Run-length distribution of multivariate control charts for covariance matrix with runs rules using finite Markov chain imbedding

Run-length distribution of multivariate control charts for covariance matrix with runs rules using finite

Markov chain imbedding

Yuri Seo ¹ · Gyo-Young Cho ²

12 Department of Statistics, Kyungpook National University

Received 2 June 2021, revised 9 July 2021, accepted 9 July 2021

Abstract

Hotelling’s (1947) chi-square chart has a disadvantage in that it does not react sensitively to very small changes in the process, so we propose a multivariate chart with rules of various auxiliary runs to compensate for this. We use the finite Markov chain imbedding method and the rule of an auxiliary run to find the run-lengths distribution in a multivariate control chart. In this paper, we find the run-lengths distribution for the covariance matrix. At this time, it is assumed that the correlation coefficient according to the change of the covariance matrix does not change. We try to help evaluate the performance of the control chart by indicating not only the average run length but also the quartile.

Keywords: Average run length, Markov chain imbedding, multivariate control charts, run-length distribution.

1. Introduction

Control charts were initiated by Walter A. Shewhart at Bell Labs in 1924. Shewhart em- phasized that bringing the production process to statistical control and maintaining control are necessary to predict future production and to manage the process economically. The control charts have an upper and lower limit to which a process can be controlled. This Shewhart control chart has its limitations. To expedite detection of a small shift in process is not efficiency. To overcome this problem, Woodall and Montgomery (1999) attempted several approaches. First, they developed exponentially weighted moving average (EWMA) chart and cumulative sum (CUSUM) chart. Compared to Shewhart control chart, EWMA chart and CUSUM chart are effective on detection of small mean shifts. But, using these charts means that it is complex and difficult to use, and they had tried second method, that is applying supplementary stopping rules that are sensitive to small shift of changes. In 1959, the Western Electric Company proposed the runs rules that increase the sensitivity of the

1

Graduate student, Department of Statistics, Kyungpook National University, Daegu 41566, Korea

2

Corresponding author: Professor, Department of Statistics, Kyungpook National University, Daegu

41566, Korea. E-mail: [email protected]

Shewhart control charts. Champ and Woodall (1987) tried to draw an accurate average run length (ARL) through the method combining two or more rules by using the Markov chain imbedding. We will present the new method to compute an accurate run-length distribution by using the finite Markov chain imbedding method. It has same results as these of Champ and Woodall (1987) and Shmueli and Cohen (2003).

We want to expand these theory to multivariate control charts. In reality, there are many situation that two or more involved quality characteristics simultaneously monitoring and controlling such problems are referred in the theory as multivariate quality control problems.

In 1947, Hotelling initiated this work. The Hotelling χ ² control chart has used widely as multivariate chart and is the Shewhart-type control chart. To monitor process variance- covariance matrix, we will calculate the ARL and derive run-length distribution.

The contents of this paper are as follows. In Section 2, we introduce the theoretical con- cepts about the finite Markov chain imbedding. In Section 3, we present performance of multivariate control charts for covariance matrix with runs-rules. In Section 4, we provide concluding remarks.

2. Finite Markov chain imbedding

2.1. Run

The term runs, is often used in the area of probability and statistics. Balakrichnan and Koutras (2002) described how to be defined and used for runs. A run means a subgroup of sequential points. Traditionally, within a sequence of Bernoulli, a run suggests a consecutive sequence.

For example, if we have the binary sequence 111000110, we know the following runs. First, we have a run of the three 1’s, and then a run of three 0’s, a run of two 1’s run, finally a run of one 0. As a result, the total binary sequence we have is four runs.

2.2. Finite Markov chain imbedding

Applying the finite Markov chain imbedding technique to random variable X n (Λ) was initiated by Fu and Koutras, in 1994. Let Γ _n = {1, 2, · · · , n} be an index set and Ω = {a 1 , · · · , a _m } be a finite state space. It is practicable to imbed the finite Markov chain, if a nonnegative integer random variable meets the following conditions.

First, a finite Markov chain {Y _t : t ∈ Γ _n } have to be defined within a finite state space Ω.

Next, a finite partition {C _x , x = 1, 2, · · · , l _n } exists on the finite state space Ω. Third, for every x = 0, 1, · · · , l n , we have

P (X n (Λ) = x) = P (Y n ∈ C x |ξ 0 ), (2.1) where ξ ₀ is the initial probability vector on the state space.

Fu and Koutras (1994) presented to calculate the probability using the following theorem.

If X n (Λ) is finite Markov chain imbeddable, P (X n (Λ) = x) = ξ 0 ( Y Y Y

M t )U ⁰ (C x ), (2.2)

where {M t , t = 1, 2, · · · , n} are the transition probability matrices of the imbedded Markov

chain and U ⁰ (C _x ) = Σ _r:a

_∈C

e _r , e _r is a 1 × m unit vector corresponding to state a _r of the

state space Ω.

2.3. Waiting-time distribution

Waiting time distributions of runs and patterns for Bernoulli trials such as geometric, geometric of order k, negative binomial etc, have been studied by Aki (1985), Koutras (1996), Feller (1968), Philippou (1986) and others.

Assume that there is a simple pattern of k uninterrupted successes. Then the number of trials until a simple pattern first appears is called the waiting time random variable W (Λ), i.e.

W (Λ) = inf{n : X _n−k+1 = X _n−k+2 = · · · = X _n = S}. (2.3) For instance, given k = 3, W (Λ) = 6 imply that the pattern of SSS first occurs after six trials, like FSFSSS. The waiting-time distribution for Bernoulli’s basic trials of binary tests is often regarded as the geometric distribution of order k.

Fu and Lou (2003) derived explanation and result of the waiting-time distribution and average of waiting time using Markov chain imbedding. The followings are its contents.

Fu assumed that the pattern length k is greater than 1 and the Bernoulli trials are {X t }, the distribution of W (Λ) is

P (W (Λ) = n) = ξN ⁿ⁻¹ (Λ)(I − N(Λ))1, (2.4) where ξ = (1, 0, . . . , 0) is a 1 × k row vector, 1 = (1, 1, . . . , 1) ⁰ is a k × 1 column vector and N(Λ) is the k × k important transition probability submatrix of

M(Λ) = 0 1 .. . .. . k − 1

α





















q p 0 · · · 0 0 q 0 p · · · 0 0 .. . .. . .. . . . . .. . .. . .. . .. . .. . . . . .. . .. . q 0 · · · · · · 0 p 0 0 · · · · · · 0 1





















(k+1)×(k+1)

=

 N(Λ) C

0 1



. (2.5)

As the waiting-time satisfies the average of W (Λ), the formula is EW (Λ) = X

ξN ⁿ⁻¹ 1 = ξ(I − N(Λ)) ⁻¹ 1. (2.6) The average waiting time and the average run length (ARL) have the same value. By the ARL, we can verify the performance of the Shewhart chart.

3. Run-length distribution of multivariate control charts

3.1. Runs rules and notation

The Shewhart control chart has the drawback of being insensitive to small changes. So we try to add runs rules and combine two rules; first rule and other one rule. The Western Electric Handbook(1956) set out four runs rules. That is the followings.

Rule 1: One point is out of the three sigma control limits

Rule 2: Two of three continuous points are out of the two sigma limits

Rule 3: Four of five continuous points are out of the one sigma limits Rule 4: Eight continuous points fall on one side of the center line

Runs rules are divided the area in the control chart and applied four rules. There are seven zones divided as follows.

Figure 3.1 Seven zones for control chart

The rule signals if k of the last m standardized points fall in zone Z, denoted by T (k, m, Z), by using the notation of the Champ and Woodall (1987) for runs rules. The followings are the six rules which are considered.

Rule 1: R 1 = {T (1, 1, S)}

Rule 2: R ₂ = {T (2, 3, A ₂ ), T (2, 3, A ₁ )}

Rule 3: R ₃ = {T (4, 5, A ₂ ∪ B 2 ), T (4, 5, A ₁ ∪ B 1 )}

Rule 4: R ₄ = {T (8, 8, A ₂ ∪ B ₂ ∪ C ₂ ), T (8, 8, A ₁ ∪ B ₁ ∪ C ₁ )}

Additional rules suggested by Duncan (1974):

Rule 5: R 5 = {T (2, 2, A 2 ), T (2, 2, A 1 )}

Rule 6: R 6 = {T (5, 5, A 2 ∪ B 2 ), T (5, 5, A 1 ∪ B 1 )}

We combine the two rules that are Rule 1 and other one rule to make supplementary runs

rules. Until now, the runs have been developed on the basis of Bernoulli trials. Especially,

Aki (1985) and Hirano (1986) had studied the waiting-time distribution of Bernoulli trials

in accordance with the geometric distribution of order k.

3.2. Run-length distribution for multivariate control charts

Suppose that the objective is to monitor Σ where the target values Σ 0 and µ 0 are known.

It is assumed that the in-control process covariance matrix is as follows:

Σ 0 =











σ 011 σ 012 · · · σ 01p

σ 021 σ 022 · · · σ 02p

.. . .. . . . . .. . σ _0p1 σ _0p2 · · · σ _0pp











. (3.1)

We consider the case in which the primary purpose is to detect changes in the variances, not correlation.

Several different control charts for Σ will be presented since different statistics can be used to describe variability. In the univariate case, µ = µ ₀ is known, the S ² -chart is used to control the variance under the normality assumption. The S ² -chart signals for large values S ² or equivalently for large values of V _k

V k = nS _k ² /σ ² ₀ , (3.2)

where S _k ² = P n

j=1 (X kj − µ 0 ) ² /n and σ ₀ ² is the target value for σ ² . One possible multivariate version of V k is

Y k = X

(X kj − µ 0 ) ⁰ Σ ⁻¹ ₀ (X kj − µ 0 ) = ntr( c Σ k Σ ⁻¹ ₀ ), (3.3) where c Σ k = P n

j=1 (X kj − µ 0 )(X kj − µ 0 ) ⁰ /n is the maximum likelihood estimator of Σ.

Hotelling (1947) proposed the use of the Lawley-Hotelling Y _k statistic for monitoring the process variance-covariance matrix. The distribution of Y _k was studied by Lawley (1938) and Hotelling (1951). Hotelling proposed the following control statistic for monitoring Σ,

Y _k = X

(X _kj − µ ₀ ) ⁰ Σ ⁻¹ ₀ (X _kj − µ ₀ ) = ntr( c Σ _k Σ ⁻¹ ₀ ). (3.4) When Σ = Σ ₀ , Y _k has a chi-square distribution with np degrees of freedom.

Though this, we try to obtain the ARL and run-length distribution when changing the variance-covariance matrix Σ.

We will divide the zone into four to apply the above rules to the chi-square distribution.

Each zone is S, A, B, C as following and the probability is approximately 0.27%, 4.28%, 27.18% and 68.27% respectively.

Then we use the notation {T (1, 1, S)} by using the notation of the Champ and Woodall (1987) for runs rules as followings:

Rule 1: R 1 = {T (1, 1, S)}

Rule 2: R ₂ = {T (2, 3, A)}

Rule 3: R ₃ = {T (4, 5, A ∪ B)}

Rule 4: R ₄ = {T (2, 2, A)}

Figure 3.2 Four zones for multivariate control chart

Rule 5: R 5 = {T (5, 5, A ∪ B)}

Application of Rule 4 in Chapter 3.1 is impossible because of including the all range of the chi-square distribution. To achieve the prime purpose in this paper, which is to obtain run-length distribution of multivariate control chart for covariance matrix, we apply the combination rules.

If we combine R ₁ and R ₂ , the possible states are S, CS ₁ S ₁ , S ₁ CS ₁ , and S ₁ S ₁ where S ₁ = A and C = B ∪ C in Figure 3.2. The probability of transition matrix M is p s = δ = 0.0027, p s

= P (A) = 0.0428 and p c = P (B) + P (C) = 0.9545.

If we combine R ₁ and R ₃ , the possible states are S, CS ₁ S ₁ S ₁ S ₁ , S ₁ CS ₁ S ₁ S ₁ , S ₁ S ₁ CS ₁ S ₁ , S ₁ S ₁ S ₁ CS ₁ and S ₁ S ₁ S ₁ S ₁ where S ₁ = A ∪ B in Figure 3.2. The probability of transition matrix M is p _s = δ = 0.0027, p _s

= P (A) + P (B) = 0.3146 and p _c = P (C) = 0.6827.

If we combine R 1 and R 4 , the possible states are S and S 1 S 1 where S 1 = A and C = B ∪ C in Figure 3.2. The probability of transition matrix M is p s = δ = 0.0027, p s

= P (A) = 0.0428 and p c = P (B) + P (C) = 0.9545.

If we combine R 1 and R 5 , the possible states are S and S 1 S 1 S 1 S 1 S 1 where S 1 = A ∪ B in Figure 3.2. The probability of transition matrix M is p s = δ = 0.0027, p s

= P (A) + P (B) = 0.3146 and p c = P (C) = 0.6827.

Figures for above the four combination are shown in Figures 3.3 and 3.4.

Figure 3.3 Zone of Rule 1 and Rule 2 (or Rule 1 and Rule 4)

Figure 3.4 Zone of Rule 1 and Rule 3 (or Rule 1 and Rule 5)

For n = 3, 5 and p = 2, 3, Tables 3.1, 3.2, 3.3 and 3.4 show ARLs of control chart in each cell and quartiles of run-length when variances are changed and correlations are not changed, respectively. Here variance-covariance matrices are changed from Σ = Σ 0 to Σ = Σ 1 = cΣ 0 , for c = 1.21, 1.44, 1.69, · · · , 6.25.

As shown in Tables 3.1, 3.2, 3.3 and 3.4, the multivariate control charts for monitoring the variance-covariance matrix are effective in detecting changes in variances.

Table 3.1 ARL and quartiles of run length with R

and R

p 2 3

n c ARL Q

Med Q

ARL Q

Med Q

1.00 166.59 49 116 230 166.58 49 116 230

1.21 41.54 13 29 57 33.93 11 24 47

3 1.44 16.22 5 12 22 12.11 4 9 16

2.56 2.97 1 2 4 2.23 1 2 3

5.76 1.30 1 1 2 1.12 1 1 1

1.00 166.58 49 116 230 166.57 49 116 230

1.21 32.02 10 23 44 25.10 8 18 34

5 1.44 11.17 4 8 15 8.06 3 6 11

2.56 2.07 1 2 3 1.61 1 1 2

5.76 1.09 1 1 1 1.02 1 1 1

Table 3.2 ARL and quartiles of run length with R

and R

p 2 3

n c ARL Q

Med Q

ARL Q

Med Q

1.00 53.28 17 38 73 53.28 17 38 73

1.21 19.23 7 14 26 16.20 6 12 21

3 1.44 10.09 5 8 13 8.07 4 6 10

2.56 3.09 1 3 4 2.42 1 2 4

5.76 1.34 1 1 2 1.13 1 1 1

1.00 53.28 17 38 73 53.28 17 38 73

1.21 15.45 6 11 20 12.71 5 10 17

5 1.44 7.60 4 6 10 6.04 4 5 8

2.56 2.26 1 2 3 1.73 1 1 2

5.76 1.10 1 1 1 1.02 1 1 1

Table 3.3 ARL and quartiles of run length with R

and R

p 2 3

n c ARL Q

Med Q

ARL Q

Med Q

1.00 224.44 65 156 311 224.42 65 156 311

1.21 54.42 16 38 75 44.35 13 31 61

3 1.44 20.25 6 14 28 14.93 5 11 20

2.56 3.17 1 2 4 2.31 1 2 3

5.76 1.31 1 1 2 1.12 1 1 1

1.00 224.42 65 156 311 224.42 65 156 311

1.21 41.83 13 29 58 32.60 10 23 45

5 1.44 13.71 5 10 19 9.68 3 7 13

2.56 2.14 1 2 3 1.63 1 1 2

5.76 1.09 1 1 1 1.02 1 1 1

Table 3.4 ARL and quartiles of run length with R

and R

p 2 3

n c ARL Q

Med Q

ARL Q

Med Q

1.00 207.56 61 145 287 207.54 61 145 287

1.21 50.70 16 36 70 40.62 13 29 56

3 1.44 19.77 7 14 27 14.56 6 11 20

2.56 3.55 1 3 5 2.60 1 2 4

5.76 1.34 1 1 2 1.13 1 1 1

1.00 207.54 61 145 287 207.54 61 145 287

1.21 38.14 12 27 52 29.33 10 21 40

5 1.44 13.39 5 10 18 9.63 5 7 13

2.56 2.40 1 2 3 1.77 1 1 2

5.76 1.10 1 1 1 1.02 1 1 1

Figures 3.5 - 3.20 show the probability and run-length distribution by applying combina- tion rules respectively for n = 3, 5 and p = 2, 3, according to c change; c = 1.00, 1.44, 2.56 and 5.76. Figures 3.5 - 3.8 show that for combination of Rule 1 and Rule 2, Figures 3.9 - 3.12 show that for combination of Rule 1 and Rule 3, Figures 3.13 - 3.16 show that for combination of Rule 1 and Rule 4 and finally Figures 3.17 - 3.20 show that for combination of Rule 1 and Rule 5.

0 10 20 30 40 50

0.00300.00450.0060

c=1 (ARL=166.59)

N

P[RL<n]

N

P[RL<n] 0.150.25

0 10 20 30 40 50

0.010.030.05

c=1.44 (ARL=16.22)

N

P[RL<n]

N

P[RL<n] 0.50.9

0 10 20 30 40 50

0.000.100.20

c=2.56 (ARL=2.97)

N

P[RL<n]

N

P[RL<n] 0.50.7

0 10 20 30 40 50

0.00.20.40.6

c=5.76 (ARL=1.3)

N

P[RL<n]

N

P[RL<n] 0.91

Figure 3.5 Run-length distribution with R

and R

(df=6)

0 10 20 30 40 50

0.00300.00450.0060

c=1 (ARL=230)

N

P[RL<n]

N

P[RL<n] 0.150.25

0 10 20 30 40 50

0.000.020.040.060.08

c=1.44 (ARL=22)

N

P[RL<n]

N

P[RL<n] 0.50.95

0 10 20 30 40 50

0.00.10.20.3

c=2.56 (ARL=4)

N

P[RL<n]

N

P[RL<n] 0.751

0 10 20 30 40 50

0.00.20.40.60.8

c=5.76 (ARL=2)

N

P[RL<n]

N

P[RL<n] 0.951

Figure 3.6 Run-length distribution with R

and R

(df=9)

0 10 20 30 40 50

0.00300.00450.0060

c=1 (ARL=166.57)

N

P[RL<n]

N

P[RL<n] 0.150.25

0 10 20 30 40 50

0.000.020.040.060.08

c=1.44 (ARL=14.56)

N

P[RL<n]

N

P[RL<n] 0.51

0 10 20 30 40 50

0.00.10.20.30.4

c=2.56 (ARL=2.65)

N

P[RL<n]

N

P[RL<n] 0.751

0 10 20 30 40 50

0.00.20.40.60.8

c=5.76 (ARL=1.22)

N

P[RL<n]

N

P[RL<n] 0.961

Figure 3.7 Run-length distribution with R

and R

(df=10)

0 10 20 30 40 50

0.00300.00450.0060

c=1 (ARL=49)

N

P[RL<n]

N

P[RL<n] 0.150.25

0 10 20 30 40 50

0.000.040.080.12

c=1.44 (ARL=5)

N

P[RL<n]

N

P[RL<n] 0.51

0 10 20 30 40 50

0.00.10.20.30.40.5

c=2.56 (ARL=1)

N

P[RL<n]

N

P[RL<n] 0.751

0 10 20 30 40 50

0.00.20.40.60.81.0

c=5.76 (ARL=1)

N

P[RL<n]

N

P[RL<n] 0.991

Figure 3.8 Run-length distribution with R

and R

(df=15)

0 10 20 30 40 50

0.0050.0150.025

c=1 (ARL=53.28)

N

P[RL<n]

N

P[RL<n] 0.30.6

0 10 20 30 40 50

0.000.050.100.15

c=1.44 (ARL=10.09)

N

P[RL<n]

N

P[RL<n] 10.5

0 10 20 30 40 50

0.000.100.20

c=2.56 (ARL=3.09)

N

P[RL<n]

N

P[RL<n] 10.5

0 10 20 30 40 50

0.00.20.40.6

c=5.76 (ARL=1.34)

N

P[RL<n]

N

P[RL<n] 10.9

Figure 3.9 Run-length distribution with R

and R

(df=6)

0 10 20 30 40 50

0.0050.0150.025

c=1 (ARL=73)

N

P[RL<n]

N

P[RL<n] 0.30.6

0 10 20 30 40 50

0.000.050.100.150.20

c=1.44 (ARL=13)

N

P[RL<n]

N

P[RL<n] 10.5

0 10 20 30 40 50

0.00.10.20.3

c=2.56 (ARL=4)

N

P[RL<n]

N

P[RL<n] 10.7

0 10 20 30 40 50

0.00.20.40.60.8

c=5.76 (ARL=2)

N

P[RL<n]

N

P[RL<n] 10.95

Figure 3.10 Run-length distribution with R

and R

(df=9)

0 10 20 30 40 50

0.0050.0150.025

c=1 (ARL=53.28)

N

P[RL<n]

N

P[RL<n] 0.30.6

0 10 20 30 40 50

0.000.050.100.150.20

c=1.44 (ARL=9.28)

N

P[RL<n]

N

P[RL<n] 10.5

0 10 20 30 40 50

0.00.10.20.30.4

c=2.56 (ARL=2.82)

N

P[RL<n]

N

P[RL<n] 10.7

0 10 20 30 40 50

0.00.20.40.60.8

c=5.76 (ARL=1.24)

N

P[RL<n]

N

P[RL<n] 10.95

Figure 3.11 Run-length distribution with R

and R

(df=10)

0 10 20 30 40 50

0.0050.0150.025

c=1 (ARL=17)

N

P[RL<n]

N

P[RL<n] 0.30.6

0 10 20 30 40 50

0.000.100.20

c=1.44 (ARL=5)

N

P[RL<n]

N

P[RL<n] 10.5

0 10 20 30 40 50

0.00.10.20.30.40.5

c=2.56 (ARL=1)

N

P[RL<n]

N

P[RL<n] 10.8

0 10 20 30 40 50

0.00.20.40.60.81.0

c=5.76 (ARL=1)

N

P[RL<n]

N

P[RL<n] 10.99

Figure 3.12 Run-length distribution with R

and R

(df=15)

0 10 20 30 40 50

0.00300.0040

c=1 (ARL=224.44)

N

P[RL<n]

N

P[RL<n] 0.10.2

0 10 20 30 40 50

0.010.030.05

c=1.44 (ARL=20.25)

N

P[RL<n]

N

P[RL<n] 0.90.5

0 10 20 30 40 50

0.000.100.20

c=2.56 (ARL=3.17)

N

P[RL<n]

N

P[RL<n] 10.7

0 10 20 30 40 50

0.00.20.40.6

c=5.76 (ARL=1.31)

N

P[RL<n]

N

P[RL<n] 10.9

Figure 3.13 Run-length distribution with R

and R

(df=6)

0 10 20 30 40 50

0.00300.0040

c=1 (ARL=311)

N

P[RL<n]

N

P[RL<n] 0.10.2

0 10 20 30 40 50

0.000.020.040.06

c=1.44 (ARL=28)

N

P[RL<n]

N

P[RL<n] 0.950.5

0 10 20 30 40 50

0.00.10.20.3

c=2.56 (ARL=4)

N

P[RL<n]

N

P[RL<n] 10.7

0 10 20 30 40 50

0.00.20.40.60.8

c=5.76 (ARL=2)

N

P[RL<n]

N

P[RL<n] 10.95

Figure 3.14 Run-length distribution with R

and R

(df=9)

0 10 20 30 40 50

0.00300.0040

c=1 (ARL=224.41)

N

P[RL<n]

N

P[RL<n] 0.10.2

0 10 20 30 40 50

0.000.020.040.06

c=1.44 (ARL=18.11)

N

P[RL<n]

N

P[RL<n] 0.950.6

0 10 20 30 40 50

0.00.10.20.30.4

c=2.56 (ARL=2.8)

N

P[RL<n]

N

P[RL<n] 10.7

0 10 20 30 40 50

0.00.20.40.60.8

c=5.76 (ARL=1.22)

N

P[RL<n]

N

P[RL<n] 10.95

Figure 3.15 Run-length distribution with R

and R

(df=10)

0 10 20 30 40 50

0.00300.0040

c=1 (ARL=65)

N

P[RL<n]

N

P[RL<n] 0.10.2

0 10 20 30 40 50

0.000.040.08

c=1.44 (ARL=5)

N

P[RL<n]

N

P[RL<n] 0.990.6

0 10 20 30 40 50

0.00.10.20.30.40.5

c=2.56 (ARL=1)

N

P[RL<n]

N

P[RL<n] 10.8

0 10 20 30 40 50

0.00.20.40.60.81.0

c=5.76 (ARL=1)

N

P[RL<n]

N

P[RL<n] 10.99

Figure 3.16 Run-length distribution with R

and R

(df=15)

0 10 20 30 40 50

0.002400.002550.00270

c=1 (ARL=207.56)

N

P[RL<n]

N

P[RL<n] 0.1250.07

0 10 20 30 40 50

0.0100.0200.030

c=1.44 (ARL=19.77)

N

P[RL<n]

N

P[RL<n] 0.80.4

0 10 20 30 40 50

0.000.100.20

c=2.56 (ARL=3.55)

N

P[RL<n]

N

P[RL<n] 10.6

0 10 20 30 40 50

0.00.20.40.6

c=5.76 (ARL=1.34)

N

P[RL<n]

N

P[RL<n] 10.9

Figure 3.17 Run-length distribution with R

and R

(df=6)

0 10 20 30 40 50

0.002400.002550.00270

c=1 (ARL=287)

N

P[RL<n]

N

P[RL<n] 0.1250.07

0 10 20 30 40 50

0.0050.0150.0250.035

c=1.44 (ARL=27)

N

P[RL<n]

N

P[RL<n] 0.850.5

0 10 20 30 40 50

0.00.10.20.3

c=2.56 (ARL=5)

N

P[RL<n]

N

P[RL<n] 10.7

0 10 20 30 40 50

0.00.20.40.60.8

c=5.76 (ARL=2)

N

P[RL<n]

N

P[RL<n] 10.95

Figure 3.18 Run-length distribution with R

and R

(df=9)

0 10 20 30 40 50

0.002400.002550.00270

c=1 (ARL=207.53)

N

P[RL<n]

N

P[RL<n] 0.1250.07

0 10 20 30 40 50

0.010.020.030.04

c=1.44 (ARL=17.65)

N

P[RL<n]

N

P[RL<n] 0.90.5

0 10 20 30 40 50

0.00.10.20.30.4

c=2.56 (ARL=3.15)

N

P[RL<n]

N

P[RL<n] 10.7

0 10 20 30 40 50

0.00.20.40.60.8

c=5.76 (ARL=1.24)

N

P[RL<n]

N

P[RL<n] 10.96

Figure 3.19 Run-length distribution with R

and R

(df=10)

0 10 20 30 40 50

0.002400.002550.00270

c=1 (ARL=61)

N

P[RL<n]

N

P[RL<n] 0.1250.07

0 10 20 30 40 50

0.010.030.05

c=1.44 (ARL=6)

N

P[RL<n]

N

P[RL<n] 0.950.5

0 10 20 30 40 50

0.00.10.20.30.40.5

c=2.56 (ARL=1)

N

P[RL<n]

N

P[RL<n] 10.8

0 10 20 30 40 50

0.00.20.40.60.81.0

c=5.76 (ARL=1)

N

P[RL<n]

N

P[RL<n] 10.99

Figure 3.20 Run-length distribution with R

and R

(df=15)

4. Conclusion

We suggest a finite Markov chain imbedding method for more easily deriving the ARLs and quartiles of run-length distribution. Run-length distribution of the multivariate control charts can be obtained using runs rules and finite Markov chain imbedding and it presents more accurately performance such as easy and fast calculation and clear distribution.

In this paper, we could see the values according to the change of the covariance matrix through multivariate control chart. For the small changes in covariate matrix, the ARL of the run-length distribution has been shown to decrease rapidly.

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