T 2 chart for monitoring variance of the residuals

we are interested. Let the vector of estimate of coefficients in the profile model (2.10) for the ith subgroup which is denoted as

𝛿̂_𝑖 = {𝑐̂_𝑖, 𝜑̂_1𝑖, … , 𝜑̂_𝑝𝑖, 𝜙̂_1𝑖, … , 𝜙̂_𝑃𝑖, 𝜃̂_𝑞𝑖, . . , 𝜃̂_1𝑖, 𝛩̂_1𝑖, … 𝛩̂_𝑄𝑖), i=1,2,…,N We defined the Hotelling’s T² test as

𝑇_1𝑖² = (𝛿̂_𝑖 − 𝛿̅)^𝑇𝛥⁻¹(𝛿̂_𝑖− 𝛿̅), 𝑖 = 1,2, … , 𝑁 (2.19)

where 𝛿̅ is the averages of all 𝛿̂_𝑖 ’s ,and 𝛥 = ∑^𝑁_𝑖=1(𝛿̂_𝑖 − 𝛿̅)(𝛿̂_𝑖 − 𝛿̅)^𝑇/(𝑁 − 1). The Kalman Filter approach can give the estimates, 𝛿̂_𝑖 and σ̂_ai² by using the arima function in R. The results (2.19) follow to F-distribution with degrees of freedom r and T-r as the test statistic (2.18), where r is the number of parameters in 𝛿̂. By the book of Montgomery (2009), the phase I control limits for the statistic 𝑇_1𝑖² are given by

𝑈𝐶𝐿_{4_1} =^{(𝑇−1)𝑟}_𝑇−𝑟 𝐹_{𝛼,𝑟,𝑇−𝑟} (2.20)

In the phase II, when the chart is used for monitoring the coefficients of the new profiles, the control limits are as follows:

𝑈𝐶𝐿 =𝑟(𝑇+1)(𝑇−1)

𝑇(𝑇−𝑟) 𝐹_{𝛼,𝑟,𝑇−𝑟} (2.21)

2.4.4 T

²

chart for monitoring variance of the residuals

To monitor the possible deviation due to 𝜎_𝑎² (see Soleimani et al. (2009)), 𝑒_𝑖 denotes T× 1 residual vector for the ith subgroup, 𝜎̂_ai² is the corresponding estimate of 𝜎_𝑎² for subgroup i ,use the test statistics:

𝑇_2𝑖² = (𝑒_𝑖− 0)^𝑇𝛴_𝑒⁻¹(𝑒_𝑖− 0), 𝑖 = 1,2,3, … , 𝑁 (2.22) where 𝛴_𝑒 = 𝜎̅_𝑎²𝐼，𝜎̅_𝑎² is the average of all 𝜎̂_𝑎𝑖² ′s. 𝑇₂² follows an 𝜒² distribution with T-1 degrees of freedom. The upper control limits for (2.22) are as follows:

𝑈𝐶𝐿_{4_2} = 𝜒_{𝛼,𝑇−1}² . (2.23)

We use these two charts simultaneously to detect a change in parameters of profile model and monitor the variance of residuals. If statistics 𝑇₁² and/ or 𝑇₂² of the subgroups is

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

detected by either the 𝑇₁² chart or 𝑇₂² chart, the subgroup may be identified to be

out-of-control. We then need to find the cause of out-of-control which makes the subgroup to be out-of-control.

There are several restrictions when using Hotelling T² control charts. For phase I scheme, if a subgroup is in-control and fit the profile (2.10), but the 𝑇₂² statistic exceeds the UCL due to the variance. There may be a false alarm case occurring in the phase I control chart. For Phase II scheme, if one of the subgroups can’t be fitted by profile model (2.10), then we can’t get the estimate of coefficients and residuals of the in-control profile to detect whether it is out-of-control. Because we can find a better time series model for the subgroup which is different from the one we determined, it indicates that the profile is out-of-control. In Chapter 3, we will illustrate an example how to use our four methods to construct the control bands, confidence bands, and control charts to monitor the

out-of-control profiles.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y Chapter 3 Real data analysis: Power

Consumption in National Chengchi University

3.1 Introduction

Electricity is now an indispensable thing in our life. Without electricity, it is inconvenient to do everything. Recently with the concern of energy issue, we want to know how to reduce the use of power. Being a College of Commerce student in National Chengchi University, we concern about the electricity consumption for College of Commerce building. We collect the hourly data from the electricity monitor system in the Office of the General Affair in Figure 3.1. The College of Commerce building is named “GCB5”. The technician in the Office of the General Affair mentioned that 60% of total consumption is for the air-conditioning, 15% is for the lighting, 15% is for the use of outlets, and 10% is for the other electrical consumption (public electricity, water, etc.). Therefore, the electricity consumption is mainly affected by the use of air-conditioning.

Figure 3.1 Plot of Electricity Monitor System

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

We use the hourly data from 2008, February to 2009, January to construct the control and confidence bands and control charts. There are 52 weeks including the Spring Semester (from February to June ), Summer Vacation, Fall Semester(from September to January ), and Winter Vacation. The following we just discuss the days when students attend school.

It is reasonable that we exclude the weekend (Saturday and Sunday), Summer Vacation and Winter Vacation data. Because we focus on the time when the electricity consumption will exceed the limits we determined, and try to find the cause that rises the consumption.

3.2 Data classification

Air-conditioning electricity consumption accounts for about 60% percent of the total consumption, while the use of the air conditioning in winter is different from the use in summer, which contribute to the difference in power consumption. In general, the highest consumption in winter is approximately 800 kilowatt hour (kWh), while in summer is 1000 kWh as Figure 3.2 shows the electricity consumption of a weekdays in March and June. It indicates that the electricity consumption contains a seasonal phenomenon due to weather.

Figure 3.2 Plot of the electricity consumption and corresponding temperature in March and June

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

It shows that the highest consumption of the two months differ 200 kWk from part(a), while the temperature differs about 5℃ to 10℃ from part (b), so it can be inferred that temperatures may affect the electricity consumption. We use the temperature as the classification rule to divide our data into “Semester without Air-conditioner data”

(denoted as WOAC data) and “Semester with Air-conditioner data” (denoted as WAC data). According to the Executive Yuan’ policy of energy conservation and carbon reduction, if the temperature doesn’t exceed 26℃, the office may not open the air conditioner. Due to temperature or climate changes (such as typhoons, cold), it may be suddenly cold or hot. Also the technician in the Office of the General Affairs of NCCU also takes 26℃ as a standard to decide to open the Air conditioner. So we use the

rule ,“temperature exceed 26℃ from Monday to Friday”, as our premier grouping criteria.

Based on the Central Weather Bureau in Taipei station statistical data, if the data in the semester (including Fall and Spring Semester), which depends on when the highest daily temperatures are all below 26℃ everyday in a weekday, is listed as WOAC; the remains are divided to WAC data. Table 3.1 shows the period of WOAC data and WAC data.

Regroup the data by using the procedures repeatedly until all the points fall in the control bands, confidence bands and control limits, which means that all the weekdays’ data are in-control. We will illustrate them in section 3.3.

‧

Note: In Table 3.1, “ ※” means that the weekday is from “No-Event Weekdays’ data”.

The remaining are from ”Special-Event Weekdays’ data”.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

According to calendar of National Chengchi University, after excluding the weekday that data exists the missing values, we divide the weekday data into “Special-Event Weekdays’

data” (denoted as SE data) and “No-Event Weekdays’ data”. (denoted as NE data) For SE data, it means that events occur such as national holidays, exam week, typhoons, etc. The occurrence of the events may affect the use of the electricity. For example, when the national holiday is coming, the number of students in the college of commerce building decrease and reduce the consumption of the electricity. Furthermore, when the day is hotter than usual, it is reasonable to see that the electricity consumption for the air-conditioning will be higher. For NE data, it means that no-special events could be found in that weekday, so the consumption of electricity may not be affected by unusual incident. The

establishment of the bands is based on the NE data, and use the bands to detect the SE data.

The following lists the plots for the WOAC data and WAC data. Figures 3.3 and 3.4 represent all the weekdays’ electricity consumption and the corresponding temperatures for NE data and SE data in WOAC dataset and WAC dataset, respectively.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

Figure 3.3 Plot of power and corresponding temperature for NE data in WOAC data

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

Figure 3.4 Plot of power and corresponding temperature for SE data in WOAC data

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

Figure 3.5 Plot of power and corresponding temperature for NE data in WAC data

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

Figure 3.6 Plot of power and corresponding temperature for SE data in WAC data

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

3.3 Phase I and Phase II control scheme

When control and confidence bands and control charts are applied to data, there are two phases, Phase I and Phase II. Phase I is the period of building bands, including obtaining a set of in-control data and using them to construct the bands such that they can be used in Phase II. In this stage, “no-event weekdays” data are treated as in-control data and used to construct the confidence bands and control charts. Phase II is the period of using bands and control limits completed in Phase I to monitor the interested processes to understand whether they are in control. The confidence bands and limits we construct in Phase I are applied to monitor the SE data, which is regard as the out-of-control data. According to four monitoring methods in Section 2, we use WOAC data and WAC data to construct the control and confidence bands and control charts, respectively. The procedure of

constructing the charts and monitoring SE data are illustrated as follows.

3.3.1 Model assumption and identification

There are N weekdays in one group, which can be expressed “ith weekday ” , i=1, 2 ,…,N.

We collect hourly electricity consumption data and all the subgroups contain 120 observations, which can be denoted as 𝑥_𝑡𝑖, t=1,2,…,120. Obviously, there is seasonal influence and exists a daily cycle as shown in Figure 3.7, so we fit the seasonal ARMA model with period s=24. We can regard one peak as one day because of the daily cycle.

Figure 3.7 Plot of the electricity consumption

‧

In order to fit the suitable time series model, several models are selected as candidates. One of the most commonly used to determine the most “adequate” model is based on Akaike Information Criteria (AIC) ,which is defined as

𝐴𝐼𝐶 = −2 𝑙𝑜𝑔(𝑚𝑎𝑥. 𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑) + 2𝑛_𝑖,

where 𝑛_𝑖 is the number of estimated model parameters. In the case of the ARMA(p,q) model we get

𝐴𝐼𝐶 = 𝑇 𝑙𝑜𝑔(𝜎̂_𝑎²) + 2(𝑝 + 𝑞),

By AIC, it is found that SARMA(1,0)×(2,0)₂₄ is the best model for WOAC data : (1 − 𝜑₁𝐵)(1 − 𝛷₂₄𝐵²⁴− 𝛷₄₈𝐵⁴⁸)𝑥_𝑡 = 𝑐 + 𝑎_𝑡, 𝑎_𝑡𝑖. 𝑖. 𝑑

~ 𝑊𝑁(0, 𝜎_𝑎²) (3.1) We use model (3.1) to fit WOAC data. Table 3.2 lists the parameter estimation for NE data, while Table 3.3 represents the parameter estimation for SE data. And SARMA(1,0)×(1,0)₂₄ is a better model for each weekday for WAC data :

(1 − 𝜑₁𝐵)(1 − 𝛷₂₄𝐵²⁴)𝑥_𝑡 = 𝑐 + 𝑎_𝑡, 𝑎_𝑡𝑖. 𝑖. 𝑑

~ 𝑊𝑁(0, 𝜎_𝑎²) (3.2) We use model (3.2) to fit WAC data. Table 3.4 lists the parameter estimation for NE data, while Table 3.5 represents the parameter estimation for SE data.

Model (3.1) and (3.2) are applied to analysis these weekdays for WOAC data and WAC data, respectively. After model fitting, diagnostic check of residuals is performed to assess the suitability of the residuals. The results indicate that residuals follow a normal distribution.

Figure 3.8 and Figure 3.9 show the plot of the autocorrelation function (ACF) and partial autocorrelation function (PACF) for the residuals by fitting model (3.1) and (3.2). Only the results of the four weekday’s observations are shown here, and the remains show similar patterns in the ACF and PACF plots.

‧

Table 3.2 Parameter Estimation of SARMA model for NE data in WOAC dataset

i wd 𝝋_𝟏 𝜱_𝟐𝟒 𝜱_𝟒𝟖 𝒄 𝝈_𝒂^𝟐 AIC

(123.006) 100.964 1019.04

2 5 0.904

(189.703) 163.677 1067.42

3 10 0.861

(142.446) 286.384 1116.39

4 39 0.834

(131.995) 119.422 1050.35

5 42 0.870

(117.086) 189.657 1077.23

7 47 0.866 Table 3.3 Parameter Estimation of SARMA model for SE data in WOAC dataset

i wd 𝝋_𝟏 𝜱_𝟐𝟒 𝜱_𝟒𝟖 𝒄 𝝈_𝒂^𝟐 AIC

1 “NA” means that data can’t fit time series model we determined, and it exists parameter estimation problem

‧

Table 3.4 Parameter Estimation of SARMA model for NE data in WAC dataset

j Weekday 𝝋_𝟏 𝜱_𝟐𝟒 c 𝝈_𝒂^𝟐 AIC

(117.018) 615.7216 1194.15

2 13 0.836

(0.05)

0.991 (0.003)

535.237

(127.565) 200.553 1081.27

3 16 0.780

(0.062)

0.965 (0.012)

584.756

(107.503) 1002.518 1242.68

4 17 0.802

(0.057)

0.975 (0.008)

625.030

(125.592) 771.8483 1219.77

5 18 0.771

(0.062)

0.988 (0.004)

606.336

(110.34) 381.8266 1152.17

6 36 0.830

(0.051)

0.985 (0.005)

646.989

(156.404) 506.166 1181.88

7 37 0.788

(0.059)

0.989 (0.004)

656.502

(130.405) 393.5577 1159.19

Table 3.5 Parameter Estimation of SARMA model for SE data in WAC dataset

j Weekday 𝝋_𝟏 𝜱_𝟐𝟒𝒋 c 𝝈_𝒂^𝟐 AIC

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

Figure 3.8 The ACF and PACF plots of residuals in WOAC data

Figure 3.9 The ACF and PACF plots of residuals in WAC data

‧

We introduce four methods which are (1) control bands under each time unit, (2) confidence bands based on Kalman Filter approach, (3) confidence bands based on Bootstrap approach, and (4) Hotelling T² control charts. By using electricity consumption data in NCCU, we construct the Phase I control charts by NE data and monitor SE data for WOAC data and WAC data, respectively, and let α =0.0027 in the Phase I scheme. In Phase II, we apply the bands to monitor SE data, which is regard as out-of-control. SE data means that one or more events occur in the weekday such as national holidays, typhoons, abnormally high

temperature, etc.. Table 3.6 represents the special event occurs in the weekday. We use this kind of data to evaluate the capability of four monitoring methods. When special events occur, we detect whether the electricity would be abnormal or not.

Table 3.6 Table of special events

Group weekday Period Special Event

WOAC data

1 2/18-2/22 Beginning weekday of Spring Semester

2 2/25-2/28 228 Memorial Day

6 3/24-3/28 none

7 3/31-4/4 Tomb Sweeping Day

8 4/7-4/11 Abnormal Temperature

9 4/14-4/18 Abnormal Temperature

11 4/28-5/2 Abnormal Temperature

46 12/29-1/2 New Year’s Day

48 1/12-1/16 Exam Week

WAC data

14 5/19-5/23 NCCU Anniversary

15 5/26-5/30 Abnormal Temperature

31 9/15-9/19 Beginning weekday of Fall Semester

32 9/22-9/26 Abnormal Temperature

33 9/29-10/3 Typhoon

34 10/6-10/10 Double Tenth Day

35 10/13-10/17 none

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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

‧

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 𝛼̂_{𝑡𝑖|𝑇} and 𝑃̂_{𝑡𝑖|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.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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

‧

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 𝜎̂_𝑎𝑖 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

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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 𝜎̅_𝑎² 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

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

3.3.4 Example of Semester with AC data (WAC data)

I. Control bands under each hour

In Phase I, we calculate the control bands by equations (2.7) and (2.8). Figure 3.16 part (a) shows the control bands under each time unit and the electricity consumption in weekday32 from SE data. All the observations from NE data fall in the bands, which means that the subgroups are in-control. We can use this chart to monitor the data.

In phase II, we plot the SE data together with control bands (2.7) and (2.8). Table 3.11 shows the time points that electricity consumption are out of the control bands. It represents

In phase II, we plot the SE data together with control bands (2.7) and (2.8). Table 3.11 shows the time points that electricity consumption are out of the control bands. It represents

在文檔中 追蹤季節性時間數列模型之流程資料 - 政大學術集成 (頁 26-0)