臺灣博碩士論文加值系統

　　工業界常用統計管制圖作為監控製程穩定性的工具，近年來管制圖亦被應用於環境品質，如空氣品質及工業廢棄物污染的改善上。一般而言，當製程資料呈現自我相關時，使用傳統管制圖可能會導致誤判之狀況。因此，發展一種有別於傳統的特殊管制圖有其必要，雖然此種適合相關資料的單變量管制圖已廣受討論，但環境資料常呈現長期記憶(long memory)之特性，我們應該使用自我迴歸移動平均部份整合模式(ARFIMA)，而非自我迴歸移動平均整合模式(ARIMA)去配適它。為了監控呈現長期記憶的環境資料，本論文提出使用ARFIMA模式的管制圖去監控呈自我相關的空氣品質資料。最後，我們以南台灣地區空氣品質資料比較使用ARFIMA模式與ARIMA模式在監控空氣品質之間的差異。結果顯示使用ARFIMA模式的殘差管制圖較使用ARIMA模式的殘差管制圖適當(可及早偵測出異常點)。
　　至於多變量累和管制圖(MCUSUM)與多變量指數平滑與移動平均管制圖(MEWMA)則常用於偵測多變量製程的微量移動(small shift)上。不過，此種多變量管制圖皆假設製程服從常態分配，而並不符合現況。為了發展一個簡單而且實用的多變量管制圖，本論文提出了多變量舒華特累和管制圖(SMCUSUM)。SMCUSUM管制圖與單變量舒華特累和管制圖(Shewhart-like CUSUM)類似，其優點在於無需給定目標值與參數我們即可偵測製程的微量移動。模擬結果顯示SMCUSUM管制圖之偵測表現就平均串長度而言較MCUSUM管制圖與MEWMA管制圖均佳。
　　由於多變量製程資料常呈現自我相關，因此模式選擇的正確與否會直接影響多變量製程監控的表現。我們以一組美國水文的資料為例，說明SMCUSUM管制圖可適用於監控多變量自我相關資料。實例分析結果顯示，SMCUSUM管制圖在偵測製程異常變動的表現亦較MCUSUM管制圖與MEWMA管制圖為佳。

　　The statistical control chart is commonly used in industry to help ensure stability of the manufacturing process. It can also be used to monitor environmental quality; such as a change in air quality, industrial pollution, etc. When process data are autocorrelated, traditional control charts may lead to false results. Hence, nontraditional control charts are needed and this area has been widely discussed for the univariate case.
　　Sometimes processes have not only short memory, but long memory as well. Instead of following the autoregressive integrated moving-average (ARIMA) models, these processes with long memory follow the so called fractionally integrated autoregressive moving-average (ARFIMA) models. In this thesis, a control chart for autocorrelated data using the ARFIMA model is proposed to monitor the longmemory air quality data. Finally, we use the air quality data for Taiwan to compare the difference between the ARFIMA and ARIMA models. The results show that
residual control charts using the ARFIMA models are more appropriate than those using the ARIMA models.
　　For multivariate processes, various control charts including the multivariate cumulative sum (MCUSUM) and multivariate exponentially weighted moving average
(MEWMA) charts were proposed to monitor small process shifts. However, the calculation of optimal parameters assume that the distribution of a manufacturing process is known. This may not be true in practice. In order to develop a simple yet practical multivariate control chart, a Shewhart-like multivariate CUSUM (SMCUSUM) control chart is proposed. Similar to the univariate Shewhart-like
CUSUM chart, it has the advantage of detecting the small process shift without requiring target values and parameters. The results of our simulation study show that our proposed SMCUSUM chart outperforms the MEWMA and MCUSUM charts in terms of in-control and out-of-control ARLs.
　　Similar to univariate processes, the data for multivariate processes may also be autocorrelated. Thus, choosing an appropriate model is also critical for monitoring autocorrelated processes. To demonstrate the application of SMCUSM chart to autocorrelated processes, an example of hydrological data is used to compare the
　　performance of SMCUSUM, MEWMA, and MCUSUM charts. The results show that the SMCUSUM chart is more sensitive than MEWMA and MCUSUM charts.

Contents
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Process Outputs are Correlated . . . . . . . . . . . . . . . . . . . . . 2
1.3 The Calculation of ARLs for Control Charts . . . . . . . . . . . . . . 3
1.4 Development of a Sensitive Multivariate Control Chart . . . . . . . . 4
2 Literature Review 6
2.1 Long Memory Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Hotelling’s T2 Control Chart . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 MCUSUM Control Chart . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 MEWMA Control Chart . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Shewhart-like CUSUM control chart . . . . . . . . . . . . . . . . . . 20
3 Development of Univariate Control Charts for Monitoring Long
Memory Data 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 The Properties of Air Quality Data . . . . . . . . . . . . . . . . . . . 24
3.3 Development of Statistical Control Charts for Long Memory Data . . 27
3.4 Comparison of ARFIMA and ARIMA Models . . . . . . . . . . . . . 30
3.4.1 Example of Using the PM10 Data of Nantsz . . . . . . . . . . 30
3.4.2 Example of Using the PM10 data of Tsoying . . . . . . . . . . 35
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Development of a Shewhart-likeMultivariate CUSUMControl Chart 41
4.1 Development of Shewhart-like MCUSUM Control Charts . . . . . . . 42
4.1.1 SMCUSUM chart for individual observations (n = 1) . . . . . 43
4.1.2 SMCUSUM chart for multiple observations (n > 1) . . . . . . 48
4.2 Comparison of SMCUSUM and Other Multivariate Control Charts . 51
4.2.1 The Comparison of Detecting Performance of Various Multivariate
Control Charts . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Application of SMCUSUM Control Chart to Hydrological Data . . . 55
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 Conclusions and Future Research 67

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