隨著市場競爭，企業為了降低內部及外部的不良品成本，品質管制越來越受到重視。因此大多企業會使用不同的方法來監控生產製程的穩定性，例如統計製程管制(SPC)即是一套常用的工具，主要用來監控製程有無發生變異。傳統在建構管制圖時常會假設資料的參數已知且來自於特定分配，其中最常被假設的即為常態分配，但在實務上常態分配的基本假設不一定會被滿足，尤其在生產初期，因為樣本數不足，更不容易符合常態分配的基本假設，一旦使用錯誤的分配，將會影響所建構管制圖的績效。而為了避免分配假設錯誤的情況發生，無母數管制圖的方法被提出來，其主要的優勢為當母體分配未知時，可以維持一定的監控能力。資料深度(Data Depth)的方法可以辨別一筆資料在整體資料中是靠近中心或位於邊緣。其中深度值越高即代表越接近中心，反之則越靠近邊緣，這方法不只可以使用在單變量的資料，更適合使用於多變量型態資料的判讀。本研究以資料深度的概念結合無母數管制圖，目的是使用穩定狀態所收集的參考資料與新加入的測試資料進行比較，主要用來監控兩組資料是否具有相同的離散程度。其執行方法為從穩定狀態收集一組參考資料，並將所要監控的資料採用由最新一期往前選取的方式加入測試資料中，目的是能有效的利用最新樣本的資訊，再將參考資料與測試資料結合，計算出每筆資料的資料深度值，並透過無母數Wilcoxon rank-sum統計量進行標準化後來建構無母數管制圖。而本研究探討四種不同的資料深度，分別為Simplicial Depth、Tukey Depth、 Spatial Depth、Projection Depth，最後在比較哪一個資料深度在本論文所建構的管制圖有較好的績效表現。由結果分析顯示以Projection深度建構之管制圖在監控離散程度的績效表現上均優於其他深度函數，且與比較之管制圖相比，在離散程度發生小變動時有較好的監控能力。

In this study, we construct the nonparametric control chart based on data depth(NP-DDC). This method uses steady state data to compare with newly added data. Consider two samples from two univariate distributions which are identical except for a possible dispersion difference. The execution method is to collect the reference data from steady state and add the test data from the latest phase forward. This methed make effective use of the latest sample information. The reference data is combined with the test data to calculate the depth value of each data, and the Wilcoxon rank-sum statistic is used to construct the nonparametric control chart. This study compares the four depth functions of Simplicial Depth, Tukey Depth, Spatial Depth, and Projection Depth on the NP-DDC control chart. The simulation results show that the NP-DDC control chart constructed with Projection depth is superior to other depth functions in monitoring the dispersion, and has better monitoring ability than the CPM-Mood if the scale changes slightly. However, the NP-DDC control chart is less sensitive to the monitoring of the location compared with the EWMA chart proposed by Zhou et al. and the CUSUM chart proposed by Wang et al.

中文文獻:
鄭旻紘，應用無母數統計方法於建構資訊理論管制圖用於製程監控，國立
成功大學工業與資訊管理研究所碩士論文，民國一百零七年七月。
楊瑋欣，應用幾何分佈於監控努力過程之資訊理論管制圖，國立成功大學
工業與資訊管理研究所碩士論文，民國一百零五年六月。
英文文獻:
Alloway Jr, J. A., & Raghavachari, M. (1991). Control chart
based on the Hodges-Lehmann estimator. Journal of
Quality Technology, 23(4), 336-347.
Amin, R. W., & Searcy, A. J. (1991). A nonparametric
exponentially weighted moving average control scheme.
Communications in Statistics-Simulation and Computation
, 20(4), 1049-1072.
Amin, R. W., Reynolds Jr, M. R., & Saad, B. (1995). Nonparametric quality control charts based on the sign
statistic. Communications in Statistics-Theory and
Methods, 24(6), 1597-1623.
Bakir, S. T., & Reynolds, M. R. (1979). A nonparametric
procedure for process control based on within-group
ranking. Technometrics, 21(2), 175-183.
Bellera, C. A., Julien, M., & Hanley, J. A. (2010). Normal
approximations to the distributions of the Wilcoxon
statistics: accurate to what N? Graphical insights.
Journal of Statistics Education, 18(2).
Brown, B. and Hettmansperger, T. (1989). The affine invariant
bivariate version of the sign test. Journal of the
Royal Statistal Society B, 51, 117-125.
Chakraborti, S., Van der Laan, P., & Bakir, S. T. (2001).
Nonparametric control charts: an overview and some
results. Journal of Quality Technology, 33(3), 304-315.
Chatterjee, S., & Qiu, P. (2009). Distribution-free
cumulative sum control charts using bootstrap-based
control limits. The Annals of Applied Statistics, 3(1),
349-369.
Christmann, A. (2002). Classification based on the SVM and
on regression depth. In Dodge, Y. (editores),
Statistical data analysis based on the L1 norm and
related methods, Basel: Birkh¨ Ouser, 341-352.
Chenouri, S., Small, C. G., & Farrar, T. J. (2011). Data
depth‐based nonparametric scale tests. Canadian Journal
of Statistics, 39(2), 356-369.
Donoho, D. L., & Gasko, M. (1992). Breakdown properties of
location estimates based on halfspace depth and
projected outlyingness. The Annals of Statistics,
20(4), 1803-1827.
Donoho, D. L. (1982). Breakdown properties of multivariate
location estimators. Ph.D. qualifying paper, Dept.
Statistics, Harvard Univ.
Eto, M. R. The concept of depth in statistics. Tech. Rep.
(cited on page 31).
Fraiman, R., Liu, R. Y., & Meloche, J. (1997). Multivariate
density estimation by probing depth. Lecture Notes-
Monograph Series, 415-430.
Graham, M. A., Chakraborti, S., & Human, S. W. (2011). A
nonparametric EWMA sign chart for location based on
individual measurements. Quality Engineering, 23(3),
227-241.
Hackl, P., & Ledolter, J. (1991). A control chart based on
ranks. Journal of Quality Technology, 23(2), 117-124.
Hawkins, D. M., Qiu, P., & Kang, C. W. (2003). The
changepoint model for statistical process control. Journal of Quality Technology, 35(4), 355-366.
Hettmansperger, T. (1984). Statistical inference based on
ranks. Wiley, New York.
Hettmansperger, T., Nyblom, J. and Oja, H. (1992). On
multivariate notions of sign and rank. In Dodge, Y.
(editores), L-1 Statistics and Related Methods.
Amsterdam: Elsevier , 267-278.
Hettmansperger, T. and Oja, H. (1994). Affine invariant
multivariate multisample sign tests. Journal of the
Royal Statistal Society B, 56, 235-249.
Jörnsten, R. (2004). Clustering and classification based on
the L1 data depth. Journal of Multivariate Analysis,
90, 67-89.
Kvam, P. H., & Vidakovic, B. (2007). Nonparametric
statistics with applications to science and engineering
(Vol. 653). John Wiley & Sons.
Li, S. Y., Tang, L. C., & Ng, S. H. (2010). Nonparametric
CUSUM and EWMA control charts for detecting mean
shifts. Journal of Quality Technology, 42(2), 209-226.
Liu, R. Y. (1990). On a notion of data depth based on
random simplices. The Annals of Statistics, 405-414.
Liu, R., Parelius, J. M. and Singh, K. (1999). Multivariate
analysis by data depth descriptive statistics, graphics
and inference. Annals of Statistics, 27, 783-858.
Liu, R. Y. (1992). Data depth and multivariate rank tests.
L1-statistical analysis and related methods, 279-294.
Liu, R., & Singh, K. (2003). Rank tests for comparing
multivariate scale using data depth: Testing for
expansion or contraction. Unpublished Manuscript.
Liu, R. Y. (1995). Control charts for multivariate
processes. Journal of the American Statistical
Association, 90(432), 1380-1387.
McDonald, D. (1990). A CUSUM procedure based on sequential
ranks. Naval Research Logistics (NRL), 37(5), 627-646.
Mood, A. M. (1954). On the asymptotic efficiency of certain
nonparametric two-sample tests. The Annals of
Mathematical Statistics, 25(3), 514-522.
Li, J., & Liu, R. Y. (2004). New nonparametric tests of
multivariate locations and scales using data depth.
Statistical Science, 686-696.
Orban, J., & Wolfe, D. A. (1982). A class of distribution-
free two-sample tests based on placements. Journal of
the American Statistical Association, 77(379), 666-672.
Pappanastos, E. A., & Adams, B. M. (1996). Alternative
designs of the Hodges-Lehmann control chart. Journal of
Quality Technology, 28(2), 213-223.
Park, C., Park, C., Reynolds Jr, M. R., & Reynolds Jr, M.
R. (1987). Nonparametric procedures for monitoring a
location parameter based on linear placement
statistics. Sequential Analysis, 6(4), 303-323.
Rousseeuw, P. J., & Hubert, M. (1999). Regression depth.
Journal of the American Statistical Association,
94(446), 388-402.
Ruts, I. and Rousseeuw, P. (1996). Computing depth contours
of bivariate point clouds. Computational Statistics and
Data Analysis, 23, 153-168.
Ross, G. J., Tasoulis, D. K., & Adams, N. M. (2011).
Nonparametric monitoring of data streams for changes in
location and scale. Technometrics, 53(4), 379-389.
Serfling, R. (2002). A depth function and a scale curve
based on spatial quantiles. In Statistical Data
Analysis Based on the L1-Norm and Related Methods (pp.
25-38). Birkhäuser, Basel.
Serfling, R. (2004). Nonparametric multivariate descriptive
measures based on spatial quantiles. Journal of
Statistical Planning and Inference, 123, 259-278.
Serfling, R. (2006). Depth functions in nonparametric
multivariate inference. DIMACS
Series in Discrete Mathematics and Theoretical Computer
Science, 72, 1-16.
Shewhart, W. A. (1931). Economic control of quality of
manufactured product. ASQ Quality Press.
Small, C. G. (1990). A survey of multidimensional medians.
International Statistical Review/Revue Internationale
de Statistique, 263-277.
Stahel, W. A. (1981). Breakdown of covariance estimators.
Research Report 31, Fachgruppe für Statistik, ETH,
Zürich.
Tukey, J.(1975). Mathematics and the picturing of data.
Proceedings of the International Congress of
Mathematics, 2,Vancouver, 523-531.
Wang, J. and Serfling, R. (2005). Nonparametric multivariate
kurtosis and tailweight measures. Journal of
Nonparametric Statistics, 17, 441-456 .
Wang, D., Zhang, L., & Xiong, Q. (2017). A non parametric
CUSUM control chart based on the Mann–Whitney
statistic. Communications in Statistics-Theory and
Methods, 46(10), 4713-4725
Wilcoxon, F. (1945). Individual comparisons by ranking
methods. Biometrics Bulletin, 1(6), 80-83.
Woodall, W. H. (2000). Controversies and contradictions in
statistical process control. Journal of Quality
Technology, 32(4), 341-350.
Woodall, W. H., & Montgomery, D. C. (1999). Research issues
and ideas in statistical process control. Journal of
Quality Technology, 31(4), 376-386.
Yang, S. F., Lin, J. S., & Cheng, S. W. (2011). A new
nonparametric EWMA sign control chart. Expert Systems
with Applications, 38(5), 6239-6243.
Yashchin, E. (1992). Analysis of CUSUM and other Markov-
type control schemes by using empirical distributions.
Technometrics, 34(1), 54-63.
Zhang, J. (2002). Some extensions of Tukey’s depth
function. Journal of Multivariate Analysis, 82, 134-
165.
Zhou, C., Zou, C., Zhang, Y., & Wang, Z. (2009).
Nonparametric control chart based on change-point
model. Statistical Papers, 50(1), 13-28.
Zuo, Y., & Serfling, R. (2000). General notions of
statistical depth function. Annals of Statistics, 461-
482.