GLR control charts for monitoring the covariance matrix of bivariate normal processes

GLR control charts for monitoring the covariance matrix of bivariate normal processes

Jiayi Zhow ¹ · Gyo-Young Cho ²

12 Department of Statistics, Kyungpook National University

Received 20 August 2018, revised 13 September 2018, accepted 17 September 2018

Abstract

If we want to detect both small shifts and large shifts in means and variances, we use the generalized likelihood ratio (GLR) control chart, in which the range of shift sizes in the parameter does not need to be specified, but can be estimated from the pro- cess data. There have been some works on developing GLR control charts specifically for the problem of monitoring mean and variance. GLR control charts for monitoring the process mean of normal distribution were investigated. Also GLR control charts for monitoring the Bernoulli process were investigated. In the multivariate case, GLR control charts for monitoring the process mean vector of a bivariate normal distribu- tion were investigated. In this paper, we will investigate the GLR control chart for monitoring the covariance matrix of bivariate normal process.

Keywords: Bivariate normal distribution, change point, covariance matrix, GLR control chart, SSATS.

1. Introduction

Statistical process control (SPC) is the important concept of statistical quality control.

In SPC, when we want to detect the quality characteristic of some variables, it is usually necessary to monitor means and variances of the quality characteristic. In this situation, we use control charts. The purpose of control charts is to detect the assignable causes through the mean and variance. A good control chart produces a few false alarms when the production process is in-control and detects the shift as soon as possible when the production process is out-of-control.

The first control chart is named Shewhart control chart. It can be monitoring the large shifts of means and variances. But if the shifts are small or moderate, it has less effective.

So some people developed the cumulative sum (CUSUM) control chart (Page, 1954) and the exponentially weighted moving average (EWMA) control chart (Robert, 1958). But if the shift that occurs is not close the specified size, these control charts are also have less effective. If we want to detect both small shifts and large shifts in means and variances, we

1

Graduate student, Department of Statistics, Kyungpook National University, Daegu, 702-701, Korea.

2

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

702-701, Korea. E-mail: [email protected]

shift does not need to be specified but can be estimated from the process data.

There have been some works on developing GLR control charts specifically for the prob- lem of monitoring means and variances. Reynolds and Lou (2010), Choi and Lee (2014) investigated the GLR control charts for monitoring the process means of normal distribu- tion. Reynolds et al. (2013) investigated the GLR control charts for monitoring the process means and variance. Xu et al. (2012) investigated the GLR control charts for monitoring the Bernoulli process. Reynolds and Wang (2013) investigated the GLR control charts for monitoring the process mean vector of a bivariate normal distribution. Han et al. (2018) studied a Bernoulli GLR chart based on Bayes estimator.

The multivariate control charts for monitoring covariance matrix were studied by Reynolds and Cho (2006, 2011), Chang and Heo (2011), Jeong and Cho (2012) and Kwon and Cho (2017).

In this paper, we will investigate the GLR control chart for monitoring the covariance matrix of bivariate normal process.

2. GLR control charts

2.1. Notations and assumptions

Suppose that the process of interest has p quality characteristics represented by X, then p × 1 vector X = [x 1 , x 2 , · · · , x p ] ⁰ is the p quality measurements of an item from the process.

Let X k = [x 1k , x 2k , · · · , x pk ] ⁰ represent the observation at the kth sampling time point.

Assume that X has the multivariate normal distribution with mean vector µ and covari- ance matrix Σ. The covariance matrix Σ is as follows;

Σ =











σ 11 σ 12 · · · σ 1p

σ 21 σ 22 · · · σ 2p

.. . .. . . . . .. . σ _p1 σ _p2 · · · σ _pp









 .

Let µ 0 , Σ 0 , and σ 0 represent the in-control values for µ, Σ, and σ respectively. We assume that (µ 0 , Σ 0 ) is the known target process value for (µ, Σ), so 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









 .

Let Σ ₁ and σ ₁ represent the out-of-control values for Σ and σ respectively. We assume that

Σ ₁ is the known process value for Σ and the out-of-control covariance matrix is as follows;

Σ ₁ =











σ ₁₁₁ σ ₁₁₂ · · · σ _11p σ ₁₂₁ σ ₁₂₂ · · · σ _12p .. . .. . . . . .. . σ 1p1 σ 1p2 · · · σ 1pp









 .

Consider that the shift from the in-control normal process to the out-of-control normal process has occurred at a time between τ and τ + 1 where τ < k. Thus the change point τ c is satisfied τ ≤ τ c < τ + 1 and τ c ∼ U (τ, τ + 1). It means that a random sample before time point τ is in-control process, while the random sample after τ comes from the out-of- control process. For example, if τ c = 7.9 then τ = 7 and τ + 1 = 8. Thus X 1 , X 2 , · · · , X 7

are in-control which are multivariate normal process with mean vector µ 0 and covariance matrix Σ 0 , while X 8 , X 9 , · · · are out-of-control which are multivariate normal process with mean vector µ 0 and covariance matrix Σ 1 . Note that the value of change point τ c can not be observed.

In this paper, we investigate the generalized likelihood ratio (GLR) control charts for monitoring the covariance matrix of bivariate normal process. Thus our in-control covariance matrix Σ ₀ which is from the bivariate normal distribution (µ ₀ , Σ ₀ ) is as follows;

Σ ₀ = σ ₀₁₁ σ ₀₁₂ σ ₀₂₁ σ ₀₂₂

 .

Thus the correlation coefficient ρ ₀ is

ρ ₀ = σ ₀₁₂

√ σ 011

√ σ 022

.

The out-of-control covariance matrix Σ 1 which is from the bivariate normal distribution (µ 0 , Σ 1 ) is as follows;

Σ 1 = σ 111 σ 112

σ 121 σ 122

 .

Thus the correlation coefficient ρ 1 is

ρ 1 = σ 112

√ σ 111

√ σ 122

.

2.2. Generalized likelihood ratio control charts

Since the change point τ _c can not be observed, we must estimate it. We assume that the

process has changed prior to time k, then the maximum likelihood estimator of τ can be

obtained by maximizing the profile likelihood

ˆ

τ = t = arg max

0≤t<k L(τ, µ 0 , Σ|x 1 , x 2 , · · · , x k )

= arg max

0≤t<k t

Y

i=1

f (x _i |µ ₀ , Σ ₀ ) ×

k

Y

i=t+1

f (x _i |µ ₀ , Σ ₁ )

or we will get the τ as follows;

ˆ

τ = t = arg max

0≤τ <k

L(τ, µ 0 , Σ 1 |x 1 , x 2 , · · · , x k ) L(∞, µ ₀ , Σ ₀ |x ₁ , x ₂ , · · · , x _k )

= arg max

0≤τ <k R _k .

Note that in the null hypothesis, there has no change in process, the likelihood function does not depend on τ . Hence the two equations are equal.

Suppose there has been a shift between samples, then we have the likelihood function at sample k is as follows:

L(τ, µ 0 , Σ 1 |x 1 , x 2 , · · · , x k )

=(2π) ^−npk/2 |Σ 0 | ^{−nτ /2} |Σ 1 | ^{−n(k−τ )/2} exp



− 1 2





τ

X

i=1 n

X

j=1

(X ij − µ 0 ) ^T Σ ⁻¹ ₀ (X ij − µ 0 )

− 1 2





k

X

i=τ +1 n

X

j=1

(X ij − µ 0 ) ^T Σ ⁻¹ ₁ (X ij − µ 0 )











 ,

where the bivariate vector X _ij = (x _i1j , x _i2j ) ^T , i = 1, 2, · · · , k, j = 1, 2, · · · , n.

Suppose there has been no shift between samples, then we have the likelihood function at sample k is

L(µ ₀ , Σ ₀ , x ₁ |x 2 , · · · , x _k )

= (2π) ^−npk/2 |Σ 0 | ^−nk/2 exp



− 1 2





k

X

i=1 n

X

j=1

(X _ij − µ 0 ) ^T Σ ⁻ ₀ 1(X _ij − µ 0 )







 .

Let sample covariance matrix S τ,k be defined by

S τ,k = 1 n(k − ˆ τ )

k

X

i=ˆ τ +1 n

X

j=1

(X ij − µ 0 )(X ij − µ 0 ) ^T .

If there has been no restriction of value Σ ₁ , then S _τ,k will be the maximum likelihood

estimate of Σ ₁ . Under the restriction |Σ ₁ | ≥ |Σ 0 |, the maximum likelihood estimate of Σ 1 is

Σ ˆ ₁ =

( S τ,k if |S τ,k | ≥ |Σ 0 | Σ 0 if |Σ 0 | > |S τ,k |.

We will get the log likelihood ratio statistic for testing whether the statistic has been changed in the past k samples is

R k = log max 0≤τ <k,Σ

L(τ, µ 0 , Σ 1 |x 1 , x 2 , · · · , x k ) L(∞, µ ₀ , Σ ₀ |x 1 , x ₂ , · · · , x _k )

= max

0≤τ <k

k − τ 2

"

tr(S _τ,k Σ ⁻¹ ₀ ) − tr(S _τ,k Σ ˆ ⁻¹ ₁ ) − log | ˆ Σ 1 |

|Σ 0 |

#

= n(k − ˆ τ ) 2

"

tr(S τ ,k ˆ Σ ⁻¹ ₀ ) − tr(S τ ,k ˆ Σ ˆ ⁻¹ ₁ ) − log | ˆ Σ 1 |

|Σ ₀ |

# .

At a time k when a new observation is observed, the generalized likelihood ratio statistic is calculated using only the observations at times with max(0, k − m) ≤ t < k. The modified control statistic then becomes

R m,k = log max max(0,k−m)≤τ <k,Σ

L(τ, µ 0 , Σ 1 |x 1 , x 2 , · · · , x k ) L(∞, µ 0 , Σ 0 |x 1 , x 2 , · · · , x k )

= max

max(0,k−m)≤τ <k

n(k − τ ) 2

"

tr(S τ,k Σ ⁻¹ ₀ ) − tr(S τ,k Σ ˆ ⁻¹ ₁ ) − log | ˆ Σ 1 |

|Σ ₀ |

#

= n(k − ˆ τ ) 2

"

tr(S τ ,k ˆ Σ ⁻¹ ₀ ) − tr(S τ ,k ˆ Σ ˆ ⁻¹ ₁ ) − log | ˆ Σ 1 |

|Σ 0 |

# .

2.3. Performance measurements

The GLR control chart for monitoring the covariance matrix of bivariate normal process is the same as the hypothesis test as follows;

H 0 : x k ∼ N (µ 0, Σ 0 ), k ≥ 1,

H 1 : x k ∼

( N (µ 0 , Σ 0 ), 1 ≤ k ≤ τ c

N (µ 0 , Σ 1 ), k > τ c .

If the null hypothesis is accepted, then the produce is normal. In this situation, we hope that the control chart has few false alarm. On the other hand, if the alternative hypothesis is accepted, then we hope that the control chart will detect the shift as quickly as possible.

In the statistical process control, if the sample size are not changed and the sampling

interval are changed, then we say that the control chart is variable sampling interval (VSI)

signal.

When the process is in-control, we hope that the ATS is as large as best. When the process is out-of-control, we hope that the ATS is as small as best. So ATS is the important indicator for comparing the control charts. Usually we fixed the in-control ATS and compare the out-of-control ATS.

ATS can be calculated in the control chart which the out-of-control is happened at start.

But in many really situation, the process is in-control at start, and the out-of-control will be happened randomly. In this situation, we use the steady-state ATS (SSATS) to monitoring it.

The SSATS based on the assumption that the process is in control when monitoring begins, and a change occurs at a random time τ _c after the control-chart statistic reaches its steady-state distribution. The SSATS is believed to be a more realistic metric for measuring the ability of a control chart to detect a change in the process.

In this paper, we simulate 10,000 iterations and note that the SSATS is expected time to signal from change point τ _c .

3. Numerical results

3.1. Introduction

In order to calculate and compare the performance of each GLR control charts, we must have some kinds of standards. In this paper, our in-control bivariate normal process is distributed by mean vector 0 and covariance matrix are as follows respectively;

Σ 01 =  1 0.8 0.8 1



, Σ 02 =  1 0.3 0.3 1

 .

The window size m is 25, 50, 100, 200, 400 and 12,000. For each m, when the ATS is approximately to 400, we get the control limit. Note that for an in-control ATS of 400, the GLR control chart with m = 12, 000 is presumably identical to the GLR chart without a window.

Table 3.1 Control limits m

25 50 100 200 400 12,000

Control 7.7841 7.9101 7.9609 7.9956 8.0066 8.0066

limits

When the production processes are changed, the following types of shifts in the covariance matrix with bivariate normal processes are considered;

(1) Two variances are changed, covariance is changed and correlation coefficient is not changed.

(2) Two variances are changed, covariance is not changed and correlation coefficient is changed.

(3) Two variances are changed, covariance is changed and correlation coefficient is changed.

3.2. Simulation results

In this paper, we defined the in-control bivariate normal process for GLR control chart with mean vector 0 and the covariance matrix is defined as follows;

Σ 0 = σ ₀₁₁ σ 012

σ 021 σ 022

 .

Note that the means are always to be 0 whether the bivariate normal process is in-control or not. And σ 011 and σ 012 are equal to 1.

When the process is out-of control, the covariance matrix is defined as follows;

Σ 1 = σ ₁₁₁ σ 112

σ 121 σ 122

 .

Tables 3.2 to 3.4 investigate the in-control covariance matrix with variance 1 and corre- lation coefficient 0.8 (thus the covariance is 0.8). For the out-of-control covariance matrix, we investigate the change for standard deviation which is 1.1, 1.2, 1.3, 1.5, 2.0, 2.5, 3.0 respectively, and correlation coefficient which is 0.72, 0.56, 0.40 respectively. The window size is 25, 50, 100, 200, 400, 12,000 respectively.

Tables 3.5 to 3.7 investigate the in-control covariance matrix with variance 1 and corre- lation coefficient 0.3 (thus the covariance is 0.3). For the out-of-control covariance matrix, we investigate the change for standard deviation which is 1.1, 1.2, 1.3, 1.5, 2.0, 2.5, 3.0 respectively and correlation coefficient which are 0.27, 0.21, 0.15 respectively. The window size is 25, 50, 100, 200, 400, 12,000 respectively.

Tables 3.2 and 3.5 show that the standard deviation is changed and the correlation coef- ficient is not changed. Thus the covariance must be changed. Tables 3.3 and 3.6 show that the standard deviation is changed and the covariance is not changed. Thus the correlation coefficient must be changed. Tables 3.4 and 3.7 show that the standard deviation is changed and the correlation coefficient is changed. Thus the covariance must be changed.

Table 3.2 SSATS when two variances are changed, covariance is changed and correlation coefficient is not changed (σ

= σ

= 1, σ

= 0.8, ρ

= 0.8)

σ

= σ

σ

m

25 50 100 200 400 12000

1.21 0.97 99.33 93.85 90.96 86.27 86.93 86.93

1.44 1.15 41.06 39.40 37.75 36.63 37.04 37.04

1.69 1.35 22.81 21.63 21.14 20.70 20.95 20.95

2.25 1.80 10.10 10.07 10.03 9.86 9.98 9.97

4.00 3.20 3.88 3.95 3.93 3.86 3.89 3.89

6.25 5.00 2.33 2.35 2.35 2.32 2.35 2.35

9.00 7.20 1.63 1.64 1.64 1.63 1.63 1.63

Control limit 7.78 7.91 7.96 8.00 8.01 8.01

changed (σ

= σ

= 1, σ

= σ

= 0.8, ρ

= 0.8)

σ

= σ

ρ

m

25 50 100 200 400 12000

1.21 0.66 23.26 22.57 21.98 21.63 21.80 21.80

1.44 0.56 9.76 9.83 9.73 9.60 9.69 9.69

1.69 0.47 6.06 6.12 6.10 5.99 6.05 6.05

2.25 0.36 3.46 3.49 3.49 3.44 3.47 3.47

4.00 0.20 1.78 1.79 1.79 1.78 1.79 1.79

6.25 0.13 1.27 1.26 1.26 1.25 1.25 1.25

9.00 0.09 1.02 1.02 1.02 1.01 1.02 1.02

Control limit 7.78 7.91 7.96 8.00 8.01 8.01

Table 3.4 SSATS when two variances are changed, covariance is changed and correlation coefficient is changed (σ

= σ

= 1, σ

= 0.8, ρ

= 0.8)

ρ

σ

= σ

m

25 50 100 200 400 12000

0.72 1.21 38.18 35.73 34.63 33.70 34.20 34.20

1.44 20.61 20.01 19.68 19.36 19.49 19.49

1.69 13.48 13.34 13.28 13.07 13.19 13.19

2.25 7.11 7.21 7.17 7.06 7.15 7.15

4.00 3.13 3.15 3.15 3.11 3.13 3.13

6.25 1.97 1.99 1.99 1.96 1.98 1.98

9.00 1.43 1.43 1.43 1.42 1.42 1.42

0.56 1.21 13.84 13.68 13.50 13.29 13.38 13.38

1.44 9.92 9.95 9.90 9.77 9.86 9.86

1.69 7.34 7.43 7.44 7.32 7.34 7.34

2.25 4.81 4.86 4.85 4.77 4.80 4.80

4.00 2.40 2.41 2.41 2.38 2.39 2.39

6.25 1.60 1.61 1.61 1.59 1.60 1.60

9.00 1.24 1.24 1.25 1.23 1.24 1.24

0.40 1.21 8.63 8.65 8.60 8.37 8.53 8.53

1.44 6.66 6.69 6.64 6.53 6.59 6.59

1.69 5.30 5.37 5.34 5.26 5.31 5.31

2.25 3.66 3.70 3.71 3.66 3.68 3.68

4.00 2.01 2.03 2.04 2.02 2.03 2.03

6.25 1.42 1.42 1.42 1.40 1.41 1.41

9.00 1.12 1.13 1.13 1.12 1.13 1.13

Control limit 7.78 7.91 7.96 8.00 8.01 8.01

Table 3.5 SSATS when two variances are changed, covariance is changed and correlation coefficient is not changed (σ

= σ

= 1, σ

= 0.3, ρ

= 0.3)

σ

= σ

σ

m

25 50 100 200 400 12000

1.21 0.36 99.33 93.85 90.96 87.82 86.93 86.93

1.44 0.43 41.06 39.40 37.75 37.07 37.04 37.04

1.69 0.51 22.82 21.63 21.14 20.86 20.95 20.95

2.25 0.68 10.10 10.07 10.03 9.95 9.97 9.97

4.00 1.20 3.88 3.95 3.93 3.88 3.89 3.89

6.25 1.88 2.33 2.35 2.35 2.34 2.35 2.35

9.00 2.70 1.63 1.64 1.64 1.63 1.63 1.63

Control limit 7.78 7.91 7.96 8.00 8.01 8.01

Table 3.6 SSATS when two variances are changed, covariance is not changed and correlation coefficient is changed (σ

= σ

= 1, σ

= σ

= 0.3, ρ

= 0.3)

σ

= σ

ρ

m

25 50 100 200 400 12000

1.21 0.25 89.75 85.07 81.25 78.54 78.50 78.50

1.44 0.21 35.04 33.17 31.81 31.52 31.56 31.56

1.69 0.18 18.91 18.63 18.35 18.10 18.18 18.18

2.25 0.13 9.01 9.04 8.99 8.90 8.89 8.89

4.00 0.08 3.52 3.56 3.55 3.54 3.54 3.54

6.25 0.05 2.14 2.16 2.16 2.14 2.14 2.14

9.00 0.03 1.53 1.53 1.53 1.52 1.52 1.52

Control limit 7.78 7.91 7.96 8.00 8.01 8.01

Table 3.7 SSATS when two variances are changed, covariance is changed and correlation coefficient is changed (σ

= σ

= 1, σ

= 0.3, ρ

= 0.3)

ρ

σ

= σ

m

25 50 100 200 400 12000

0.27 1.21 95.11 90.24 85.95 83.56 82.77 83.94

1.44 38.93 36.89 35.83 35.12 35.28 35.28

1.69 22.11 21.17 20.75 20.33 20.36 20.36

2.25 9.87 9.96 9.85 9.73 9.73 9.26

4.00 3.82 3.87 3.88 3.83 3.84 3.84

6.25 2.30 2.32 2.31 2.30 2.30 2.30

9.00 1.62 1.62 1.63 1.62 1.63 1.63

0.21 1.21 80.65 75.81 71.58 69.45 69.32 69.82

1.44 35.21 33.49 31.90 31.58 31.64 31.64

1.69 19.79 19.47 19.19 18.95 18.99 18.99

2.25 9.44 9.45 9.41 9.34 9.36 8.87

4.00 3.69 3.73 3.72 3.69 3.70 3.70

6.25 2.25 2.26 2.25 2.25 2.25 2.25

9.00 1.59 1.60 1.60 1.59 1.59 1.59

0.15 1.21 69.85 63.37 61.01 59.13 58.37 59.21

1.44 31.27 29.86 28.74 28.46 28.52 28.52

1.69 18.24 17.81 17.70 17.54 17.58 17.58

2.25 9.10 9.07 9.01 8.93 8.96 8.57

4.00 3.60 3.64 3.65 3.62 3.62 3.62

6.25 2.21 2.22 2.22 2.21 2.21 2.21

9.00 1.56 1.57 1.57 1.56 1.57 1.56

Control limit 7.78 7.91 7.96 8.00 8.01 8.01

4. Conclusions

In this paper, we investigate the GLR control charts for monitoring the covariance matrix of bivariate normal processes.

From Table 3.2 to Table 3.7, the GLR control charts for monitoring the covariance matrix are effective in terms of SSATS. And the GLR control charts are very effective for monitoring the correlation coefficient. When the changes in the covariance matrix are small, the GLR control charts with large window sizes are effective in terms of SSATS. When the changes in the covariance matrix are larger, the SSATS for all window sizes are almost same.

From Table 3.5 to Table 3.7, the GLR control charts with in-control correlation coefficient

0.3 have the same properties as the GLR control charts with in-control correlation coefficient

As shown in Table 3.2 and Table 3.5 when two variances are changed and the correlation coefficient is not changed, the SSATS of the GLR control charts which have the same changes of variances and the same window size are equal. When the variances are changed and correlation coefficient is not changed, the GLR control charts with the in-control correlation coefficient 0.8 are more effective than the GLR control charts with the in-control correlation coefficient 0.3 in terms of SSATS. Except small changes in the covariance matrix, The SSATS of the GLR control charts are almost same for all window sizes.

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