Chapter 3 Real data analysis: Power Consumption in National
3.4 Performance Comparison
In this section, we compare the performance of four methods. In Phase I, as we mention in Section 3.3, for WOAC and WAC data, all the points fall in the bands (or limits) based on four different methods. Therefore, we can conclude that NE data are in-control, so we can use these four charts to detect the special-event weekdays’ data. As one can see, both of the confidence bands based on Kalman Filter approach and Bootstrap approach are similar. We plot out-of-control data together with the confidence bands, if the fitted points fall outside the band at some points, we issue an alarm. As Table (3.15) and Table (3.16) show, all the weekday data can be identified to be out-of-control except weekday6 and weekday35. It indicates that four methods are capable of identifying any change from the model.
The following represents some special-event weekdays’ data to show the capable of identifying to out-of-control. Figure 3.23 represent that some electricity consumption points fall outside the bands due to the abnormal temperature. For weekday8, temperature exceeds 30℃ that rise the electricity consumption. The possible reason to the abnormal temperature is seasonal change in this week. Figure 3.24 shows that when there is a national holiday in a weekday, the electricity consumption becomes lower than the lower band. We can infer that some of students return to their hometown that makes the electricity consumption decrease.
Figure 3.25 part (b) and part (c) represent that the charts can detect two sharp points at time 60 (Wed. 12 am) and time 67 (Wed. 19 pm), while part (a) can only detect one of the two points (time 67 ). From Figure 3.26, weekday 31 is the first weekday in Fall Semester, but the first two days in the chart are during the summer vacation that make the electricity
consumption lower, while third, fourth, fifth peaks are higher than UCB. Because September is an extreme hot month that rises the consumption of the electricity and the use of power is related to temperature, some values are exceed the UCB, which are also shows in part (a) to (c). The first peak in Figure 3.27 is lower than the other peaks due to Super Typhoon Jangmi
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(2008/09/27). By the Tables (3.15) and (3.16), the performance of the control bands under each time unit is slightly worse than the confidence bands based on Kalman Filter approach and bootstrap approach. But by this real data case it can’t tell which confidence bands is better, we can only say that these two methods have a good performance to monitor the process with seasonal time series model.
However, there are some restrictions to our methods. For control bands under each time unit, it is a simple and fast way to monitor our data without fitting a time series model.
But this method can only detect the observations which are extremely abnormal. Our empirical study of electricity consumption in National Chengchi University shows that when the electricity consumption is higher, the chart can’t immediately detect it, while model-based methods can detect this kind of situation (see Figure 3.25 and Figure 3.26). It is reasonable that the confidence bands under each time units are wider than other methods.
Conversely, model-based approaches require fitting time series models and use the profile model to monitor our data. But when the data can’t fit the in-control time series model, which means that there exist a estimation problem so that we can’t use the data to establish control charts in Phase I, such as weekday6. But when monitoring this weekday, as Figure 3.22 shows, no points is out of the confidence bands based on the methods we construct. It indicates that this weekday is a type II error case. For Hotelling T2 control charts, if the monitoring data can’t apply to a profile model, we can only say that the data has another adequate model to apply and regard the data as out-of-control. In this empirical, it tells that confidence bands based on the Kalman Filter approach and bootstrap approach are two good charts to monitor the process data with a seasonal time series model, and we can also know which point is out of bands, while the Hotelling T2 control charts can identify
out-of-control data.
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Table 3.15 Comparison Table for SE data in WOAC data2
weekday CB under
Table 3.16 Comparison Table for SE data in WAC data
weekday CB under each hour
2 The number of points that the electricity consumption exceed the UCB or LCB in a weeday.
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Figure 3.22 Plot of CB for weekday 6 under out-of-control for WOAC
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Figure 3.23 Plot of CB for weekday8 under out-of-control for WOAC
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Figure 3.24 Plot of CB for weekday46 under out-of-control for WOAC
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Figure 3.25 Plot of CB for weekday15 under out-of-control for WAC
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Figure 3.26 Plot of CB for weekday31 under out-of-control for WAC
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Figure 3.27 Plot of CB for weekday33 under out-of-control for WAC
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N a tio na
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When process data are autocorrelated, using traditional control charts may lead to a false alarm or detection. This paper proposes approaches to monitoring the process data with a seasonal time series model. For the model-free method, we construct the confidence bands under each time unit. For the model-based methods, we use the profile model to build the confidence bands based on Kalman Filter and bootstrap approaches. We use a real example to compare these three methods by plotting the monitoring graphs in Phase II together.
When a special cause occurs, although all these three methods issue a alarm, the
performance of the model-based methods are absolutely better than model-free method. For Hotelling T2 control charts, the first test statistic diagnoses the shift in the coefficients of the SARMA model. The second test monitors the variance of random errors. Also there are some restrictions for fitting profile model. A real data exemplifies the proposed procedure that how we use these four monitoring methods, and graphical analyses enhance the results.
This research considers both the Phase I scheme and Phase II monitoring application.
Several issues remains for further investigation. We can take the electricity data from Saturday and Sunday into account. And we should conduct a simulation study to verify the performance of our four different methods based on the average run length (ARL) criterion.
In our real data example, we only compare the methods by counting how many points are out of the limits, so the simulation study is required.
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