Adaptive process monitoring using scale cusum for serially correlated processes

Computers ind. Engng Vol. 33, Nos 3-4, pp. 737-740, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-8352/97 $17.00 + 0.00

Pergamon PII: S 0 3 6 0 - 8 3 5 2 ( 9 7 ) 0 0 2 3 5 - 0

ADAPTIVE PROCESS MONITORING USING SERIALLY CORRELATED

SCALE CUSUM FOR

PROCESSES

Sanghoon Lee and Sungwoon

Choi

Dept. of Industrial Engineering, Kyung Won University Seong Nam, Korea

Abstract

We present an adaptive monitoring approach for serially correlated data. This algorithm uses the adaptive linear prediction lattice filter (ALPLF) which makes it compute process parameters and prediction errors in real time and recursively update their estimates. We propose to apply a scale CUSUM control chart to prediction errors as an omnibus method for detecting changes in process parameters. Results of computer simulations demonstrate that the proposed adaptive monitoring approach has great potentials for real-time induslxial applications which vary frequently in their control environment. © 1997 Elsevier Science Ltd Kevwords : adaptive monitoring; prediction errors; lattice filter; scale CUSUM 1. Introduction

conventional Shewhart chart to the residuals. English et al. [2] proposed a similar approach using the forecasted errors from Kahnan filtering to monitor a continuous flow process. They modelled the flow process as an autoregressive (A_R) process. Given the correct order of AR model, the Kalman filter makes the control chart use directly" autocorrelated data. The Box-Jenkins methodology of time series analysis is currently one of the most useful approaches to autocorrelated data. However, it requires an extensive amount of past observations to develop an acceptable time series model. Since the model selected is fixed to fit to the observations in the control chart procedure, the application of Box-Jenkins approach has no capability of improving the model parameters as more observations of the process being collected and may need to reformulate the model whenever a change in data properties occurs in the continuous flow process. In this study, the Im~posed control chart scheme for continuous flow processes employes the adaptive linear prediction lattice filter (ALPLF) [3] which is designed for adaptive prediction of time series as an on-line process. The problems related to the Box-Jenkins methodology can be resolved by using the ALPLF algorithm, which provides on-line update on the model by "automatic learning." It is very important when uncertainty- about the process is high. The approach of English et al. using the recursive Kalman filter [2] is conceptually quite simple, but requires a fair

Statistical process control (SPC) techniques have been widely applied in industry for process irr~ovement and for estimating parameters or monitoring the variability of a given process. In the typical application of the SPC charts, it is traditionally assumed that the observations are uncorrelated. However, this assumption is generally invalid in many industrial processes. The presence of autocorrelation in the processes gives a profound effect on control charts developed for identically and independently distributed (I/D) observations, thereby resulting in increasing the frequency of false signals. Approaches for dealing with autocorrelated data in the SPC environment have been developed by fitting an al~opriate time series models to the observations and the applying control charts to the stream of residuals from this model. These methods axe based on the assumption that the residuals are white noise when there is no special cause in the process and can then utilize any of the conventional tools for SPC. Alwan and Roberts [1] proposed two separate charts to monitor the process: common-cause chart and slx~:ial-cause chart. The common cause chart is a plot of fitted values using the autoregressive integrated moving average (ARIMA) model and provides information on the systematic variation of the process. The special cause chart is to apply a

737

Proceedings of 1996 ICC&IC

738

ego

y(t)

el(O ................ ',,

_ KI~ /

/

e.-l(O .......i

~i

e.C t)

/

\

'\•

/

\

\, .~ r . ( t )

to(t)

r.-l(t)

.

.

.

.

Figure 1. Linear Prediction Lattice Filter

amount of computation and need to reinitialize the filter whenever the errors do not behave as a white-noise sequence if this disturbance results from the change in the process parameter uncontrolled. The lattice prediction technique is more efficient for on-line computation than the Kalman filtering and can easily update the model order without reinitialization. The purpose of this paper is to present the adaptive approach to monitor for the change in process parameters of AR processes. Bagshaw and Johnson [4] suggested that a scheme for monitoring the variance of the prediction errors would provide an omnibus for detecting any changes in the process parameters. Nishina [5] recommended the CUSUM chart for change-point estimation. Hawkins [6] introduced a scale CUSUM procedure for controlling the variance for IID normal ~ocesses. If any of the parameters changes, the identified model will no longer be correct at the time point when the change occurs and the filter will have a transient period to adapt the new environment. The model misslx~cification in the transient period will be transferred to the prediction errors, and will then result in shifts in error variance. This study suggests a scale CUSIYM control chart using the l~-ediction errors which are recursively obtained by the ALPLF and investigates performance of the new adaptive chart for various cases of the change in the process parameters.

where

A~ ~)

e(t) - N ( 0 , 02).

is

the

ith

AR

coefficient

and

The forward linear predictor and

prediction error of the pth order are then written: P

; , ( t ) = - ,~ti(')y( t - z) i=l

e~.) = y ( t ) - ;p(t) where

l~P~n.

The coefficients

optimal predictor are

{A} *)} of the

uniquely determined by

the

second-order statistics of the process, that is, by the autocorrelation

coefficients

Ri= E [ y ( t ) y ( t - z)].

(R3

where

Using the Levinson algorithm

[3], the predictor coefficients can be efficiently computed from the correlation sequence of the process. It involves computation of the backward predictors and prediction error: P

p)

;, ( t - P - 1) = - E B / , - i + t ) y ( t - ,) i=1

rp(t-1) = y ( t - p - 1 ) -

L(t-p-1)

The second order statistics of the process, the forward and backward mean-~uare errors are then given by

R~=E[e~(t)]

and

R;=E[~(t-1)].

Figure

1

outlines the ALPLF algorithm. The transfer function of the lattice filter in Figure 1 is determined by the values of the parameters {K~) which are referred to

2. Adaptive Linear Prediction Lattice Filter

as reflection coefficients and are determined by the

A serially-con'elated processes {y(t)} can be modeled with a discrete AR zero-mean time series of order n if the orocess mean is known:

coefficients can be defined as a cross correlation of the forward and backward prediction errors:

autocorrelation

sequence

(Ri}.

K~p+l = E[ e~orp( t - 1 ) ]/ li~p 3(t) =

- ~ A(")y(t - z) +e(t) i~l

K~p+l = E[ e~or>( t - 1 ) ]/ R'~

The

reflection

Proceedings of 1996 ICC&IC For a time varying system, it is assumed that the second orde~ statistics are varying over time. As in the recursive least squares method, a f~getting factor A is introduced in the time updates of the second

Table 1. ARLs according to Ixrtult~on in afc$ AR(1). CUSUM ~ o n ina

order statistics, R~ and R~ to track a time varying system. This univariate case will be easily extended to the multivariate one.

3. Adaptive Process Monitoring Fitting of the AR model makes it possible by study of its residuals to isolate the departures from control that may be traceable to special causes. If the adaptive filter estimates the appropriate model, the sequence of prediction errors from the filter will then behave as white noise. Therefore, conventional control charts can be applied to the stream of the prediction errors. The CUSUM procedure for a scale parameter, VCX was proposed by Hawkins [6]. If

zt~N(O,~),

then

[zt[ 1/2 closely approximates an

739

scale of Zt affect the location of

[zt[ux.

Given a

sequence of observation {at}, the CUSUM is operated for a given reference value cumulative sums

k

by

forming the

wt = (Iwt~ 1/a_0.822)/0.349 s ,+ = m ~ sT

The CUSUMs

{o,st_~ + w,-

k} .

= rain {O,ST_~ + z,+ k} {S +} and

{St+} represent variability

in the upward and downward directions, respectively. The control chart signals an out-of-control condition when

vcx= ,,~(st,

s ; } > h.

4. Exlmriments

In this section, we considered AR(n) time series models with the following form:

f~

e(t) from

~1=0.0 ~1=0.0 ~1=0.5 ~1=-0.5 -50% -40% -30% -20% -10°/6 0 10% 26% 30°/0 40% 50%

Table

1

16 24 39 77 162 200 iii 58 37 26 20

16 ~ 38 73 159 217 120 62 38 27 21

contains

perturbation of N(.822, .349) 2 distribution, and that changes in the

CUSUM

for y(t)

the

a=l

ARL

16 24 39 73 161 217 119 62 38 27 20

results

16 38 73 161 213 120 62 38 27 20

for

various

when the chart VCX was

directly applied to the simulated sequences of IID normal data and the same scheme was applied to the prediction errors which were generated by the ALPLF for the same data sequences. The ALPLF requires a transient period to be stable, that is, the prediction errors will be correctly estimated from some initial transient period after. Aft~ initiating the ALPLF for 40 steps with initial model parameter of 0.1, we started to apply the VCX to the prediction errors. The results of adaptive approach is slightily different to the direct application of the VCX, but it is not significant enough to reject the use of the adaptive filter. When using the first order AR time series simulated, the adaptive scheme shows the same Ix~rformance. It indicates that the prediction errors from the ALPLF is almost normal. Next, the VCX was applied to the prediction errors from the ALPLF for the simulated sequences of combining two different AR time series. The AR Table 2. ARLs of signaling out-of-control from the change-lmint

for various

as

when

AR(1)

parameter changes at the 261th step.

y(t)

=

- ~ ~;y(t- 0 +e(0

0.5

i=1

-0.5 where e ( t ) ~ / ~ 0 . o 2) and all the results were obtained by 10,000 Monte Carlo simulation runs. The performance of VCX using k = 0.25 and h = 6.8460 for IID normal data was illustrated in Lucas [7]. He shows the average run length (ARL) of VCX is approximately 200 for the standard normal data.

1.00 111.44

~ 21 18 14

-0.5

0.2

0.8

0.4

0.6

0.5

0.8

0.2

0.6

0.4

43 36 33 ~

45 43 32, ~

61 6O 57 41

65 61 59 42

9.4 _~ 17 13

Proceedings of 1996 ICC&IC

740 Table 3. Average values of ~

S for eve~¢ 40 step

when AR(1) l~-ameter changes at the 201th step.

TilT~ Interval

0.5 - 0 . 5 -0.5

41 81 121 161 201 241 281 321 361 -

80 120 160 200 240 280 320 360 400

0.5

0.2

0.8

0.4

0.6

0.8

0.2

0.6

0.4

5.57 5.56 5.58 5.61 7.58 8.06 7.44 6.61 6.08

5.55 5.56 5.56 5.58 6.13 5.88 5.56 5.52 5.60

5.57 5.56 5.57 5.58 6.13 5.95 5.64 5.59 5.60

5.56 5.56 5.55 5.56 5.59 5.55 5.57 5.56 5.56 5.58 5.56 5.59 12.78 12.88 8.51 14.38 14.21 7.56 12.01 11.77 6.13 R89 8.74 5.67 6.96 6.95 5.63

model parameter or parameters in the data series were changed after 200 time steps. Table 2 shows the ARLs for detecting a change in the model parameter for AR(1) time series using h = 9.0. This value is the control limit to give signals before the 200th time step for approximately 20% of 10,000 standard normal data series simulated. The lengths in Table 2 were obtained by counting until signaling out-of-control from the 201th time step which is the change-point of the model parameter. When the positive relation of serially-correlated data changes to the negative or reversely, the negative to the positive, the detecting performance of the adaptive scheme is invariable ff the absolute levels are same. The chart shows better performance when the correlation level increases than when it decreases. The behavior of VCX for the sequential data can be ilhs~ated by a series of the CUSUM statistics. Table 3 shows the average values Table ~ Average values d rnaxinama S for every 40 step when AR(2) parameter changes at the 201~ step.

Time Interval

41 81 121 161 201 241 -

80 120 160 200 2A0 280

¢1:0.8 - 0 0 . 0 ~ 0.0~0.8

Cv 0.5--* 1.4 Cz: 0 . 0 ~ - 0 . 9

6.20 5.73 5.59 5.49 13.95 12.60

6.16 5.69 5.46 5.47 16.46 16.59 11.46 7.9 6.3

281 - 320

8.42

321 - 360

6.40

361 - 400

5.74

of the maximum S for every 40 time steps for 10,000 simulation runs. The adaptive scheme may fail to detect a small variation in the AR(1) coefficient. It results in quick adaption of the ALPLF. If this variation should be conlxolled, the common cause chart can be implemented to examine the estimated model parameters. Two sets of AR(2) time series whose parameters change at the 201th time step were generated and the adaptive process monitoring method using the ALPLF and the VCX was applied to them. The average statistics of Maximum S for every 40 time steps are contained in Table 4. It shows the adaptive scheme is successful of identifying the process change for our exemplary cases. 5. Conclusions This paper presented an adaptive monitoring approach for the detection of changes in process parameters such as white noise variance and model parameters for serially correlated data. This scheme ern~loys the adaptive linear prediction lattice filter and the scale CUSUM control chart. Although the lattice filter is conceptually easy, its implementation is quite simple and the a l g o r i t h m is computationally efficient to eliminate the systematic pattern and generate white noise prediction error. In our experiments, the proposed scheme demonstrates a good prospect of monitoring both c o m m o n and special causes simultaneously. References

1. Alwa~ L. and Roberts, H. V., "Time Series Modeling for Statistical Process Control," Journal of Business & Economic Statistics, Vol. 6, 87-95, 1988. 2. English, J. R, Krishnamurth, M. and Sastxi, T., "Quality Monitoring of Continuous Flow Processes," Computers and Industrial Engineering, Vol. 20, 251-260, 1991. 3. Friedlander, B., "Lattice Filters for Adaptive Processing," Proceedings of the IEEE, VoL 70, 829-867, 1982. 4. Bagshaw, M., and Johnson, R. A., "Sequential Procedures for Detecting Parameter Changes in a Time-Series ModeL" Tecl~mrnetrics, VoL 72, 593-597, 1977. 5. Nishina, K., "A Comparison of Control Charts from the Viewpoint of Change-point Estimation,"

Quality and Reliability Engineering I n t e ~ o n a l , V o l . 8, 537-541, 1992.

6.

Hawkins, D. M., "A CUSUM for a Scale Parameter," Journal of Quality Technology, Vol. 13, 228-231, 1981 7. Lucas, J. M., "The Design and Use of V-Mask control Schemes," Journal of Quality Ted~ology, Vol. 22, 173-186, 1976.