Chapter 3 Real data analysis: Power Consumption in National
3.3 Phase I and Phase II control scheme
3.3.3 Example of Semester without AC data (WOAC data)
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3.3.3 Example of Semester without AC data (WOAC data) I. Control bands under each hour
In Phase I, we calculate the control bands by equations (2.8) and (2.9). Figure 3.10part (a) shows the control bands under each time unit and the electricity consumption in weekday8 from SE data. All the observations from NE data fall in the bands, which means that the subgroups are from in-control. We can use this chart to monitor the data. In phase II, we plot the SE data into control bands (2.8) and (2.9). Table 3.7 shows the time points that
electricity consumption are out of the control bands. It represents that eight of nine
weekdays are detected to be out-of-control due to the special events. Only weekday 6 can’t be detected. We will discuss it in Section 3.4.
To monitor the variance, we calculate the control bands of MS-bar by equation (2.9).
Figure 3.10 part (b) shows the control bands of MS under each time unit and the MS of weekday8 from SE data. All the MS from NE data fall in the bands, so we can use this chart to monitor. In phase II, we plot the SE data into control bands. It represents that all the weekdays are detected to be out-of-control due to the special events.
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Figure 3.10 Plot of control bands under each hour in Phase I for WOAC data Table 3.7 Table of the OCC time point by control bands under each hour for WOAC data
Weekday Out-of-control time point (t)
Weekday 1 21, 40, 43, 44, 45, 46, 68, 69 Weekday 2 1, 2, 47, 81~96
Weekday 6 None
Weekday 7 43, 81~95, 105~ 119 Weekday 8 38, 43, 57, 60, 62~69 Weekday 9 62, 68, 69
Weekday 11 62,81, 87~93, 105~109, 116, 117 Weekday 46 67~70, 80~96, 103, 105~119
Weekday 48 17,18, 20, 21, 22, 37~46, 54, 58~70, 81, 85, 87~89, 91~93,102 , 103, 105~118
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II. Confidence bands based on Kalman Filter approach
To construct the chart in Phase I scheme, we use the model (3.1) to apply to NE data. To the Kalman Filter method, the State-Space equation for each weekday can be expressed by (2.11) and (2.12) where functions “Kfilter0” and “Ksmooth0”, we get the estimate of the state variable 𝛼̂𝑡𝑖|𝑇 and 𝑃̂𝑡𝑖|𝑇, i=1,2,…,7, t=1,2,…,120. Confidence bands can be calculated by (2.13) and (2.14). All the fitted values fall in the confidence bands based on Kalman Filter approach. Figure 3.11 shows the confidence bands based on Kalman Filter approach and the electricity
consumption in weekday8 from SE data.
In phase II, we apply model (3.1) to the SE data, and plot the fitted values of a profile into equation (2.13) and (2.14). According to Table 3.3, the data of weekday1, weekday6, weekday9 weekday11 can’t be applied to model (3.1), so we plot the observations into the confidence bands. Table 3.8 shows the time points that fitted value of a profile are out of the bands. It represents that eight of nine weekdays are detected to be out-of-control due to some special events.
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Figure 3.11 Plot of CB based on Kalman Filter approach in Phase I for WOAC data
Table 3.8 Table of the OOC time point by CB based on Kalman Filter approach for WOAC
Weekday Out-of-control time point (t)
Weekday 1 12~22,37~ 41, 43~46, 63, 64, 67, 68, 69 Weekday 2 80~95
Weekday 6 None
Weekday 7 32,33,43,45, 80~95, 104~ 119 Weekday 8 37,38, 40~44, 57~69
Weekday 9 57, 61, 62, 64, 65, 66, 68, 69
Weekday 11 57, 61, 62, 63, 81, 84, 86~93, 105~109, 112, 113, 114, 117 Weekday 46 67~70, 80~95, 104~119
Weekday 48 12~22, 34~46, 57~70, 81~94, 105~118
III. Confidence bands based on Bootstrap approach
To construct the confidence bands in Phase I scheme, we use the model (3.1) to apply to NE data. We calculate the confidence bands by equations (2.15) and (2.16). Figure 3.12 shows the confidence bands based on Bootstrap approach and the electricity consumption in weekday8 from SE data. All the fitted values of profile for NE data fall in the bands. It means that the subgroups are in-control. We can use this confidence bands to monitor the future data of subgroups. In phase II, as we do in Kalman Filter approach, we plot the fitted
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value of profile for SE data into equations (2.15) and (2.16). Table 3.9 shows the time points that the electricity consumption are out of the bands. It represents that eight of nine
weekdays are detected to be out-of-control due to some special events.
Figure 3.12 Plot of CB based on Bootstrap approach in Phase I in WOAC data Table 3.9 Table of the OOC time point by CB based on Bootstrap approach for WOAC
Weekday Out-of-control time point (t)
Weekday 1 12~22,38~ 41, 43~46, 63, 64, 65, 68, 69 Weekday 2 23, 32, 43, 45, 80~95 ,106, 109
Weekday 6 None
Weekday 7 32,33,43,45, 81~95, 104~ 118 Weekday 8 37, 39~43, 58, 59, 60, 62~69 Weekday 9 62~67, 69
Weekday 11 62, 63, 81, 82, 84, 86~93, 105~109, 112, 113, 114, 117 Weekday 46 23, 67~70, 80~95, 105~118
Weekday 48 12~23, 34~46, 58~70, 81~94, 105~118
To monitor the variance of residuals, Figure 3.13 part(a) shows the plots of the values of 𝜎̂𝑎𝑖 for each subgroup. The UCL is 15.71. One of the points exceeds the UCL from NE data.
It is a false alarm case in the Phase I control chart. In phase II, from Figure 3.13 part (b), five of the nine weekdays’ data exceeds the UCL in s control charts, which indicate that these data are out-of-control. Fitting these data to the in-control model (3.1) leads to violating the stationary condition and reach no estimation results. Due to the parameter estimate problem
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for the remaining four weekdays’ data (we denote “NA” in the control chart), we can’t fit them with the same in-control profile model in (3.1).
Figure 3.13 Plot of standard deviation of the residuals in WOAC data IV. Hotelling T2 control charts
Due to some of the values are very large, we take the logarithm to Hotelling T2 statistics in this section. Figure 3.14 shows the plots of the values of log (𝑇12) and log (𝑇22) statistics to monitor the coefficients of the profile and variance of residuals for NE data. The UCL for log (𝑇12) is 2.414 and the UCL for log (𝑇22) is 5.1396. All the statistics are under the upper limit, except for that the log (𝑇22) of weeday10 (wd10) exceeds UCL. It is a false alarm case in the Phase I control chart. So we can conclude that model (3.1) can be used as an in-control
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profile data, and six of the seven weekday’s data are in-control data.
To compute the values of log (𝑇12) and log (𝑇22) statistics, we use the corresponding
estimates δ̅ and 𝜎̅𝑎2 in the Phase I scheme. From Figure 3.15, six of the nine weekdays’ data exceeds the UCL in both log (𝑇12) and log (𝑇22) control charts, which indicates that these data are out-of-control. The reason is that fitting these data to the in-control model (3.1) leads to violating the stationary condition and reach no estimation results. Due to the parameter estimate problem for the remaining three weekdays’ data (we denote “NA” in log (𝑇12) and log (𝑇22) control chart), we can’t fit the same in-control profile model (3.1) to get the estimation of coefficients. It is reasonable to infer that these weekdays are not suitable for model (3.1), so we can’t calculate log (𝑇12) and log (𝑇22) statistics. They should have the better time series model which is different from the in-control model (see Table 3.10), so they can be identified to be out-of-control.
Table 3.10 Table of the best time series model in Weekday1,6,9,11 for WOAC data wd Best seasonal time series model
1 Sarima(1,0)×(1,0)24
6 Sarima(2,0)×(1,0)24
9 Sarima(3,2)×(1,0)24
11 Sarima(1,0)×(1,0)24
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Figure 3.14 Plot of the values of log(T2) statistic for each weekday in Phase I for WOAC data
Figure 3.15 Plot of the values of log(T2) statistic for Special-Event Weekdays Data for WOAC data