comparison of process capability indices under autocorrelated data - guevara

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Revista Colombiana de Estadstica

Volumen 30 No. 2. pp. 301 a 316. Diciembre 2007

Comparison of Process Capability Indices underAutocorrelated Data

Comparacin de ndices de capacidad de procesos con datosautocorrelacionados

Rubn Daro Guevara 1, a , Jos Alberto Vargas 2, b

1 Programa de Ciencias Bsicas, Universidad de Ciencias Aplicadas y Ambientales,Bogot, Colombia

2 Departamento de Estadstica, Facultad de Ciencias, Universidad Nacional deColombia, Bogot, Colombia

Resumen

The process capability indices provide a measure of how a process tswithin the specication limits. In calculating indices is usual to assume thatthe process data are independent. However, in industrial applications dataare often autocorrelated. This paper deals with the indices C p , C pk , C pm andC pmk when data are autocorrelated. Variances for their estimators are deri-ved and coverage probabilities of some condence intervals are calculated.

Palabras clave : Autocorrelation, Process analysis, Estimation, Process ca-pability indices, SPC.

Abstract

Los ndices de capacidad de un proceso suministran una informacin nu-mrica acerca de cmo el proceso se ajusta a unos lmites de especicacin.En el clculo de estos ndices se asume que las observaciones son indepen-dientes; sin embargo, en aplicaciones industriales frecuentemente los datosestn autocorrelacionados. Este artculo analiza los ndices C p , C pk , C pm yC

pmk cuando los datos presentan autocorrelacin, se encuentran las varian-zas para sus estimadores cuando los procesos son gaussianos y se calculanlos porcentajes de cobertura para algunos intervalos de conanza.

Key words : autocorrelacin, anlisis de procesos, estimacin, ndice de ca-pacidad de procesos, SPC.

a Profesor. E-mail: [email protected] Profesor titular. E-mail: [email protected]

301


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302 Rubn Daro Guevara & Jos Alberto Vargas

1. Introduction

Process capability analysis is an important tool, widely used in industrial qua-lity improvement programs. A basic assumption in process capability analysis isthat process data are independent and identically normally distributed. Howe-ver, in several industrial processes, data exhibit some degree of autocorrelation.Though the literature on process capability indices is voluminous, there is notsignicant research when data are autocorrelated, as can be observed in Kotz &Johnson (2002) and in Spiring et al. (2003). Zhang (1998) studied the indices C pand C pk for autocorrelated data. We extend Zhangs study to the capability in-dices C pm and C pmk and compare the four indices C p , C pk , C pm and C pmk forstationary gaussian processes. Additionally, some results have been taken fromGuevara (2005) in his masters thesis. In the rst part of this article we introducethe denition of the capability indices C p , C pk , C pm and C pmk . Subsequently, we

present: 1) some relevant terminology about stationary gaussian processes, and 2)the expected values and variances of X , S 2 and S in these processes. In Section4, we calculate the variances of the estimators of C pm and C pmk , in stationarygaussian processes. Section 5, shows why the autocorrelation structure of processdata, when it is present, should not be ignored for calculating any of the fourindices. This is carried out through a simulation study with AR(1) processes. InSection 6, inferential procedures are performed, by measuring the coverage proba-bility of condence intervals constructed for some of the indices studied. Finally,some conclusions are given.

2. Process Capability Indices

Among the several capability indices dened in the literature, the most exten-sively used in the industry are C p , C pk , C pm and C pmk , dened as follows:

C p = USL LSL

6

where U SL and LSL are the upper and lower specication limits respectively and is the process standard deviation.

C pk = mnUSL

3 ,

LSL3

=a | b|

3 =

d | 2 m |6

where is the process mean, a = ( USL LSL )/ 2, b = ( USL + LSL )/ 2,d = U SL LSL and m = ( USL + LSL ).

C pm = USL LSL6 2 + ( T )2 =

C p

1 + 2Revista Colombiana de Estadstica 30 (2007) 301316


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Comparison of Process Capability Indices under Autocorrelated Data 303

where T is the target value and = T .

C pmk = mn USL 3 2 + ( T )2 ,

LSL3 2 + ( T )2 =C pk

1 + 2 = a | b|

3 2 + ( T )23. Stationary Gaussian Processes

If {X t } is a process such that V ar(X t ) < for each t W R , where R isthe set of real numbers, then the autocovariance function X (, ) of {X t } is:

X (r, s ) = Cov(X r , X s ) = E (X r EX r )(X s EX s ) , r, s W

The time series {X t , t Z }, with index set Z , is said to be stationary if E |X t |2 < , t Z , E (X t ) = m, t Z and X (r, s ) = X (r + t, s + t),r,s,t Z ; therefore X ( t ) is independent of t , and X (t + h, t ) is independent of t for each h Z .

If {X t } is a stationary process, the autocovariance function (ACVF) of {X t }is

X (h) = Cov(X t + h , X t )

and the autocorrelation function (ACF) is given by

X (h) = X (h) X (0)

= Cor(X t + h , X t )

(see Brockwell & Davis 1996).{X t } is a process gaussian if all of its joint distributions are multivariate nor-

mal. A process is said to be stationary gaussian if it is stationary and gaussiansimultaneously (see Brockwell & Davis 1996).

Let {X t } be a stationary gaussian process. Let {X 1 , X 2 , . . . , X n } be a sample

of size n from the process {X t }. Let X = ni=1X in , and S

2 =P ni =1 (X i X )2

n 1be the sample mean and the sample variance respectively. Zhang (1998) gives theexpected values and variances of X , S 2 and S :

E X = x

V ar X = 2Xn

g(n, i ) (1)

E S 2 = 2X f (n, i ) (2)

V ar S 2 = 24X(n 1)2

F (n, i ) (3)

E (S ) E S 2 1/ 2 = X f (n, i )1/ 2 (4)

Revista Colombiana de Estadstica 30 (2007) 301316


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and

V ar(S ) V ar S 2

4E S 2 =2 4X

(n 1)2 F (n, i )

42X f (n, i ) = 2X F (n, i )2(n 1)2f (n, i )

where i = X (i), for i = 1 , . . . , n , is the autocorrelation of X , at lag i,

f (n, i ) = 1 2

n(n 1)

n 1

i=1

(n i)i

F (n, i ) = n + 2n 1

i=1

(n i)2i + 1n2

n + 2n 1

i=1

(n i)i

2

2n

n 1

i=0

n i

j =0

(n i j )i j

and

g(n, i ) = 1 + 2n

n 1

i=1

(n i)i

4. Variances of Process Capability IndicesEstimators

Let {X t } be a stationary gaussian process. Let {X 1 , X 2 , . . . , X n } be a sampleof size n from {X t }. The usual estimators of C p , C pk , C pm y C pmk are:

C p =

USL LSL6S

C pk = mnUSL X

3S ,

X LSL3S

=a | X b|

3S =

d | 2X m |6S

C pm = USL LSL

6 S 2 + X T 2 = C p

1 + 2where = X T

S , and

C pmk = mnUSL X

3 S 2 + X T 2 , X LSL

3 S 2 + X T 2 =

C pk

1 + 2=

a | X b|

3 S 2 + X T 2Zhang (1998) found the following approximations for the variances of C p and C pk : V ar C p C

2 p

F (n, i )2(n 1)2f 3(n, i )



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and

V ar

C pk

C 2 pk

f (n, i )

g(n, i )

9nC 2

pk

+ F (n, i )

2(n 1)2f

2(n, i )

For the variances of C pm and C pmk , we found the following approximations(see appendix A and B):V ar C pm C

2 p

2F (n, i )(n 1) 2 +

4g(n, i )2

n

4[f (n, i ) + 2]3

and

V ar

C pmk C 2 pk

1f (n, p i ) + 2

F (n, p i )2(n 1)2[f (n, p i ) + 2 ]2

+ g(n, p i )

9n 1C pk

+ 6

2[f (n, p i ) + 2]2

where = T . If = T then

V ar C pm C 2

pF (n, i )

2(n 1)2f 3(n, i )

which equals the variance of C p , andV ar

C pmk

C 2 pkf (n, p i )

F (n, p i )2(n 1)2f 2(n, p i )

+ 42g(n, p i )

n 1

a | 2 b|

2

=C 2 pk

f (n, p i ) F (n, p i )

2(n 1)2f 2(n, p i ) +

g(n, p i )9nC 2 pk

which equals the variance of C pk .When the observations are independent, the variances reduce to:V ar C pm C

2 pm

11 + 2

12(n 1)

+ 2

n

and

V ar C pmk C 2 pmk 12(n 1)[1 + 2 ]2 + 19n 1C pk

+ 6 2[1 + 2]

2

If, simultaneously the observations are independent and = T , we have that

V ar C pm C 2

p

2(n 1)

4 =

C 2 p2(n 1)

V ar C pRevista Colombiana de Estadstica 30 (2007) 301316


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and

V ar

C pmk C 2 pk

1

2(n 1) +

1

9nC 2 pk Var

C pk

These last two results are equal to those found by Bissel (1990).To compare the variances of these estimators, a simulation study was carried

out for a rst order stationary autorregressive process with parameter . For C p =1.33, = 0 , 5, 10, = 0 .25, 0.50, 0.75 and n = 10 , 20, . . . , 200, Figure 1 shows greatvariability of V ar C pm for n < 100. Fixing C p , n and as || increases, V ar C pmincreases. Similar results are obtained for V ar C pmk , substituting C pm for C pmkand C p for C pk as can be seen in Figure 2. Now, if and are xed values andn increases, V ar C pm and V ar C pmk decreases. For xed values of and n,when increases, this is, the target value is far away from the mean of the process,V ar

C pm and V ar

C pmk decreases.

5. Autocorrelation Effects on Process CapabilityIndices

We consider the example given in Shore (1997). A quality characteristic isnormally distributed with mean 40 and standard deviation 7. The specicationlimits are USL = 61 and LSL = 19. Different target values are considered: 40,41, 42, 45 and 50. We then compare two processes. A process with independentobservations and a process with observations following an AR(1) model, X t =X t 1 + et , where {et } is a series of uncorrelated errors, et N (0, 2e ) and e = 7 .For each process, the mean, the standard deviation and the capability indices C

p,

C pk , C pm and C pmk are calculated. We do not show the values of C pk and C pmkbecause C p = C pk and C pm = C pmk see Table 1.

We observe in Table 1 that the higher the autocorrelation level the lower thecapability index value.

Tabla 1: Mean, standard deviation (STD), C p and C pm of a process not autocorrelatedvs. a process following an AR (1) model.

C pm| | Mean STD C p *d = 0 *d = 1 *d = 2 *d = 3 *d = 4 *d = 5

No auto 40 7.00 1.000 1.000 0.990 0.962 0.919 0.868 0.8140.25 40 7.23 0.968 0.968 0.959 0.933 0.894 0.847 0.7960.50 40 8.08 0.866 0.866 0.859 0.841 0.812 0.776 0.7370.75 40 10.58 0.661 0.661 0.659 0.650 0.636 0.619 0.598

*d = T

Through a simulation study we analyze the effect of the autocorrelation in theexpected value of the sample mean and in the expected value of the standard error.We generated 1000 samples from a no autocorrelated model and 1000 samples froman AR(1) model for each of the following cases: n = 15 , 50, 100, 200; T = 40 , 41,



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0 20 40 60 80 100 120 140 160 180 2000.0

0.1

0.2

0.3

0.4

0.5

n

= 0

V a r `

b C p m

:

+ 0.75 0.50

0.25

+

+

+

++

+ + + + + + + + + + + + + +

0 20 40 60 80 100 120 140 160 180 200n

= 5

V a r `

b C p m

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

:+ 0.75

0.50 0.25

+

+

+

+

++

++

+ + + + + + + + + + +

0 20 40 60 80 100 120 140 160 180 200n

= 10

V a r `

b C p m

0.00000

0.00001

0.00002

0.00003

0.00004

0.00005

:+ 0.75

0.50

0.25

+

+

+

+

++

++

+ + + + + + + + + + +

Figura 1: Variance of b

C pm in function of the sample size with C p = 1 .33, = 0 , 5, 10and = 0 .25, 0.50, 0.75.



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0 20 40 60 80 100 120 140 160 180 2000.0

0.1

0.2

0.3

0.4

0.5

n

= 0

V a r `

b C p m k

:+ 0.75

0.50 0.25

+

+

+

++

++ + + + + + + + + + + + +

0 20 40 60 80 100 120 140 160 180 200

n

= 5

V a r `

b C p m k

0.000

0.001

0.002

0.003

0.004

:+ 0.75

0.50 0.25

+

+

+

++

++ + + + + + + + + + + + +

0 20 40 60 80 100 120 140 160 180 200n

= 10

V a r `

b C p m k

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

:+ 0.75

0.50

0.25

+

+

+

+

++

++

+ + + + + + + + + + +

Figura 2: Variance of b

C pmk in function of the sample size with C pk = 1 .33, = 0 , 5, 10and = 0 .25, 0.50, 0.75.



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respectively. As n increases, the estimated expected value of the standard errorincreases slightly for autocorrelated data. For example, for = 0 .25 the estimated

expected values for n = 15 , 50, 100 are 7.05, 7.18 and 7.23 respectively.Through other simulation study we analyze the performance of the capabilityindices estimators. Comparing the estimated expected values of the capabilityindices estimators shown in Table 3 with the theoretical values shown in Table1, it can be observed that for autocorrelated processes the estimators are slightlybiased, bias that decreases as n increases. For example, for = 0.75 and n = 15 ,50, 100 the expected values of C p are 0.703, 0.673 and 0.668 respectively, while thetrue value is 0.661. For = 0 .25 and n = 15 , 50 y 100 the expected values of C pkare 0.937, 0.935 and 0.938 while the true value is 0.968. For n = 15 and = 0 .50,the expected values of C pm are 0.84, 0.84, 0.83 and 0.81 when T = 0 , 1, 2 and3 respectively, while the true values are 0.866, 0.859, 0.841 and 0.812.6. Condence Intervals for the Capability Indices

under Stationary Gaussian Processes

Let us assume that = 50 , U SL = 3 and LSL = 3. For sample sizes n = 25 ,50, 100 and standard deviation values = 0 .5, 1.0, 2.0, AR(1) processes withnormally distributed white noise are generated with = 0.75, 0.50, 0.25,0.25, 0.50 and 0.75. For each combination of n, and , 5000 random samplesfrom a normal distribution were generated. For each sample, the values of X , S ,

C p , C pk , C pm , C pmk , cp , cpk , cpm and cpmk are obtained. We calculated theproportion of times that the true index is contained in the interval

capability index k( capability index )

The simulations showed a coverage probability for C p of about 95% for k = 2and 99% for k = 3 . For C pk the coverage probability was of around 90 % fork = 2 . Tables 4 and 5 show the coverage probabilities of the intervals for C p andC pk . The coverage probabilities for C pm and C pmk are not shown here becausethese were very low.

In Tables 4 and 5 we observe that as n increases, the coverage probabilitiesincrease. In Table 5 it can be observed that coverage percentage for C pk decreaseswhen increases, being more evident for large values of . We observe that for = 0.75 the coverage probabilities are very similar for different values of .

We also calculate through simulations the coverage probabilities of the con-dence interval for C pm , proposed by Boyles (1991). This interval is dened asfollows:

C pm 2bf,/ 2 f , C pm 2bf , 1 / 2 f where 2bf , denotes the 100 % percentile of a chi-square distribution with f de-grees of freedom, for f = n 1 +

2 1 + 2 , = X T 2

2 and

2 =



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Tabla 7: Interval estimation average coverage rate of C pmk for AR (1) processes with = 0 .50, = 1 .5, n = 50 .

k2

2.0 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3.0 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 3,9 4.02.0 3 3 33 35 36 37 38 39 41 41 42 44 44 44 45 46 46 46 46 46 49 492,1 3 4 34 36 37 38 39 40 42 42 43 45 45 45 46 47 47 47 47 47 50 502,2 3 4 34 36 37 38 39 40 42 42 43 45 45 45 46 47 47 47 47 47 50 502,3 3 4 34 36 37 38 39 40 42 42 43 45 45 45 46 47 47 47 47 47 50 502,4 3 5 35 37 38 39 40 41 43 43 44 46 46 46 47 48 48 48 48 48 51 512,5 3 5 35 37 38 39 40 41 43 43 44 46 46 46 47 48 48 48 48 48 51 512,6 3 5 35 37 38 39 40 41 43 43 44 46 46 46 47 48 48 48 48 48 51 512,7 3 6 36 38 39 40 41 42 44 44 45 47 47 47 48 49 49 49 49 49 52 522,8 3 7 37 39 40 41 42 43 45 45 46 48 48 48 49 50 50 50 50 50 53 532,9 3 7 37 39 40 41 42 43 45 45 46 48 48 48 49 50 50 50 50 50 53 53

k 1 3.0 3 7 37 39 40 41 42 43 45 45 46 48 48 48 49 50 50 50 50 50 53 533,1 3 8 38 40 41 42 43 44 46 46 47 49 49 49 50 51 51 51 51 51 54 543,2 3 8 38 40 41 42 43 44 46 46 47 49 49 49 50 51 51 51 51 51 54 543,3 3 8 38 40 41 42 43 44 46 46 47 49 49 49 50 51 51 51 51 51 54 543,4 3 8 38 40 41 42 43 44 46 46 47 49 49 49 50 51 51 51 51 51 54 543,5 4 0 40 42 43 44 45 46 48 48 49 51 51 51 52 53 53 53 53 53 56 563,6 4 0 40 42 43 44 45 46 48 48 49 51 51 51 52 53 53 53 53 53 56 563,7 4 1 41 43 44 45 46 47 49 49 50 52 52 52 53 54 54 54 54 54 57 573,8 4 1 41 43 44 45 46 47 49 49 50 52 52 52 53 54 54 54 54 54 57 573,9 4 2 42 44 45 46 47 48 50 50 51 53 53 53 54 55 55 55 55 55 58 584.0 4 2 42 44 45 46 47 48 50 50 51 53 53 53 54 55 55 55 55 55 58 58

7. Conclusions

We found approximations of the variances of

C pm and

C pmk for stationary

gaussian processes. In particular, we show expressions for the variance of theseestimators when the observations are independent, for = T and = T .Through a simulation study, we show that the higher the autocorrelation level

the lower the capability index value. We also observed that for autocorrelatedprocesses the estimators are slightly biased, bias that decreases as n increases.

The autocorrelation does not affect the expected value of the sample mean of the capability indices estimators but affect the estimated expected value of thestandard error, that increases slightly for autocorrelated data when n increases.

Recibido: julio de 2007 Aceptado: octubre de 2007

Referencias

Bissel, A. F. (1990), How Reliable is your Capability Index?, Applied Statistics 39 , 331340.

Boyles, R. A. (1991), The Taguchi Capability Index, Journal of Quality Techno-logy 23 , 1726.



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Brockwell, P. J. & Davis, R. A. (1996), Introduction to Time Series and Forecas-ting , Springer, New York.

Guevara, R. D. (2005), Comparacin de los ndices de capacidad c p , c pk , c pm y c pmkcon datos autocorrelacionados, Tesis de Maestra, Estadstica, UniversidadNacional de Colombia, Facultad de Ciencias, Departamento de Estadstica,Bogot.

Kotz, S. & Johnson, N. L. (2002), Process Capability Indices A Review, 1992-2000, Journal of Quality Technology 34 , 219.

Mood, Graybill & Boes (1974), Introduction to the Theory of Statistics , McGraw-Hill, New York.

Shore, H. (1997), Process Capability Analysis when Data are Autocorrelated,Quality Engineering 9(4), 615626.

Spiring, F. A., Leung, B., Cheng, S. & Yeung, A. (2003), A Bibliography of ProcessCapability Papers, Quality and Reliability Engineering International 19 , 445460.

Zhang, N. F. (1998), Estimating Process Capability Indexes for AutocorrelatedData, Journal of Applied Statistics 25 , 559574.

Apndice A.

In this appendix, we derive an approximation of the variance of

C pm .

Let {X t } be a stationary gaussian process. Using the estimator of C pm :

C pm = USL LSL

6 S 2 + X T 2the variance of C pm , V ar C pm is

V ar C pm =USL LSL

6

2

V ar S 2 + X T 2 12

Using the approximation

V ar h(X, Y ) V ar[X ] x

h(x, y ) x, y

2

+

V ar[Y ] y

h(x, y ) x, y

2

+ 2 Cov[X, Y ] x

h(x, y ) x, y

, y

h(x, y ) x, y

(see Mood et al. 1974):



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comparison of process capability indices under autocorrelated data - guevara

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