ORIGINAL ARTICLE

Control charts for individual observations of a bivariatePoisson process

Linda Lee Ho & Antonio Fernando Branco Costa

Received: 2 March 2008 /Accepted: 3 September 2008 / Published online: 1 October 2008# Springer-Verlag London Limited 2008

Abstract In this paper, three single-control charts are pro-posed to monitor individual observations of a bivariate Poissonprocess. The specified false-alarm risk, their control limits, andARLs were determined to compare their performances fordifferent types and sizes of shifts. In most of the cases, thesingle charts presented better performance rather than twoseparate control charts (one for each quality characteristic).A numerical example illustrates the proposed control charts.

Keywords Control charts . Bivariate Poisson distribution .

ARL

1 Introduction

In many production processes such as printed circuitboards, textile, or paper manufacturer, monitoring thenumber of defects or non-conformities is preferable ratherthan a fraction of non-conformities. In this case, Poissondistribution is assumed for such a variable and c or ucontrol charts have been used for such purpose. In order toassure the quality of the products, controlling more thanone type of defect in the same inspected unit is necessary insome situations. For example, two types of defects can beobserved in the same circuit board if the soldering processis not well calibrated: excess of solder and surface defects.One can suggest two separate control charts (c or u control

charts) for each type of defect. This may be a good solutionfor independent events; however, if they may be positivelycorrelated (as the number of defect of excess solderincreases, the same may be observed in the number ofsurface defects too), the two separate control charts do nottake into account this possibility.

Multivariate process control is one of the most rapidlydeveloping sections of statistical process control. However,most of the work has been devoted to normally distributedvariables. Hotelling’s T2 control chart has been presented tomonitor shifts in process mean vector. Alternatively, exten-sions of cumulative sum and EWMA control charts formultivariate version were the subjects of research of someauthors such as Crosier [3]; Pignatiello and Runger [13] andLowry et al. [11]. Alt [1] presented two interesting proceduresto monitor shifts in covariance matrix (one is an extension ofS2 control chart and the second is based on the generalizedvariance |S|). Few contributions related to quality control formultivariate binomial and Poisson can be found in theliterature. Patel [12] has presented a scheme for multivariatebinomial and Poisson as an extension of Hotelling’s proposal;and recently, Skinner et al. [15] considered a generalizedlinear model to monitor multiple count data. So, there is a lackof control charts for this type of multivariate distributions.

The aim of this paper is to propose single charts as analternative to two univariate control charts for monitoring two(correlated) types of defects in manufactured units. Threemonitoring statistics were considered: SM=X1+X2; MX=max(X1, X2); DF=X1−X2 and two separate charts (one for X1and one for X2—hereafter referred as control chart 2C) beingX1 and X2 the times these defects are observed in eachmanufactured unit. It is assumed that the random vector (X1,X2) follows a bivariate Poisson distribution. After specifyingthe false-alarm risks, the control limits were determined andthe ARLs of the SM, MX, DF, and 2C charts were calculated

Int J Adv Manuf Technol (2009) 43:744–755DOI 10.1007/s00170-008-1746-4

L. Lee Ho (*)University of São Paulo,São Paulo, Brazile-mail: lindalee@usp.br

A. F. Branco CostaSão Paulo State University,São Paulo, Brazile-mail: fbranco@feg.unesp.br

in order to compare their performances. The paper isorganized as follows: a brief revision of the bivariate Poissondistribution is in section 2; the proposed charts arepresented in section 3 where their performances are studiedand their ARLs are compared for a large set of inputparameters. Illustrative examples are in section 4. Conclu-sions and final remarks are in section 5.

2 Brief revision of a bivariate Poisson distribution

The bivariate Poisson distribution was firstly presented byHolgate [5] as a sum of three independent Poissonvariables. Let Y1, Y2, and Y3 be independent Poissonvariables with parameters λ1, λ2, λ3, respectively. So, thevector X1 ¼ Y1 þ Y3;X2 ¼ Y2 þ Y3, follows a bivariatePoisson distribution and the probability function is given by

P X1 ¼ x1;X2 ¼ x2ð Þ ¼ exp � l1 þ l2 þ l3ð Þ½ �

�Xmin x1;x2ð Þ

i¼0

lx1�i1 lx2�i

2 li3x1 � ið Þ! x2 � ið Þ!i! ð1Þ

Manipulating Eq. 1, it can be written as

P X1 ¼ x1;X2 ¼ x2ð Þ ¼ exp � l1 þ l2 þ l3ð Þ½ � lx11 l

x22

x1!x2!

�Xmin x1;x2ð Þ

i¼0

x1ið Þ x2

ið Þ i!l3l1l2

� �i

ð2Þ

The next recurrence expressions

x1P x1; x2ð Þ ¼ l1P x1 � 1; x2ð Þ þ l3P x1 � 1; x2 � 1ð Þx2P x1; x2ð Þ ¼ l2P x1; x2 � 1ð Þ þ l3P x1 � 1; x2 � 1ð Þ ð3Þ

facilitate the calculation of the probability values. Someimportant parameters as mean, variance, covariance, andcorrelation are given below

E X1ð Þ ¼ Var X1ð Þ ¼ l1 þ l3E X2ð Þ ¼ Var X2ð Þ ¼ l2 þ l3Cov X1;X2ð Þ ¼ l30 � Corr X1;X2ð Þ ¼ l3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l1þl3ð Þ l2þl3ð Þp � l3 l3 þmin l1; l2ð Þ½ ��0:5

ð4ÞNote that the correlation of X1 and X2 assumes only

positive values. The conditional distribution function of X1|X2 is expressed as

P X1 ¼ x1jX2 ¼ x2ð Þ ¼ exp �l1ð ÞXmin x1;x2ð Þ

i¼0

x2i

� �i!

l3l3 þ l2

� �i l2l3 þ l2

� �x2�ilx1�i1

x1 � ið Þ! ð5Þ

The expression 5 is a convolution of two independentvariables, a Poisson variable with parameter λ1 and abinomial one with parameters x2;

l3l3þl2

� �. The conditional

mean and variance are given as

E X1jX2ð Þ ¼ l1 þ l3l3 þ l2

x2

Var X1jX2ð Þ ¼ l1 þ l3l2l3 þ l2ð Þ2 x2

If l3 þ l2 ! 1; l3 þ l1 ! 1; and l3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil1þl3ð Þ l2þl3ð Þ

p !r,

then Z1; Z2ð Þ ¼ X1� l1þl3ð Þffiffiffiffiffiffiffiffiffiffiffiffil1þl3ð Þ

p ; X2� l2þl3ð Þffiffiffiffiffiffiffiffiffiffiffiffil2þl3ð Þ

p� �

follows a standardized

normal bivariate distribution and asymptotically, Z21�2rZ1Z2þZ2

2ð Þ1�r2

is a Chi-squared distribution with two degrees of freedom.According to Rayner and Best [14], this approximation doesnot provide good results in high-correlation cases. Otherbivariate Poisson count data model using conditionalprobabilities were introduced by Berkhout and Plug [2].

Table 1 Description of the parameters and control limits

Control limits

Case λ1 λ2 λ3 E(X1) E(X2) ρ s/a SM MX DF 2C

1 1 1 1 2 2 0.50 s [0;12] [0;7] [−5;4] [0;7],[0;7]2 1 1 2 3 3 0.67 s [0;17] [0;9] [−5;4] [0;9],[0;9]3 1 2 1 2 3 0.41 a [0;14] [0;9] [−7;4] [0;8],[0;9]4 1 2 2 3 4 0.58 a [0;18] [0;11] [−7;4] [0;10],[0;11]5 2 1 1 3 2 0.41 a [0;14] [0;9] [−5;6] [0;9],[0;8]6 2 1 2 4 3 0.58 a [0;18] [0;11] [−5;6] [0;11],[0;10]7 2 2 1 3 3 0.33 s [0;15] [0;9] [−7;6] [0;9],[0;9]8 2 2 2 4 4 0.50 s [0;19] [0;11] [−7;6] [0;11],[0;11]9 1 3 1 2 4 0.35 a [0;15] [0;11] [−9;3] [0;7],[0;11]10 1 4 1 2 5 0.32 a [0;17] [0;12] [−11;3] [0;7],[0;13]11 1 3 2 3 5 0.52 a [0;19] [0;12] [−9;3] [0;9],[0;13]12 1 4 2 3 6 0.47 a [0;21] [0;14] [−11;3] [0;9],[0;14]

Int J Adv Manuf Technol (2009) 43:744–755 745

Table 2 ARL values of the control charts (for λ1=λ2=λ3=1)

λ1 λ2 λ3 Control limits

SM [0;12] MX [0;7] DF [−5;4] 2C [0;7],[0;7] 2c* [0;7],[0;7]SM (ARL) MX (ARL) DF (ARL) 2C (ARL) 2C* (ARL)

1 1 1 377.64 478.01 545.83 477.97 456.162 1 1 124.44 78.52 37.77 78.52 76.993 1 1 48.85 19.32 9.45 19.32 19.173 2 1 22.35 16.54 16.36 16.54 16.024 1 1 22.35 7.47 4.18 7.47 7.444 2 1 11.68 7.10 6.64 7.10 6.961 2 1 124.44 78.52 115.49 78.52 76.991 3 1 48.85 19.32 21.77 19.32 19.171 3 2 9.63 7.29 21.77 7.29 6.961 3 3 3.97 3.76 21.77 3.76 3.401 4 1 22.35 7.47 7.65 7.47 7.441 4 2 6.10 3.87 7.65 3.87 3.781 4 3 2.97 2.46 7.65 2.46 2.321 4 4 1.93 1.80 7.65 1.80 1.652 2 1 48.85 44.05 59.28 44.05 42.252 3 1 22.35 16.54 32.42 16.54 16.022 4 1 11.68 7.10 12.39 7.10 6.962 4 2 4.17 3.63 12.39 3.63 3.402 4 3 2.34 2.31 12.39 2.31 2.083 1 2 9.63 7.29 9.45 7.29 6.963 1 3 3.97 3.76 9.45 3.76 3.403 3 1 11.68 10.52 21.93 10.52 10.033 4 1 6.85 5.86 15.90 5.86 5.634 1 2 6.10 3.87 4.18 3.87 3.784 1 3 2.97 2.46 4.18 2.46 2.324 1 4 1.93 1.80 4.18 1.80 1.654 2 2 4.17 3.63 6.64 3.63 3.404 2 3 2.34 2.31 6.64 2.31 2.084 3 1 6.85 5.86 10.19 5.86 5.634 4 1 4.43 4.22 12.44 4.22 4.021 1 2 31.02 49.43 545.83 49.43 42.251 1 3 8.36 12.90 545.83 12.90 10.031 1 4 3.81 5.45 545.83 5.45 4.021 2 2 16.50 17.56 115.49 17.56 16.021 2 3 5.59 6.68 115.49 6.68 5.631 2 4 2.92 3.51 115.49 3.51 2.811 3 4 2.33 2.40 21.77 2.40 2.082 1 2 16.50 17.56 37.77 17.56 16.022 1 3 5.59 6.68 37.77 6.68 5.632 1 4 2.92 3.51 37.77 3.51 2.812 2 2 9.63 11.34 59.28 11.34 10.032 2 3 3.97 4.85 59.28 4.85 4.022 2 4 2.33 2.79 59.28 2.79 2.242 3 2 6.10 6.20 32.42 6.20 5.632 3 3 2.97 3.27 32.42 3.27 2.812 3 4 1.93 2.14 32.42 2.14 1.802 4 4 1.66 1.71 12.39 1.71 1.513 1 4 2.33 2.40 9.45 2.40 2.083 2 2 6.10 6.20 16.36 6.20 5.633 2 3 2.97 3.27 16.36 3.27 2.813 2 4 1.93 2.14 16.36 2.14 1.803 3 2 4.17 4.48 21.93 4.48 4.023 3 3 2.34 2.60 21.93 2.60 2.24

746 Int J Adv Manuf Technol (2009) 43:744–755

The parameters of the bivariate Poisson distribution can beestimated by different methods as the well-known methods ofmoments and maximum likelihood. However, some methodswere specifically developed to estimate the covarianceparameter as the double-zero proportion and even point (seeKocherlakota and Kocherlakota [9]). For more details aboutthe bivariate Poisson distribution, see Johnson et al. [6] andKocherlakota and Kocherlakota [9, 10].

The use of the EM algorithm for maximum likelihoodestimation of the parameters of the multivariate Poissondistribution is described in Karlis [8]. And, recently, Karlisand Ntzoufras [7] presented an R package called bivpois formaximum likelihood estimation of the parameters ofbivariate Poisson regression models.

Some interesting distributions can be derived from thebivariate Poisson distribution, for example the distributionof DF=X1−X2, the distribution of SM=X1+X2 and thedistribution of MX=max(X1, X2). The probability functionof DF=X1−X2 is

P DF ¼ yð Þ ¼X1j¼0

P X1 ¼ j;X2 ¼ j� yð Þ ¼ e� l1þl2ð ÞX1j¼y

lj�y2 lj1

j! j� yð Þ!ð6Þ

Note that Eq. 6 does not depend on the covarianceparameter. Its mean and variance are, respectively,

E DFð Þ ¼ l1 � l2

Var DFð Þ ¼ l1 þ l2

Or the distribution function of SM=X1+X2 is

P SM ¼ yð Þ ¼Xyj¼0

P X1 ¼ j;X2 ¼ y� jð Þ

¼Xyj¼0

exp � l1 þ l2 þ l3ð Þ½ � lj1

j!

l y�jð Þ2

y� jð Þ!

�Xmin j;y�jð Þ

i¼0

j

i

� �y� j

i

� �i!

l3ðl1l2

� �i

:

ð7Þ

Its mean and variance are, respectively

E SMð Þ ¼ l1 þ l2 þ 2l3

Var SMð Þ ¼ l1 þ l2 þ 4l3

The distribution of MX=max(X1,X2) is

P MX ¼ yð Þ ¼ P X1 ¼ y; X2 ¼ yð Þ þXy�1

j¼0

P X1 ¼ j;X2 ¼ yð Þ þ P X1 ¼ y; X2 ¼ jð Þ ¼

¼ e½� l1 þ l2 þ l3ð Þ�y!

Xy�1

j¼0

lj1ly2 þ ly1l

j2

j!

!Xji¼0

ji

� �yi

� �i!

l3l1l2

� �i

þ l1l2ð Þyy!

Xyi¼0

yi

� �2

i!l3l1l2

� �i( ) ð8Þ

Table 2 (continued)

λ1 λ2 λ3 Control limits

SM [0;12] MX [0;7] DF [−5;4] 2C [0;7],[0;7] 2c* [0;7],[0;7]SM (ARL) MX (ARL) DF (ARL) 2C (ARL) 2C* (ARL)

3 3 4 1.66 1.83 21.93 1.83 1.563 4 2 3.04 3.09 15.90 3.09 2.813 4 3 1.92 2.04 15.90 2.04 1.803 4 4 1.46 1.56 15.90 1.56 1.374 2 4 1.66 1.71 6.64 1.71 1.514 3 2 3.04 3.09 10.19 3.09 2.814 3 3 1.92 2.04 10.19 2.04 1.804 3 4 1.46 1.56 10.19 1.56 1.374 4 2 2.35 2.46 12.44 2.46 2.244 4 3 1.64 1.75 12.44 1.75 1.564 4 4 1.33 1.41 12.44 1.41 1.26

Int J Adv Manuf Technol (2009) 43:744–755 747

Table 3 ARL values of the control charts (for 11=13=1 and λ2=2)

λ1 λ2 λ3 Control limits

SM [0;14] MX [0;9] DF [−7;4] 2C [0;8],[0;9] 2c* [0;8],[0;9]SM (ARL) MX (ARL) DF (ARL) 2C (ARL) 2C* (ARL)

1 2 1 572.41 874.89 847.08 758.04 454.961 4 1 74.57 31.39 35.11 31.27 30.401 4 2 13.65 11.87 35.11 11.71 10.552 4 1 33.61 30.57 53.70 28.57 23.072 6 1 9.75 5.88 9.15 5.82 5.572 6 2 3.94 3.50 9.15 3.44 3.123 6 1 6.10 5.72 13.62 5.44 4.723 8 1 3.01 2.41 4.51 2.37 2.263 8 2 1.91 1.82 4.51 1.79 1.664 2 2 8.42 11.31 6.67 6.41 3.824 2 3 3.79 5.51 6.67 3.59 2.381 8 1 6.10 2.42 2.40 2.42 2.422 2 1 190.62 461.04 75.28 207.46 76.963 2 1 74.57 109.57 16.83 44.92 19.164 2 1 33.61 30.57 6.67 14.53 7.441 6 1 17.15 5.90 5.92 5.90 5.871 6 2 5.58 3.53 5.92 3.52 3.431 6 3 2.90 2.42 5.92 2.41 2.261 6 4 1.93 1.84 5.92 1.83 1.661 8 2 2.95 1.84 2.40 1.84 1.831 8 3 1.92 1.52 2.40 1.51 1.481 8 4 1.48 1.32 2.40 1.32 1.272 8 1 4.13 2.42 3.25 2.42 2.382 8 2 2.32 1.84 3.25 1.83 1.772 8 3 1.65 1.51 3.25 1.50 1.422 8 4 1.34 1.32 3.25 1.31 1.221 2 2 45.95 113.39 847.08 91.51 50.151 2 3 11.54 28.26 847.08 22.64 12.291 2 4 4.90 10.65 847.08 8.71 4.851 4 3 5.19 5.85 35.11 5.72 4.721 4 4 2.85 3.49 35.11 3.39 2.642 2 2 23.97 66.50 75.28 36.68 16.992 2 3 7.52 18.50 75.28 11.78 6.212 2 4 3.66 7.66 75.28 5.43 3.142 4 2 8.42 11.31 53.70 10.28 7.652 4 3 3.79 5.51 53.70 4.98 3.572 4 4 2.31 3.28 53.70 2.99 2.142 6 3 2.31 2.39 9.15 2.34 2.042 6 4 1.67 1.82 9.15 1.78 1.523 2 2 13.65 26.60 16.83 13.77 7.123 2 3 5.19 9.92 16.83 6.08 3.583 2 4 2.85 4.95 16.83 3.44 2.213 4 1 17.15 25.67 38.32 19.62 12.293 4 2 5.58 9.43 38.32 7.44 4.853 4 3 2.90 4.69 38.32 3.85 2.623 4 4 1.93 2.87 38.32 2.46 1.753 6 2 2.95 3.37 13.62 3.18 2.643 6 3 1.92 2.30 13.62 2.17 1.783 6 4 1.48 1.75 13.62 1.66 1.383 8 3 1.46 1.49 4.51 1.47 1.343 8 4 1.24 1.30 4.51 1.28 1.174 2 4 2.31 3.28 6.67 2.38 1.714 4 1 9.75 16.42 16.94 10.55 6.21

748 Int J Adv Manuf Technol (2009) 43:744–755

When MX=max(X1,X2), the expressions of E(MX) andVar(MX) are quite complicated, see the Appendix. In anyof the three random variables derived from bivariatePoisson distribution, the next results follow

E Yð Þ ¼ f l1; l2; l3ð ÞVar Yð Þ ¼ g l1; l2; l3ð Þ ð9Þ

Y=MX, SM, or DF. So estimates of Eq. 9 are availablereplacing the parameters by its estimators (as the maximumlikelihood estimator for example) as

E Yð Þ ¼ f l1; l2;l3� �

Var Yð Þ ¼ g l1; l2; l3� � ð10Þ

And asymptotically Y � N E Yð Þ; Var Yð Þn

� �, n is the

sample size.

3 Control charts for individual observationsof a bivariate Poisson process

In this section, control charts based on individual observa-tions of a bivariate Poisson will be introduced. Specifically,single-control charts based on SM=X1+X2; MX=max(X1,X2); DF=X1−X2 and two separate charts (one for X1 andone for X2—hereafter referred as 2C) were considered, thevector (X1,X2) follows a bivariate Poisson distribution. Fora fixed level of type I error (α=0.0027), the control charts’limits were determined as

P C < cð Þ ¼ a or

P cL < C < cUð Þ ¼ a

where C is the monitoring statistic (in our case: SM, MX,DF, or X1 and X2), c is the control limit for unilateralcontrol charts (SM; MX; X1 and X2) and cL and cU are,

respectively, the lower and the upper control limits forbilateral control charts [for DF].

Due to the nature of the monitoring statistics (they arediscrete ones), the type I error are reduced to a set of values,and, in general, 0.0027 is not one of them, consequently, it isimpossible to obtain an exact ARL0 ¼ 0:0027ð Þ�1� 370:4for all charts, so the values of c, cL, and cU (the controllimits) are determined by search (employing expressions 6–8) that lead to the closest ARL0’s to the desired level (370.4)in order to make the comparison among charts as fair aspossible.

Table 1 presents the in-control mean of X1 and X2, theirrespective correlation and the control limits for the SM,MX, DF, and 2C charts for twelve combinations of λ1, λ2,and λ3. The symmetrical cases [when E(X1)=E(X2)] weremarked by “s” and asymmetrical ones by “a”. Controllimits which yield the largest ARL0 are also bolded inTable 1. The first eight cases follow a factorial design; thelast four sets are included to analyze cases with |E(X1)−E(X2)|>1. A total of 64 shifts (given by the combinations ofkλ1; kλ2, kλ3; with k=1,…, 4) were considered to comparethe control charts performance. The ARLs of the SM, MX,DF, and 2C control charts for the in-control values: λ1=λ2=λ3=1 and (λ1=λ3=1 and λ2=2) are respectively in Tables 2and 3. The first one is a symmetrical case since E(X1)=E(X2) with ρ=0.50 and the second one is an asymmetricalone with E(X1)=2 and E(X2)=3 and ρ=0.41. Similar tablesfor other sets of parameters were built but not reproducedhere. In both tables, the ARLs for the 2C control charts(denoted as 2C*) were also computed assuming (wrongly)that the variables X1 and X2 are independents (see lastcolumns of Tables 2 and 3). The lowest ARL for each shiftis in bold.

For instance, in Table 2, when λ1 quadruplicates (λ1=4)and λ2 and λ3 remain unaltered (λ2=λ3=1) the lowest ARLis 4.18 (the fifth row of Table 2) and the DF chart is the fast

Table 3 (continued)

λ1 λ2 λ3 Control limits

SM [0;14] MX [0;9] DF [−7;4] 2C [0;8],[0;9] 2c* [0;8],[0;9]SM (ARL) MX (ARL) DF (ARL) 2C (ARL) 2C* (ARL)

4 4 2 3.94 6.72 16.94 4.83 3.144 4 3 2.31 3.64 16.94 2.85 1.994 4 4 1.67 2.38 16.94 1.99 1.484 6 1 4.13 5.21 16.48 4.57 3.574 6 2 2.32 3.07 16.48 2.73 2.144 6 3 1.65 2.12 16.48 1.93 1.544 6 4 1.34 1.64 16.48 1.52 1.264 8 1 2.33 2.34 6.25 2.25 2.044 8 2 1.63 1.77 6.25 1.70 1.524 8 3 1.33 1.46 6.25 1.41 1.264 8 4 1.17 1.27 6.25 1.24 1.12

Int J Adv Manuf Technol (2009) 43:744–755 749

one to detect this disturbance. In cases which DF chart isthe best option (see Table 2—from second to sixth rows),the value of λ3 remains unaltered. The next rows were thedisturbances (different sizes of shifts in all parameters) thatthe MX and 2C charts detect fast (with equal performance).The out-of-control average values of λ1, λ2, and λ3 forthese cases are respectively 2.62; 2.81; 2.10 that provideE(X1)=4.71, E(X2)=4.90, and ρ=0.43 (these values are inthe second block of columns of Table 4). Note that this

average value of correlation is lower than its in-controlvalue. And finally disturbances (different sizes of shifts inall parameters) that SM chart detects faster. The averagevalues of λ1, λ2, and λ3 are, respectively, 2.50, 2.50, and3.12 yielding E(X1)=5.62, E(X2)=5.62, and ρ=0.56 (higherthan its in-control value).

A similar presentation is found in Table 3. For example,when λ1 remains unaltered (λ1=1), but λ2 triplicates (λ2=6) and λ3 duplicates (λ3=2), the charts MX and 2C are

Table 4 The best chart control chart

750 Int J Adv Manuf Technol (2009) 43:744–755

faster to detect this disturbance with the lowest ARL equalto 3.53. Out-of-control average values of λ1, λ2, and λ3(when MX and 2C are the best options) are, respectively,1.36, 7.27, and 2.64 that result E(X1)=4.00, E(X2)=9.91,and ρ=0.41. Note that the average correlation is equal to itsin-control value but larger shifts are observed on λ2 andconsequently larger shifts on E(X2).

To identify an appropriate chart to quickly detect somespecific disturbance, Table 4 was built. The in-controlvalues are in the first block of columns. Out-of-controlaverage values of the main parameters are in the secondblock of columns and, in the last block of columns, amosaic was drawn to identify which chart is more adequateto detect fast some specific disturbance. The entries are theratios of out-of-control and in-control parameters denotedas li

l0i¼ 1; 2; 3; 4 and i=1, 2, 3. Analyzing Table 4, some

remarks can be made:

& The DF chart is the best option when the assignablecause only increases λ1, except when the in-controlvalues of λ1 and λ3 are small in comparison to λ2. Inthese cases, E(X2) remains unaltered or changes slightly,E(X1) changes substantially. As l3

l03¼ 1, the correlation

decreases.& The MX and the 2C charts (shortly M2 charts) are the

best options when the in-control values of λ1 and λ2 areequal (the symmetric cases 1, 2, 7, and 8) and theassignable cause increases both λ1 and λ3 largely orboth moderately holding λ2 unaltered or symmetrically,increases both λ2 and λ3 largely or both moderatelyholding λ1 unaltered. In these cases, the assignablecause only increases the mean of X1 or the mean of X2.In general, the correlation is slightly or not affected.

& Considering yet the symmetrical cases, when l3l03>

1; l1l01� 2 and l2

l02� 2, the SM chart has the best

performance.& Regarding the asymmetric cases, when the assignable

cause changes slightly or holds the correlation unal-

tered, the MX and the 2C chart are the best option.When the assignable cause increases (decreases) thecorrelation the SM (DF) chart is the best option.

4 Numerical example

In this application, we consider the data extracted fromHo & Singer [4] of two types of defects counted in 230samples of 100 g textiles fibers produced by two machinesfrom manufacturer A (the data are reproduced in Table 5).In this context, each machine defined a stratum and, aftercomparing the expected rates of defects between strata andevaluating the intensity and homogeneity of the associationbetween the frequencies of two types of defects, therelevant parameters (by maximum likelihood method) areestimated and reproduced in Table 6. Note that thecorrelation between the two types of defects in textilesfibers from manufacturer A is positive (correlation of0.295). This fact may suggest that increases on theincidence of one type of defect might lead to a moreoccurrences of the other one. Control limits and ARL0

values are summarized in Table 7. Control charts SM andMX identify one observations (X1=3; X2=7, one frommachine A1).

Figure 1 illustrates the proposed control charts in asimulated data set. The first ten samples were taken from

Table 5 Frequency of defects per 100 g of textile fibers

Machine # of defects of type 1 # of defects of type 2 Total

0 1 2 3 4 5 6 7

A1 0 13 26 22 16 3 2 . . 821 3 9 12 7 5 2 1 . 392 . 3 1 . . 1 . . 53 . . 1 2 . . . 1 5

Total 16 38 36 25 8 5 1 1 130A2 0 15 14 11 10 3 1 . . 54

1 2 12 6 7 5 3 1 . 362 . 1 2 1 . 2 . . 63 . . . 1 1 1 1 . 4

Total 17 27 19 19 9 7 2 . 100

Table 6 Estimation of the parameters of the numerical example

Parameter Estimate Standarddeviation

Expected type 1 defect rate 0.5243 0.0476Expected type 2 defect rate 1.9914 0.0928Covariance between the frequenciesof two types of defects

0.3013 0.0629

Int J Adv Manuf Technol (2009) 43:744–755 751

λ1=0.2230, λ2=1.6901, and λ3=0.3013 (to produce theaverage rates presented in Table 6) and the last ten fromλ3=4.3013, keeping λ1 and λ2 unaltered. Employing thecontrol limits from Table 7, the SM chart signals at thethirteenth sample (three samples after the change point);the MX control chart at the sixteenth sample (six samplesafter the process change), and the DF chart at the nineteenthsample. Finally, the separate control charts signal at thenineteenth sample, the signal is given by X1 chart. Theseresults are coherent since shift in λ3 from 0.3013 to 4.3013keeping unaltered the other parameters will make increasesnot only in both expected rates but also the correlation from0.295 (in control) to 0.826 (out of control). And, in that

case, the SM chart is the fastest one to detect this type ofdisturbance.

5 Conclusion and discussions

This paper presented single charts as an alternative totwo univariate control charts for monitoring two types ofdefects in manufactured units under a bivariate Poissondistribution. Three single-control charts based on themonitoring statistics SM=X1+X2; MX=max(X1, X2); DF=X1−X2 were considered and two separate charts (one for X1

and one for X2—referred in the text as 2C chart). Thecontrol charts were designed to have the false-alarm risk≤0.0027. There is not a single chart that always offers afaster detection of any size and type of shift. The main gainin the proposed charts SM, MX, or DF is the simplicity tojudge if the process is in control or out of control by asingle chart rather than by two separate control charts. Themain difficulty to apply the proposed single-control chartsmay be a correct or best choice for an adequate controlchart. Some guidelines may be drawn here: if increases are

# of Sample

# of Sample

# of Sample

SM

2018161412108642

2018161412108642

2018161412108642

# of Sample

2018161412108642

14

12

10

8

6

4

2

0

8

MX

9

8

7

6

5

4

3

2

1

0

66

Control chart SM Control chart MX

DF

2

1

0

-1

-2

-3

-4

-5

-6

-7 -7

1

X1;

X2

9

8

7

6

5

4

3

2

1

0

33

77

Variable

X1

X2

Control chart DF Control chart 2C

Fig. 1 Control charts for bivariate Poisson process—simulated data

Table 7 Control limits of the numerical example

Control chart Control limits ARL0

SM [0;8] 282.05MX [0;6] 229.5DF [−7;1] 209.12C [0;3];[0;7] 334.9

752 Int J Adv Manuf Technol (2009) 43:744–755

observed in only one type of defects, the DF chart seems tobe the best option. However, if increases are observed inboth types of defects, the rule plays on the correlation: ifthe correlation remains stable, the MX or 2C charts are thebest options; if the correlation increases, the SM chart is thebest option. For future research, EWMA and CUSUM maybe the possible extensions for the proposed control charts tomonitor bivariate Poisson processes.

Acknowledgements The authors gratefully acknowledge financialsupport from FAPESP and CNPq. The authors would also like tothank the referees for their valuable comments and suggestions.

Appendix

A.1 Probability function of MX=max(X1,X2)

P MX ¼ yð Þ ¼Xy�1

j¼0

P X1 ¼ j;X2 ¼ yð Þ þXy�1

j¼0

P X1 ¼ y;X2 ¼ jð Þ þ P X1 ¼ y;X2 ¼ yð Þ

¼ e � l1 þ l2 þ l3ð Þ½ �y!

Xy�1

j¼0

lj1ly2 þ ly1l

j2

j!

!Xji¼0

ji

� � yið Þi!

l3l1l2

� �i

þ l1l2ð Þyy!

Xyi¼0

yið Þ2i!

l3l1l2

� �i( )

A.2 Obtaining E(MX)

E MXð Þ ¼X1y¼0

yXy�1

j¼0

P X1 ¼ j;X2 ¼ yð Þ þXy�1

j¼0

P X1 ¼ y;X2 ¼ jð Þ þ P X1 ¼ y;X2 ¼ yð Þ !

¼X1y¼0

Xy�1

j¼0

yP X1 ¼ j;X2 ¼ yð Þ þXy�1

j¼0

yP X1 ¼ y;X2 ¼ jð Þ þ yP X1 ¼ y;X2 ¼ yð Þ ! ðA:1Þ

Employing the recurrence expressions (3) in (A.1) thenext equalities follow:

E MXð Þ ¼X1y¼0

Xy�1

j¼0

l2P X1 ¼ j;X2 ¼ y� 1ð Þ þ l1P X1 ¼ y� 1;X2 ¼ jð Þð"

þl3 P X1 ¼ y� 1;X2 ¼ j� 1ð Þ þ P X1 ¼ j� 1;X2 ¼ y� 1ð Þð ÞÞþl1P X1 ¼ y� 1;X2 ¼ yð Þ þ l3P X1 ¼ y� 1;X2 ¼ y� 1ð Þ�

ðA:2Þ

After some manipulations the equation (A.2) can beexpressed as

E MXð Þ ¼X1y¼0

l1 þ l2 þ l3ð Þ P X1 ¼ y;X2 ¼ yð Þ þ P X1 ¼ yþ 1;X2 ¼ yð Þ½ �

þX1y¼0

l1 þ l3ð Þ P X1 > y;X2 ¼ yð Þ½ � þX1y¼0

l2 þ l3ð Þ P X1 ¼ y;X2 > yþ 1ð Þ½ �ðA:3Þ

Int J Adv Manuf Technol (2009) 43:744–755 753

A.3 Obtaining E(MX2)

E MX2� � ¼X1

y¼0

y2Xy�1

j¼0

P X1 ¼ j;X2 ¼ yð Þ þXy�1

j¼0

P X1 ¼ y;X2 ¼ jð Þ þ P X1 ¼ y;X2 ¼ yð Þ !

¼X1y¼0

Xy�1

j¼0

y2P X1 ¼ j;X2 ¼ yð Þ þXy�1

j¼0

y2P X1 ¼ y;X2 ¼ jð Þ þ y2P X1 ¼ y;X2 ¼ yð Þ ! ðA:4Þ

Applying the recurrence relations in (A.4), E(MX2) maybe expressed as

¼X1y¼0

Xy�1

j¼0

l2P X1 ¼ j;X2 ¼ y� 1ð Þ þ l3P X1 ¼ j� 1;X2 ¼ y� 1ð Þ þ l22P X1 ¼ j;X2 ¼ y� 2ð Þ

þ2l2l3P X1 ¼ j� 1;X2 ¼ y� 2ð Þ þ l23P X1 ¼ j� 2;X2 ¼ y� 2ð ÞÞþX1y¼0

Xy�1

j¼0

l1P X1 ¼ y� 1;X2 ¼ jð Þ þ l3P X1 ¼ y� 1;X2 ¼ j� 1ð Þ þ l21P X1 ¼ y� 2;X2 ¼ jð Þ

þ 2l1l3P X1 ¼ y� 2;X2 ¼ j� 1ð Þ þ l23P X1 ¼ y� 2;X2 ¼ j� 2ð ÞÞþl1l2P X1 ¼ y� 1;X2 ¼ y� 1ð Þ þ 2l1l3P X1 ¼ y� 2;X2 ¼ y� 1ð Þ þ l3P X1 ¼ y� 1;X2 ¼ y� 1ð Þ

þl23P X1 ¼ y� 2;X2 ¼ y� 2ð Þ

ðA:5Þ

After some algebraic manipulations the expression (A.5)results

E MX2� � ¼ l3 1þ l3ð Þ þ

X1y¼0

l2 1þ l3ð ÞP X1 � y;X2 ¼ yð Þ þ l22P X1 � yþ 1;X2 ¼ yð Þ

þ l1P X1 ¼ y;X2 � yð Þ þ l1 1þ l3ð ÞP X1 ¼ y;X2 ¼ yþ 1ð Þ þ l1l2P X1 ¼ y;X2 ¼ yð ÞðA:6Þ

A.4 Var(MX)

With the expressions (A.3) and (A.6), Var(MX) can beobtained.

References

1. Alt FB (1985) Multivariate quality control in Encyclopedia ofStatistics Sciences, vol. 6 Johnson NL & Kotz S (eds), Wiley,New York

2. Berkhout P, Plug E (2004) A bivariate Poisson count data modelusing conditional probabilities. Stat Neerl 58(3):349–364.doi:10.1111/j.1467-9574.2004.00126.x

3. Crosier RB (1988) Multivariate generalizations of cumulative sumquality control schemes. Technometrics 30:291–303. doi:10.2307/1270083

4. Ho LL, Singer JM (2001) Generalized least squares methods forbivariate Poisson regression. Comm Statist Theory Methods30:263–277. doi:10.1081/STA-100002030

5. Holgate P (1964) Estimation for the bivariate Poisson distribution.Biometrika 51:241–245

6. Johnson N, Kotz S, Balakrishnan N (1997) Discrete multivariatedistributions. Wiley, New York

7. Karlis D, Ntzoufras I (2005) Bivariate Poisson and diagonalinflated bivariate Poisson regression models in R. J Stat Softw 14(10). http://www.jstatsoft.org/

8. Karlis D (2003) An EM algorithm for multivariate Poissondistribution and related models. J Appl Stat 30(1):63–77.doi:10.1080/0266476022000018510

9. Kocherlakota S, Kocherlakota K (1992) Bivariate discretedistributions. Marcel Dekker, New York

754 Int J Adv Manuf Technol (2009) 43:744–755

10. Kocherlakota S, Kocherlakota K (2001) Regression in thebivariate Poisson distribution. Comm Statist Theory Methods30:381–393. doi:10.1081/STA-100002259

11. Lowry CA, Woodall WH, Champ CW, Rigdon SE (1992) Amultivariate exponentially weighted moving average control chart.Technometrics 34:46–53. doi:10.2307/1269551

12. Patel HI (1973) Quality control methods for multivariate binomialand Poisson distribution. Technometrics 15:103–112. doi:10.2307/1266828

13. Pignatiello JJ Jr, Runger GC (1990) Comparison of multivariateCUSUM charts. J Qual Technol 22(3):173–186

14. Rayner JCW, Best DJ (1995) Smooth tests for the bivariatePoisson distribution. Aust J Stat 37(2):233–245. doi:10.1111/j.1467-842X.1995.tb00656.x

15. Skinner KR, Montgomery DC, Runger GC (2003) Processmonitoring for multiple count data using generalized linearmodel-based control charts. Int J Prod Res 41(6):1167–1180.doi:10.1080/00207540210163964

Int J Adv Manuf Technol (2009) 43:744–755 755