as shown in Table 1 and figure 1.   Table 3 gives results showing the model parameters and the model performance parameters namely r 2 and RMSE for the autoregressive moving average (ARMA) model. Based on the results, the autoregressive moving average (ARMA) model for predicting the industrial electricity consumption in Nigeria is given as: The results also show that the ARMA model accounted for 66 percent of the variation in industrial electricity consumption (r 2 = 0.66 = 66 %). The actual and the ARMA model predicted values of industrial electricity consumption are as shown in Table 4 and figure 2.   Table 5 reveals that the Autoregressive Moving Average (ARMA) model with r 2 = 66.0% and RMSE = 68.628 gives better prediction performance than the Moving Average with r 2 =42.6% and RMSE = 84.749. As such, the ARMA model is recommended for forecasting industrial electricity consumption in Nigeria. However, r 2 value of 66% is not particularly adequate for acceptable prediction accuracy. In that case, for better prediction accuracy, other models may need to be examined apart from the two models considered in this paper.

Conclusion
Moving Average Model and Autoregressive Moving Average Model (ARMA) are evaluated for their suitability in the prediction of the industrial electricity consumption in Nigeria. Industrial electricity consumption data from 1979-2014 is used to determine the model parameters and prediction performance in terms of RMSE and values. In all, ARMA model gives better prediction performance than the Moving Average model. However, the relatively low values for the two models shows that the models are not particularly adequate for prediction of the industrial electricity consumption in Nigeria. In that case, for better prediction accuracy, other models may need to be examined.