A side-sensitive synthetic chart for the multivariate coefficient of variation.

Control charts for the coefficient of variations (γ) are receiving increasing attention as it is able to monitor the stability in the ratio of the standard deviation (σ) over the mean (μ), unlike conventional charts that monitor the μ and/or σ separately. A side-sensitive synthetic (SS) chart for monitoring γ was recently developed for univariate processes. The chart outperforms the non-side-sensitive synthetic (NSS) γ chart. However, the SS chart monitoring γ for multivariate processes cannot be found. Thus, a SS chart for multivariate processes is proposed in this paper. A SS chart for multivariate processes is important as multiple quality characteristic that are correlated with each other are frequently encountered in practical scenarios. Based on numerical examples, the side-sensitivity feature that is included in the multivariate synthetic γ chart significantly improves the sensitivity of the chart based on the run length performance, particularly in detecting small shifts (τ), and for small sample size (n), as well as a large number of variables (p) and in-control γ (γ0). The multivariate SS chart also significantly outperforms the Shewhart γ chart, and marginally outperforms the Multivariate Exponentially Weighted Moving Average (MEWMA) γ chart when shift sizes are moderate and large. To show its implementation, the proposed multivariate SS chart is adopted to monitor investment risks.

1. Introduction

Synthetic charts are among the charts proposed to increase the sensitivity for the detection of changes in process parameters. The first synthetic chart was proposed by Wu and Spedding [1] to monitor the process mean. Synthetic chart is different from the traditional Shewhart chart, where, unlike the traditional Shewhart chart which immediately classifies a process as out-of-control when a sample is not within the control limits, the synthetic chart only classifies a process as out-of-control when there are less than L samples between two successive samples that plot beyond the control limits. Note that the L samples must fall within the control limits. Synthetic charts were shown to outperform the traditional Shewhart chart. A recent review of synthetic-type charts is available in Rakitzis et al. [2]. Some of the more recent studies on synthetic charts are Lee et al. [3], Haq and Khoo [4], Hu et al. [5], Haq [6] and many others.

Synthetic charts are then proposed by Calzada and Scariano [7] to monitor γ, where shifts in the ratio of the standard deviation (σ) to the mean (μ) are monitored. This enables processes with an inconsistent μ and/or σ but a consistent ratio to be monitored, and allows the detection of special cause(s) that shifts the ratio . The chart outperforms the Shewhart γ chart by Kang et al. [8], but not the EWMA γ chart by Castagliola et al. [9]. Numerous studies are extended for synthetic γ charts. Recently, Tran et al. [10] developed one with measurement errors, and Yeong et al. [11] designed synthetic γ charts to reduce cost.

For the synthetic charts discussed in the preceding paragraph, samples falling beyond the control limits can fall on either side of the control limits. This has led Yeong et al. [12] to develop a side-sensitive synthetic γ chart, where successive samples need to be on the same side of the limits. In this paper, this chart is referred to as the SS chart, while the non-side-sensitive synthetic γ chart is referred to as the NSS chart. The SS chart is shown to significantly outperform the NSS chart [12]. Yeong et al. [12] evaluated the SS chart through the average run length (ARL) and expected ARL (EARL) criteria. Subsequently, Yeong et al. [13] evaluated the SS chart through the median run length (MRL) and expected MRL (EMRL) criteria. Run lengths are commonly used to evaluate the performance of control charts, where run lengths measure the number of samples until the chart gives an out-of-control signal. There are two types of run lengths, the in-control run length which measures the number of samples collected until the chart gives a false out-of-control signal (i.e. the chart gives an out-of-control signal when the process is in-control) and the out-of-control run length which measures the number of samples until an out-of-control condition is detected by the chart. A chart is said to show good performance if it has a large in-control run length and a small out-of-control run length. Common measures of run length include the ARL, MRL and SDRL, where these measures evaluate the average, median and standard deviation of the run lengths, respectively. However, the ARL, MRL and SDRL requires the exact value of the shift size to be unknown, which is not possible in certain scenarios [9]. In these cases, the performance of the chart will be measured through the EARL and EMRL, which measures the expected value of the ARL and MRL over a range of shift sizes.

A SS chart for multivariate processes is not available. To fill this gap, a multivariate SS chart is proposed in this paper. Multivariate charts are more useful in practice as most processes usually involve several quality characteristics which are correlated to each other, hence, they have to be jointly monitored. Dubious conclusions will be obtained if different univariate charts are used to monitor these quality characteristics, as the correlation between these quality characteristics are ignored. The first multivariate γ chart can be found in Yeong et al. [14]. Subsequently, Lim et al. [15] proposed the multivariate run sum γ chart; Abbasi and Adegoke [16] studied the phase-I implementation of multivariate γ charts; Khaw et al. [17], Chew et al. [18], Nguyen et al. [19] and Ayyoub et al. [20] varied the charting parameters of multivariate γ charts; Khatun et al. [21] proposed multivariate γ charts for short production runs; Giner-Bosch et al. [22], and Haq and Khoo [23] developed a multivariate EWMA (MEWMA) chart to monitor γ; Chew et al. [24] and Chew et al. [25] proposed multivariate run rules γ charts; finally, Ayyoub et al. [26], Ayyoub et al. [27] and Nguyen et al. [28] proposed multivariate γ charts that consider measurement errors.

Although several multivariate charts are available in the literature to monitor γ, a multivariate SS chart cannot be found. A multivariate SS chart will be proposed in this paper. Section 2 gives a list of notations and abbreviations that are used throughout the paper. Next, Section 3 gives a description of the properties of the sample γ . Subsequently, a description of how the proposed multivariate SS chart operates is provided in Section 4, together with the formulae for the ARL, standard deviation of the run length (SDRL) and EARL, and the algorithms to optimize its performance. These algorithms are implemented on several numerical examples in Section 5, while Section 6 compares the multivariate SS chart with the multivariate NSS, MEWMA and Shewhart γ charts. Next, Section 7 shows the implementation of the proposed chart through an illustrative example, followed by the conclusion in Section 8.

2. List of abbreviations and notations

Table 1 shows the list of abbreviations and notations that are used throughout the paper.

Table 1

List of abbreviations and notations.

Abbreviations / Notations	Description
μ	Mean
μ	Mean vector
X¯	Sample mean vector
σ	Standard deviation
Σ	Covariance matrix
S	Sample covariance matrix
γ	Coefficient of variation
γ^	Sample coefficient of variation
γ 0	In-control coefficient of variation
γ 1	Out-of-control coefficient of variation
τ	Shift size
μ(γ^^2)	Mean of γ^^2
μ_0(γ^^2)	In-control mean of γ^^2
σ(γ^^2)	Standard deviation of γ^^2
σ_0(γ^^2)	In-control standard deviation of γ^^2
μ1'(F')	First moment of F’
μ˜1'(F')	Adjusted first moment of F’ for p = 2
μ2'(F')	Second moment of F’
μ˜2'(F')	Adjusted second moment of F’ for p∈{2,3,4}
K	Control limit coefficient
L	Threshold for the CRL sub-chart
n	Sample size
p	Number of variables
ARL	Average run length
ARL 0	In-control average run length
ARL 1	Out-of-control average run length
CRL	Conforming run length
EARL	Expected average run length
EMRL	Expected median run length
EWMA	Exponentially weighted moving average
LCL	Lower control limit
MEWMA	Multivariate exponentially weighted moving average
MRL	Median run length
NSS	Non-side-sensitive synthetic chart
SDRL	Standard deviation of the run length
SDRL 0	In-control standard deviation of the run length
SDRL 1	Out-of-control standard deviation of the run length
SS	Side-sensitive synthetic chart
UCL	Upper control limit
0	Sample where LCL<γ^<UCL
1	Sample where γ^<LCL
1	Sample where γ^>UCL

3. Properties of the sample multivariate coefficient of variation

Let X = (X1,X2,…,X) be p quality characteristics from a multivariate normal distribution with mean vector μ = (μ1,μ2,…,μ) and covariance matrix . The multivariate γ is defined as [29]

To monitor γ, samples of size n are collected and measured at regular intervals. We denote the measurement of the jth quality characteristic of the ith unit in the sample as X. The sample mean vector can be obtained as while the sample covariance matrix is given as where X = (X,…,X). The sample γ can then be computed as where is the (biased) natural estimator of γ. From Yeong et al. [14], i.e. follows a non-central F distribution with p and (n−p) degrees of freedom (df) and a non-centrality parameter (ncp) of , with n>p. From Eq (5), the cumulative distribution function (cdf) for is obtained as where is the cdf for the non-central F distribution in Eq (5).

By letting , it follows that . Thus, the mean and standard deviation of can be obtained through the first and second moments of F’ as follows and where and are the first and second moments of F’. Since from Eq (5) F’ is a non-central F variable with n−p and p df and ncp of 0 and , i.e., . From Giner-Bosch et al. [22], the first and second moments of F’ can be obtained as follows: where C(a,z) in Eqs (9) and (10) is obtained as where C(a,z) will converge with sufficient accuracy with 300 nested fractions [22]. Thus, this paper will adopt the same number of nested fractions.

Note that is undefined for p≤2 and is undefined for p≤4 [22]. For these cases, Giner-Bosch et al. [22] suggested the following alternative versions of and : and where ε is a small value (for example 10−4), , is the inverse cdf for F’, and f(.) is the probability density function (pdf) for F’. Eqs (12) and (13) can be numerically integrated [30].

For p = 2, and can be computed from Eqs (7) and (8) by replacing and with and , respectively, while for p∈{3,4}, since is finite, only needs to be replaced with in Eq (8) to obtain .

4. A multivariate side-sensitive synthetic chart for monitoring

This section describes the multivariate SS chart for . The same approach as that in Yeong et al. [12] is adopted, but by adapting it for of multivariate processes, since the SS chart proposed by Yeong et al. [12] monitors for univariate processes.

The synthetic γ chart is made up of the Shewhart γ and conforming run length (CRL) sub-charts. For the Shewhart sub-chart of the NSS chart, when or , where UCL and LCL are the upper and lower control limits, then that sample is non-conforming; conversely, it is conforming. The CRL sub-chart then defines the CRL as the number of conforming samples between two successive non-conforming samples, inclusive of the most recent non-conforming sample. For example, if there are five conforming samples between two successive non-conforming samples, then CRL = 6. If CRL≤L, with L being a threshold set by the user, the process is considered to have gone out-of-control. In other words, if there are less than L conforming samples between two successive non-conforming samples, the chart will produce an out-of-control signal. The SS chart includes an additional feature where successive non-conforming samples must belong to the same side of the centreline (CL). Hence, if the first non-conforming sample is above the UCL (below the LCL), then only samples that are above the UCL (below the LCL) are non-conforming.

Figs 1 and 2 illustrate the difference between the NSS and SS charts. From Fig 1, Sample 3 is the first non-conforming sample, and it falls above the UCL, while Sample 7 is the second non-conforming sample, and it falls below the LCL. For the NSS chart, both samples are considered to be non-conforming samples, although they fall on different sides of the CL. Thus the CRL = 4. By comparison, for the SS chart in Fig 2, although Samples 2, 5 and 7 falls outside the region between LCL and UCL, the CRL = 5. This is because the first sample to fall outside the region between LCL and UCL, Sample 2, falls above the UCL. Although Sample 5 falls outside the region between LCL and UCL, it is not considered to be a non-conforming sample as it falls below the LCL, which is on the opposite side of the CL from Sample 2. Instead, the next non-conforming sample is Sample 7, since similar with Sample 2, it also falls above the UCL. As a result, CRL = 5. In short, successive non-conforming samples for the SS chart needs to fall on the same side of the CL, whereas successive non-conforming samples for the NSS chart do not have to fall on the same side of the CL.

Fig 1

The CRL sub–chart of the Non–side–sensitive Synthetic–γ (NSS) chart.

Fig 2

The CRL sub–chart of the Side–sensitive Synthetic–γ (SS) chart.

The proposed multivariate SS chart monitors , instead of , due to the availability of the mean and standard deviation of from Giner-Bosch et al. [22]. The following are the LCL and UCL of the proposed multivariate SS chart and where and are the in-control mean and standard deviation of which is obtained from Eqs (7) and (8), respectively, by evaluating the first and second moments of F’ by letting , with γ0 being the in-control value of γ, while K is the control limit coefficient that controls the width of the region between LCL and UCL. The last two paragraphs of this section describe the methodology in determining the value of K.

A Markov chain approach similar to that by Yeong et al. [12] is adopted to obtain the ARL, SDRL and EARL values, but modified for the case of multivariate processes. The states of the Markov chain are defined as in Yeong et al. [12] based on a string of L successive samples, where each sample is defined as either 0, 1 or 1, which denote samples between the LCL and UCL, samples below the LCL and samples above the UCL, respectively. The states of the Markov chain are defined as follows:

State 1: 100..0

State 2: 010..0

State 3: 001..0

⋮

State L: 000..1

State L + 1: 00..00

State L + 2: 0..001

State L + 3: 0..010

⋮

State 2L: 010..0

State 2L + 1: 100..0

State 2L + 2: Signaling state (i.e. the state where the chart signals an out-of-control condition when CRL≤L)

A (2L+2)×(2L+2) transition probability matrix is then formed as follows: where

The ARL and SDRL can be obtained from the Markov chain in Eq (16) by evaluating the expected and standard deviation for the number of transitions until the Markov chain reaches the out-of-control state (State 2L+2), as follows and where q is a (2L+1)×1 vector of initial transient state probabilities, I is the identity matrix, and 1 is a vector of ones. The derivations for Eqs (20) and (21) are shown in Yeong et al. [12]. A zero-state condition is considered, so the (L+2)th element of q is one, and all other elements are zeros, in order to give the proposed chart a head-start. The out-of-control ARL (ARL1) and SDRL (SDRL1) are obtained by substituting γ = γ1 = τγ0 into Eqs (20) and (21), where τ, γ1 and γ0 denote the shift size, out-of-control γ and in-control γ, respectively, while the in-control ARL (ARL0) and SDRL (SDRL0) are computed by substituting γ = γ0 into Eqs (20) and (21).

To evaluate the ARL and SDRL, the exact value of τ must be known. This is not possible in some practical scenarios [9]. For such cases, the EARL is adopted to measure the performance of the chart, as follows: with f(τ) being the pdf of τ. In most scenarios, there is a lack of available data to estimate the actual distribution of τ, hence, this paper assumes that τ follows a uniform distribution over the interval (τmin,τmax) [9]. To evaluate the integral in Eq (22), the Gauss-Legendre quadrature is adopted [31].

Two approaches will be adopted so that the optimal charting parameters (L*,K*) are obtained. Firstly, (L*,K*) is obtained to minimize the ARL1 for pre-determined values of (τ,n,p,γ0), subject to satisfying constraints in the ARL0, i.e., subject to where ξ is the pre-determined ARL0 value. In this paper, we consider L∈{1,2,…,100}, and for each of these values of L, the value of K that satisfies Eq (24) will be obtained through numerical methods. Among all the combinations of (L,K), the combination with the smallest ARL1 will be the optimal (L*,K*). The optimal (LCL*,UCL*) is then obtained from Eqs (14) and (15). Subsequently, the smallest ARL1 value is obtained by substituting (L*,LCL*,UCL*) into Eq (20).

In the second approach, (L*,K*) is obtained based on minimizing the EARL value for pre-determined values of (τmin,τmax,n,p,γ0), subject to satisfying constraints in the ARL0. A similar approach to that described in the preceding paragraph is adopted, with the exception that (L*,K*) minimizes the EARL value, and the shift is the range (τmin,τmax), instead of an exact value τ.

5. Numerical examples

The optimal SS chart for several numerical examples will be obtained in this section. As described in Section 3, two approaches will be adopted, where the first approach minimizes ARL1, and the second minimizes EARL. In both of these approaches, the ARL0 constraint is set as 370.4. In this paper, we consider p∈{2,3,5,8}, τ∈{1.10,1.20,1.25,1.50,2.00} and γ0∈{0.10,0.20,0.30,0.50}. For p∈{2,3}, p = 5 and p = 8, we consider n∈{5,10}, n∈{6,10} and n∈{10,15}, respectively. Different n are considered for different p as n needs to be larger than p. Tables 2 and 3 show the optimal charting parameters and their ARL1 and SDRL1 values for p∈{2,3} and p∈{5,8}, respectively.

Table 2

Optimal charting parameters and the corresponding ARL1 and SDRL1 values of the multivariate SS Chart for p∈{2,3}, n∈{5,10}, τ∈{0.10,1.20,1.25,1.50,2.00} and γ0∈{0.10,0.20,0.30,0.50}.

τ	γ0 = 0.10				γ0 = 0.20				γ0 = 0.30				γ0 = 0.50
	L*	K*	ARL 1	SDRL 1	L*	K*	ARL 1	SDRL 1	L*	K*	ARL 1	SDRL 1	L*	K*	ARL 1	SDRL 1
	p = 2
	n = 5
1.10	47	3.60	74.72	97.85	48	3.70	76.78	100.55	49	3.85	80.43	105.35	49	3.95	94.08	123.16
1.20	27	3.37	26.36	33.71	28	3.46	27.47	35.17	29	3.58	29.48	37.85	31	3.62	37.39	48.49
1.25	22	3.29	17.90	22.41	22	3.35	18.73	23.56	23	3.46	20.25	25.59	25	3.48	26.33	33.80
1.50	10	2.95	5.36	5.92	10	2.99	5.66	6.36	11	3.09	6.23	7.05	13	3.04	8.55	10.14
2.00	5	2.64	2.10	1.75	5	2.67	2.22	1.92	6	2.78	2.45	2.16	7	2.65	3.38	3.39
τ	n = 10
1.10	31	3.04	44.09	57.46	32	3.11	46.07	60.07	33	3.23	49.46	64.55	35	3.54	60.80	79.48
1.20	15	2.80	12.50	15.49	16	2.87	13.28	16.50	17	2.97	14.66	18.33	19	3.24	19.59	24.96
1.25	12	2.73	8.07	9.57	12	2.77	8.61	10.32	13	2.87	9.56	11.57	15	3.12	13.01	16.20
1.50	5	2.43	2.41	2.19	5	2.46	2.56	2.42	6	2.57	2.84	2.72	7	2.75	3.88	4.11
2.00	3	2.25	1.22	0.55	3	2.27	1.27	0.63	3	2.30	1.35	0.78	4	2.48	1.69	1.23
τ	p = 3
	n = 5
1.10	53	3.87	88.50	115.98	54	3.95	90.70	118.88	56	4.08	94.62	124.03	56	4.17	109.31	143.16
1.20	34	3.66	34.01	43.73	34	3.72	35.34	45.54	35	3.82	37.75	48.78	38	3.89	47.33	61.58
1.25	27	3.55	23.67	30.00	28	3.62	24.71	31.37	29	3.72	26.61	33.91	31	3.75	34.28	44.27
1.50	13	3.21	7.40	8.52	14	3.27	7.82	9.02	15	3.35	8.61	10.03	17	3.33	11.93	14.51
2.00	7	2.91	2.81	2.58	7	2.92	2.99	2.84	8	3.01	3.33	3.24	9	2.92	4.82	5.25
τ	n = 10
1.10	33	3.10	47.95	62.55	33	3.16	50.04	65.32	34	3.27	53.63	70.05	37	3.57	65.65	85.84
1.20	16	2.86	14.01	17.50	17	2.93	14.88	18.62	18	3.02	16.40	20.65	21	3.29	21.89	27.98
1.25	13	2.79	9.09	10.91	13	2.83	9.70	11.75	14	2.92	10.76	13.16	16	3.15	14.67	18.41
1.50	6	2.52	2.68	2.49	6	2.55	2.86	2.75	6	2.59	3.18	3.21	8	2.81	4.41	4.77
2.00	3	2.27	1.29	0.67	3	2.29	1.35	0.77	3	2.31	1.45	0.94	4	2.48	1.87	1.50

Table 3

Optimal charting parameters and the corresponding ARL1 and SDRL1 values of the multivariate SS chart for p∈{5,8}, n∈{6,10,15}, τ∈{0.10,1.20,1.25,1.50,2.00} and γ0∈{0.10,0.20,0.30,0.50}.

τ	γ0 = 0.10				γ0 = 0.20				γ0 = 0.30				γ0 = 0.50
	L*	K*	ARL 1	SDRL 1	L*	K*	ARL 1	SDRL 1	L*	K*	ARL 1	SDRL 1	L*	K*	ARL 1	SDRL 1
	p = 5
	n = 6
1.10	71	4.46	112.56	147.69	71	4.50	115.14	151.09	72	4.57	119.65	157.02	74	4.68	135.50	177.77
1.20	47	4.23	49.41	63.96	48	4.27	51.28	66.45	49	4.33	54.58	70.88	52	4.42	66.83	87.27
1.25	39	4.12	35.90	45.98	40	4.16	37.44	48.03	42	4.23	40.20	51.68	46	4.33	50.65	65.71
1.50	21	3.76	12.39	14.78	22	3.80	13.13	15.73	23	3.85	14.49	17.54	27	3.94	19.97	24.86
2.00	11	3.37	4.73	4.93	12	3.43	5.09	5.35	13	3.48	5.77	6.21	16	3.56	8.64	9.97
τ	n = 10
1.10	37	3.26	58.24	76.13	39	3.33	60.57	79.20	41	3.44	64.60	84.51	42	3.67	78.11	102.20
1.20	20	3.04	18.36	23.21	21	3.10	19.45	24.64	22	3.18	21.37	27.21	25	3.41	28.31	36.49
1.25	16	2.96	12.11	14.86	17	3.02	12.90	15.87	17	3.08	14.28	17.80	20	3.30	19.41	24.66
1.50	7	2.65	3.53	3.62	7	2.67	3.79	3.99	8	2.77	4.25	4.54	10	2.96	6.03	6.87
2.00	4	2.44	1.53	0.99	4	2.45	1.62	1.12	4	2.47	1.79	1.37	5	2.62	2.46	2.27
τ	p = 8
	n = 10
1.10	57	3.89	88.72	116.28	56	3.92	91.57	120.04	56	3.97	96.43	126.41	59	4.13	112.44	147.33
1.20	33	3.64	34.14	43.98	36	3.71	35.88	46.17	36	3.75	38.91	50.29	39	3.90	49.62	64.62
1.25	27	3.55	23.78	30.17	28	3.59	25.14	31.96	29	3.64	27.53	35.18	34	3.82	36.19	46.71
1.50	13	3.20	7.45	8.58	14	3.25	8.00	9.27	15	3.31	9.00	10.58	19	3.49	12.88	15.66
2.00	7	2.90	2.83	2.61	7	2.91	3.07	2.95	8	2.99	3.51	3.49	10	3.13	5.29	5.83
τ	n = 15
1.10	31	3.08	48.01	62.65	34	3.14	50.24	65.57	37	3.23	54.07	70.61	38	3.41	66.71	87.25
1.20	17	2.88	14.03	17.45	17	2.91	14.97	18.75	18	2.97	16.61	20.94	20	3.14	22.48	28.84
1.25	13	2.78	9.11	10.93	14	2.84	9.76	11.75	14	2.88	10.92	13.37	16	3.05	15.12	19.04
1.50	6	2.52	2.69	2.50	6	2.53	2.88	2.78	6	2.56	3.24	3.29	8	2.75	4.59	5.03
2.00	3	2.27	1.29	0.67	3	2.28	1.35	0.78	3	2.30	1.47	0.98	4	2.46	1.96	1.63

From Tables 2 and 3, the proposed chart shows a better performance for larger values of n, with smaller values of L* and K*. For example, for (p,n,τ,γ0) = (2,5,1.10,0.10), (L*,K*,ARL1,SDRL1) = (47,3.60,74.72,97.85), but for (p,n,τ,γ0) = (2,10,1.10,0.10), (L*,K*,ARL1,SDRL1) = (31,3.04,44.09,57.46). Moreover, it is easier for larger values of τ to be detected, since larger values of τ show a larger shift from γ0. Thus, smaller ARL1 and SDRL1 values are associated with larger τ. Larger τ also results in smaller values of L*, and a smaller conforming region through smaller values of K*. For example, for (p,n,τ,γ0) = (2,5,1.10,0.10), (L*,K*,ARL1,SDRL1) = (47,3.60,74.72,97.85), but for (p,n,τ,γ0) = (2,5,2.00,0.10), (L*,K*,ARL1,SDRL1) = (5,2.64,2.10,1.75). Tables 2 and 3 also show that larger γ0 is associated with larger L*, K*, ARL1 and SDRL1, which shows a larger conforming region and weaker performance.

Table 4 illustrates the optimal (L*,K*) with its EARL for p∈{2,3,5,8} and γ0∈{0.10,0.20,0.30,0.50}, for scenarios in which the exact value of τ is unknown. Similar to Tables 2 and 3, for p∈{2,3}, p = 5 and p = 8, values of n∈{5,10}, n∈{6,10} and n∈{10,15}, respectively, are considered. To account for the uncertainty in the value of τ, we consider (τmin,τmax) = (1,2].

Table 4

Optimal charting parameters and the corresponding EARL values of the multivariate SS chart for p∈{2,3,5,8}, n∈{5,6,10,15} and γ0∈{0.10,0.20,0.30,0.50}.

γ 0	L*	K*	EARL	L*	K*	EARL	L*	K*	EARL	L*	K*	EARL
	p = 2						p = 3
	n = 5			n = 10			n = 5			n = 10
0.10	26	3.36	19.63	21	2.91	11.75	30	3.60	23.68	22	2.97	12.68
0.20	27	3.44	20.24	21	2.97	12.24	30	3.66	24.41	22	3.02	13.20
0.30	26	3.53	21.33	21	3.05	13.08	30	3.73	25.71	22	3.10	14.11
0.50	25	3.48	25.55	20	3.26	15.96	29	3.70	30.88	21	3.29	17.25
γ 0	p = 5						p = 8
	n = 6			n = 10			n = 10			n = 15
0.10	38	4.10	31.82	23	3.09	15.23	30	3.59	23.76	20	2.93	12.69
0.20	37	4.12	32.85	24	3.15	15.85	31	3.64	24.70	21	2.98	13.26
0.30	38	4.17	34.70	23	3.20	16.94	29	3.64	26.35	20	3.01	14.24
0.50	38	4.19	41.69	23	3.37	20.75	31	3.77	32.16	20	3.14	17.58

From Table 4, larger values of p result in larger L*, K* and EARL. This is consistent with the results in Tables 2 and 3. Similar to Tables 2 and 3, larger values of n result in smaller L*, K* and EARL, while larger γ0 results in larger EARL. In most cases, larger γ0 shows larger K*, however similar values of L* are observed for different values of γ0.

6. Comparisons

This section compares the proposed multivariate SS chart with the multivariate NSS, MEWMA and Shewhart γ charts. Similar numerical examples in Section 4 are adopted, but due to space constraint, we only consider γ0∈{0.10,0.50} in the comparison with the multivariate NSS chart, while γ0 = 0.10 is considered in the comparison with the MEWMA and Shewhart γ charts. Table 5 shows the ARL1 and SDRL1 comparisons with the multivariate NSS chart, while Table 6 compares with the MEWMA and Shewhart γ charts. Finally, Tables 7 and 8 show the EARL comparisons.

Table 5

A comparison of the ARL1 and SDRL1 of the multivariate SS and NSS charts for p∈{2,3,5,8}, n∈{5,6,10,15}, τ∈{1.10,1.20,1.25,1.50,2.00} and γ0∈{0.10,0.50}.

τ	p = 2								p = 3
	γ0 = 0.10				γ0 = 0.50				γ0 = 0.10				γ0 = 0.50
	SS		NSS		SS		NSS		SS		NSS		SS		NSS
	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1
	n = 5
1.10	74.72	97.85	127.23	167.02	94.08	123.16	157.61	206.57	88.50	115.98	143.19	188.00	109.31	143.16	172.06	225.50
1.20	26.36	33.71	45.77	58.98	37.39	48.49	67.79	88.52	34.01	43.73	57.65	74.72	47.33	61.58	82.26	107.65
1.25	17.90	22.41	30.11	38.21	26.33	33.80	47.18	61.15	23.67	30.00	39.40	50.36	34.28	44.27	59.65	77.65
1.50	5.36	5.92	7.57	8.56	8.55	10.14	13.47	16.40	7.40	8.52	10.66	12.54	11.93	14.51	19.02	23.63
2.00	2.10	1.75	2.49	2.21	3.38	3.39	4.51	4.82	2.81	2.58	3.42	3.28	4.82	5.25	6.68	7.50
τ	n = 10
1.10	44.09	57.46	82.94	108.44	60.80	79.48	117.18	153.76	47.95	62.55	89.09	116.59	65.65	85.84	124.11	162.88
1.20	12.50	15.49	21.66	27.23	19.59	24.96	36.87	47.51	14.01	17.50	24.42	30.88	21.89	27.98	41.21	53.31
1.25	8.07	9.57	13.10	15.97	13.01	16.20	23.27	29.45	9.09	10.91	14.91	18.33	14.67	18.41	26.40	33.62
1.50	2.41	2.19	3.08	2.97	3.88	4.11	5.59	6.25	2.68	2.49	3.49	3.55	4.41	4.77	6.47	7.40
2.00	1.22	0.55	1.31	0.70	1.69	1.23	2.01	1.61	1.29	0.67	1.40	0.86	1.87	1.50	2.27	1.99
τ	p = 5								p = 8
	γ0 = 0.10				γ0 = 0.50				γ0 = 0.10				γ0 = 0.50
	SS		NSS		SS		NSS		SS		NSS		SS		NSS
	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1
	n = 6								n = 10
1.10	112.56	147.69	167.13	219.54	135.50	177.77	195.18	256.02	88.72	116.28	141.81	186.24	112.44	147.33	175.16	229.82
1.20	49.41	63.96	79.65	103.79	66.83	87.27	107.38	140.77	34.14	43.98	57.73	74.83	49.62	64.62	85.19	111.45
1.25	35.90	45.98	57.32	74.02	50.65	65.71	82.28	107.43	23.78	30.17	39.47	50.46	36.19	46.71	62.19	80.89
1.50	12.39	14.78	18.05	21.90	19.97	24.86	31.30	39.53	7.45	8.58	10.73	12.64	12.88	15.66	20.39	25.28
2.00	4.73	4.93	5.97	6.40	8.64	9.97	12.24	14.47	2.83	2.61	3.45	3.32	5.29	5.83	7.33	8.32
τ	n = 15								n = 15
1.10	58.24	76.13	104.27	136.71	78.11	102.20	140.18	183.96	48.01	62.65	89.02	116.48	66.71	87.25	125.32	164.50
1.20	18.36	23.21	32.19	41.08	28.31	36.49	52.72	68.46	14.03	17.45	24.39	30.85	22.48	28.84	42.21	54.56
1.25	12.11	14.86	20.21	25.21	19.41	24.66	35.08	45.07	9.11	10.93	14.90	18.31	15.12	19.04	27.21	34.66
1.50	3.53	3.62	4.78	5.10	6.03	6.87	9.19	10.86	2.69	2.50	3.49	3.47	4.59	5.03	6.77	7.72
2.00	1.53	0.99	1.72	1.27	2.46	2.27	3.14	3.05	1.29	0.67	1.40	0.86	1.96	1.63	2.40	2.18

Table 6

A comparison of the ARL1 and SDRL1 of the multivariate SS chart with the MEWMA and Shewhart γ charts for γ0 = 0.10, p∈{2,3,5,8}, n∈{5,6,10,15} and τ∈{1.10,1.20,1.25,1.50,2.00}.

τ	p = 2						p = 3
	SS		MEWMA		Shewhart		SS		MEWMA		Shewhart
	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1
	n = 5
1.10	74.72	97.85	56.54	47.47	171.28	170.78	88.50	115.98	70.37	61.35	187.38	186.88
1.20	26.36	33.71	220	16.94	75.36	74.86	34.01	43.73	29.57	23.03	90.65	90.15
1.25	17.90	22.41	16.90	12.09	52.50	52.00	23.67	30.00	22.08	16.56	65.53	65.03
1.50	5.36	5.92	6.74	4.62	13.79	13.29	7.40	8.52	8.96	6.35	19.02	18.52
2.00	2.10	1.75	2.86	1.92	3.79	3.25	2.81	2.58	3.80	2.70	5.36	4.84
τ	n = 10
1.10	44.09	57.46	31.68	23.75	122.37	121.87	47.95	62.55	34.48	26.30	129.25	128.75
1.20	12.50	15.49	11.86	7.94	40.66	40.16	14.01	17.50	13.01	8.84	44.94	44.44
1.25	8.07	9.57	8.66	5.61	25.68	25.18	9.09	10.91	9.51	6.25	28.78	28.28
1.50	2.41	2.19	3.37	2.09	5.51	4.99	2.68	2.49	3.70	2.34	6.32	5.80
2.00	1.22	0.55	1.52	0.81	1.69	1.08	1.29	0.67	1.65	0.92	1.87	1.27
τ	p = 5						p = 8
	SS		MEWMA		Shewhart		SS		MEWMA		Shewhart
	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1	ARL 1	SDRL 1
	n = 6						n = 10
1.10	112.56	147.69	94.06	83.90	209.40	208.90	88.72	116.28	70.52	61.14	184.80	184.30
1.20	49.41	63.96	43.80	35.31	116.90	116.40	34.14	43.98	29.69	23.13	90.59	90.09
1.25	35.90	45.98	33.57	26.25	88.35	87.85	23.78	30.17	22.17	16.65	65.52	65.02
1.50	12.39	14.78	14.38	10.53	30.53	30.03	7.45	8.58	9.01	6.38	19.14	18.63
2.00	4.73	4.93	6.48	4.53	9.42	8.91	2.83	2.61	3.82	2.72	5.41	4.89
τ	n = 10						n = 15
1.10	58.24	76.13	42.40	33.70	145.84	145.34	48.01	62.65	34.58	26.44	128.92	128.42
1.20	18.36	23.21	16.37	11.52	56.46	55.95	14.03	17.45	13.04	8.87	44.78	44.28
1.25	12.11	14.86	12.02	8.16	37.44	36.94	9.11	10.93	9.53	6.27	28.69	28.19
1.50	3.53	3.62	4.71	3.08	8.79	8.27	2.69	2.50	3.71	2.34	6.33	5.81
2.00	1.53	0.99	2.04	1.24	2.46	1.89	1.29	0.67	1.65	0.92	1.28	1.28

Table 7

A comparison of the EARL of the multivariate SS and NSS charts for p∈{2,3,5,8}, n∈{5,6,10,15} and γ0∈{0.10,0.50}, where (τmin,τmax) = (1,2].

γ 0	p = 2				p = 3
	n = 5		n = 10		n = 5		n = 10
	SS	NSS	SS	NSS	SS	NSS	SS	NSS
0.10	19.63	30.64	11.75	19.61	23.68	35.86	12.68	20.95
0.50	25.55	40.76	15.96	27.46	30.88	47.65	17.25	29.35
γ 0	p = 5				p = 8
	n = 6		n = 10		n = 10		n = 15
	SS	NSS	SS	NSS	SS	NSS	SS	NSS
0.10	31.82	45.93	15.23	24.53	23.76	35.75	12.69	20.99
0.50	41.69	61.06	20.75	34.30	32.16	49.25	17.58	29.80

Table 8

A comparison of the EARL of the multivariate SS chart with the MEWMA and Shewhart γ charts for γ0 = 0.10, p∈{2,3,5,8} and n∈{5,6,10,15}, where (τmin,τmax) = (1,2].

n	SS	MEWMA	Shewhart	n	SS	MEWMA	Shewhart
	p = 2				p = 3
5	19.63	17.20	42.54	5	23.68	21.28	49.47
10	11.75	10.13	27.48	10	12.68	10.92	29.30
n	p = 5			n	p = 8
6	31.82	29.59	62.36	10	23.76	21.36	49.19
10	15.23	13.16	34.14	15	12.69	10.95	29.34

Table 5 shows that the multivariate SS chart has a smaller ARL1 and SDRL1 values than the multivariate NSS chart, particularly when τ is small. This results in quicker detection of the assignable cause(s) and less variability in the run lengths. For example, for (p,n,τ,γ0) = (2,5,1.10,0.10), the (ARL1,SDRL1) = (74.72,97.85) for the multivariate SS chart, while the (ARL1,SDRL1) = (127.23,167.02) for the multivariate NSS chart. A smaller improvement is shown for larger τ. For example, for (p,n,τ,γ0) = (2,5,2.00,0.10), the (ARL1,SDRL1) = (2.10,1.75) for the multivariate SS chart, while (ARL1,SDRL1) = (2.49,2.21) for the multivariate NSS chart. Thus, the improvement is not as large as that for τ = 1.10. The multivariate SS chart also shows a larger improvement for smaller values of n, and larger values of p and γ0.

Tables 7 and 8 show the EARL comparisons, where (τmin,τmax) = (1,2]. Table 7 shows the multivariate SS chart has a smaller EARL than the multivariate NSS chart. For example, for (p,n,γ0) = (2,5,0.10), the EARL = 19.63 for the multivariate SS chart, while EARL = 30.64 for the multivariate NSS chart. Similar to Table 5, smaller n, and larger p and γ0 shows larger improvement. Compared to the MEWMA and Shewhart γ charts, as shown in Table 8, the multivariate SS chart significantly outperforms the Shewhart γ chart, whereas the MEWMA γ chart slightly outperforms the multivariate SS chart.

7. An illustrative example

The implementation of the multivariate SS chart on an illustrative example that was also adopted by Giner-Bosch et al. [22] is shown in this section. In this example, the γ for the investment returns from p = 3 industrial sectors S1 (automotive), S2 (aeronautic) and S3 (electronic) for n = 5 regions R1 (Africa), R2 (North America), R3 (South America), R4 (Asia) and R5 (Europe) are monitored. Table 9 shows the rates of return from years 2000 to 2016, and for each of these years, the , S and are shown.

Table 9

Rates of return (in %) from 2000 to 2016 for p = 3 industrial sectors {S1,S2,S3} and n = 5 regions {R1,R2,R3,R4,R5}.

		R 1	R 2	R 3	R 4	R 5	X¯	S			γ^^2
2000	S 1	17.8	25.2	18.1	19.0	19.0	19.82	9.3320	1.3540	0.1205	0.004082
	S 2	42.0	40.7	35.5	42.0	40.5	40.14	1.3540	7.2230	1.1710
	S 3	8.3	9.4	8.6	10.5	12.1	9.78	0.1205	1.1710	2.4070
2001	S 1	21.5	22.5	22.0	18.1	19.1	20.64	3.7180	3.6455	0.0435	0.001739
	S 2	40.5	36.9	42.0	36.2	35.1	38.14	3.6455	8.7530	4.1410
	S 3	11.9	8.5	12.8	11.4	9.3	10.78	0.0435	4.1410	3.2770
2002	S 1	17.5	18.9	19.1	21.8	22.7	20.00	4.7000	-0.5675	0.3900	0.000539
	S 2	39.9	38.2	38.0	39.8	37.8	38.74	-0.5675	1.0480	-0.0355
	S 3	8.9	11.0	7.8	9.6	9.5	9.36	0.3900	-0.0355	1.3530
2003	S 1	19.1	18.7	21.4	20.8	18.9	19.78	1.5170	0.5765	0.4550	0.001422
	S 2	38.8	42.4	42.5	39.6	39.5	40.56	0.5765	3.0730	0.6325
	S 3	9.7	10.3	11.0	10.4	10.1	10.3	0.4550	0.6325	0.2250
2004	S 1	19.0	21.6	19.5	19.0	19.2	19.66	1.2180	1.2920	-0.1385	0.002000
	S 2	39.4	40.8	35.9	35.0	41.6	38.54	1.2920	8.6780	-2.0265
	S 3	8.2	9.7	12.6	10.5	10.9	10.38	-0.1385	-2.0265	2.6070
2005	S 1	18.9	19.1	21.3	17.0	20.2	19.30	2.5750	3.3900	-0.6550	0.001470
	S 2	41.1	38.1	42.8	36.2	39.2	39.48	3.3900	6.5970	-0.1985
	S 3	10.8	9.5	8.7	9.6	9.2	9.56	-0.6550	-0.1985	0.6030
2006	S 1	17.9	20.0	20.5	18.5	19.4	19.26	1.1330	-1.6770	-0.5400	0.000603
	S 2	43.0	41.2	36.5	39.1	41.0	40.16	-1.6770	6.0930	2.0675
	S 3	8.2	9.6	6.3	9.6	9.8	8.70	-0.5400	2.0675	2.2100
2007	S 1	20.6	18.7	18.5	23.6	19.7	20.22	4.2770	3.6525	3.8680	0.001834
	S 2	40.3	36.9	35.4	40.8	41.6	39.00	3.6525	7.2650	3.9925
	S 3	9.0	8.7	6.8	12.4	9.5	9.28	3.8680	3.9925	4.0870
2008	S 1	19.0	20.4	21.6	20.3	18.4	19.94	1.5880	1.0640	1.3085	0.001383
	S 2	37.3	44.8	40.5	38.9	40.6	40.42	1.0640	7.8170	4.2605
	S 3	8.3	11.7	10.7	7.0	8.2	9.18	1.3085	4.2605	3.7870
2009	S 1	21.2	16.5	18.2	21.2	21.2	19.66	4.8080	1.9075	0.2025	0.001305
	S 2	38.9	39.6	36.8	40.6	41.6	39.50	1.9075	3.3200	-0.9650
	S 3	10.9	8.6	9.1	7.0	8.9	8.90	0.2025	-0.9650	1.9350
2010	S 1	9.6	8.8	8.4	6.9	7.4	8.22	1.1720	-2.0755	-2.4445	0.000499
	S 2	19.5	17.9	18.9	23.7	21.6	20.32	-2.0755	5.4020	4.0830
	S 3	2.2	5.0	5.3	8.9	6.0	5.48	-2.4445	4.0830	5.7470
2011	S 1	11.0	11.8	15.6	11.2	10.3	11.98	4.3820	-1.7770	-0.5615	0.002599
	S 2	18.9	21.6	19.0	20.1	22.5	20.42	-1.7770	2.5370	0.7490
	S 3	4.6	6.2	4.8	6.1	5.5	5.44	-0.5615	0.7490	0.5330
2012	S 1	9.5	10.0	8.3	8.8	12.3	9.78	2.4070	2.2845	-0.6710	0.007852
	S 2	22.1	17.1	19.6	23.4	25.2	21.48	2.2845	10.1570	1.0565
	S 3	4.5	3.8	6.2	5.8	5.0	5.06	-0.6710	1.0565	0.9380
2013	S 1	10.0	8.0	8.2	11.5	7.1	8.96	3.1230	-1.5155	-1.8780	0.001588
	S 2	21.1	21.0	21.2	17.9	20.0	20.24	-1.5155	1.9430	2.0330
	S 3	4.4	7.0	7.3	3.0	4.0	5.14	-1.8780	2.0330	3.6380
2014	S 1	12.2	9.6	7.7	11.3	13.2	10.80	4.7550	-1.3900	-0.4325	0.004144
	S 2	20.0	18.2	18.4	19.6	14.8	18.20	-1.3900	4.2000	0.2850
	S 3	4.6	4.5	4.4	3.5	3.8	4.16	-0.4325	0.2850	0.2330
2015	S 1	11.6	9.8	12.4	11.0	9.4	10.84	1.5480	0.6550	0.7060	0.003456
	S 2	17.2	20.1	21.5	18.6	18.1	19.10	0.6550	2.9050	1.8425
	S 3	4.6	6.4	6.4	4.7	3.3	5.08	0.7060	1.8425	1.7570
2016	S 1	11.2	5.4	9.4	8.4	6.9	8.26	4.9980	-0.5210	1.9820	0.006183
	S 2	20.0	22.1	24.5	17.2	20.6	20.88	-0.5210	7.2470	1.8835
	S 3	5.8	4.0	7.3	4.7	4.9	5.34	1.9820	1.8835	1.6130

The coefficient of variation measures the volatility (standard deviation) of investment returns compared to its expected return. Hence, monitoring γ allows investors to monitor the relative risk of investments, in order to make a fair comparison between different investments. Suppose the company feels that the rates of return and relative risk for years 2000 to 2009 are satisfactory. Thus, the rates of return from years 2000 to 2009 are considered as the Phase I samples, and are estimated from the average of the from years 2000 to 2009, i.e.,

The company would like to monitor whether there is any shift in the relative risks of the investments from years 2010 to 2016. Suppose the company is not sure what is the size of shift that needs to be detected. In this case, the optimal (L*,K*) for the multivariate SS chart will be determined from the second approach as described in Section 3, i.e., the (L*,K*) in minimizing the EARL, subject to constraints in the ARL0, will be adopted to monitor the relative risks of the investment returns from years 2010 to 2016.

By adopting the second approach for , (L*,K*) = (30,3.59) is obtained, with an EARL of 23.49. Fig 3 shows the γ2 sub-chart of the multivariate SS chart that monitors the of the investment returns from years 2010 to 2016.

Fig 3

γ2 Sub–chart of the SS chart for monitoring the investment returns from years 2010 to 2016.

From Fig 3, the for years 2012, 2014 and 2016 is larger than the UCL. Hence, they are non-conforming samples. The CRL for each of these samples are: CRL1 = 3, CRL2 = 2 and CRL3 = 2, all of which are less than L*. Thus, the samples for years 2012, 2014 and 2016 are out-of-control samples. The multivariate SS chart shows an increase in the relative risk for the companies’ investments in years 2012, 2014 and 2016. This agrees with the results from Giner-Bosch et al. [22] who showed that there is a change in the rates of returns from years 2012 onwards. Note that Giner-Bosch et al. [22] monitored the investment returns through the MEWMA chart.

8. Conclusion

A multivariate SS chart to monitor γ is proposed in this paper. Formulae for the ARL, SDRL and EARL criteria are derived, and algorithms are proposed for the optimization of the proposed multivariate SS chart. Tables of optimal charting parameters and performance are shown for numerical examples with different p, n, τ and γ0 values, and also for unknown τ. The multivariate SS chart is shown to outperform the multivariate NSS chart. A larger improvement is shown for smaller τ and n, and larger p and γ0. The multivariate SS chart significantly outperforms the Shewhart γ chart, and shows marginally better performance than the MEWMA chart for moderate and large τ. The proposed multivariate SS chart provides a good alternative for practitioners.

The proposed multivariate SS chart adopts fixed charting parameters. In the future, a multivariate SS chart with adaptive charting parameters can be developed. Another possible area of research is to evaluate the multivariate SS chart through its MRL and run length percentiles, to account for skewed run length distributions.