ACTAUNIVERSITATISUPSALIENSISUPPSALA2007

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Social Sciences 25

Essays on Capability Indices forAutocorrelated Data

ERIK WALLGREN

ISSN 1652-9030ISBN 978-91-554-6835-4urn:nbn:se:uu:diva-7761

Papers summarized in the dissertation

Wallgren, E. (1998). Properties of the Taguchi Capability Index for Markov Dependent Quality Characteristics.

Wallgren, E. (2001). Confidence Limits for the Process Capability Index Cpk. for Autocorrelated Quality Characteristics.

Wallgren, E. (2001). A generalization of the Taguchi capability index for data generated by a first order moving average process.

Wallgren, E. (2000). Confidence limits for a generalized Cpk index for data. from an MA(1) process.

Wallgren, E. (2002). Confidence limits for Process Capability Indices Cpkand Cpm for Stationary Time-series Data.

Contents

Introduction .......................................................................................... 7

Summary of the papers ....................................................................... 18

Reports using an AR(1) model................................................................18 Report 1. Properties of the Taguchi Capability Index for Markov Dependent Quality Characteristics..................................19 Report 2. Confidence Limits for the Process Capability Index Cpk. for Autocorrelated Quality Characteristics ............................21

Reports using an MA(1) model...............................................................23 Report 3. A generalization of the Taguchi capability index for data generated by a first order moving average process ...............24 Report 4. Confidence limits for a generalized Cpk index for data from an MA(1) process..................................................................26

Report 5. Confidence limits for Process Capability Indices Cpk and. Cpm for Stationary Time-series Data...................................................27

Acknowledgements…………………..………………………………….... 31

References…………………..………………………………….…….........32

7

Introduction

The history of process capability indices, PCIs, or process capability ratios,

PCRs, originates in the 1920s when the statistician Walter A. Shewhart first

used lot sampling and significance theory in process control. In his book

from 1931, Economic Control of Quality of Manufactured Products, She-

whart introduced a strategy for monitoring and controlling variation which

‘brought together the disciplines of statistics, engineering and economics’

(Deleryd 1996).

In the works of Shewhart, control charts play a most important role. Using

charts is a graphical method to monitor the variability of a process, and con-

trol charts for sample statistics, such as means, standard deviations or ranges,

are used as well as control charts for individual observations. Originally, the

charts were constructed in such a way that all personnel involved in the

process were able to do the measurements and to make notations in the

charts. Today, however, charts are commonly presented with control limits

formulated from concepts based on statistical theory. Measurements are of-

ten made automatically and the charts are often printed by special quality

software, such as quality tool packs included in e.g. SAS or MINITAB.

For a single quality characteristic, QC, a control chart for the sample mean,

x , is constructed as

nLCL

lineCenternUCL

/3:

:/3:

8

where UCL and LCL are the upper and lower control limits, and are

the mean and standard deviation and n is the sample size used. Commonly,

the parameters and are unknown and many alternative estimates are

used, often based on a series of samples from the process. The constant 3 is a

traditional choice and is probably explained by the fact that, if the sample

mean is an observation of a normally distributed variable and if the actual

process mean and standard deviation equal the anticipated and , almost

all (99.7 %) sample means will fall inside the control limits.

If the sample mean was included in the control limit interval the process was

considered to be under ‘statistical control’, a situation defined by Shewhart

(1931) as:

A phenomenon will be said to be controlled when, through the use

of past experience, we can predict, at least within limits, how the

phenomenon may be expected to vary in the future.

However, a control chart for x only monitors one part of the process vari-

ability. For the same process, the sample means may stay almost constant

over time while the sample standard deviations, monitored in another control

chart, indicate an increased variability. Hence, a simultaneous registration of

the sample mean and standard deviation undoubted

9

ly gives a more efficient monitoring of the process compared to a control

chart for a single statistic. As an example of a combined sx, control chart,

Fig. 1 is based on a data set from Montgomery (2001) where each sample

point is x or s in a sample of size n=5 from diameter measurements in an

automobile manufacturing. In both charts, x and s stay inside their respec-

tive control limits and hence indicate a process under control.

To be able to predict the future the process must behave in such a way that it

is possible to assign a probability model to the QC, and a process with this

property is then said to be stable. Deming (1986) strengthens this further and

also requires that the quality characteristic, at least with respect to its mean

and standard deviation, must stay constant over time. Deming also discusses

the advantages of a stable process and says, ‘it gives an identity to the proc-

ess, it has a measurable, communicable capability.’ In the language of time

series analysis, Deming’s requirements correspond to weak stationarity of a

time series.

10

Figure 1 An sx / , control chart for a data set from an automobile manufac-

turer. (Chart constructed in MINITAB).

In Juran’s Quality Control Handbook, Juran(1988), process capability is

defined as

6capabilityProcess

where is the process standard deviation ‘under a state of statistical con-

trol’. Juran further defines the capability ratio as

capabilityprocesswidthtoleranceratioCapability

where the tolerance width refers to the maximal variability accepted by the

product customer. With the commonly used notation Cp, and upper and

lower specification limits for the tolerance, USL and LSL respectively, a

basic capability ratio is

252015105Subgroup 0

74.015

74.005

73.995

73.985

Sam

ple

Mea

n

Mean=74.00

UCL=74.01

LCL=73.99

0.02

0.01

0.00

Sam

ple

StD

ev

S=0.009443

UCL=0.01973

LCL=0

Xbar/S Chart for x1-x5

11

36dLSLUSLC p

where d is the half-width of the specification interval, (USL-LSL)/2. With the

process capability in the denominator, a large Cp-value is desirable, and val-

ues of Cp 3/4 are sometimes regarded as a requisite for a well-behaved

process. The capability ratio Cp, today commonly referred to as a capability

index, is sometimes defined in reciprocal form, which gives Cp-values near 0

as favourable.

After the introduction of the fundamental capability index Cp, many alterna-

tive indices have been suggested. The two most common indices in today’s

industry are Cpk, introduced by Kane (1986) but used in Japan already in the

70s, Kane (1986), and Cpm, introduced by Hsiang and Taguchi (1985), and

independently by Chan et al. (1988)

Cpk is defined by Kane as

33),min( MdLSLUSLC pk

where is the process mean and M is the midpoint of the specification

interval. While Cp, for a fixed specification interval only varies with , Cpk

is also sensitive to the deviation of from M. This can be seen in Figure 2,

where the two lines (a) combine points ( dM / , )/ d where Cpk= Cp

=1 and the two lines (b) combine points where Cpk= Cp =2. When differs

from M, i.e. 3./ dM , the standard deviation d/ must be reduced

from approximately .33 to .23 to keep an index value Cpk=1 and from ap-

proximately .17 to .12 to keep an index value Cpk=2.

12

Figure 2 Loci of points ddM /,/ such that: (a) 1pkp CC ;

(b) 2pkp CC . For each pair, pC = solid line and pkC = dashed line.

In this form of the index, positive and negative deviations of from M are

treated equivalently but by writing the index

plpupk CCLSLUSLC ,min3

,3

min

i.e by defining two indices, one upper, Cpu, and one lower, Cpl, deviations

from M can be evaluated individually.

The index Cpm is defined as

2222 36 T

d

T

LSLUSLC pm

where T is the process target, most commonly equal to M. Compared to pkC ,

a deviation from T has a larger impact on Cpm. For example, for a given de-

viation, except for small values, an index value Cpm=1 demands a smaller

relative variance than pkC =1, see Figure 3.

0.40.30.20.10.0

0.5

0.4

0.3

0.2

0.1

0.0-M|/d

/d

(a)

(b)

|

13

Figure 3 Loci of points ddM /,/ such that: (a) 1pmpk CC ;

(b) 2pmpk CC . For each pair, pkC = solid line and pmC = dashed line.

In addition to the three indices above, many index modifications have been

proposed but here we only mention the superstructure of the PCIs introduced

by Vännman (1995):

223),(

Tv

MudvuC p

from which the three indices above can be formulated as special cases:

pp CC )0,0( , pkp CC )0,1( and pmp CC )1,0( .

The PCIs mentioned above, and many others, have been studied in many

aspects and by many authors. Kotz & Johnson (1993) and Kotz & Lovelace

(1998) give comprehensive expositions and Kotz & Johnson (1999) study

‘delicate relations’ between the three indices. In a recent report, Kotz &

Johnson (2002) give a review of PCIs for the period 1992-2000.

0.40.30.20.10.0

0.5

0.4

0.3

0.2

0.1

0.0| -M|/d

/d

(a)

(b)

14

For an industrial process, many different QCs can be of interest, and for each

QC, any of the three indices mentioned above may be used. Whatever PCI is

chosen, the index includes at least one of the parameters and , and in

general, the parameters are unknown. Hence the PCIs have to be estimated.

Consider one specific QC. Let n successive observations on this specific QC

be denoted X1, . . ., Xn. In studies of the properties of estimated PCIs, the

following three assumptions, together notated IIN, are commonly used:

1. Identically distributed random variables. To make the PCIs interpretable,

the process has to be stable, with at least a constant mean and a constant

standard deviation, Deming (1931). This can be strengthened and most

commonly, the variables X1, . . ., Xn are assumed to be identically distributed.

2. Independently distributed random variables. The variables X1, . . ., Xn are

considered to be a random sample from the process, i.e. the variables are

assumed to be independently distributed.

3. Normally distributed random variables. To facilitate inference concerning

the true PCIs, a commonly used assumption is that the variables X1, . . ., Xn

are normally distributed, in particular when the inference is based on small

samples. Traditionally, a sample of five observations is used which makes

the normal distribution assumption important. For larger sample sizes, this

assumption is of course less important.

If and are estimated respectively by the sample mean n

i i nXX1

/

and by the sample standard deviation 2/1

12 )1/(n

i i nXXS , or

with an n in the denominator, straightforward estimators for the three indices

are obtained by replacing and by X and S respectively. Using the IIN

assumptions above, exact or approximate sampling distributions for the es-

15

timators can more or less easily be found. In addition, confidence intervals,

or lower confidence limits for the indices can be deduced.

Confidence limits for the pC index are easily obtained by observing that,

222 /ˆ/ SCC pp which is distributed as )1/(21 nn . See e.g. Kane

(1986), Chou et al. (1990).

For pmC , we have, see Boyles (1991),

22222 /ˆ/ TTXSCC pmpm

This ratio follows a scaled non-central 2 distribution and hence confidence

limits can be found, either directly, or via an approximation to a central 2 variable.

For the third PCI, pkC , a compact analytic form for the probability distribu-

tion of pkC can not be derived, and useful confidence limits have to be

found from suitable approximations. Heavlin (1988), Zhang et al. (1990) and

Bissel (1990), among others, have given results in this area and Kushler and

Hurley (1992) give an expose over these results. In a simulation study they

compare the empirical coverage rates for different approximate confidence

limits by means of empirical coverage rates and they conclude that the re-

sults of Bissel are to be recommended.

Even if not all of the three IIN assumptions are fulfilled, it may still be pos-

sible to formulate probability models. Several reports published during the

last decade deal with problems in the area, and several alternative indices are

proposed.

The case of non-normality is dealt with in different ways. Franklin and

Wasserman (1992) use a bootstrap methodology for estimating pC , pkC

16

and pmC . They conclude that the actual distribution of the QC is crucial, but

from large samples, reliable confidence limits can be found by means of the

bootstrap technique.

A common statistical technique for non-normal data is the use of a suitable

transformation, e.g. using logarithms, to achieve approximate normality,

Kane (1986) and Gunther (1989). On the other hand, if the QC can be mod-

eled by some non-normal distribution, e.g. Gamma or Weibull, an estimated

PCI can be found from quantile estimation, where the sample is used to es-

timate the parameters of the actual distribution, Kotz & Lovelace (1998). For

example, the estimate of the index pC can be written

00135.99865.ˆˆ

ˆXXLSLUSLC p

where 99865.X and 00135.X are the estimated 99.865 % and .135 % quantiles

of the QC which leaves 99.7 % of the distribution between the quantiles.

The requirement of identically distributed variables is highly important for a

process to be stable, and violations of this requirement are often difficult to

deal with. In a recent report, Carlsson (2002) studies a case with randomly

occurring shifts in the process level, a state that increases the process vari-

ability and reduces the PCI values.

This dissertation deals with the case when the assumption of independent

variables is not fulfilled. Though this situation is well known in the industry

and recognized by many authors, Kotz and Lovelace (1998), reports in the

area are not frequent. The various effects of ignoring autocorrelation on es-

timates of the process mean and standard deviation, are discussed by Shore

(1997), while Christoffersson (1999) estimates the asymptotic distribution of

),( vuC p for dependent data by means of bootstrap methods. Zhang (1998)

17

derives approximate confidence limits for pkC for samples from a process

with a known autocorrelation function.

In the present dissertation, the indices pmC and pkC are studied in situations

where the variables X1, . . ., Xn are dependent and observations from the pro-

duction process can be regarded as observations on a stationary time series

ntX t ,...,2,1, , with an unknown autocorrelation function.

The indices pkC and pmC are studied in five reports. In reports 1-4, the time

series modeling is restricted to an AR(1) or an MA(1), but in the last report,

weak stationarity is the only restriction. See Table 1

Table 1 Classification of the reports according to PCI

and model assumptions.

Model

restrictionpmC pkC

AR(1) Report 1 Report 2

MA(1) Report 3 Report 4

Weak

stationarity Report 5

For each model assumption, generalizations of pkC and pmC are proposed

and confidence limits are derived.

All reports include extensive Monte Carlo simulations used to evaluate the

confidence limits. Random samples are generated from an AR(1) or an

MA(1) process, and, in the fifth report, from more complex ARMA-models.

The confidence limits are calculated in two ways:

1. The time series structure is recognized and the confidence limits are calcu-

lated according to the derived results.

18

2. The time series structure is not recognized (ignored) and the confidence

limits are calculated according to earlier results derived for independent

samples.

For all cases the empirical percentage coverage rates are calculated and

compared to a pre-determined nominal confidence level.

Note that the five reports included in this thesis are written over a period of

years and the notations have changed. In the last report, Report 5, notations

are adapted to modern time series literature and these notations are also used

in the following summary.

Summary of the papers

Reports using an AR(1) model

For many industrial processes, like oil refinery, paper production etc, it is

well known that the level of individual quality characteristics often varies

with a wave-like pattern. Observations on such a characteristic, made at

equal time intervals, are then supposed to be dependent and the outcome of

such a process can be modeled in many ways.

In this report we choose the well-known AR(1) model,

ttt ZXX 1

where tX is the time series with parameters and , and tZ is white

noise with variance 2 . An AR(1) model, sometimes known as a low pass

filter or Markov dependency, is probably the most commonly used single

ARIMA(p,q)-model in industry. With the commonly accepted ‘keep it sim-

ple’-attitude to time series analysis, the AR(1) is often a reasonable model

and furthermore easy to estimate.

19

Report 1. Properties of the Taguchi Capability Index for Markov Dependent

Quality Characteristics.

For independent samples, confidence limits, and intervals for Cpm were given

by Boyles (1991). Using the technique discussed in the introduction, an

)%1(100 lower confidence limit for Cpm is

ˆ)ˆ(ˆ

2

pmC

where ˆ is the estimated number of degrees of freedom for a central 2 -

distribution given by

2

22

ˆ21ˆ1

ˆn

where 222 /ˆ STX is a non-centrality parameter estimate.

For a sample from an AR(1) process we propose an index denoted CpmAR.

Following the derivations made by Boyles, and using X and 2S as estima-

tors for the process level and the total process variance respectively, we

show that an )%1(100 lower confidence limit for CpmAR can be calcu-

lated as for an independent sample if ˆ is estimated by

ˆ1ˆ1ˆ2ˆ1

ˆ1ˆ1

ˆ2

2

2

22n

where 2ˆ is the same as for an independent sample and where ˆ is the esti-

mated first order sample autocorrelation coefficient.

20

For dependent samples, as well as independent samples, and for large values

of ˆ , the central 2 -distribution can be approximated by a normal distribu-

tion, which gives the )%1(100 lower confidence limit

))ˆ2/(1(ˆ 2/11zC pmAR

where z is the lower % percentile of a standard normally distributed

variable. Of course, ˆ , being the estimated number of degrees of freedom,

strongly depends on the sample size n. But it can also be noted that a strong

positive autocorrelation reduces the value of ˆ which probably makes a

normal approximation impossible.

In extensive Monte Carlo simulations samples are generated from an AR(1)

process and the samples are used to estimate lower confidence limits for

CpmAR . Estimates are calculated in two ways: By taking account of the auto-

correlation, using the confidence limits developed in this report, and

by ignoring the autocorrelation, using the confidence limits developed for

independent samples by Boyles (1991). Since, from an industrial point of

view, the higher capability index the better, only a lower confidence limit for

the index is calculated.

For both estimation methods, empirical coverage rates are calculated and

compared to nominal confidence levels.

The simulations are made for a range of and -values and for autocor-

relations 1 . From the simulations it was found that, except for -

values close to 1, the coverage rates given by the lower confidence limits

given in the report are always closer to a chosen nominal confidence level

than the coverage rates given by confidence limits when autocorrelations are

ignored.

21

Report 2. Confidence Limits for the Process Capability Index Cpk for Auto-

correlated Quality Characteristics.

From a statistical point of view, Cpk, with parameters to be estimated in the

numerator as well as in the denominator, is less tractable than Cpm. Further-

more, Cpm =(tolerance width)/(process capability), has a more natural inter-

pretation than Cpk. Nevertheless Cpk is more commonly used.

For independent samples, a compact analytic expression for the sampling

distribution of pkC has not been found and only approximate results for

)ˆ( pkCE and )ˆ( pkCVar are available. As an approximate lower confidence

limit for Cpk estimated from an independent sample, Bissel (1990) suggests

12

ˆ

91ˆ

2

1 nC

nzC pk

pk

which is also recommended by Franklin and Wasserman (1992).

This confidence limit, however, is not valid for observations on dependent

variables and in this second report, observations from an AR(1) process are

used to derive point and interval estimates of Cpk .The index estimator is the

same for independent and dependent data but the variance of the estimator

depends on the strength of the dependence. For the variance, we make a

Taylor linearization of pkC and, for large samples, a compact formula for

the variance is derived.

A Monte Carlo simulation indicates that pkC for dependent data is approxi-

mately normally distributed, at least for large samples. In this report, the

proposed lower confidence limit for pkC based on data from an AR(1) proc-

ess is

22

2

22

11 ˆ1ˆ1

2

ˆˆ1ˆ1

91ˆ)ˆ(ˆˆ

nC

nzCCarVzC pk

pkpkpk

where ˆ is the estimated first order autocorrelation coefficient, commonly

denoted ˆ . It can be noted that the estimated variance of pkC increases

with increasing ˆ .

The actual coverage rate of the proposed lower confidence limit for pkC is

estimated from Monte Carlo simulations. The empirical coverage rates for

the proposed confidence limits are compared to a nominal confidence level

and also to the empirical coverage rates for the confidence limits proposed

by Bissel, which assume independent observations.

Samples are generated from an AR(1) process with process mean and

noise variance 2 and confidence limits are calculated in two ways:

1. By taking account of the autocorrelation, using the confidence limits

developed in this report and

2. By ignoring the autocorrelation, using the confidence limits developed

for independent samples by Bissel.

3. The simulations are made for a range of and -values and for auto-

correlations 1 .

The results from the simulation study indicate that, if the assumption of an

AR(1) process is adequate,

(i) when the process mean equals the target, Cpk can be used as a

PCI for 5. , approximately, and

(ii) when the process mean does not equal the target, Cpk can be used

as a PCI for for 5. .

23

(iii) furthermore, the study indicates that large samples are necessary

to achieve coverage rates close to a nominal confidence level.

Finally, by comparing the results of the first two reports, it is found that Cpm

can be used in a wider interval of autocorrelations than Cpk. It is also found

that the empirical coverage rates for Cpm are generally closer to a nominal

confidence level than Cpk. Together with earlier discussed differences be-

tween the two indices, it is concluded that the Cpm index is preferable to Cpk.

Reports using an MA(1) model

In reports 3 and 4, an MA(1) model is used, i.e.

1ttt ZZX

which is as simple as the AR(1) and can be of interest for some applications,

although AR(1) is more often used in the industry.

Here and are the time series parameters, while tZ and 1tZ are white

noise with variance 2 .

In time series analysis, parameters of ARMA models are often estimated

through a recursive procedure, e.g. the Innovation Algorithm, see Brockwell

and Davis (1988). But with an additional assumption of invertibility, the

parameter of an MA(1) model can be estimated through the estimation of

the first order autocorrelation coefficient .

24

Report 3. A generalization of the Taguchi capability index for data generated

by a first order moving average process

In this report an index Cpm is proposed to be used for data from an MA(1)

process. Approximate confidence limits for the index are derived in a similar

manner as for an AR(1) model in the first report. A generalization of Boyles’

(1991) results gives as a )%1(100 lower confidence limit for CpmMA

ˆ)ˆ(ˆ

2

pmMAC

where ˆ is the estimated number of degrees of freedom for a central 2 -

distribution and given by

ˆ21ˆ2ˆ21ˆ1

ˆ 22

22n

and where, as before, 222 /ˆ STX is an estimated non-centrality

parameter.

For large values of ˆ , an )%1(100 lower confidence limit for Cpm is

))ˆ2/(1(ˆ 2/11zC pm

Monte Carlo simulations are made to compare the empirical coverage rates

of these approximate lower confidence limits to the coverage rates based on

lower confidence limits when autocorrelations are ignored, or not recog-

nized.

The samples, from which the Cpm are estimated, are generated and treated in

two ways:

1. Samples are generated from an MA(1)-process and the confidence limits

are based on the assumptions of an MA(1)-process, i.e. a correct model

assumption.

25

2. Samples are generated from an AR(1) process but the confidence limits

are based on the assumptions of an MA(1) process, i.e. an incorrect

model assumption.

The simulations are made for a range of and -values and for autocorre-

lations 5. .

From the simulation study the following conclusions are made:

1. Autocorrelations are included in the calculations.

If samples are generated from an MA(1)-process, the empirical coverage

rates of the confidence limits proposed in this report are very close to the

nominal confidence level for all parameter settings. If the model is misspeci-

fied, samples are actually generated from an AR(1)-process, the coverage

rates can differ substantially from the nominal confidence level, especially

when the process level is out of target.

2. Autocorrelations are ignored in the calculations.

Irrespective of generating process, an MA(1) or an AR(1)-process, the em-

pirical coverage rates of the confidence limits calculated in this report differ

considerably from a nominal confidence level. The size of the difference

depends on both the size of the autocorrelation and on whether the process

level is on target or not.

Finally, it is observed that increasing the sample size from 50 to 200 does

not give any substantial increase in the coverage rates.

26

Report 4. Confidence limits for a generalized Cpk index for data from an

MA(1) process.

This fourth report completes the generalization of PCIs for simple ARMA-

models. The index Cpk, is estimated from MA(1) process data in almost the

same way as in report 2. With an assumption of an MA (1)-process, and as-

ymptotic normality of pkC , the following )%1(100 lower confidence

limit for CpA is proposed:

22

1 ˆ212

ˆˆ21

91ˆ

nC

nzC pk

pk

Monte Carlo simulations are made to compare the empirical coverage rates

of the approximate lower confidence limits to the coverage rates based on

lower confidence limits for independent samples given by Bissel.

The samples from which the Cpk are generated are treated in two different

ways:

1. Samples are generated from an MA(1) process and the confidence limits

are based on the assumptions of an MA(1) process, i.e. a correct model

assumption.

2. Samples are generated from an AR(1) process but the confidence limits

are based on the assumptions of an MA(1) process, i.e. an incorrect

model assumption.

The simulations are made for a range of and -values and for autocorre-

lations 5. .

The analyses of the simulation results are based on four-way ANOVAs with

the four factors n, , and . To distinguish different parts of the cover-

age rate errors, three response variables are defined:

27

1. The coverage rate error defined as the difference between the empirical

coverage rate and the nominal confidence level when the model assump-

tion is correct and autocorrelations are taken into account.

2. The corresponding coverage rate error that results from ignoring the

autocorrelations, i.e. samples are generated from an MA(1) process but

treated as simple random samples.

3. The corresponding coverage rate error that results from an erroneous

time series assumption, i.e. samples are generated from an AR(1) process

but treated as samples from an MA(1) process.

From the simulations it can be concluded that the estimation approach pro-

posed in the report gives coverage rates close to nominal confidence levels

when the time series is correctly specified. If the time series structure is not

recognized and the samples are regarded as simple random samples, the cov-

erage rate errors increase with increasing autocorrelation. But for the models

under study, the error sizes are not notably large.

On the other hand, if the time series is misspecified, the coverage rate errors

may be very large and it is reasonable to believe that, for time series with a

strong serial dependency, the characteristics of the time series must be ac-

knowledged, either by finding a suitable time series model or by studying the

complete autocorrelation function.

Report 5. Confidence limits for Process Capability Indices Cpk and Cpm for Stationary Time-series Data

In the last report, the earlier assumption of a simple AR- or MA-model is

relaxed and weak stationarity of the time series is now the only restriction.

To be used for data from a stationary time series, the use of the indices Cpk

and Cpm, are generalized and two indices, CpkS and CpmS, are proposed. As in

28

the earlier reports, the proposed indices are identical to Cpk and Cpm for inde-

pendent data.

The variance of pkSC is approximated through a Taylor expansion, which

leads to the proposed )%1(100 lower confidence limit for CpkS

hh

pkS

hhpkS n

Cn

zC ˆ2

ˆˆ

91ˆ

22

1

where hˆ is the sample autocorrelation at lag h. This confidence limit cov-

ers the corresponding confidence limits in reports 2 and 4 as special cases.

For pmSC , the procedure adopted in reports 1 and 3, used by Boyles (1991),

is generalized and an )%1(100 lower confidence limit for pmSC is

ˆ)ˆ(ˆ

2

pmC

where ˆ is the estimated number of degrees of freedom for a central 2 -

distribution and given by

h hh h

n

ˆˆ2ˆ

ˆ1ˆ

22

22

and where 222 /ˆ STX is a non-centrality parameter estimator.

For large values of ˆ , an )%1(100 lower confidence limit for Cpm is

))ˆ2/(1(ˆ 2/11zC pmS

This confidence limit covers the corresponding confidence limits in reports 1

and 3 as special cases.

For both indices, the approximate confidence limits include 1 ˆh h and

12ˆh h , and each h is estimated by the traditional sample autocorrela-

tion coefficient. For each sum, the number of autocorrelation coefficients

29

included is determined by sequential tests and a general test procedure for

stationary data is proposed.

In a limited Monte Carlo simulation the reliability of the proposed limits is

exemplified. As examples of stationary time series, two contrasting

ARMA(p,q) are used: an ARMA(2,1) and an ARMA(1,2) with parameters

chosen to achieve both causality and invertibility, i.e. for both models, both

parameter polynomials have roots well outside the unit circle. The only pa-

rameters allowed to vary in the simulations are the process mean and the

noise variance 2 .

Samples are generated from each process and lower confidence limits for

CpkS and CpmS are estimated in two ways:

1. By taking account of the autocorrelations, using formulas proposed in

the report.

2. By ignoring the autocorrelations, using formulas proposed by Bissel

(1990) and Boyles (1991) respectively.

For both CpkS and CpmS the results of the simulations indicate that, when the

autocorrelations are taken into account, the coverage rates are generally

closer to the nominal confidence level than when the autocorrelations are

ignored. Comparing the indices, the simulations also indicate that CpmS is less

influenced by changes in 2 and than CpkS .

One noteworthy result is found in the estimation of Cpk: if the autocorrela-

tions are ignored, the coverage rates strongly depend on , but if autocorre-

lations are included, the coverage rates do not vary with .

Finally in this report, a data set from an oil refinery is studied. Observations

on a gas concentration are made each minute during a 10-hour period and the

30

series shows an apparent time series behavior. Lower confidence limits for

CpkS and CpmS are estimated: both for the frequency one observation each

minute and one observation each three or five minutes. The confidence lim-

its for CpkS and CpmS diverge and clearly depend on the observation fre-

quency. The results make it obvious that if a passage regarding a capability

index is included in a contract, the estimation procedure must be clearly

specified.

31

Acknowledgements.

This thesis has been written rather late in my career as a university teacher;

for many years the plans of a dissertation was not on my agenda. But when

professor Olle Carlsson was engaged at the staff of statisticians of Örebro

University, he introduced to me the problem of capability index calculations

for dependent observations and slowly the possibility of producing a thesis

became realistic. I owe him a dept of gratitude for all the time he spent with

me, discussing my problems and reading my papers.

At Uppsala University, professor Anders Christoffersson and the board of

supervisors at the department of statistics have read and commented on my

thesis and I am very grateful for their help. Valuable suggestions on the Eng-

lish text were made by Mr Gray Gatehouse. The financial support provided

by Örebro University is gratefully acknowledged.

I am also indebted to my colleagues at the university for their friendly atti-

tude to my work and last but not least, I am grateful to my wife Lotta for

here support and patience.

Örebro, February 2007

Erik Wallgren

32

REFERENCES:

Bissel, A.F. (1990). How Reliable is Your Capability Index? Applied Statistics, 39(3), pp 331-340.

Boyles, R.A. (1991). The Taguchi Capability Index, Journal of Quality Technology, 23, pp 17-26.

Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (1994). Time Series Analysis. Prentice Hall, New Jersey.

Brockwell, P.J. and Davis, R.A. (1988). Time Series. Theory and Methods. 2nd ed. Springer, New York.

Carlsson, O. (2002). A Generalization of the Cpk Index for Production Processes with Varying Process Levels. Private Communications.

Chan, L.K., Cheng, S.W. and Spiring, F.A. (1988). A New Measure of Process Capability: Cpm . Journal of Quality Technology, 20(3), pp 163-175.

Chou, Y-M. and Owen, D.B. (1989). On the Distribution of the Estimated Process Capability Indices. Communications in Statistics-Theory and Methods, 18(12), pp 4549-4560.

Chou, Y-M., Owen, D.B. and Borrego, S.A. (1990). Lower Confidence Limits on Process Capability Indices. Journal of Quality Technology,

22(3), pp 223-229.

Christoffersson, J. (1999). Process Capability Indices and Block Bootstrap for Serially Correlated Data. Research Report 1999:4. Luleå University of Technology

Deleryd, M. (1996). Process Capability Studies in Theory and Practice. Licentiate Thesis. Luleå University of Technology.

Deming, W.E. (1986). Out of the Crisis. Cambridge University Press, Cambridge.

Franklin, L.A. and Wasserman, G.S. (1992). Bootstrap Lower Confidence Limits for Capability Indices. Journal of Quality Technology, 24, pp 196-210.

33

Franklin, L.A. and Wasserman, G.S. (1992). A Note on the Conservative Nature of the Tables of Lower Confidence Limits for Cpk, with a Suggested Correction. Communications in Statistics-Simulation and Computation, 23, pp 1165-1169.

Gunther, B. (1989). The Use and Abuse of Cpk, Parts 1-4. Quality Progress, 22(1), pp 72-73.

Heavlin, W.D. (1988). Statistical properties of capability indices. Technical Report No. 320, Tech. Library, Advanced Micro Devices, Inc., Sunnyvale, California.

Hsiang, T.C. and Taguchi, G. (1985). A Tutorial on Quality Control and Assurance- The Taguchi Methods. ASA, Annual Meeting, Las Vegas, USA.

Juran’s Quality Control Handbook, (1988). Juran, J.M. (Ed). McGraw- Hill, New York.

Kane, V.E. (1986). Process Capability Indices. Journal of Quality Technology, 18, pp 41-52.

Kotz, S. and Johnson, N.L. (1993). Process Capability Indices. Chapman & Hall, London.

Kotz, S. and Johnson, N.L. (1999). Delicate Relations Among the Basic Process Capability Indices Cp, Cpk and Cpm, and their Modifications.

Communications in Statistics-Simulation and Computation., 28(3), pp 849-866.

Kotz, S. and Johnson, N.L. (2002). Process Capability Indices- A Review, 1992-2000. Journal of Quality Technology, 34, pp 2-19.

Kotz, S. and Lovelace,C.R. (1998). Process Capability Indices in Theory and Practice. Arnold, London.

Kushler, R.H and Hurley, P. (1992). Confidence Bounds for Capability Indices. Journal of Quality Technology, 24, pp 188-195.

Montgomery, C.D. (2001). Introduction to Statistical Quality Control. Wiley, New York.

Priestley, M.B. (1981). Spectral Analysis and Time Series, Volume 1. Academic Press, London

34

Shewhart, W.A. (1931). Economic Control of Quality of Manufactured Product. Van Nostrand, New York.

Shore, H. (1997). Process Capability Indices when Data are Autocorrelated. Quality Engineering 9, pp 615-626.

Wallgren, E. (1996). Properties of the Taguchi Capability Index for Markov Dependent Quality Characteristics. Lic.Thesis, Uppsala University.

Wallgren, E. (2000). Confidence Limits for a Generalized Cpk Index from an MA(1) Process. Working Paper No 4, ESA, Örebro University.

Wallgren, E. (2001). Confidence limits for the process capability index Cpk for autocorrelated quality Characteristics. Frontiers in Statistical Quality Control. Ed Lenz, H.J. and Wilrich, P.Th. Physica-Verlag, Heidelberg.

Wallgren, E. (2001). A Generalization of the Taguchi Capability Index for Data Generated by a First Order Moving Average Process. Statistical Methods, 3(1) pp 1-15.

Vännman, K. (1995). A Unified Approach to Capability Indices. Statistica Sinica 5(2), pp 805-822.

Zhang, N.F., Stenback, G.A. and Wardrop, D.M. (1990). Interval Estimation of Process Capability Index Cp.. Communications in Statistics-Theory and Methods, 19, pp 4455-4470.

Zhang, N.(1998). Estimating Capability Indices for Autocorrelated Data.Journal of Applied Statistics, 25(4), pp 559-574.

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