Abstract
In many manufacturing applications, the monitoring of categorical event series is required, i. e., of processes, where the quality characteristics are measured on a qualitative scale. We survey three groups of approaches for this task. First, the categorical event series might be transformed into a count process (e. g., event counts, discrete waiting times). After having identified an appropriate model for this count process, diverse control charts are available for the monitoring of the generated counts. Second, control charts might be directly applied to the considered categorical event series, using different charts for nominal than for ordinal data. The latter distinction is also crucial for the respective possibilities of analyzing and modeling these data. Finally, also rule-based procedures from machine learning might be used for the monitoring of categorical event series, where the generated rules are used to predict the occurrence of critical events. Our comprehensive survey of methods and models for categorical event series is complemented by two real-data examples from manufacturing industry, about nominal types of defects and ordinal levels of quality.
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Notes
- 1.
Although the presented approaches for episode mining do not explicitly use stochastic assumptions, several connections to models from Sect. 3.1 have been established in the literature, namely to Markov models by Gwadera et al. [16], to Hidden-Markov models by Laxman et al. [27], and to variable-length Markov models by Weiß [55].
References
Agrawal R, Srikant R (1994) Fast algorithms for mining association rules. In: Proceedings of the 20th international conference on very large databases, pp 487–499
Agresti A (2010) Analysis of ordinal categorical data, 2nd edn. John Wiley & Sons Inc., Hoboken
Bai K, Li J (2021) Location-scale monitoring of ordinal categorical processes. Naval research logistics, forthcoming
Bashkansky E, Gadrich T (2011) Statistical quality control for ternary ordinal quality data. Appl Stoch Models Bus Ind 27(6):586–599
Baum LE, Petrie T (1966) Statistical inference for probabilistic functions of finite state Markov chains. Ann Math Stat 37(6):1554–1563
Bersimis S, Sachlas A, Castagliola P (2017) Controlling bivariate categorical processes using scan rules. Methodol Comput Appl Probab 19(4):1135–1149
Blatterman DK, Champ CW (1992) A Shewhart control chart under 100% inspection for Markov dependent attribute data. In: Proceedings of the 23rd annual modeling and simulation conference, pp 1769–1774
Bourke PD (1991) Detecting a shift in fraction nonconforming using run-length control charts with 100% inspection. J Qual Technol 23(3):225–238
Brook D, Evans DA (1972) An approach to the probability distribution of CUSUM run length. Biometrika 59(3):539–549
Bühlmann P, Wyner AJ (1999) Variable length Markov chains. Ann Stat 27(2):480–513
Duncan AJ (1950) A chi-square chart for controlling a set of percentages. Ind Qual Control 7:11–15
Duran RI, Albin SL (2009) Monitoring and accurately interpreting service processes with transactions that are classified in multiple categories. IIE Trans 42(2):136–145
Ferland R, Latour A, Oraichi D (2006) Integer-valued GARCH processes. J Time Ser Anal 27(6):923–942
Gan FF (1990) Monitoring observations generated from a binomial distribution using modified exponentially weighted moving average control chart. J Stat Comput Simul 37(1–2):45–60
Göb R (2006) Data mining and statistical control – a review and some links. In: Lenz HJ, Wilrich PT (eds) Frontiers in statistical quality control 8, pp 285–308. Physica-Verlag, Heidelberg
Gwadera R, Atallah MJ, Szpankowski W (2005) Markov models for identification of significant episodes. In: Proceedings of the 2005 SIAM international conference on data mining, pp 404–414
Höhle M (2010) Online change-point detection in categorical time series. In: Kneib T, Tutz G (eds) Statistical modelling and regression structures. Festschrift in Honour of Ludwig Fahrmeir, pp 377–397. Physica-Verlag, Heidelberg
Höhle M, Paul M (2008) Count data regression charts for the monitoring of surveillance time series. Comput Stat Data Anal 52(9):4357–4368
Jacobs PA, Lewis PAW (1983) Stationary discrete autoregressive-moving average time series generated by mixtures. J Time Ser Anal 4(1):19–36
Klein I, Doll M (2021) Tests on asymmetry for ordered categorical variables. J Appl Stat 48(7):1180–1198
Klemettinen M, Mannila H, Toivonen H (1999) Rule discovery in telecommunication alarm data. J Netw Syst Manag 7(4):395–423
Koutras MV, Bersimis S, Antzoulakos DL (2006) Improving the performance of the chi-square control chart via runs rules. Methodol Comput Appl Prob 8(3):409–426
Koutras MV, Maravelakis PE, Bersimis S (2008) Techniques for controlling bivariate grouped observations. J Multivar Anal 99(7):1474–1488
Kvålseth TO (1995) Coefficients of variation for nominal and ordinal categorical data. Percept Mot Skills 80(3):843–847
Lambert D, Liu C (2006) Adaptive thresholds: monitoring streams of network counts. J Am Stat Assoc 101(473):78–88
Laxman S, Sastry PS (2006) A survey of temporal data mining. Sādhanā 31(2):173–198
Laxman S, Sastry PS, Unnikrishnan KP (2005) Discovering frequent episodes and learning hidden Markov models: a formal connection. IEEE Trans Knowl Data Eng 17(11):1505–1517
Li J, Tsung F, Zou C (2014) A simple categorical chart for detecting location shifts with ordinal information. Int J Prod Res 52(2):550–562
Li J, Xu J, Zhou Q (2018) Monitoring serially dependent categorical processes with ordinal information. IISE Trans 50(7):596–605
Mannila H, Toivonen H, Verkamo AI (1997) Discovery of frequent episodes in event sequences. Data Min Knowl Disc 1(3):259–289
Marcucci M (1985) Monitoring multinomial processes. J Qual Technol 17(2):86–91
McKenzie E (1985) Some simple models for discrete variate time series. Water Resour Bull 21(4):645–650
Montgomery DC (2009) Introduction to statistical quality control, 6th edn. John Wiley & Sons Inc., New York
Morais MC, Knoth S, Weiß CH (2018) An ARL-unbiased thinning-based EWMA chart to monitor counts. Seq Anal 37(4):487–510
Mousavi S, Reynolds MR Jr (2009) A CUSUM chart for monitoring a proportion with autocorrelated binary observations. J Qual Technol 41(4):401–414
Mukhopadhyay AR (2008) Multivariate attribute control chart using Mahalanobis \(D^2\) statistic. J Appl Stat 35(4):421–429
Nembhard DA, Nembhard HB (2000) A demerits control chart for autocorrelated data. Qual Eng 13(2):179–190
Page E (1954) Continuous inspection schemes. Biometrika 41(1):100–115
Perry MB (2020) An EWMA control chart for categorical processes with applications to social network monitoring. J Qual Technol 52(2):182–197
Raftery AE (1985) A model for high-order Markov chains. J Roy Stat Soc B 47(3):528–539
Rakitzis AC, Weiß CH, Castagliola P (2017) Control charts for monitoring correlated counts with a finite range. Appl Stoch Models Bus Ind 33(6):733–749
Reynolds MR Jr, Stoumbos ZG (1999) A CUSUM chart for monitoring a proportion when inspecting continuously. J Qual Technol 31(1):87–108
Roberts SW (1959) Control chart tests based on geometric moving averages. Technometrics 1(3):239–250
Ryan AG, Wells LJ, Woodall WH (2011) Methods for monitoring multiple proportions when inspecting continuously. J Qual Technol 43(3):237–248
Spanos CJ, Chen RL (1997) Using qualitative observations for process tuning and control. IEEE Trans Semicond Manuf 10(2):307–316
Steiner SH (1998) Grouped data exponentially weighted moving average control charts. J Roy Stat Soc C 47(2):203–216
Steiner SH, Geyer PL, Wesolowsky GO (1996) Grouped data-sequential probability ratio tests and cumulative sum control charts. Technometrics 38(3):230–237
Szarka JL III, Woodall WH (2011) A review and perspective on surveillance of Bernoulli processes. Qual Reliab Eng Int 27(6):735–752
Tucker GR, Woodall WH, Tsui K-L (2002) A control chart method for ordinal data. Am J Math Manag Sci 22(1–2):31–48
Vasquez Capacho JW, Subias A, Travé-Massuyès L, Jimenez F (2017) Alarm management via temporal pattern learning. Eng Appl Artif Intell 65:506–516
Wang J, Li J, Su Q (2017) Multivariate ordinal categorical process control based on log-linear modeling. J Qual Technol 49(2):108–122
Wang J, Su Q, Xie M (2018) A univariate procedure for monitoring location and dispersion with ordered categorical data. Commun Stat Simul Comput 47(1):115–128
Weiss GM (1999) Timeweaver: a genetic algorithm for identifying predictive patterns in sequences of events. In: Banzhaf et al (eds) Proceedings of the genetic and evolutionary computation conference, pp 718–725. Morgan Kaufmann, San Francisco
Weiß CH (2009) Group inspection of dependent binary processes. Qual Reliab Eng Int 25(2):151–165
Weiß CH (2011) Rule generation for categorical time series with Markov assumptions. Stat Comput 21(1):1–16
Weiß CH (2012) Continuously monitoring categorical processes. Qual Technol Quant Manag 9(2):171–188
Weiß CH (2015) SPC methods for time-dependent processes of counts – a literature review. Cogent Math 2(1):1111116
Weiß CH (2017) Association rule mining. In: Balakrishnan et al (eds) Wiley StatsRef: statistics reference online. John Wiley & Sons Ltd., Hoboken
Weiß CH (2018a) An introduction to discrete-valued time series. John Wiley & Sons Inc., Chichester
Weiß CH (2018b) Control charts for time-dependent categorical processes. In: Knoth S, Schmid W (eds) Frontiers in statistical quality control 12. Physica-Verlag, Heidelberg, pp 211–231
Weiß CH (2020) Distance-based analysis of ordinal data and ordinal time series. J Am Stat Assoc 115(531):1189–1200
Weiß CH (2021) Stationary count time series models. WIREs Comput Stat 13(1):e1502
Weiß CH, Zhu F, Hoshiyar A (2022) Softplus INGARCH models. Statistica Sinica 32(3), forthcoming
Ye N (2003) Mining computer and network security data. In: Ye N (ed) The handbook of data mining. Lawrence Erlbaum Associations Inc., New Jersey, pp 617–636
Ye N, Borror C, Zhang Y (2002a) EWMA techniques for computer intrusion detection through anomalous changes in event intensity. Qual Reliab Eng Int 18(6):443–451
Ye N, Masum S, Chen Q, Vilbert S (2002b) Multivariate statistical analysis of audit trails for host-based intrusion detection. IEEE Trans Comput 51(7):810–820
Yuan Y, Zhou S, Sievenpiper C, Mannar K, Zheng Y (2011) Event log modeling and analysis for system failure prediction. IIE Trans 43(9):647–660
Zimmermann A (2014) Understanding episode mining techniques: benchmarking on diverse, realistic, artificial Data. Intell Data Anal 18(5):761–791
Acknowledgments
The author thanks the referee for useful comments on an earlier draft of this article. The author is grateful to Professor Jian Li (Xi’an Jiaotong University, China) for providing the flash data discussed in Sect. 5.2.
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Weiß, C.H. (2022). On Approaches for Monitoring Categorical Event Series. In: Tran, K.P. (eds) Control Charts and Machine Learning for Anomaly Detection in Manufacturing. Springer Series in Reliability Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-83819-5_5
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