# The effects of autocorrelation on real-time statistical process control with solutions for forest products manufacturers. (Management).

TIMOTHY M. YOUNG (*)PAUL M. WINISTORFER (*)

ABSTRACT

The use of statistical methods for continuous improvement is emerging as a low-risk business strategy for lowering manufacturing costs and improving product value for the forest products industry. One of the primary tools used by the practitioner to promote continuous improvement is the Shewhart control chart. As forest products manufacturing evolves toward high speed, continuous systems, certain limitations of the Shewhart control chart become apparent. Positive autocorrelation is inherent to automated data collected from continuous manufacturing systems and may cause false signals of statistical control. In this study, two techniques for adjusting the X-Individuals control chart for positive autocorrelation were compared with the outcomes of unadjusted X-Individuals control charts. Comparisons were based on 1,000 samples (10 sets of 100) of fiber moisture from a continuous medium density fiberboard (MDF) manufacturing process. The samples were derived at time intervals varying from 1 to 10 minutes. False si gnals of special-cause variation increased as positive autocorrelation (rl) increased. A technique for adjusting the exponentially weighted moving average (EWMA) control chart in the presence of positive autocorrelation was also examined. One hundred and sixty-three samples of within-panel, out-of-press thicknesses as measured by laser scanning technology were obtained from a different MDF manufacturer. In the presence of positive autocorrelation, the traditional EWMA control chart for independent data resulted in 10 suspect false signals of special-cause variation. An EWMA chart adjusted for autocorrelation for the same data had no signals of special-cause variation.

The notion of plotting data in some manner as soon as it becomes available and of observing trends and changes in the data is very basic, and yet practitioners often ignore it. Shewhart (20) suggested plotting the data sequentially in time on a control chart. (1) The control chart, or Shewhart control chart, relies on sampling and probability theory to separate special-cause variation from natural or common-cause variation. The improvement of manufacturing processes and product quality is initiated from the detection of special-cause variation.

The spirit of using the control chart is to prevent the manufacture of defective product. Preventing the manufacture of defective products is in stark contrast to traditional quality control that relies on inspection to sort good products from bad products. Prevention of the manufacture of defective products using control charts comes from the ability to detect problems early and "upstream" in the manufacturing process. The basic idea behind the Shewhart control chart is that when the process is "in control," the average should be independently and normally distributed about the target and the variance should be constant. The purpose of the Shewhart control chart is not simply to provide a feedback process control mechanism. The Shewhart control chart enables one to quantify variation and examine carefully process data to find assignable causes for special-cause variation. Continuous improvements can be made from the elimination and prevention of future assignable causes. If the ability to detect special-caus e variation is hindered, the continuous improvement process may suffer due to the inability to assign events to special-cause variation, preventing further root-cause analysis of special-cause variation.

Shewhart (20) originally developed and tested his control chart for non-continuous manufacturing processes that generated independent data. The Shewhart control chart was not developed and tested for continuous processes such as the manufacture of paper, oriented strandboard, medium density fiberboard (MDF), and particleboard. The ability to detect special-cause variation using the Shewhart control chart may no longer apply for continuous processes that produce time-ordered data that are not independent.

Statistical process control (SPC) has evolved beyond the control charting of historical data from yesterday's or last week's production runs. Collecting real-time data in an SPC format gives machine operators immediate knowledge of key process variables that may influence future production. The use of real-time SPC in operator control rooms has greatly enhanced Shewhart's concept of preventing the manufacture of defective products.

A key issue with the use of real-time SPC is that continuous processes produce data that may have positive autocorrelation. Positive autocorrelation means that real-time samples tend to take similar values, resulting in process drift or a wandering process average. Positive autocorrelation may affect Shewhart control charts by producing control limits that are too narrow, which send false signals to machine operators. False signals not only inhibit the detection of special-cause variation but may lead to an over adjustment of the process by machine operators. Positive autocorrelation becomes more of an issue as the sampling intensity increases and may occur due to inertial effects of continuous processes (28).

The objective of this analysis was to determine the effect that positive autocorrelation had on the detection of special-cause variation. The X-Individuals and EWMA control charts for independent data were examined. Eleven different data sets from two different MDF manufacturers were used in the study.

METHODS

AUTOCORRELATION

As defined by Wheeler (25), two variables, X and Y, are said to be positively correlated if a sequence of paired observations of these two variables displays the following property: large values of X occur in conjunction with large values of Y, and small values of X occur in conjunction with small values of Y. The positive correlation between X and Y is measured by the sample correlation coefficient (r). If the ith value of X is denoted by [X.sub.i] and [Y.sub.i] denotes the ith value of Y, then the sample correlation coefficient r is defined as:

r = [summation over (n/i=1)]([X.sub.i] - X)([Y.sub.i] - Y)/[square root of([summation over n/i=1)][([X.sub.i] - X).sup.2] [([Y.sub.i] - Y).sup.2])] (1)

where -1 [less than or equal to] r [greater than or equal to] 1; X = sample average of all values of X; Y = sample average of all values of Y; [X.sub.i] = the ith sample observation of X; [Y.sub.i] = the ith sample observation of Y; n = sample size; i = the ith sample.

In many continuous forest products processes, data are obtained sequentially using electronic measurement devices. Sometimes relationships exist in the sequential data. Just as the sample correlation coefficient may be used to characterize the relationship between two variables, the autocorrelation coefficient may be used to characterize relationships between observations in the same sequence of values (25).

Let a sequence of observations be represented by the symbols {[t.sub.1], [t.sub.2], [t.sub.3], ... [t.sub.n]}. By pairing successive values, this sequence of n observations will yield n - 1 moving subgroups of the form {([t.sub.1], [t.sub.2]), ([t.sub.2], [t.sub.3]), ([t.sub.3], [t.sub.4]), ..., ([t.sub.n-1], [t.sub.n])}. The sample correlation coefficient (r) obtained from these pairs will be the autocorrelation coefficient with a lag of one, i.e., each pairing was obtained one time period apart and the autocorrelation coefficient is defined as [r.sub.1].

Positive autocorrelation produces correlation coefficients between zero and one. Data that have positive autocorrelation will exhibit a "roller-coaster" type pattern when plotted on an XY graph. Negative autocorrelation produces correlation coefficients between negative one and zero. Data with negative autocorrelation will exhibit a "saw-tooth" pattern when plotted on an XY graph.

AUTOCORRELATION AND SPC

The effect of autocorrelation on the statistical performance of the Shewhart control chart has been studied by Harris and Ross (10), Cook (6), Aiwan and Radson (1), Maragah and Woodall (13), Padgett et al. (16), Woodall and Faltin (28), Amin et al. (2), Gilbert et al. (9), Schmid and Shoan (18), Lu and Reynolds (11) and others. The consensus from previous research is that positive autocorrelation results in an increase in the number of special-cause variation signals on control charts. All types of control charts for monitoring the process average are affected by positive autocorrelation, including the Shewhart control chart, the X-Individuals chart, the cumulative sum (CUSUM), and the exponentially weighted moving average (EWMA) chart (28).

The primary purpose of Shewhart and other types of control charts is to recognize departures from statistical control, i.e., observations beyond the control limits are special-cause variation that have assignable causes. The spirit of using the control chart is to remove or prevent special-cause variation from occurring in the future. This will reduce long-term process and product variability.

Two methods are presented for adjusting traditional control charts when positive autocorrelation exists in forest products processes. Even though theoretical studies have been conducted for autocorrelated processes and autocorrelated control charting (10-13,16,18,21,22,27,28), we feel the two methods presented in this paper represent practical solutions for the quality improvement practitioner working in the forest products industry.

AUTOCORRELATION ADJUSTED CONTROL CHARTS FOR INDIVIDUAL OBSERVATIONS

A theoretical approach to charting autocorrelated data for individual observations commonly cited in the statistical literature is to develop control charts that are based on a first-order stationary autoregressive model, or AR(1) model (4,9,28). (2) The form of this model is:

[z.sub.t+1] = [PHI][z.sub.t] + [a.sub.t+1] [2]

where [z.sub.t] the disturbance of some quality characteristic from a target value in time period t; [z.sub.t] + 1 = the disturbance of some quality characteristic from a target value in time period t + 1; [a.sub.t] = white noise or the error term at time t. The [a.sub.t] are independent, identically distributed random variables with mean zero and standard deviation = [[sigma].sub.a]; [PHI] = the autoregression or positive autocorrelation coefficient, 0 < [PHI] < 1.

As Box and Luceno (4) and Gilbert et al. (9) noted, even though the AR (1) model is a relatively simple model, the mathematical expressions required to develop control charts from this model are burdensome. We feel the control limits necessary to calculate control charts using this model are impractical for the practitioner in the forest products industry and have not been well documented in the literature (21).

Wheeler (25) and Gilbert et al. (9) provide two practical solutions for developing control charts for individual observations for autocorrelated data. Wheeler (25) proposed the following solution for calculating control limits of individual observations for the AR(1) model: (3)

X [+ or -] 3 x (mR)/[d.sub.2] x [square root of (1 - [r.sup.2.sub.1])] [3]

where X = the process average; mR = the average moving range; [r.sub.1] = the lag-1 autocorrelation coefficient (Eq. [1]); [d.sub.2] = the bias correction factor, e.g., for the X-Individuals control chart with subgroup size of 2, [d.sub.2] = 1.128.

Gilbert et al. (9) proposed the following solution for calculating control limits of individual observations for the AR(1) model:

X [+ or -] 3 x (mR)/2 x [square root of (1 - [r.sub.1]/[pi]))] [4]

where X = the process average; mR = the average moving range; [r.sub.1] = the lag-1 autocorrelation coefficient (Eq. [1]).

Note that in the case where [r.sub.1] = 0, both the Wheeler (25) and Gilbert et al. (9) models have values of [d.sub.2] that agree with the value of [d.sub.2] computed under the assumption of independent, identically distributed observations for Shewhart control charts (n = 2, [d.sub.2] = 1.128).

EWMA CONTROL CHARTS

It has been well documented that the EWMA model was developed to forecast real-time data (8,18,23,24,26). The EWMA can also be applied to control charts. The EWMA statistic is:

[z.sub.t] + [lambda] x [X.sub.t] + (1 - [lambda]) x [z.sub.t - 1] [5]

where [X.sub.t] = sample observation at time t; [z.sub.t] = weighted average of the current observation and all the past observations at time t; [z.sub.t -1] = weighted average of the current observation and all the past observations at time t - 1; [lambda] = the weight factor, 0 < [lambda] [less than or equal to] 1.

Montgomery (14) observed that if the observations [X.sub.t] are independent, the control limits for the EWMA control chart for individual observations are:

UCL = X + 3[sigma][square root of ([lambda][(2 - [lambda])])] [6]

LCL = X - 3[sigma][square root of ([lambda][(2 - [lambda])])] [7]

where UCL = the upper control limit; LCL = the lower control limit; X = the process average; [sigma] = the process standard deviation; [lambda] = the weight factor, 0 < [lambda] [less than or equal to] 1.

When the data are positively autocorrelated Montgomery and Mastrangelo (15) suggested selecting the value of [lambda] that minimizes the mean square error. Cox (7) developed an estimate for [lambda] based on the lag-1 autocorrelation coefficient [r.sub.1] Cox (7) observed that the EWMA will provide a minimum mean square error (4) for the stationary autoregressive model (Eq. [2]), if:

[lambda] = 1 - 1/2[(1-[r.sub.1])/[r.sub.1]] [8]

where [lambda] = the derived weight factor, 0 < [lambda] [less than or equal to] 1; [r.sub.1] = the lag-1 autocorrelation coefficient (Eq. [1]); 1/3 [less than or equal to] [r.sub.1] [less than or equal to] 1.

The control limits that Montgomery and Mastrangelo (15) suggest when the data are positively autocorrelated are that the observation [X.sub.t + 1] should be compared to the following limits to test for statistical control:

[UCL.sub.t + 1] = [z.sub.t] + [U.sub.a/2] x [sigma] [9]

[LCL.sub.t + 1] = [z.sub.t] - [U.sub.a/2] x [sigma] [10]

where [UCL.sub. t + 1] = the upper control limit in time period t + 1; [LCL.sub.t + 1] = the lower control limit in time period t + 1; [z.sub.t] = the center line (Eq. [5]); [U.sub.a/2] = the upper [alpha]/2 percentage point of the unit normal distribution; [sigma] = the process standard deviation.

Methods for control charting subgrouped data that exhibit autocorrelation were not explored in this paper. Existing methods for control charting autocorrelated-subgrouped data are not based on any specific underlying model or autocorrelation pattern in the data.

"REAL-TIME" DATA SETS FROM MDF MANUFACTURERS

Ten time-ordered data sets were obtained from a North American MDF manufacturer. The manufacturer currently uses a continuous press to produce MDF. One hundred, real-time moisture samples were obtained for each data set from an electronic infrared moisture meter that measured fiber moisture for 3/4-inch-thick MDF production. The samples were obtained at varying intervals (1 min. apart ... 10 min. apart). Therefore, the total time span covered in the data sets ranged from 100 minutes to 1,000 minutes. The data sets were used for the X-Individuals control charts assuming the property of stationarity in the first-order autoregressive model, i.e., the time series varies in a stable manner about the mean (target) and machine operators will always attempt to return moisture to its mean (target). There were some minor shifts in the process average and standard deviation as the sample interval changed, e.g., the process average declined by 0.2 percent when the sample interval changed from 100 to 300 minutes. The mino r changes in the manufacturing process were not great enough to violate the assumption of stationarity in the first-order autoregressive model.

The other time-ordered data set was obtained from a different North American MDF manufacturer. The manufacturer's process used a multi-opening press to produce MDF. One hundred and sixty-three samples of within-panel MDF thicknesses were obtained from a laser thickness gauge on the production line. The 163 thickness samples were obtained from one panel approximately 196 inches in length. The sampling intensity of the laser thickness gauge was one sample every 1.2 inches along the length of the panel, taken at a sampling rate of less than 1 second per sample. A constant was subtracted from the original data given the confidentiality wishes of the manufacturer. The subtraction of a constant did not influence the level of autocorrelation of the data. The data set was used for the EWMA control charts assuming the property of a nonstationary time series model, i.e., it is not feasible for a machine operator of a conventional multi-opening press to control within-panel thickness variation after the pressing cycle i s initiated (4).

RESULTS AND DISCUSSION

X-INDIVIDUALS CHARTS ADJUSTED FOR LAG-1 AUTOCORRELATION

X-Individuals control charts were generated for the 10 different data sets of electronically measured fiber moisture. The 10 data sets had varying levels of lag-1 positive autocorrelation (Table 1). The lag-1 positive autocorrelation coefficient ([r.sub.1]) tended to vary as the sample interval varied. The lag-1 autocorrelation coefficient was significant [alpha] = 0.05) for [r.sub.1] [greater than or equal] 0.578.

X-Individuals control charts as defined by Shewhart (20) were generated for all 10 data sets. Signals of special-cause variation ("out-of-control" signals) increased for the 10 data sets as lag-1 positive autocorrelation increased (Table 1). Data sets with [r.sub.1] <0.600 produced fewer signals of special-cause variation (0 to 5 signals) than data sets with [r.sub.1] [greater than or equal] 0.600 (10 to 18 signals). The increase in special-cause signals for the same data set as [r.sub.1] increased may indicate false signals of special-cause variation. Special-cause signals occurred on Shewhart control charts as [r.sub.1] became statistically significant [alpha] = 0.05). The false signals of special-cause variation were more pronounced when the Shewhart control charts were compared with control charts adjusted for lag-1 autocorrelation (9,25).

A comparison of the Shewhart control chart with the autocorrelated adjusted control charts developed by Wheeler (25) and Gilbert et al. (9) revealed identical signals of special-cause variation for [r.sub.1] <0.300 (Fig. 1, Table 1). As [r.sub.1] increased beyond 0.600 the Gilbert et al. (9) control charts detected the least amount of special-cause variation (Table 1). Even at moderate levels of lag-1 autocorrelation ([r.sub.1] = 0.7 17), Gilbert et al. (9) control charts were less sensitive to detecting special-cause variation than Wheeler (25) autocorrelated adjusted control charts (Fig. 2).

Shewhart's (20) philosophy of continuous improvement was based on the premise that statistical methods, primarily the control chart, should be used to accurately and objectively detect problems to prevent the manufacture of defective products. In the spirit of Shewhart's philosophy, Wheeler's (25) approach may be more appropriate for the practitioner in the detection of special-cause variation in the presence of positive lag-1 autocorrelation. Shewhart's continuous improvement philosophy as related to the control chart was for practitioners to detect manufacturing problems using the control chart and not have the control chart hide or mask problems. In this spirit, Wheeler's (25) autocorrelated adjusted control chart detected more special-cause signals in the presence of auto-correlation than did the Gilbert et al. (9) autocorrelated adjusted control chart.

EWMA CONTROL CHARTS ADJUSTED FOR LAG-1 AUTOCORRELATION

An X-Individuals control chart was generated for the 163 samples of within-panel MDF thickness. Shewhart's X-Individuals control for independent data had 37 signals of special-cause variation for the 163 data points (Fig. 3). The data set had an [r.sub.1] = 0.808.

In a manufacturing setting, if the X-Individuals control chart was used to assess the statistical control of within-panel MDF thickness, it seems highly likely that press operators may adjust the out-of-press thickness target given such signals of poor statistical control. Any false signal of special-cause variation may lead to an adjustment of the process when an adjustment is not necessary. False signals of statistical control also prevent the isolation of actual special-cause variation.

Montgomery (14) proposed a method for control charting data using the EWMA model (Eqs. [6] and [7]). Montgomery's method was for independent observations assuming the nonstationary time series model (3,5,17,19,26). Montgomery's EWMA control charts of the within-panel MDF thickness data set had 10 signals of special-cause variation (Fig. 3). Note, that [lambda] was set to 0.5 for Montgomery's EWMA control chart given a lack of prior knowledge of the process.

Many practitioners in the forest products industry employ EWMA control charts for independent data assuming that it is the best model for a continuous process with a slow moving average and because of the misperception that the traditional EWMA control chart may be more appropriate for autocorrelated data. Montgomery's EWMA control chart method of the within-panel MDF thickness data set produced 10 signals of special-cause variation. These signals of special-cause variation were suspect false signals given an [r.sub.1] = 0.808.

Montgomery and Mastrangelo (15) developed a method of employing the EWMA model for autocorrelated data (Eqs. [9] and [10]). The method developed by Montgomery and Mastrangelo used the lag-1 autocorrelation coefficient ([r.sub.1]) to estimate the weight factor ([lambda]) for the EWMA model (Eq. [8]). For the within-panel MDF thickness data set, [r.sub.1] = 0.808 and [lambda] was derived to equal 0.881.

Montgomery and Mastrangelo EWMA control charts of the within-panel MDF thickness data set had no signals of special-cause variation (Fig. 3). All of the variation for this data set was attributed to natural or common-cause variation. The common-cause variation for within-panel MDF thickness may be due to variations within a press-platen, variation in mat forming, or variations within press closing. The common-cause variation in this example is inherent to the manufacturing system of the producer. Reductions in common-cause variation can only occur from improvements to the overall multi-opening press.

CONCLUSIONS

As competition among producers in the forest products industry increases and customers of forest products demand higher quality, it will be imperative for successful businesses of the future to focus on statistical methods for continuous improvement. Excessive manufacturing variability and product variation lead to higher manufacturing costs and inferior product quality, which increases business risk in a highly competitive marketplace.

Statistical methods for continuous improvement are emerging in the manufacturing sector of the forest products industry and it is imperative that such methods lead to the accurate detection of special-cause variation. Many practitioners in the forest products industry use the Shewhart control chart as a method for distinguishing between common-cause and special-cause variation. However, many forest products manufacturing processes such as paper, oriented strandboard, MDF, particleboard, etc., are continuous manufacturing systems and such systems may have positive autocorrelation in measured variables. In the presence of positive autocorrelation the Shewhart control chart may not be appropriate for detecting special-cause variation and the continuous improvement effort may be compromised.

The results from this study suggest that Shewhart control charts give false signals of special-cause variation in the presence of positively autocorrelated data. Detection of special-cause variation in the presence of positive autocorrelation may be dependent on the overall degree of positive autocorrelation. Control charts as adjusted for positive autocorrelation may be necessary for detecting special-case variation. Other results from this study suggest that the EWMA control chart for independent data also gives false signals of special-cause variation in the presence of positively autocorrelated data. Adjusting the weight factor ([lambda]) in the EWMA control as a function of [r.sub.1] may be necessary to detect special-cause variation in the presence of positive autocorrelation.

The X-Individuals and EWMA control charts as adjusted for positive autocorrelation provide the practitioner in the forest products industry with an improved method for detecting special-cause variation. The model presented by Wheeler (25) may be more appropriate for the practitioner applying the X-Individuals chart for processes that exhibit autocorrelation and stationarity in the first-order autoregressive model, i.e., the time series varies in a stable manner about the mean or target. The EWMA control chart adjusted for autocorrelation may be more appropriate for the practitioner with a process that exhibits autocorrelation and the property of a nonstationarity time series model, i.e., it is not feasible to have a process with controlled variation about the mean or target.

The improved methods for detecting special-cause variation reduce the risk of machine operators over-adjusting a process that is in statistical control, and also enhance the ability of practitioners to isolate and identify special-cause variation. Autocorrelated adjusted control charts are a powerful diagnostic tool for root-cause analysis and should be used by the practitioner when autocorrelation is present in the process.

A realistic limitation for the use of the X-Individuals and EWMA control charts as adjusted for positive autocorrelation is the lack of availability of this technique on commercial software in machine control centers. However, most of the leading commercial human-machine-interface platforms allow for customization and encoding. The algorithms presented in this paper for adjusting X-Individuals and EWMA control charts for positive autocorrelation can be easily encoded into commercial human-machine-interface platforms such as Wonderware [C] and Intellusions [C].

(*.) Forest Products Society Member.

(1.) The Shewhart control chart consists of time-ordered data plotted with the average and control limits. The control limits are approximately plus and minus three standard deviations from the average. Approximations of three standard deviations are used for calculating control limits given that there is a very small probability (0.003) that data would lie outside the control limits and still represent natural or common-cause variation in the process.

(2.) The property of stationarity implies that the generated time series varies in a stable manner about the fixed mean (4).

(3.) Recall that control limits for X-Individuals control charts for data that are not autocorrelated are; the process average, plus or minus, the product of 2.66 and the average moving range.

(4.) The mean square error (MSE) is the long run average of the squared deviations from target value. MSE = [[sigma].sup.2] + [([micro] - T).sup.2] where [[sigma].sup.2] = the process variance; [micro] = the process mean; T = the target value.

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TABLE 1. Signals of special-cause variation by level of lag-1 positive autocorrelation and type of control chart. Shewart Autocorrelation Sample control chart coefficient ([r.sub.1]) interval special cause signals (min.) 0.119 4 0 0.212 3 0 0.314 2 1 0.578 (*) (a) 5 3 0.593 (*) 10 5 0.717 (*) 1 10 0.762 (*) 9 12 0.828 (*) 8 12 0.844 (*) 6 18 0.908 (*) 7 14 Wheeler (25) Gilbert et al. (9) Autocorrelation control chart control chart coefficient ([r.sub.1]) special cause signals special cause signals 0.119 0 0 0.212 0 0 0.314 0 0 0.578 (*) (a) 0 0 0.593 (*) 1 0 0.717 (*) 2 1 0.762 (*) 7 6 0.828 (*) 10 8 0.844 (*) 10 10 0.908 (*) 8 8 False signal False signal Autocorrelation detection assuming detection assuming coefficient ([r.sub.1]) Wheeler (25) Gilbert et al.(9) 0.119 0 0 0.212 0 0 0.314 1 1 0.578 (*) (a) 3 3 0.593 (*) 4 5 0.717 (*) 8 9 0.762 (*) 5 6 0.828 (*) 2 4 0.844 (*) 8 8 0.908 (*) 6 6 (a)(*)= significant at [alpha] = 0.05.

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Title Annotation: | Shewhart chart |
---|---|

Author: | Young, Timothy M.; Winistorfer, Paul M. |

Publication: | Forest Products Journal |

Article Type: | Statistical Data Included |

Geographic Code: | 1USA |

Date: | Nov 1, 2001 |

Words: | 5079 |

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