A NEW CHART FOR DETECTING THE PROCESS MEAN AND VARIABILITY
Jiujun Zhang† , Changliang Zou‡ and Zhaojun Wang‡1 †
Department of Mathematics, Liaoning University, Shenyang 110036, China
‡
LPMC and School of Mathematical Sciences, Nankai University, Tianjin 300071, China
Key Words: average run length; Shiryaev-Roberts test; statistical process control. ABSTRACT Since the earliest days of statistical process control, it has been recognized that it is essential to monitor both mean and variability and to trigger an alarm if either characteristic shows sign of a special cause. It is, however, possible to monitor both mean and variance in a single chart responsive to shifts in either. Traditionally, this has been done by using ¯ and S or R charts. The purpose of this paper is to propose a new single chart separate X with the Shiryaev-Roberts (SR) procedure to monitor both mean and variance. It is shown that, in addition to the simplicity of a single chart rather than two, the new chart has some advantages over many other charts in detecting process changes. 1. INTRODUCTION Ever since Shewhart introduced control charts, it has become a common practice for practitioners to use various control charts to monitor different processes. When we deal with variable data, the charting technique usually employs a chart to monitor the process mean and a chart to monitor the process variance. But it is not enough to monitor just the averages of our rational groups; it is not enough to monitor just their standard deviations. Rather, proper statistical process control (SPC) involves simultaneously monitoring both the process mean and the process variance with a single chart (Hawkins and Deng 2009). When a single chart is used, the design and operation of the monitoring scheme can be greatly simplified. Efforts have been made to use single control chart to monitor both the process mean and the process variance at the same time. Among others, Domangun ¯ r test statistics with an EWMA chart. Based on and Patch (1991) proposed to use |X| 1
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the inverse normal transformation, Chen et al. (2001) proposed an EWMA chart which takes the maximum of two test statistics, named as MaxEWMA chart (MEW chart). Costa and Rahim (2004) proposed to use a non-central chi-square statistic (NCS chart) and Wu and Tian (2005) developed a CUSUM chart based on the weighted loss function (WLC chart). Costa and Rahim (2006) proposed a single EWMA chart (CRE chart) which is the extension to the chart studied by Chen et al. (2001), but it is not very sensitive to the small shifts in the process mean without changing the process variability. Hawkins and Deng (2009) proposed two new combination charts which integrate the CUSUM procedure with the GLR and the Fisher statistics, and it is shown that the proposed charts have significant ¯ and S chart pair. But the main disadvantage of the performance advantages over the X Fisher chart, as they pointed out, is that the Fisher method is biased for variance decrease without concomitant shifts in mean. Zhang et al. (2010) proposed an ELR chart with the generalized likelihood ratio (GLR) test statistics. Due to the good properties of the GLR test and EMWA procedure, this new chart is very effective for diverse cases, including the detection of the decrease in variability and the individual observation case which is also very important in many practical applications. The Shiryaev-Roberts test was proposed by Shiryaev (1961) for the Brownian motion case and it has been proved to be optimal for detecting a change that occurs at distant time horizon when the observations are i.i.d. and the pre- and post-change distributions are known (Pollak 1987; Srivastava and Wu 1993). In the present problem of monitoring both the mean and variance, we will show that the SR charting technique leads to a single control chart and allows a convenient recursive representation as well. Some numerical results regarding the comparison between the SR chart and some other competing charts demonstrate its good detection ability. The rest of this paper is organized as follows. In the next section, we describe how to construct the SR chart. The numerical comparisons with other procedures are carried out in Section 3. Several remarks conclude the paper in Section 4. 2. DESCRIPTION OF THE SR CHART Let xt = (xt1 , . . . , xtn ) denotes a sample of size n ≥ 1 taken on a quality characteristic x. 2
The monitoring problems with n > 1 and n = 1 are usually referred to as group observations case and individual observations case, respectively. In industrial practice, sampling may be expensive, time consuming, and the sample interval may be relatively long. In such cases, individual observation at sampling points is usually preferred. In what follows, we assume that the xt for t ≥ 1, the observations collected over time, come from the following process model xti = µ + εti ,
i = 1, . . . , n,
t = 1, 2, . . . ,
where εt1 , . . . εti are identically and independently distributed (i.i.d) normal variables with mean 0 and standard deviation σ. When the process is in-control, µ = µ0 and σ = σ0 . In this paper, we consider the Phase II case in which the in-control (IC) µ0 and σ0 , are assumed to be known, i.e., it is assumed that the IC data set used in Phase I is enough to estimate the parameters well. When a process shift occurs, µ = µ1 and/or σ = σ1 where µ1 = µ0 + δσ0 , σ1 = γσ0 , and δ 6= 0 and/or γ 6= 1 and the pdf is denoted as g(x). The values of δ and γ are known before monitoring. Without loss of generality, we assume µ0 = 0 and σ0 = 1 and the probability density function (pdf) is denoted as f (x). If the in-control mean equals µ0 , which is not 0, and the in-control variance equals σ0 , which is not 1, one can transform the random variable such that the distribution of the transformed variable is N (0, 1). Let xt = (xt1 , . . . , xtn ) P denotes a sample of size n ≥ 1 taken on a quality characteristic x. Let x¯t = nj=1 xtj /n and P St2 = nj=1 (xtj − x¯t )2 /n be the t-th sample mean and sample variance. 2.1 THE CHARTING STATISTIC For t ≥ 1, define Λt =
g(xt ) , f (xt )
to be the instantaneous likelihood ratio between the post-
change and pre-change hypotheses. Under the assumption, the likelihood ratio is given by ) ( n n 2 2 X X g(xt ) γ − 1 δ nδ Λt = = γ −n exp x2tj + 2 xtj − 2 . (1) 2 f (xt ) 2γ j=1 γ j=1 2γ The SR procedure stops and raises an alarm if Rt > h, where Rt is the SR detection statistic defined as Rt =
Pt
k=1
Qt
j=k
Λj and h > 0 is chosen
to achieve a specified IC average run length (ARL). It is easily verified that the SR statistic 3
allows the following convenient recursive representation Rt = (1 + Rt−1 )Λt ,
(2)
where R0 = 0. Pollak and Tartakovsky (2009) show that the SR procedure is exactly optimal in the sense of minimizing the relative integral average detection delay (see more details in Pollak and Tartakovsky 2009). In the literatures, some asymptotic results regarding the ARL of the SR chart have been provided; see Pollak (1987) and Yakir (1995). Next, we will present a Markov chain approach for numerically calibrating the IC and out-of-control (OC) ARLs of the SR chart. 2.2 MARKOV CHAIN CALIBRATIONS THE ARL OF THE SR CHART When evaluating and comparing the performances of statistic control charts, the zerostate and steady-state ARLs are considered. Similar to the procedure proposed by Zhang et al. (2010), we develop a Markov chain model for calibrating the zero-state and steadystate ARLs of the SR chart. The model described below can be regarded as an extension of Brook and Evans (1972), and hence we only briefly describe the approximation method, but highlight some necessary modifications and formulas. For more details on the Markov chain approximation for the conventional charts, readers may refer to Lucas and Saccucci (1990) and again Brook and Evans (1972). Define the m by m transition probability matrix, P = (pij ), where the element pij denotes the transition probability of Λt from state i to state j, and m is the number of states along the axis over the range (0, h). Then the width of each segment is ω = 2h/(2m − 1), except that the width of the first segment is ω/2. The states along the axis are labeled by i = 0, 1, · · · , (m − 1). The center point of state i along the axis is iω. Therefore, the IC region is divided into a number of m regions. Define b 2 n(µ + 2a ) γ2 − 1 δ nδ 2 1 a= , b = , c = , d = , e= 2 2 2 2 2 2γ γ 2γ σ aσ
and aj =
1 [c aσ 2
+
nb2 4a
+ ln(
(j− 12 )wγ n )], 1+iw
bj =
1 [c aσ 2
+
nb2 4a
+ ln(
(j+ 12 )wγ n )], 1+iw
4
nb2 wγ n c+ + ln , 4a 2(1 + iw)
When j 6= 0 and γ > 1, (i.e., a > 0) then ) ( n n 1 n X X (j − 12 )wγ n (j + )wγ 2
