Time series structures, which are common occurrences with data in many industrial processes, complicate a quality practitioner s e orts to accurately position control chart limits. ARIMA modeling and a variety of control charting methods have been recommended for monitoring process data with a time series structure. Estimates of ARIMA model parameters may not be reliable, however, if assignable causes of variation are present in the process data used to fit the time series model. Control limits may also be misplaced if the process inputs are dynamic and exhibiting a time series structure. Our purpose in this paper is to explore the ability of a transfer function model to identify assignable causes of variation and to model dynamic relationships between process inputs and outputs. A transfer function model is developed to monitor biochemical oxygen demand output from a wastewater treatment process, a process with dynamic inputs. This model is used to identify periods of disturbance to the wastewater process and to capture the relationship between the variable nature of the input to the process and the resulting output. Simulation results are included in this study to measure the sensitivity of transfer function models and to highlight conditions where transfer function modeling is critical.
Key Words: Autocorrelation, Statistical Process Control, Time Series Analysis.
By DAVID WEST and SCOTT DELLANA, East Carolina University, Greenville, NC 27858-4353
JEFFREY JARRETT, The University of Rhode Island, Kingston, RI 02881
IN an extensive survey, Alwan and Roberts (1995) found that more than 85% of industrial process control applications resulted in charts with possibly misplaced control limits. In many instances, the misplaced control limits result from autocorrelation of the process observations, which violates a basic assumption often associated with the Shewhart chart (Woodall (2000)). Autocorrelation of process observations has been reported in many industries, including cast steel (Alwan (1992)), blast furnace operations (Notohardjono and Ermer (1986)), wastewater treatment plants (Berthouex, Hunter, and Pallesen (1978)), chemical processes industries (Montgomery and Mastrangelo (1991)), semiconductor manufacturing (Kim and May (1994)), injection molding (Smith (1993)), and basic rolling operations (Xia, Rao, Shan, and Shu (1994)).
Several models have been proposed to monitor processes with autocorrelated observations. Alwan and Roberts (1988) suggest using an autoregressive integrated moving average (ARIMA) residuals chart, which they referred to as a special cause chart. For subsample control applications, Alwan and Radson (1992) describe a xed limit control chart, where the original observations are plotted with control limit distances determined by the variance of the sub-sample mean series. Montgomery and Mastrangelo (1991) use an adaptive exponentially weighted moving average (EWMA) centerline approach, where the control limits are adaptive in nature and determined byasmoothed estimate of process variability. Lu and Reynolds (2001) investigate the steady state average run length of cumulative sum (CUSUM), EWMA, and Shewhart control charts for autocorrelated data modeled as a first order autoregressive process plus an additional random error term.
A problem with all of these control models is that the estimate of the process variance is sensitive to outliers. If assignable causes are present in the data used to fit the model, the model may be incorrectly identified and the estimators of model parameters may be biased, resulting in loose or invalid control limits (Boyles (2000)). To justify the use of these methods, researchers have made the assumption that aperiod of "clean data" exists to estimate control limits. Therefore, methods are needed to assure that parameter estimates are free of contamination from assignable causes of variation. Intervention analysis, with an iterative identification of outliers, has been proposed for this purpose. The reader interested in more detail should see Alwan (2000, pp. 301–307), Atienza, Tang, and Ang (1998), and Box, Jenkins, and Reinsel (1994, pp. 473–474). Atienza, Tang, and Ang (1998) recommend the use of a control procedure based on an intervention test statistic, , and show that their procedure is more sensitive than ARIMA residual charts for process applications with high levels of positive autocorrelation. They limit their investigation of intervention analysis, however, to the detection of a single level disturbance in a process with high levels of rst order autocorrelation. Wright, Booth, and Hu (2001) propose a joint estimation method capable of detecting outliers in an autocorrelated process where the data available is limited to as few as 9 to 25 process observations. Since intervention analysis is crucial to model identification and estimation, we investigate varying levels of autocorrelation, autoregressive and moving average processes, different types of disturbances, and multiple process disturbances.
The ARIMA and intervention models are appropriate for autocorrelated processes whose input streams are closely controlled. However, there are quality applications, which we refer to as dynamic input processes, where this is not a valid assumption. The treatment of wastewater is one example of a dynamic process that must accommodate highly fluctuating input conditions. In the health care sector, the modeling of emergency room service must also deal with highly variable inputs. The dynamic nature of the input creates an additional source of variability in the system, namely the time series structure of the process input. For these applications, modeling the dynamic relationship between process inputs and outputs can be used to obtain improved process monitoring and control as discussed by Alwan (2000, pp. 675–679).
We propose a more general transfer function: an ARIMA model that accounts for both outliers in process output and dynamic effects from process input. In the following section, we briefly describe the relevant theory of time series analysis used in this paper. We then analyze the transfer function model terms to identify disturbances in a wastewater treatment process. We follow in later sections with supporting empirical evidence on the sensitivity of these methods. The paper concludes with a discussion of the implications for quality practitioners who may be monitoring processes which produce data with time series structures and which have dynamic inputs. In this paper, we refer to autocorrelated processes as either autoregressive (AR) or as moving average (MA) (as defined by Box, Jenkins, and Reinsel (1994)).
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