Multivariate Cusum Control Charts Based on data ...

Multivariate Cusum Control Charts Based on data depth For Preliminary Analysis Yi Dai, Chunguang Zhou, Zhaojun Wang

∗

Department of Statistics, School of Mathematical Sciences Nankai University, Tianjin 300071, PR China

Abstract In this paper, a CUSUM control chart based on data depth is considered for detecting a shift in either in the mean vector, the covariance matrix, or both of the process. The proposed chart is preferable from a robustness point of view, has attractive detection performances and can be useful in preliminary analysis setting where there is limited knowledge about the underlying process. A diagnostic aid is also given to estimate the location of the change.Using simulation, approximate values are given for the variance values of the plotted statistics and an upper control limit.Moreover,the new control chart can detect small sample . Unlike the existing nonparametric methods,which can detect only large sample. Keywords: preliminary analysis; nonparametric test; Mann-Whitney statistic; false alarm probability; individual observations; robustness.

1

Introduction

Since its introduction by W. A. Shewhart (1931), a physicist and statistician working for Bell laboratory, the SHEWHART control chart has become a popular tool for monitoring the performance of industrial processes. Over the past century, the SHEWHART chart and its properties have received much attention in the statistical literature. Montgomery (1996) gave a detailed research of SHEWHART control chart. The same is true with the Cumulative Sum (CUSUM) control chart proposed by Page (1954) and Exponentially Weighted Moving Average (EWMA) control scheme proposed by Roberts (1959). Hawkins and Olwell (1998) gave a comprehensive and ∗

Corresponding author, his email is [email protected]

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systemic description of CUSUM chart while Lucas and Saccucci (1990) presented a detailed research of the properties of EWMA chart. SHEWHART, CUSUM and EWMA schemes are acknowledged as the most widely used control charts. In modern quality control, it is becoming common to monitor several quality characteristics of a process simultaneously. This challenge motivates attempts to extend the univariate SHEWHART, CUSUM and EWMA statistics to multivariate data. In the past decades, several kinds of multivariate control chart for the process mean have appeared, most of them are generalizations of their corresponding univariate procedures. Three of the most useful multivariate quality control statistics are Hotelling’s T 2 (Hotelling, 1947), MCUSUM proposed by Woodall and Ncube (1985), MCUSUM proposed by Crosier (1988) and MEWMA proposed by Lowry (1992) . The classical application of these three types of control schemes, namely SHEWHART, CUSUM and EWMA, for monitoring the process mean requires the assumption that the process being inspected follows a multivariate normal distribution. In some applications the normality assumption is not valid. Amin et al (1995) studied the effect of non-normality on X chart and S 2 chart, Ryan and Faddy (2001) studied the effect of non-normality on the performance of CUSUM procedures, Borror, Montgomery and Runger (1999) and Stoumbos and Sullivan (2002) respectively studied the robustness of EWMA and Multivariate EWMA control chart to nonnormality. The results showed that if the true distribution is quite different from the assumed form, the properties of the control procedure can be considerably different. From another perspective, in most industrial and service applications, even the process follows multi-normal distribution, the mean and standard deviation of the process to be monitored are unknown, a common practice is to replace them with estimates from an in-control reference sample, however, using estimated parameters usually leads to significantly deteriorated chart performance. see Jones, Champ and Rigdon (2001), Jones (2002) and Jones, Champ and Rigdon (2004). Under such circumstances it is pragmatic to consider a nonparametric control procedure which is not only less in uenced by the underlying distribution but does not need to establish the process variance. Nonparametric control schemes enjoy many advantages over parametric control 2

schemes for their distribution-free, simplicity, greater robustness . There are some literatures about nonparametric univariate control charts including SHEWHARTtype, CUSUM-type and EWMA-type control charts. The statistics used in nonparametric control charts are mostly ranks, see Hackl and Ledolter (1991), signed ranks, see Amin et al (1995) or other nonparametric statistics, such as hodges-lehmann statistics, see Alloway and Raghavachari (1991). Chakraborti, Van and Bakir (2001) gave an overview of the development of the nonparametric control schemes. Few nonparametric multivariate control schemes has been proposed. Liu (1995) proposed three nonparametric control charts based on simplicial data depth—the r chart, Q chart and S chart. These three charts can be viewed as simplicialdepth-based multivariate generalizations of the univariate X, X and CUSUM chart. However ,all of the three charts need large sample. In the literature there are Phase I and Phase II control charts need to distinguish. In Phase II the process distribution is assumed to be completely known. However, the process distribution or the process parameters are often unknown in practice. So, it’s necessary to establish that a process is statistically in control and estimate the process parameters, referred as preliminary, retrospective or Phase I analysis. In preliminary analysis, the finite historical data is used to decide if the process is statistically in control and to estimate the parameters of the process. Some times, the nature of the process may suggest rational subgroups within which the quality measurements are relatively homogeneous. On the other hand there are many situations where it is reasonable to analyze individual observations, possibly because measurement is automated and every unit is measured, because the rate of production is slow, or for other reasons (See Montgomery(1997)). The preliminary, or Phase I, situation with individual observations is the focus of this article. In the literature, there are a lot of multivariate control charts were used in the preliminary analysis for individual or grouped observations, such as the usual T 2 control charts (see Wierda (1994), Lowry and Montgomery(1995), Mason et al. (1997)), the developed T 2 control charts(T 2 D) (see Holmes and Mergen (1993), Sullivan and Woodall (1996)),and the control chart based on the likelihood ratio test (M + D) and (S&W )(see Sullivan and Woodall (2000) and Srivastava and Worsley (1986)). Sullivan and Woodall (2000) pointed out the M + D control chart is more 3

powerful than the T 2 and T 2 D control charts in detecting a shift in either the mean vector, the covariance matrix, or both for the preliminary analysis. The (S&W ) control chart has better properties to detect shift in mean vector,but it can’t detect the shift of covariance matrix. Based on data depth, a CUSUM control chart is considered for detecting a shift in either in the mean vector, the covariance matrix, or both of the process. A diagnostic aid is given to estimate the position of the change. For comparisons, the LRT control charts are taken as standard alternatives. However, in fact, there are no standard alternatives because these two methods rely on the assumption that the observations follow a normal distribution. Therefore, we choose some perhaps imperfect comparisons to show the effectiveness of our approach.

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The Effect of Non-Normality on LRT Chart

The design of LRT chart is usually based on the assumption that the distribution of the observations is normal . Stoumbos and Sullivan(2002) studied the robustness to non-normality of the MEWMA control chart. They mainly studied the robustness to non-normality of the MEWMA when the underlying distribution is p-variate t distribution tp (ν) or Γ distribution Γp (r, λ). Table 1 give the false alarm probability (FAP) α for LRT chart for the p-variate t distribution tp (ν) , Γ distribution Γp (r, λ) and Dirichlet distribution Dp (ν1 , ν1 ¦¦¦νp+1 ) (see Kotz et al.(2000)). Table P 1 Simulated false alarm probability (FAP) α for LRT chart Np (u, ) t2 (2) t6 (3) Γ2 (1, 1) Γ6 (1, 1) D2 (1) D2 (2) D6 (1) D6 (2) 0.050 0.542 0.815 0.330 0.338 0.864 0.971 0.793 0.905 • Dirichlet Distributions D2 (1) = D2 (7.5, 12, 0.1), D2 (2) = D2 (20, 0.1, 0.2), D6 (1) = D6 (0.1, 0.2, 0.3, 10, 20, 30, 40), D6 (2) = D6 (40, 0.2, 0.3, 0.1, 0.4, 0.5, 1). From Table 1 we see that, the actual FAP is larger than the calculation based on a normal distribution . The difference between the actual FAP and the FAP for the normal is particularly pronounced for the in-control situation when the distribution

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is Non-Normality. This means that ,FAP will occur much more frequently than expected when the process is operating properly . For example, when the observations has t distribution t2 (2), the FAP will be 0.542 which is almost 11 times as many as 0.05.

3

Introduction of Data Depth

Statistical depth functions have become increasingly researched as a useful tool in nonparametric inference for multivariate data. They can be used as quality index in quality control schemes. There are many types of definitions of data depth, such as Mahalanobis Depth (Mh D) (Mahalanobis, 1936), Tukey Depth (TD) (Tukey, 1975), Simplicial Depth (SD) (Liu, 1990), Majority Depth (Mj D) (Singh, 1991),Projection Depth (PD) (Zuo, 2001),Spatial Rank Depth (SRD) (Yonghong Gao, 2003). THE DEFINITIONS OF DATA DEPTH (F KNOWN) Mahalanobis depth (Mahalanobis 1936): Mh D(F, x) = (1 + d(x, µF ))−1 where d(x, µF ) = (x − µF )0 Σ−1 F (x − µF ). Tukey’s depth(Tukey 1975): T D(F, x) = inf {F (H) : H is a closed half space containing x} H

(when p = 1, T D(F, x) = min{F (x), 1 − F (x−)}) Simplicial depth(Liu 1990): SD(F, x) = PF (x ∈ s[X1 , . . . , Xp+1 ]) (when p = 1, SD(F,x)=2F(x)(1-F(x-)).) Majority depth(Singh 1991): Mj D(F, x) = PF {(X1 , . . . , Xp ) : x is in major side} [For a given sample X1 , . . . , Xp from F , let H(X1 , . . . , Xp ) denote the hyperplane containing these points. The hyperplane divides Rp into two half spaces, we say 5

a point x is in the major side if x falls inside the half space that has probability greater than or equal to 12 .] (when p = 1, Mj D(F, x) =

1 2

+ min{F (x), 1 − F (x−)}.)

Projection depth(Zuo 2001): P D(F, x) = [1 + Op (F, x)]−1 where Op (F, x) = sup||u||=1

||u0 x−µFu || , σFu

Fu denote the distribution of u0 X

Spatial Rank Depth(Yonghong Gao, 2003):

P D(F, x) = 1 − ||E(S(x − Y ))||2 where Y ∼ F and ½ S(x) =

x , ||x||

x 6= 0, 0, x ≤ 0.

THE DEFINITIONS OF SAMPLE DATA DEPTH (F UNKNOWN) Mahalanobis depth(Mahalanobis 1936): ¯ 0 S −1 (x − X)] ¯ −1 Mh D(Fm , x) = [1 + (x − X) Tukey’s depth(Tukey 1975): T D(Fm , x) = inf {Fm (H) : H is a closed half space containing x} H

Simplicial depth(Liu 1990): µ ¶−1 X ∗ m SD(Fm , x) = I(x ∈ s[Xi1 , . . . , Xip+1 ]) p+1 where summation * is taken over all possible subsets of {X1 , . . . , Xm } of size p + 1. Majority depth(Singh 1991): Mj D(Fm , x) = PFm {(X1 , . . . , Xp ) : x is in major side} Projection depth(Zuo 2001): P D(Fm , x) = [1 + Op (Fm , x)]−1 6

||u0 x−µFmu || , σFmu u0 X1 , · · · , u0 Xm .

where Op (Fm , x) = sup||u||=1 standard deviation of

µFmu , σFmu denote the mean (or dedian) and

Spatial Rank Depth(Yonghong Gao, 2003): n

1X P Dn (F, x) = 1 − || S(x − Xi )||2 n i=1 .

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A New Nonparametric Control Chart for Preliminary Analysis

In this section we propose a preliminary control chart based on data depth, which to the best of our knowledge has not appeared in the literature before and is used in detecting the shifts in either in the mean vector, the covariance matrix, or both of the process. Suppose we have n independent observations from a multivariate distribution of dimensionality p, i.e., xi ∼ Fp (µi , Σi ), i = 1, · · · , n. If the process is in control then µi = µ and Σi = Σ for all i. Assume that a step shift in the mean or variance or both occurs after n1 th observations, i.e. the mean and variance of the first n1 observations is (µ, Σ), and the last n2 = n − n1 observations have the same mean and variance (µ1 , Σ1 ). When the dimensionality p = 1 , a straightforward nonparametric test to detect a mean change would be to use the Mann-Whitney two-sample test and the Wilcoxon rank-sum test. The exact distributions of these statistics for different n1 ,n2 are tabulated, and the asymptotics are known (for example, Lehmann 1975 and Hettmansperger 1984). In the high dimension (p > 1), Liu (1993) proposed a test can be viewed as a multivariate extension of the Wilcoxon rank-sum test. Even though the asymptotic was also known (Liu 1993) and had been used in control charts ( Q chart) (Liu 1995),the Q chart can detect only large sample. We will propose a test used in control chart here. When the dimensionality p = 1, our proposed test is the same as Mann-Whitney two-sample test. As the 7

high dimension (p > 1), our proposed test statistic’s asymptotic is also the same as Liu’s. Moreover,the new control chart can not only detect large sample but also detect small sample. Consider a continuous p-variate quality vector whose distribution is F when the process is in control. Assume that an assignable cause occurs, then any resulting change in the process will be reflected by a location change and/or a scale increase and characterized as a departure from F to a continuous out-of-control distribution G, and the departure will be reflected by a decrease in the data depth. we define the statistic: n X

Q(n1 ) =

Rn1 (j)

j=n1 +1

where Rn1 (j) =#{xi |DFn1 +1 (xi ) < DFn1 +1 (xj ), i = 1, 2 · · · · · · n1 } 1 + #{xi |DFn1 +1 (xi ) = DFn1 +1 (xj ), i = 1, 2 · · · · · · n1 } 2 The standardized statistic SQ(n1 ) is defined by SQ(n1 ) =

Q(n1 ) − E(SQ(n1 )) p . Var(SQ(n1 ))

(1)

it’s easy to get the following formula: Q(n1 ) − q

n1 (n−n1 ) 2

n1 (n−n1 )(n+1) 12

D

−→N (0, 1),

as n1 → ∞, n − n1 → ∞.

(2)

We recommend to use the asymptotic expectation and variance in practice for simplicity. So, in the rest of this paper, the standardized likelihood ratio is defined by Q(n1 ) − SQ(n1 ) = q

n1 (n−n1 ) 2

n1 (n−n1 )(n+1) 12

.

(3)

The proposed CUSUM-type chart based on data depth is defined by Si = max{0, Si−1 − SQi − k}, 8

(4)

where the initial value S0 = 0 and k is the reference value. Our proposed CUSUM chart is constructed by plotting the statistics Si versus i. If a Si exceeds the given decision interval hα,n .an out of control signal is triggered. The reference value k is set equal to 2.0. Certainly one could use other different values. In general, smaller k lead to quicker detection of smaller shifts and in this paper we choose this value based on our simulation results which show that the control chart with k = 2.0 has good performance. For given false alarm probability (FAP) 0.05 and dimensionality p, let hn,p denote the decision interval of the CUSUM chart. For given various combinations of p and n, the simulated hn,p of CUSUM are shown in Table 2.

n 30 40 50 60 70 80 90 100 110 120

p=1 32.14 42.26 55.11 69.06 78.90 92.03 101.8 110.6 119.3 134.6

Table p=2 30.50 41.99 51.56 64.68 75.62 85.46 95.31 108.4 116.0 129.7

2 Simulated hn,p of CUSUM p=3 p=4 p=5 p=6 29.14 26.54 25.24 23.53 38.43 37.61 35.97 35.15 48.82 46.64 45.54 46.09 63.59 60.85 60.31 57.57 72.34 68.78 67.42 66.60 83.82 80.00 79.45 77.81 92.98 91.48 90.93 88.75 103.2 100.7 98.59 98.59 115.5 113.9 112.8 111.7 127.5 121.5 120.4 119.9

p=7 22.98 34.19 45.00 57.03 65.23 76.99 86.28 94.21 107.8 119.3

p=8 21.62 32.96 42.81 55.39 64.00 76.99 86.01 92.03 108.4 118.8

p=9 20.11 32.01 41.71 53.75 63.04 75.62 84.92 93.12 107.3 118.4

p = 10 17.92 31.05 40.62 54.29 64.14 75.35 86.01 93.94 107.8 116.0

From Table 2 we observed that hn,p increases as n increases and p decreases. As pointed out in Sullivan and Woodall (1996), it is not important for the preliminary application to find exact control limits that correspond to a specific FAP. For simplicity, we take the following line to approximate the decision interval hn,p of CUSUM: hn,p = 1.0936n − 1.4746p.

(5)

The simulated results show that the approximated decision interval in equation (7) performs very well, and it can be used for the practical application of the preliminary analysis. In the preliminary analysis, it’s necessary to give an estimation for the position of shift if the process parameters has been shifted. For our proposed CUSUM chart, 9

the estimation of the position of shift is given by τb = arg max{|SQt |},

(6)

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