4. Results and discussion
4.4 Performance of ensemble average
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4.4. Performance of ensemble average 4.4.1. Electricity load data from NCCU
Since the BPNN aims to find the minimum mean square error function, the forecasting results try to approach the optimal result (i.e. global minimum). However, the program doesn‘t find minimum mean square error function every time; the results always deviate from the best or find the local minimum. Figure 4.6 shows the sketch of the error function surface. We can see that if the program finds a local minimum on the right of the global minimum, the error may be positive. In contrast, the local minimum on the left side may be a negative error. Therefore, we can apply the ensemble average method on to the prediction result to cancel out the error. We can see that the performance by using ensemble average was better in this section.
Figure 4.6 A diagrammatic sketch of the BPNN error function surface
Here we used the load data of May 22, 2008 of GCB10 for an analysis testing. We chose this data because its variation was not so regular compared with June 6, 2008 of GCB10, which had been analyzed in the proceeding text. In the Figure 4.7, we can see that the electricity load suddenly jumps over 1000KWH at the point 300 and 320 (i.e.
Global minimum Local minimum
Configuration Error function
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14:00~16:00). It decreased gradually until about 820KWH at the point 380 (i.e.
18:00~19:00). Nevertheless, it suddenly jumped to 900KWH again the next hour.
Although we could find the precise future trend, we couldn‘t get an accurate value in the suddenly changing points. On the other hand, the smooth, stable changed points could be predicted accurately. Based on these results, we took the data to see whether or not the forecasting performance would be better after the ensemble average.
The forecasting results of the different ensemble levels are illustrated in Figure 4.7, Figure 4.8 and Figure 4.9. Figure 4.7 present forecasting results without the ensemble average. We can see that in the abruptly changed points such as 320 and 400, the large error occurs, however, in Figures 4.8 and 4.9, which apply three and five times the ensemble average respectively, we can clearly see that the forecasting results become better. In addition, Table 4.6 shows the comparison of quantity between these three different ensemble average levels. The results without ensemble average have four points at which error percentage was over 5%. The RMSE, MAE and standard deviation of error are 34.8, 21.0 and 28.4 respectively. When we apply the three times ensemble average to improve the results, the points which error percentages over 5%
were three, and the RMSE, MAE and standard deviation of error become 22.8, 17.2 and 15.4 respectively, all of which are better than the results without ensemble average. Even more, from the results using five times ensemble average, the points which error percentage was over 5% were only one. The RMSE, MAE and standard deviation of error were 18.5, 13.8 and 12.5 respectively, obviously better than the other two.
In the comparison, we concluded that the forecasting error could be improved by applying the ensemble average method. However, a crucial drawback of this method is the large amount of time spent completing the process. For this reason, we aim to
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0 50 100 150 200 250 300 350 400 450 500
500 600 700 800 900 1000 1100 1200
Time(every 3 minutes)
KWH
Load forecasting of GCB10(General Building) 2008 5/22 Thursday
predicted hour predicted data real data
reduce the program processing time in the future. The outlook will be discussed in the conclusion.
Figure 4.7 Forecasting performance for
May 22, 2008 of GCB10, without ensemble average‧
Load forecasting of GCB10(General Building) 2008 5/22 Thursday ensemble(3)
predicted hour
Load forecasting of GCB10(General Building) 2008 5/22 Thursday ensemble(5)
predicted hour predicted data real data
Figure 4.8 Forecasting performance for
May 22, 2008 of GCB10, with three times ensemble averageFigure 4.9 Forecasting performance for
May 22, 2008 of GCB10, with five times ensemble average‧ Table 4.6 The performance comparison between different ensemble average levels
no ensemble ensemble(3) ensemble(5)
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4.4.2. Gold price daily data
In this section the forecasting results of gold price daily data without the ensemble average and five times ensemble average are both discussed. The two cases are illustrated in Figure 4.9 and Figure 4.10, with four mark influential events on the graph. For comparing, the RMSE of result without the ensemble average is 25.95, the MAE is 19.46, the standard deviation of error is 17.2 and the correlation coefficient between predicted data and actual data is 0.9813. The RMSE of result with five times ensemble average is 19.18 and the MAE is 14.28, both better than the performance of cases without ensemble average. The standard deviation of error is 12.84, meaning that the error is more stable, and no large errors had occurred. Finally, the correlation coefficient was 0.9902, presenting a stronger correlation than the one without the ensemble average. The correlation between predicted price and real price of these two cases are presented in Figure 4.11 and 4.13.
The gold price has abruptly sky-rocketed in this past year, the main reason being the European debt crisis. The euro against the U.S. dollar came to the lowest level on 30 Jan 2011 meanwhile the price of gold also came to a local minimum. This data shows a strong correlation between currency and gold price. However, in April 23, 2011, Greece applied for assistance to EU and IMF, and the gold price suddenly increased. It could be interpreted as people were looking for a hedge by buying large amounts of gold. Similarly, due to S&P reducing Greece‘s credit rating to CC on July 4 and the U.S.‘s later on August 5, as well as the world‘s stock market crashing as a result of worries about the U.S. second recession, the gold price experienced a dramatic rise.
Our results present fine performance for prediction. However, after several times ensemble average, the results became even better. Upon these accurate results, maybe we can design a trading strategy in the future to see whether we can make a profit.
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4-Jan-11 1-Feb-11 1-Mar-11 1-Apr-11 3-May-11 1-Jun-11 1-Jul-11 1-Aug-11 1-Sep-11 1300
1300 1400 1500 1600 1700 1800 1900
1300
Correlation between forecasted daily data and real daily data
Figure 4.10 Forecasting performance for 2011 gold price daily data,
without ensemble averageFigure 4.11 The performance of correlation between forecasted and actual daily data
for gold price daily data without ensemble average1.30.
The euro against the U.S.
dollar came to the lowest level in six months
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Correlation between forecasted daily data and real daily data
Figure 4.12 Forecasting performance for 2011 gold price daily data,
with five times ensemble averageFigure 4.13 The performance of correlation between forecasted and actual daily data
for gold price daily data with five times ensemble average1.30.
The euro against the U.S.
dollar came to the lowest level in six months
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