DDMA-charts: Nonparametric multivariate moving average control ...

Allgemeines Statistisches Archiv 88, 235–258 c Physica-Verlag 2004, ISSN 0002-6018

DDMA-charts: Nonparametric multivariate moving average control charts based on data depth By Regina Y. Liu, Kesar Singh, and Julie H. Teng∗ Summary: This paper studies the DDMA-chart, a data depth based moving-average control chart for monitoring multivariate data. This chart is nonparametric and it can detect simultaneously location and scale changes in the process. It improves upon the existing rand Q-chart in the efficiency of detecting location changes. Both theoretical justifications and simulation studies are provided. Comparisons with some existing multivariate control charts via simulation results are also provided. Some applications of the DDMA-chart to the analysis of airline performance data (collected by the FAA) are demonstrated. The results indicate that the DDMA-chart is an effective nonparametric multivariate control chart. Keywords: DDMA-chart, MA-chart, Hotelling-T 2 , data depth, r-chart, Q-chart. JEL C10, C14.

1. Introduction Product quality is generally determined by a set of interrelated quality characteristics. As such, it is naturally multivariate, and best studied using tools of multivariate analysis rather than separate univariate analyses. The recent advances in computer technology have also helped facilitate the collection of massive multivariate data in many industries. The demand of effective multivariate analyses has never been greater, especially for quality control in industry. The goal of this paper is to propose a set of new nonparametric moving average (MA)-chart for monitoring multivariate data. Since these MA-charts are derived from the notion of data depth, they shall be referred to as DDMA-charts. Theoretical and simulation studies of DDMA-charts will be carried out. Comparisons of DDMA-charts with some existing multivariate control charts will be provided. Furthermore, the DDMA-charts will be demonstrated in an aviation safety analysis of some airline performance data collected by the FAA from 1993 to 1998. When the measurement X of a particular quality characteristic is used to monitor the quality of the product, the commonly used charts are the X¯ chart, the X-chart (or Shewhart chart), the moving average (MA)-chart, the exponentially weighted moving average (EWMA)-chart and the cumulative sum (CUSUM)-chart. For univariate data, these charts are easy to construct, visualize, and interpret. Most important, they have been proven to Received: 10.10.03/ Revised: 09.02.04 ∗ Research supported in part by grants from the National Science Foundation, the National Security Agency, and the Federal Aviation Administration. The discussion on aviation safety in this paper reflects the views of the authors, who are solely responsible for the accuracy of the analysis results presented herein, and does not necessarily reflect the official view or policy of the FAA. The dataset used in this paper has been partially masked in order to protect confidentiality.

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be effective in detecting possible process mean changes. When the measurements are multivariate, it is insufficient to simply monitor simultaneously all individual marginal variables by their univariate control charts: First, individual univariate charts do not account for the correlation among the variates; and second, there are no approaches for combining the individual charts to achieve the exact overall rate of false alarm, even with the approximations based on Bonferroni’s inequality. Thus, generalizations of the aforementioned univariate charts to the multivariate setting are required. There is an extensive literature on multivariate control charts, see for example some of the recent surveys: Alt and Smith (1988), Sparks (1992), Wierda (1994), and Fuchs and Kennett (1998). These surveys are quite thorough, although they are generally confined to normal data and focus mostly on approaches which are variations of the Hotelling-T 2 method (Hotelling, 1947). Clearly, the assumption of normal data is not always realistic, and nonparametric approaches for both univariate and multivariate data are often needed in practice. There exist relatively few nonparametric approaches in quality control. Examples include the bootstrap approach in Liu and Tang (1996) for the univariate setting, and the data depth approach in Liu and Singh (1993) and Liu (1995), and the anti-rank approach in Qiu and Hawkins (2001) for the multivariate setting. The DDMA-chart in this paper and the r- and Q-charts studied in Liu and Singh (1993) and Liu (1995) are all nonparametric multivariate control charts derived from the notion of data depth, and they can all detect simultaneously process changes in location and scale. The DDMA-chart improves on the other two charts by enhancing the ability to detect location change while retaining the same ability to detect scale change. Due to its nonparametric nature, the DDMA-chart has much broader applicability than the traditional control charts derived from the Hotelling-T 2 statistic. For example, in Section 3 the DDMA-chart is shown to be quite effective in detecting changes in Cauchy distributions where Hotelling-T 2 fails completely. As for the possible loss of efficiency in case the underlying distribution is specified, our average run length (ARL) study in Section 3 suggests that such a loss due to the use of the DDMA-chart is moderate. The rest of the paper is organized as follows. The first part of Section 2 gives a brief review of the notion of data depth, the depth induced multivariate ranking and its general usefulness in nonparametric multivariate statistics, r-charts, and Q-charts. The second part of Section 2, in Subsection 2.4, focuses on the proposed DDMA-chart and its properties. Section 3 provides simulation comparisons of DDMA-charts with some known multivariate control charts, including the standard Hotelling-T 2 , and the r- and Q-charts. Some simulation examples clearly demonstrate the improved efficiency of the DDMA-charts over the Q-charts. It contains some analysis results as well as an application of the DDMA-charts to the monitoring of some airline performance data. Some concluding remarks are provided in Section 4.

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2. Data depth and nonparametric multivariate control charts We begin by describing the basic setting and the notations needed for constructing multivariate control charts. Assume that each observation consists of k (k ≥ 1) quality measurements. The process is considered in control if the observations follow a prescribed quality distribution (required by clients or designing engineers). Let G denote the prescribed k-dimensional distribution, and let {Y1 , . . . , Ym } be m random observations from G, denoted by {Y1 , . . . , Ym } ∼ G. Each sample point is viewed as a 1 × k row vector. The sample {Y1 , . . . , Ym } is often referred to as a reference sample. Let {X1 , X2 , . . . } be the new observations to be monitored. Based on the observations Xi ’s, we need to determine whether the quality of the product has deteriorated, i.e., the Xi ’s are not meeting the prescribed G in certain ways. Assume that the Xi ’s follow a distribution F . Our task is to test if there exist differences between F and G. The differences here are often summarized as a mean difference or/and a scale difference. The task then amounts to testing the hypotheses {H0 : µF = µG versus Ha : µF 6= µG } or/and {H0 : Σ F = ΣG versus Ha : ΣF 6= ΣG }. Here µ∗ and Σ∗ denote respectively the mean vector and the covariance matrix of the distribution ∗. For a given false alarm rate α, the above test gives rise to the critical values which correspond to the upper control limit (UCL) and the lower control limit (LCL) of the control chart. The expected value of the test statistic corresponds to the center line (CL). The region outside the area between UCL and LCL is considered the out-of-control region. General discussions of control charts can be found in textbooks such as Montgomery (1998). For the reference sample {Y1 , . . . , Ym } ∼ G, and the new samples {X1 , X2 , . . . } ∼ F , the monitoring process amounts to comparing F with G. In this paper, we characterize the difference between G and F using the notion of data depth. The word ‘depth’ was first used in Tukey (1975) for picturing data, and the far reaching ramifications of depth in ordering and analyzing multivariate data are elaborated in Liu (1990), Liu et al. (1999), Mosler (2002) and others. There exist several notions of data depth, see the list in the Appendix. For convenience, the simplicial depth will be used for the demonstrations throughout the paper. However, the depth approaches described in this paper for constructing multivariate control charts are applicable to all data depths listed in the Appendix, though their performances may differ due to different distributional settings. 2.1. Data depth and ordering multivariate observations. A data depth is a measure of how deep or how central a given point is with respect to a multivariate distribution. It gives rise to a new set of parameters which quantify easily the many complex multivariate features of the underlying distribution, including location, quantiles, scale, skewness and kurtosis. It also gives rise to a center outward ordering of any set of sample points. This ordering provides a new nonparametric multivariate inference scheme (cf. Liu et al., 1999), which includes applications to multivariate statistical

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quality control. Before we show how this ordering can be used to construct multivariate control charts, we first use the simplicial depth proposed in Liu (1990) as an example to describe the general concept of data depth and its corresponding center-outward ordering. We begin with the bivariate setting, k = 2. Let 4(a, b, c) denote the triangle with vertices a, b and c. Let I(·) be the indicator function, i.e. I(A) = 1 if A occurs and I(A) = 0 otherwise. Given the sample {Y1 , . . . , Ym }, the sample simplicial depth of y is defined as −1 X m (1) I y ∈ 4(Yi1 , Yi2 , Yi3 ) , DGm (y) = 3 (∗)

which is the fraction of the triangles generated from the sample that contain the point y. Here (*) runs over all possible triplets of {Y1 , . . . , Ym }. A large value of DGm (y) indicates that y is contained in many triangles generated from the sample, and thus it is ‘relatively deep and central’ within the data cloud. On the other hand, a small DGm (y) indicates a relatively outlying position of y. Thus DGm (y) is a measure of ‘depth’ (or ‘centrality’) of y w.r.t. the data cloud {Y1 , . . . , Ym }. The above simplicial depth can be generalized to any dimension k as follows: −1 X m (2) I(y ∈ s[Yi1 , . . . , Yik+1 ]) , DGm (y) = k+1 (∗)

where (*) runs over all possible subsets of {Y1 , . . . , Ym } of size (k + 1). Here s[Yi1 , . . . , Yik+1 ] is the closed simplex whose vertices are {Yi1 , . . . , Yik+1 }, i.e., the smallest convex set determined by {Yi1 , . . . , Yik+1 }. When the distribution G is known, then the simplicial depth of y w.r.t. to G is defined as DG (y) = PG {y ∈ s[Y1 , . . . , Yk+1 ]} ,

(3)

where Y1 , . . . , Yk+1 are (k + 1) random observations from G. DG (y) measures how ‘deep’ y is w.r.t. G. A fuller motivation together with the basic properties of DG (·) can be found in Liu (1990). In particular, it is shown that DG (·) is affine invariant, and that DGm (·) converges uniformly and strongly to DG (·). The affine invariance ensures that our proposed control charts are coordinate free, and the convergence of DGm to DG allows us to approximate DG (·) by DGm (·) when G is unknown. There are many utilities of a data depth measure in multivariate statistical analysis (see, e.g., Liu et al., 1999). In particular, it provides a center-outward ordering of the data points within a given sample. Specifically, for the given sample {Y1 , Y2 , . . . , Ym } we calculate all the depth values DGm (Yi )’s and then order the Yi ’s according to their ascending depth value. Denoting by Y[j] the sample point associated with the jth smallest depth value, we obtain the sequence {Y[1] , Y[2] , . . . , Y[m] } which is the depth order

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statistics of Yi ’s, with Y[m] being the deepest or the most central point, and Y[1] the most outlying point. The implication is that a smaller rank is associated with a more outlying position w.r.t. the underlying distribution G. Note that the order statistics induced by depth are different from the usual order statistics in the univariate case, since the latter are ordered from the smallest sample point to the largest, while the former starts from the middle sample point and moves outwards in all directions. When the distribution G is known, DG (y) leads to an ordering of all points in IRk from the deepest point outward. The deepest point here is the maximizer of DG (·) (or the average of the maximizers if there are more than one), which is denoted by µ∗ . Clearly, µ∗ can be viewed as a location parameter of the distribution G, and it coincides with the mean µG if G is symmetric. The main idea behind data-depth control charts is to reduce each multivariate observation to its center-outward rank induced by a data depth. Once each multivariate observation is represented by its corresponding depth rank, we can construct control charts based on these ranks following the same principles as those for the univariate control charts. The geometric nature of the notion of data depth makes it easy to interpret the values of statistics derived from those ranks, and to visualize their plots. This approach is completely nonparametric, and thus the resulting charts are valid without parametric assumptions on the process model. Moreover, these charts can detect simultaneously the location change and the scale increase in a process as shown in Liu and Singh (1993). We now list some statistics derived from data depth which are suitable for plotting in control charts. Let rG (y) = P {DG (Y ) < DG (y)|Y ∼ G}

(4)

rGm (y) = #{Yj |DGm (Yj ) < DGm (y), j = 1, . . . , m}/m .

(5)

and

Under the distribution G, rG (y) gives the probability under G of points in the space IRk that are more outlying than the given point y. It indicates the position of y relative to all other points in terms of centering toward the deepest point of G, and is thus viewed as the relative (center-outward) rank of y. Similarly, rGm (y) gives the fraction of the sample points in the reference sample that are more outlying than y, and it provides the relative center-outward rank of y w.r.t. {Y1 , . . . , Ym }. Intuitively, an extremely small relative rank of y indicates an extremely outlying position of y, and hence suggests that y is unlikely to have come from the same distribution that produces the given Yi ’s. This is the key idea that motivates all the control charts derived from data depth. Representing each multivariate observation X by its corresponding relative rank rGm (X), we can construct control charts based on these ranks ¯ MA-, following the same procedures as those for the univariate X-, X-, EWMA- and CUSUM-charts as follows.

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2.2. r-charts. The idea of the r-chart is similar to that of the univariate X-chart, except that it monitors the relative ranks of {X1 , X2 , . . . } instead of monitoring directly the values of Xi ’s. More specifically, it plots {rG (X1 ), rG (X2 ), . . . } (or {rGm (X1 ), rGm (X2 ), . . . }, if {Y1 , . . . , Ym } are available but not G) against time i, with the center line CL = 0.5 and the lower control limit LCL = α, the preset false alarm rate. The process is declared out-of-control if rG (·) falls below α. Note that the UCL is not needed in the r-chart, since a large rGm (X) value indicates a relatively central position of X with respect to the reference sample Yi ’s, and it gives no reason to doubt that X has come from the distribution G. On the other hand, a small rGm (X) indicates a relatively outlying position of X with respect to the reference sample, and it suggests that X is unlikely to have come from G and that there may have been a process change. Note that the chart with most rGm (X)’s attaining values very close to 1 would indicate a possible reduction in dispersion in the distribution F . Although technically this indicates a process change in scale, in reality this would imply a gain in precision which is viewed as a process improvement. For this reason, we do not consider a process with reduced dispersion as out-of-control, although detection of such a change can be easily incorporated into our charts. If X ∼ F , a small value of rGm (X) suggests a possible deviation from G to F . Since rGm (·) is defined according to data depth, this possible deviation can be a shift in ‘center’ and/or an increase in scale. A detailed mathematical justification of this interpretation can be found in Section 3 of Liu and Singh (1993). The r-chart with LCL = α corresponds to an α-level test of the following hypotheses H0 : F = G v. s. Ha : there is a location shift and/or a scale increase from G to F

(6)

Note that the alternative hypothesis is particularly suitable for detecting quality deterioration in the context of quality control, since it presents a loss of accuracy and/or a loss of precision. This also justifies viewing the process as out-of-control when H0 is rejected or, equivalently, when an observation falls below α in the r-chart. The justification for choosing CL = 0.5 and LCL = α in the r-chart comes from the following proposition which was established in Liu and Singh (1993). Proposition 1 Assume that F = G and X ∼ F . Let U [0, 1] denote the uniform distribution on the unit interval [0, 1], and let the notation → L stand for the convergence in law. If DG (X) has a continuous distribution, then (1) rG (X) ∼ U [0, 1], and (2) as m → ∞, rGm (X) →L U [0, 1] along almost all {Y1 , . . . , Ym } sequences, provided that DGm (·) converges to DG (·) uniformly as m → ∞.

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The uniform convergence of DGm (·) holds for the simplicial depth if G is absolutely continuous, which is our assumption. Under H0 : F = G, the uniform distribution in Proposition 1 justifies setting CL = 0.5 and LCL = α in the r-chart. Rejecting H0 suggests a possible quality deterioration spelled in Ha , a case in which the process should be considered out-of-control. Remark 1 Even though the r-chart does not have the UCL to make its CL the true center line of the in-control region, the CL here serves as a reference point for us to see if patterns or trends are forming in a sequence of observations. Remark 2 If we are not concerned with the possibility of a location change, the r-chart can be modified to detect only the scale change. We first remove any possible location change by centering all data on the same location, i.e. subtracting the deepest point Y[m] (or µ∗ if it is known) from all data. ˜1, X ˜ 2 , . . . }. The r-chart Denote these centered data by {Y˜1 , . . . , Y˜m } and {X constructed from these centered data, with CL = 0.5, LCL = α/2, and UCL = 1 − α/2, amounts to testing H0 : F = G v. s. Ha : there is an expansion or contraction in scale from G to F. 2.3. Q-charts. The idea behind the Q-chart is similar to that behind the ¯ univariate X-chart. It plots the subgroup averages of rG (Xi )’s (or rGm (Xi )’s). Assume that subgroup size is q. The averages of the rG (Xi )’s and rGm (Xi )’s are denoted respectively by Q(G, Fqj ) and Q(Gm , Fqj ), where Fqj is the empirical distribution of the Xi ’s in the j-th subgroup, j = 1, 2, . . . . The Q-chart plots {Q(G, Fq1 ), Q(G, Fq2 ), . . . } (or {Q(Gm , Fq1 ), Q(Gm , Fq2 ), . . . }, if only Y1 , . . . , Ym are available). For a given Fq , q

Q(G, Fq ) =

1X rG (Xi ) , q i=1

(7)

and q

Q(Gm , Fq ) =

1X rG (Xi ) . q i=1 m

(8)

Note that both (7) and (8) are sample approximations of Q(G, F ) below Q(G, F ) = P {DG (Y ) ≤ DG (X)|Y ∼ G, X ∼ F }(= EF [rG (X)]) .

(9)

The Q-chart again has CL = 0.5. Its LCL depends on q. Specifically, the following are given in Liu (1995). If q is large, then LCL is approximately q

(0.5 − zα (12q)−1/2 ) for plotting {Q(G, Fqj )}’s, and 0.5 − zα

1 1 12 ( m

+ 1q )

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for plotting {Q(Gm , Fqj )}’s. If q is relatively small and α ≤ 1/q!, then LCL is exactly (q!α)1/q /q. Note that this LCL can be a reasonable approximation when α is slightly over 1/q!. Remark 3 Similar to Remark 2, if the task is to detect scale changes only, then the Q-chart should be constructed from the centered data. For the reference sample {Y1 , . . . , Ym } ∼ G and the new sample {X1 , X2 , . . . } ∼ F , Liu and Singh (1993) defined the quality index Q(G, F ) and used this to quantify the difference between the distributions G and F . Specifically, Q(G, F ) = P {DG (Y ) ≤ DG (X)|Y ∼ G, X ∼ F }(= EF rG (X)), with rG (y) = P {DG (Y ) < DG (y)|Y ∼ G}. It was established in Liu and Singh (1993) that if F differs from G in only a location shift such that F (x) = G(x − θ) for a fixed constant vector θ, then Q(G, F ) is maximized when θ = 0, provided that F is unimodal and elliptically distributed. Therefore, if Q(θ) ≡ Q(G, G(· − θ)) is sufficiently smooth, the gradient vector of Q(θ) at θ = 0 is 0. Consequently, Q(θ) − Q(0 0) = O(k θ k2 ) . The quadratic order here indicates that the control chart based on Q, the Q-chart, can be inefficient in detecting a minor location shift. The DDMAchart in the next section is devised to overcome this shortcoming. 2.4. DDMA-charts. Assume that Y = {Y1 , . . . , Ym } is the reference sample from distribution G and that X = {X1 , . . . , Xn } are new observations. We construct a DDMA-chart to monitor the moving averages with ˜ q = (X1 +. . .+Xq )/q, X ˜ q+1 = (X2 +. . .+Xq+1 )/q, . . . , X ˜n = length q, i.e., X ˜ ˜ ˜ (Xn−q+1 +. . .+Xn )/q. Let X = {Xq , . . . , Xn }. Then the corresponding ref˜i ∈ X ˜ is Y ˜ = {Y˜q , . . . , Y˜m } such that Y˜q = erence sample for monitoring X (Y1 +. . .+Yq )/q, Y˜q+1 = (Y2 +. . .+Yq+1 )/q, . . . , Y˜m = (Ym−q+1 +. . .+Ym )/q. ˜ i ∈ X, ˜ we calculate its relative rank w.r.t. {Y˜q , . . . , Y˜m }, i.e., For each X ˜i) = rG˜ m−q+1 (X

#{Y˜j |DG˜ m−q+1 (Yj ) < DG˜ m−q+1 (Xi ), j = q, . . . , m} m−q+1

. (10)

˜ m−q+1 is the empirical distribution of Y, ˜ and D ˜ Here G Gm−q+1 (·) is the ˜ empirical depth computed w.r.t. Gm−q+1 . The DDMA-chart is simply the plot of rG˜ m−q+1 (Xi ) along the indices i = q, . . . , n. Similar to the case of r-chart, the DDMA-chart has the centerline = 0.5 and LC = α. This again corresponds to an α-level test of the hypotheses in (6), treating that Xi ’s are drawn from F . The reason why the DDMAchart, in comparison to the Q-chart, is more sensitive to minor location changes and yet retains the same ability in detecting a scale change is given as follows:

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Consider the simplest case of moving averages of length 2. In this case, the underlying distributions of the reference sample and that of the moving averages are respectively G ∗ G and F ∗ F , where ∗ stands for convolution. If F and G differ only due to a location shift, so that F (·) = G(· − θ), then the location shift becomes doubled after the convolution, i.e. F ∗ F (x) = G ∗ G(x − 2θ) .

(11)

The dispersion matrix for F ∗F and G∗G is 2Σ, where Σ is the common dispersion matrix for F and G. In view of the affine-invariance property of Q, we note that Q remains the same if the populations of both reference sample and new observed sample undergo the same linear√transformation.√For charting moving averages of length 2, we regard F ∗ F ( 2·) and G ∗ G( 2·) respectively as the reference and new populations. In the √ case of a location √ shift, i.e. F (·) = G(· −√θ), the two distributions F ∗ F ( 2·) and G ∗ G( 2·) differ in location by 2θ but they retain the same dispersion √matrix Σ. Consequently, the DDMA-chart will display a location shift of 2θ which is larger than the location shift θ that occurred to the original populations. In other words, in comparison to the r- or Q-chart, the DDMA-chart accentuates the effect of location shifts. This point can be illustrated more clearly in the following case of normal distributions. Assume that Yi ∼ N (µ, Σ), and Xi ∼ N (µ + δ, Σ) for δ 6= 0, i = 1, 2. Consider the moving averages of length 2 for Yi , Xi , i = 1, 2. We have Y ≡ (Y1 + Y2 )/2 ∼ N (µ, Σ/2), X ≡ (X1 + X2 )/2√∼ N (µ the same transformation of Y √+ δ, Σ/2). Consider √ √ 2Y − 2µ+µ) and ( 2X − 2µ+µ), respectively. and X, namely, ( √ √ √ √ √ Notice that ( 2Y − 2µ+µ) ∼ N (µ, Σ) and ( 2X − 2µ+µ) ∼ N (µ+ 2δ, Σ). Both random variables retain the original dispersion matrix Σ. However, the location shift √ of δ from N (µ, Σ) to N (µ + δ, Σ) in the original data is now changed to 2δ in the moving averages Y and X. The above argument generalizes to moving averages of any length q (> 1) √ with DDMA-chart showing the effective location shift of qθ, but keeping the same original dispersion matrix Σ. This phenomenon is not shared by the Q-chart, since the linear transformation induced by the moving average does not affect Q due to its affine invariance property. In summary, if the DDMA-chart ‘dips down’ sufficiently to signal the out-of-control state, it may suggest either a location shift and/or a scale increase in the underlying distribution. If the trend of ‘dipping down’ is further accentuated as q increases, then there is strong indication of a location shift. (This phenomenon is clearly exhibited in the simulation results shown in Figures 1, 5, 7, and 9. Detailed discussions on these figures are given in Section 3.) On the other hand, if the trend is not accentuated as q increases, then there is only a scale change between F and G, as seen in Figure 3. This observation implies that the DDMA-chart improves upon the Q-chart in terms of the efficiency in detecting location change while retains the same efficiency in detecting the scale change as the Q-chart. The distribution of a moving average is a normalized convolution of its original distribution. The validity of both the r- and Q-charts requires the

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monotonicity of the underlying distribution. For the validity of our DDMAchart, we also need the similar monotonicity of the convoluted distribution. Theorem 1 below shows that the convoluted distributions inherit the ‘unimodal elliptical’ characteristics from their original distributions. Consequently, both the relative rank rG∗G (·) and the quality index Q(G∗G, F ∗F ) derived from the convoluted distributions retain all the desirable properties of the original rG (·) and Q(G, F ), as proven in Liu and Singh (1993). Theorem 1 (Unimodality of the convolution of elliptical distributions) Let {X1 , X2 , . . . , Xn } be an i.i.d. sample from the distribution F which has its density f of the form f (x) = c · h((x − θ)Σ −1 (x − θ)0 ), where c is a constant, θ a fixed k × 1 vector and Σ a k × k nonsingular matrix, and h(·) a decreasing continuous functions from IR+ to IR+ . Then Pn the distribution of the sum T ≡ i=1 Xi also belongs to the same family of distributions with the density fn (t) = cn · hn ((t − nθ)(nΣ)

−1

(t − nθ)0 ),

(12)

where hn (·) is a decreasing continuous function. Proof We provide here only the proof of the claimed monotonicity of hn (·), and omit the rest. Without loss of generality, we assume that θ = 0 and Σ=I. Step 1: For a fixed x in IRk , we may rewrite f (x) as f (x) = fm (x) + Rm (x), Pm−1 where: i)√fm (x) = i=1 aim ICim (x) with Cim denoting the sphere with radius i/ m and its center at the origin; ii) aim = fim − f(i+1)m with fim = √ f (y) for any y such that kyk = i/ m. Then, we show that Rm (x) → 0 as √ m → ∞. For any x such that kxk < m, we observe that fm (x) is a telescopic sum which reduces to (fi0 m − fmm ) with i0 = maxi {i : kxk ≥ √ i/ m}. Clearly, fi0 m → f (x) and fmm → 0 as m → ∞. Step 2: For given x and y, if kxk ≥ kyk, then the following holds: Z

fm (x − z)f (z)dz ≤

Z

fm (y − z)f (z)dz.

To prove this, we first express the above as m−1 X i=1

aim

Z

ICim (x − z)f (z)dz ≤

m−1 X i=1

aim

Z

ICim (y − z)f (z)dz.

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Applying Anderson’s Lemma (Anderson, 1955), we obtain Z R ICim (x − z)f (z)dz = ICim (z + x)f (z)dz R ≤ ICim (z + y)f (z)dz R = ICim (y − z)f (z)dz.

Thus, the claim follows. The proof of the monotonicity for F ∗ F can then be concluded using the bounded convergence theorem. To deduce the result for the n-fold convolution, we simply replace f (x) by fn−1 (x) in Step 1. 3. Simulation and comparison studies

3.1. Simulated charts. Our simulation study considers five settings: a) A bivariate standard normal distribution with a mean shift from 00 to 1 1 ; b) a bivariate standard normal distribution with the covariance matrix enlarged to 4I from I; c) a bivariate componentwise-independent exponential distribution with a ; mean shift from 22 to 2.5 2.5 d) a bivariate componentwise-independent double exponential distribution with a mean shift from 00 to 11 ; and e) a bivariate Cauchy distribution with the center being changed from 00 to 44 .

In all cases, the reference sample size is 1 000. Among the 40 new sample observations, the first half are drawn from the original distribution while the second from the distribution with the specified change. We set α = 0.05 throughout the section. The DDMA-charts in Figures 1, 3, 5, 7, and 9 correspond to the settings a) to e). Each figure contains 6 DDMA-charts with q ranging from 1 to 6. Note that the DDMA-chart with q = 1 is in fact the r-chart. All DDMA-charts detect the process change quite quickly. Moreover, in the case of a location change in Figures 1, 5, 7, and 9, the ‘dipping down’ pattern becomes more pronounced after the 20th observation as q increases. Figures 2, 4, 6, and 8 are respectively the Q-charts corresponding to Figures 1, 3, 5, and 7. They are presented here to show that the DDMAchart can be more effective than the Q-chart in detecting a location shift in the process, and they are equally effective in detecting a scale change in the process. For example, the DDMA-chart with q = 6 in Figure 1 signals out-of-control starting at the 24th observation, soon after the change took place at the 21st observation. It tenaciously clings to the out-of-control signal throughout, despite of the fact that the last observation appears to be relatively central in comparison with the reference sample, as suggested by its spiked value in the DDMA-chart with q = 1. Note that the corresponding Q-charts are slightly late in detecting the change, e.g. at the 28th point in

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the case of q = 6, and eventually turn upward and retreat from the out-ofcontrol region. Visually, the Q-charts show a clear overall downward pattern to suggest a possible process change, although they are not as effective as their DDMA counterparts. This superiority of DDMA-charts over Q-charts is even more evident in the exponential and double exponential case, Figures 5–8. In these two cases, it is worth noting that the raw datasets presented in the r-charts are hardly indicative of any process changes, and yet the DDMA-charts do prevail as q increases to successfully show strong evidence of process changes. The improved efficiency in the DDMA-chart over the Q-chart in detecting location change discussed above does not hold if the process change is due to scale changes only. This is seen in Figures 3 and 4. There both charts are equally effective in detecting the scale change. Note that the exponential and the Cauchy distributions often present difficulties in standard process control schemes, since the former is asymmetric and the latter does not admit moments. The effectiveness of our DDMA-charts are shown clearly in Figures 5 and 9. This further demonstrates the strength of the nonparametric nature of the depth-based control charts. Next, we proceed to examine the performance of DDMA-charts in comparison to the traditional Hotelling-T 2 charts. Under the assumption of multivariate normality, the well known Hotelling-T 2 chart is designed specifically to detect the mean change. Specifically, for the reference sample {Y˜1 , . . . , Y˜m }, Hotelling-T 2 charts plot ˜ − Y¯˜ ) , ˜ ) = q(X ˜ − Y¯˜ )0 S−1 (X (13) T 2 (X i

i

Y˜

i

˜ Here Y¯˜ = P Y˜i /m and S ˜ = 1 P(Y˜i − Y¯˜ )0 ×(Y˜i − Y¯˜ ), which are ˜ i ∈ X. for X Y m the sample estimates of the underlying mean and covariance matrix, µ and Σ. Since the population version of (13) follows χ2k , a χ2 distribution with the degree of freedom of k which is the dimension of the data. The HotellingT 2 chart has the CL = Y¯˜ and LCL = χ2k,(1−α) , the (1 − α)-th quantile of the χ2k distribution. To provide the Hotelling-T 2 chart in the same scale for comparison with our DDMA-charts, we plot the p-value associated with ˜ i ). Note that in the case of mean change in the normal case a), the each T 2 (X DDMA-charts in Figure 1 appear to perform equally well to the HotellingT 2 charts. The DDMA-charts for the exponential and double exponential cases provide somewhat clearer indication of out-of-control pattern than their Hotelling-T 2 charts. In the Cauchy case, the DDMA-charts in Figure 9 clearly outperform the Hotelling-T 2 charts. The Hotelling-T 2 charts break down completely as the charts fail to detect any process changes. Worse still, the p-values in this case are consistently close to 1, giving no signs of possible location change at all. We omit all Hotelling-T 2 charts to save some space. 3.2. ARL study. The DDMA-chart is completely nonparametric, and is applicable without concerning the underlying process distribution. Obviously, if the process distribution is known to be normal, then the HotellingT 2 is the most suitable. In this case, there is a loss of efficiency if the DDMA

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chart is used instead. The loss of efficiency is generally presented in terms of the average run length (ARL) needed for the first detection of change. We listed in Table 1 the ARLs for the DDMA- and Hotelling-T 2 charts, both with the moving average length 4, in case of location shifts in normal distributions. Again, α = 0.05. The ARL is the average of 1 000 runs under each specified setting. For if the process distribution is normal and a example, mean shift from 00 to 11 has occurred, then it takes on the average respectively 5.079 and 4.839 samples for the DDMA-chart and Hotelling-T 2 chart to signal out-of-control. The difference between the two ARLs is relatively small in all cases, although it grows as the mean shift decreases. Overall, the ARLs are comparable for the DDMA-chart and the Hotelling-T 2 chart, unless the size of the shift is extremely small. ARL mean shift 0 → 1.00 0 1.00 0 → 0.050 0.050 0 0 → 0.025 0.025 0 0 → 0.010 0 0.010

Hotelling-T 2

DDMA 5.079

4.839

10.963

10.057

22.892

19.218

33.752

Table 1. ARL for DDMA- and Hotelling-T shifts.

28.307 2

charts: Normal case with location

3.3. Applications to airline performance data. We apply the rand DDMA-charts to the analysis of an airline performance dataset. The data were collected by the FAA. It consists of several monthly performance measures of ten airlines from July 1993 to May 1998. The performance measures, among others, include airworthiness surveillance and operation surveillance, and they are all correlated. An acceptable sample of 500 past data points are given, which we use here as the reference sample. We use both r- and DDMA-charts to monitor simultaneously the two performance measures of each air carrier. Figure 10 contains the r- and DDMA-charts for Airlines 1 and 4. α = 0.05. For the r-charts, only few points from Airline 1 fall below the LCL, but many from Airline 4 do. The DDMA-chart for Airline 4 clearly enhances the out-of-control situation and shows the moving averages consistently below the LCL. This indicates the clear nonconformity to the desired distribution, especially the location change. This change can be seen clearly in the scatter plot in Figure 11. The convex hull in Figure 11 spans the reference sample, and the intersection of the two axes indicates the deepest point determined by the simplicial depth. The reference sample points are shown by o’s and the data points of Carriers by dark diamondshaped points. It is obvious that the reference sample as well as the data for Carriers 1 and 4 do not follow normal distributions. The data of Carrier 4 appear to shift upper-right,suggesting a possible location change at least. Many data points of Carrier 4 are outside of the convex hull in the second, third and fourth quadrants. These points correspond to most points falling

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avg1 0.1878 avg2 0.0358

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average scheme is not used Figure 4. Q-charts: moving Normal case with a scale increase.


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moving average scheme applied to data Figure 3. DDMA-charts: Normal case with a scale increase. average q=3

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avg1 0.2489 avg2 0.2192

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avg1 0.5096 avg2 0.4385

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below LCL in the r-chart in Figure 10. The r- and DDMA-charts in Figure 10 clearly indicates that Carrier 1 does but Airline 4 does not perform close to what is expected, in view of the given reference sample. In Cheng et al. (2000), the r-chart with different level of false alarm rate is further applied to set thresholds for the purpose of rating airlines in view of multiple performance measures. The same idea of thresholding should apply by using the DDMA-chart to further distinguish the ratings of airlines separately in terms of their performance consistency (i.e. measuring the scale changes) and nonconformity to desired target performance (i.e. measuring location changes).

Figure 10. r-chart and DDMA-chart for Carriers 1 and 4.

4. Concluding remarks This paper introduces the DDMA-chart and shows its effectiveness as a nonparametric multivariate control chart. It is more robust than the traditional Hotelling-T 2 chart and is applicable to a broader range of distributions. The loss in ARL in comparison with the Hotelling-T 2 approach appears negligible when the underlying process is normal. Furthermore, the DDMA-chart greatly improves upon the Q-chart in detecting location change. For general practice, we recommend that both Q- and DDMA-charts be used side by side. If we see the same effect appear in the two sets of charts, we can safely conclude that the process change is due to scale change alone. If we observe in the DDMA-chart a more pronounced pattern signaling process changes,

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Figure 11. Bivariate scatter plots for Carriers 1 and 4 versus the reference sample.

as the plotted values either decrease more sharply or stay down below LCL more consistently, then we should conclude that there is a location change in the process, in addition to a scale change. We may also evaluate the DDMA-chart by comparing it to the CUSUMchart which continuously plots the sum of the depth ranks of incoming observations (cf. Liu, 1995). Systematic comparisons of the DDMA-, CUSUMas well as EWMA-charts are lengthy and shall be presented elsewhere. We present here only a brief heuristic argument why DDMA-charts should outperform CUSUM-charts for small location changes and thus have smaller ARLs. DDMA-charts take average before depth ranking, and thus bring the underlying distribution closer to normal and make the location shift more accentuated than the original distribution. Specifically, assume that the process has a location shift δ. Following the illustration in Section 2.4, the averaging of q observations in the DDMA-chart yields an expected shift √ in the order of O((δ q)2 )(= O(δ 2 q)). In contrast, the CUSUM-chart is essentially summing of U [0, 1] samples, which, after a proper normalization, √ yields an expected shift in the order of O(δ 2 q) which is smaller than the expected shift reflected in the DDMA-chart. Moreover, we note that the LCL for the CUSUM-chart in essence is derived from a normal distribution while that for the DDMA-chart is from the uniform distribution, U [0, 1]. The fact that U [0, 1] has shorter tails than normal distributions now explains the reason why the LCL of the DDMA-chart is likely to be reached faster, for the same amount of location shift in the process. Hence, DDMA-charts are more effective than CUSUM-charts. The DDMA-chart can be constructed using any notion of data depth listed in the Appendix. Different notions of depth have different strength in capturing different features of the distribution. If a specific feature is considered likely to be in the process distribution, then we should choose the depth notion which can capture best that feature. For example, if the underlying distribution is close to elliptical, then it is more efficient to use the Mahalanobis depth. Otherwise, the more geometric type of depth such as the simplicial depth, and the halfspace depth may be more desirable since they do not require moment conditions. Different conditions may be

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needed for the Proposition 1 to apply for a specific notion of data depth. For example, the uniform convergence of DGm (·) holds for the simplicial depth if G is absolutely continuous, and for the Mahalanobis depth if G has a finite second absolute moment. Details on some of these conditions for different depths can be found in Liu and Singh (1993). In some situations the process is designed to follow as assumed model, and no reference samples are available. In this case, computing the depth for a sample observation with respect to a given distribution, especially in high dimensional settings, can be cumbersome. Although, some existing computing software such as Mathematica can be applied, a more practical approach would be simply to generate a large reference sample by simulation from the given model. An interesting new direction for multivariate control chart is in the monitoring of dependent observations which occur often in most real applications. For example, it is unrealistic to assume that the monthly airline performance data collected by the FAA are independent. To this end, the so-called moving blocks bootstrap (MBB) studied in K¨ unsch (1989) and Liu and Singh (1992) may be a useful start. This approach has already been successfully applied to account for the dependence in univariate data in Liu and Tang (1996). Appendix Assume that {Y1 , . . . , Ym } is a random sample from G which is an absolutely continuous distribution in IRk , k ≥ 1. Let Gm denote the empirical distribution of the sample {Y1 , . . . , Ym }. Here is a list of well known notions of data depth. Except for (V), all of them and their corresponding depth ranking are affine invariant. Under affine transformations, the actual depth value changes by a constant in (V), but the induced depth ranking remain invariant. (I) The Mahalanobis Depth (Mh D) (Mahalanobis, 1936) at y w.r.t. G is defined to be −1 −1 Mh DG (y) = 1 + (y − µG )ΣG (y − µG )0 , where µG and ΣG are the mean vector and dispersion matrix of G respectively. The sample version of Mh D is obtained by replacing µG and ΣG with their sample estimates. (II) The Halfspace Depth (HD) (Hodges, 1955; Tukey, 1975) at y w.r.t. G is defined to be HDG (y) = inf {P (H) : H is a closed halfspace in IRk and y ∈ H} . H

The sample version of HDG (y) is HDGm (y). The halfspace depth is also referred to as the Tukey depth in the literature. (III) The Convex Hull Peeling Depth (CD) (Barnett, 1976) at the sample point Yi w.r.t. the dataset {Y1 , . . . , Ym } is simply the level of the convex

256


layer Yi belongs to. A convex layer is defined as follows. Construct the smallest convex hull which encloses all sample points {Y1 , . . . , Ym }. The sample points on the perimeter are designated the first convex layer, and removed. The convex hull of the remaining points is constructed; those points on the perimeter are the second convex layer. The process is repeated, and a sequence of nested convex layers are formed. The higher layer a point belongs to, the deeper the point is within the data cloud. Note that several other versions of ‘convex peeling’ exist (see Tukey, 1975; Eddy, 1982). Although only the simple convex hull peeling of Barnett (1976) is specified here, the depth control charts discussed in this paper apply to all variations of convex peeling. (IV) The Projection Depth (PD) (Stahel, 1981; Donoho, 1982) at the sample point Yi w.r.t. the dataset {Y1 , . . . , Ym } is " # |y 0 u − medj (Yj0 u)| . P DGm (y) = 1/ 1 + supkuk=1 medd |(Yd0 u) − medj (Yj0 u)| Its population version is obtained by replacing Gm by G. Zuo (2003) provides a more general account for projection based depth functions. (V) The Oja Depth (OD) (Oja, 1983) at y w.r.t. G is defined to be ODG (y) = [1 + EG {volume (S[y, Y1 , . . . , Yk ])}]

−1

,

where S[y, Y1 , . . . , Yk ] is the closed simplex with vertices y, and k random observations from G. The sample version is #−1 −1 " X m 1+ {volume (S[y, Yi1 , . . . , Yik ])} . ODGm (y) = k ? Here ? indicates all k-plets (i1 , . . . , ik ) such that 1 ≤ i1 ≤ . . . ≤ ik ≤ m. (VI) The Simplicial Depth (SD) (Liu, 1990) at y w.r.t. G is defined to be SDG (y) = PG {y ∈ S[Y1 , . . . , Yk+1 ]} . Here S[Y1 , . . . , Yk+1 ] is a closed simplex formed by (k + 1) random observations from G. The sample version of SDG (y) is obtained by replacing G by Gm , or alternatively, by computing the fraction of the sample random simplices containing the point y. In other words, −1 X m SDGm (y) = I y ∈ S[Yi1 , . . . , Yik+1 ] , k+1 ? where I(·) is the indicator function, and ? indicates all (k + 1)-plets (i1 , . . . , ik+1 ) such that 1 ≤ i1 ≤ . . . ≤ ik+1 ≤ m. (VII) The Majority Depth (Mj D) (Singh, 1991) of y w.r.t. G is defined to be Mj DG (y) = PG {y is in a major side determined by (Y1 , . . . , Yk )} .

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Here a major side is the half-space bounded by the hyperplane containing (Y1 , . . . , Yk ) which has probability ≥ 0.5. The sample version of Mj DG (y) is Mj DGm (y). (VIII) The Zonoid Depth (ZD) (Mosler, 2002) of y w.r.t. {Y1 , . . . , Ym } is defined as ZDGm (y) = sup{α : y ∈ Dα (Y1 , . . . , Ym )} , where Dα (y1 , . . . , ym ) =

(m X i=1

λi yi :

m X i=1

λi = 1 ,

0 ≤ λi , αλi ≤ n−1

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)

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