P e r g a m o n

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

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

ADAPTIVE PROCESS MONITORING USING SCALE CUSUM FOR

SERIALLY CORRELATED PROCESSES

S a n g h o o n Lee a n d S u n g w o o n Choi

Dept . of I n d u s t r i a l E n g i n e e r i n g , K y u n g W o n U n i v e r s i t y

S e o n g N a m , K o r e a

A b s t r a c t

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

Statistical process control (SPC) techniques have

been widely applied in industry for process

i rr~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

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

737

738 Proceedings of 1996 ICC&IC

y(t) ego

to(t)

',, /

_ K I ~ / '\• / \

el(O ................ e.-l(O ....... i ~ i

\ / \,

r . - l ( t ) . . . .

e.C t)

.~ r . ( 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.

2. Adapt ive Linear Prediction Latt ice Filter

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:

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

where A~ ~) is the ith AR coefficient and

e(t) - N ( 0 , 02). The forward linear predictor and

prediction error of the pth order are then written:

P

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

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

where l ~ P ~ n . The coefficients {A} *)} of the

optimal predictor are uniquely determined by the

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

autocorrelation coefficients (R3 where

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

as reflection coefficients and are determined by the

autocorrelation sequence (Ri}. The reflection

coefficients can be defined as a cross correlation of

the forward and backward prediction errors:

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

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

Proceedings of 1996 ICC&IC 739

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

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

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

scale of Zt affect the location of [zt[ ux. Given a

sequence of observation {at}, the CUSUM is operated

for a given reference value k by forming the

cumulative sums

wt = (Iwt~ 1/a_0.822)/0.349

s , + = m ~ {o , s t_~ + w , - k} .

s T = rain {O,ST_~ + z,+ k}

The CUSUMs {S +} and {St +} represent variability

in the upward and downward directions, respectively.

The control chart signals an out-of-control condition when

v c x = , , ~ ( s t , s ; } > h.

4. Exlmriments

In this section, we considered AR(n) time series

models with the following form:

y(t) = - ~ ~;y(t- 0 +e(0 i = 1

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.

Table 1. ARLs according to Ixrtult~on in afc$ AR(1).

CUSUM CUSUM

~ o n for y(t) f~ e(t) from i n a

~1=0.0 ~1=0.0 ~1=0.5 ~1=-0.5

-50% 16 16 16 16

-40% 24 ~ 24

-30% 39 38 39 38

-20% 77 73 73 73

-10°/6 162 159 161 161

0 200 217 217 213

10% iii 120 119 120

26% 58 62 62 62

30°/0 37 38 38 38

40% 26 27 27 27

50% 20 21 20 20

Table 1 contains the ARL results for various

perturbation of a = l 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.

0.5 - 0 . 5 0.2 0.8 0.4 0.6

- 0 . 5 0.5 0.8 0.2 0.6 0.4

1.00 ~ 9.4 43 45 61 65

21 _~ 36 43 6O 61

1 1 1 . 4 4 18 17 33 32, 57 59

14 13 ~ ~ 41 42

740 Proceedings of 1996 ICC&IC

Table 3. Average values of ~ S for eve~¢ 40 step

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

TilT~

Interval

41 - 80

81 - 120

121 - 160

161 - 200

201 - 240

241 - 280

281 - 320

321 - 360

361 - 400

0.5 - 0 . 5 0.2 0.8 0.4 0.6

- 0 . 5 0.5 0.8 0.2 0.6 0.4

5.56 5.56 5.55 5.57 5.55 5.57

5.56 5.59 5.55 5.56 5.56 5.56

5.57 5.56 5.56 5.58 5.56 5.57

5.58 5.56 5.59 5.61 5.58 5.58

12.78 12.88 8.51 7.58 6.13 6.13

14.38 14.21 7.56 8.06 5.88 5.95

12.01 11.77 6.13 7.44 5.56 5.64

R89 8.74 5.67 6.61 5.52 5.59

6.96 6.95 5.63 6.08 5.60 5.60

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

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 ¢1:0.8 -00.0 Interval ~ 0 . 0 ~ 0 . 8

41 - 80 6.20

81 - 120 5.73

121 - 160 5.59

161 - 200 5.49

201 - 2A0 13.95

241 - 280 12.60

281 - 320 8.42

321 - 360 6.40

361 - 400 5.74

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

6.16

5.69

5.46

5.47

16.46

16.59

11.46

7.9

6.3

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 lgor i thm 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 common and special causes

simultaneously.

R e f e r e n c e s

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.