Multivariate GLR control charts for the mean vector and covariance matrix
Seong Rae Jo 1 · Gyo-Young Cho 2
12 Department of Statistics, Kyungpook National University
Received 23 October 2018, revised 14 November 2018, accepted 15 November 2018
Abstract
In statistical process control, we want to detect a change in the process accurately and quickly. The GLR (generalized likelihood ratio) control chart has problems with calculating test statistics. In modern times, however, advances in computing systems have enabled GLR chart test statistics calculation. In this paper, multivariate GLR charts were developed to find changes in the mean vector and covariance matrix. Another problem with the multivariate GLR control chart is that the covariance matrix has various forms. In particular, in order to calculate test statistics on the GLR chart, we need to assume that the determinant of covariance matrix increases, and that these assumptions are often unsatisfactory. To solve this problem, this paper suggested giving the lower limit of the covariance matrix. We showed that the GLR control chart is effective in detecting shifts in mean vector and covariance matrix. Especially, the GLR control chart is effective in detecting a wide range of shifts and the GLR control chart does not require initial parameters.
Keywords: Change point, covariance matrix, GLR control chart, mean vector, multivariate normal distribution, SSATS.
1. Introduction
Statistical process control charts are statistical tools used to detect assignable causes in a process. In the process control, detecting a shift in the mean or variance is important problem. We wish to detect shifts quickly and exactly.
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]
The traditional control charts - Shewhart, cumulative sum (CUSUM) and expo- nentially weighted moving average (EWMA) control charts have a strength and a weakness. Shewhart control chart is effective in detecting large shifts, but it is not effective in detecting small shifts, CUSUM chart or EWMA chart are effective in detecting small shifts, but there are not effective in detecting large shifts.
In traditional mutivariate control charts, the Shewhart-type multivariate control chart was developed by Hotelling (1947), Healy (1987), Crosier (1988). Pignatiello and Runger (1990) proposed a multivariate CUSUM control chart. Lowry et al.
(1992) proposed a multivariate EWMA control chart.
In applications, it is difficult to predict a size of shift. In order to detect various shifts, a combination of two or more control charts is one of options. This option has a good performance over various shifts. Shewhart control chart is effective in detecting large shifts and CUSUM chart is effective in detecting small shifts. There- fore Shewhart control chart is used in a combination with CUSUM control chart.
But this option requires many control chart parameters.
Another option is using generalized likelihood ratio (GLR) control chart. This option is effective in detecting various shifts and it does not require many control chart parameters.
In prior study, Reynolds and Lou (2010) developed the GLR control chart of the univariate normal distribution with a shift in mean. Reynolds et al. (2013) developed the GLR control chart of the univariate normal distribution with a shift in mean and variance, Choi and Lee (2014) developed the GLR control chart with a sustained shift and a linear drift in the process mean. Han et al. (2018) developed a Bernoulli GLR chart based on Bayes estimator.
Wang and Reynolds (2013) developed the GLR control chart of the multivariate normal distribution with a shift in mean vector. Zhow and Cho (2018) developed the GLR control chart of the multivariate normal distribution with a shift in covariance matrix.
The objective of this paper is assumed to be the detection of a shift in mean vector and/or variance-covariance matrix.
2. Multivariate GLR control charts 2.1. Notations and assumptions
Let the p vector X = (x 1 , x 2 , · · · , x p ) 0 be process variable and 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
covariance matrix Σ. The covariance matrix Σ is as follows;
µ =
µ 1 µ 2
.. . µ p
, Σ =
σ 11 σ 12 · · · σ 1p σ 21 σ 22 · · · σ 2p .. . .. . . .. .. . σ p1 σ p2 · · · σ pp
.
When a process is in control, the distribution of X is M N (µ 0 , Σ 0 ). The in control process mean vector and covariance matrix is as follows;
µ 0 =
µ 01 µ 02 .. . µ 0p
, Σ 0 =
σ 011 σ 012 · · · σ 01p σ 021 σ 022 · · · σ 02p .. . .. . . .. .. . σ 0p1 σ 0p2 · · · σ 0pp
.
When a process is out of control, the distribution of X is M N (µ 1 , Σ 1 ). The out of control process mean vector and covariance matrix is as follows;
µ 1 =
µ 11
µ 12
.. . µ 1p
, Σ 1 =
σ 111 σ 112 · · · σ 11p σ 121 σ 122 · · · σ 12p .. . .. . . .. .. . σ 1p1 σ 1p2 · · · σ 1pp
.
We assume that the in control values µ 0 and Σ 0 are known, and the out of control values µ 1 and Σ 1 are unknown. And n observations in a sample are taken within a sample. The distribution of sample is changed by a shift in mean vector and/or variance-covariance matrix, a shift has occurred at change point τ c between time τ and τ + 1. We assume the distribution of τ c is uniform distribution U (τ, τ + 1).
Assume that a shift in occurs at random point τ c and this means that the mean vector µ 0 changes to the unknown mean vector µ 1 , Let the noncentrality parameter δ 1 be
δ 2 1 = (µ 1 − µ 0 ) 0 Σ −1 0 (µ 1 − µ 0 ).
In this paper, we assume that a shift is a sustained shift.
2.2. Generalized likelihood ratio control charts
Suppose kth observation is in control, the maximum likelihood estimator (MLE) of
(µ 0 , Σ 0 ) is easily obtained by maximizing likelihood function of multivariate normal
distribution.
On the other hand, if the th observation is in out of control, the likelihood function is
L(τ, µ 1 , Σ 1 |X 1 , X 2 , · · · , X k ) =
τ
Y
i=1
f (X i |µ 0 , Σ 0 ) ×
k
Y
i=τ +1
f (X i |µ 1 , Σ 1 )
= (2π) −pk/2 |Σ 0 | −τ /2 |Σ 1 | −(k−τ )/2
× exp[− 1 2 (
τ
X
i=1
(X i − µ 0 ) 0 Σ −1 0 (X i − µ 0 ))]
× exp[− 1 2 (
k
X
i=τ +1
(X i − µ 1 ) 0 Σ −1 1 (X i − µ 1 ))]. (2.1)
And MLE of (µ 1 , Σ 1 ) is as follows;
ˆ
µ 1,τ,k = 1 (k − τ )
k
X
i=τ +1
X i . (2.2)
S 1,τ,k = 1 (k − τ )
k
X
i=τ +1
(X i − ˆ µ 1,τ,k )(X i − ˆ µ 1,τ,k ) 0 . (2.3)
The change point τ c is unobserved, we estimate τ instead of τ + 1. Under the assumption that the shift in mean vector and/or variance-covariance matrix occurred before time k. The MLE of τ c can be estimated by maximizing the likelihood function L(τ, µ 1 , Σ 1 |X 1 , X 2 , · · · , X k ).
For testing H 0 : the process is in control versus H 1 : the process is out-of-control, we used the log-likelihood ratio as the test statistic.
R k = log max 0≤τ <k,µ
1,Σ
1L(τ, µ 1 , 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 0,τ,k Σ −1 0 ) − tr(S 1,τ,k Σ −1 1 ) − log | ˆ Σ 1 |
| ˆ Σ 0 | ]
= (k − ˆ τ )
2 [tr(S 0,τ,k Σ −1 0 ) − tr(S 1,τ,k Σ −1 1 ) − log | ˆ Σ 1 |
| ˆ Σ 0 | ]. (2.4)
In the univariate normal distribution, X is normally distributed with the mean
µ and the variance σ. (µ 0 , σ 0 ) is in control parameter and (µ 1 , σ 1 ) is out of control
parameter. Suppose the number of observation n is small, and σ 1 decreases or µ 1
and σ 1 increase, ˆ σ 1 will produce an inflated value of R k (See Reynolds et al. (2013)).
If |Σ 1 | is closed to 0, then log |Σ |Σ
1|
0
