Increasing the Sensitivity of Multivariate EWMA Control Chart

Figures & data

Figure 1. ARL₀ vs. UCL for the asymptotic MEWMA control chart schemes for p=2.

Figure 2. ARL₀ vs. UCL for the asymptotic MEWMA control chart schemes for p=4.

Figure 3. ARL₀ vs. UCL for the asymptotic MEWMA control chart schemes for p=10.

Figure 4. ARL₀ vs. UCL for the exact MEWMA control chart schemes for p=2.

Figure 5. ARL₀ vs. UCL for the exact MEWMA control chart schemes for p=4.

Figure 6. ARL₀ vs. UCL for the exact MEWMA control chart schemes for p=10.

Table 1. A sensitivity analysis for determining the exact UCL for the asymptotic MEWMA chart based on p=2, r=0.05, and ARL₀=200

UCL	ARL0	UCL	ARL0
7.30	194.8	7.36	199.1
7.31	195.4	7.37	200.2
7.32	196.3	7.38	201.0
7.33	197.2	7.39	201.7
7.34	197.8	7.40	202.4
7.35	198.5

Table 2. Factors for the exact MEWMA statistics (f _r,t =[1−(1−r)^2t ]⁻¹)

	r
Obs. No., t	0.05	0.10	0.20	0.30	0.40	0.50
1	10.2564	5.2632	2.7778	1.9608	1.5625	1.3333
2	5.3910	2.9078	1.6938	1.3160	1.1489	1.0667
3	3.7749	2.1342	1.3553	1.1333	1.0489	1.0159
4	2.9711	1.7558	1.2016	1.0612	1.0171	1.0039
5	2.4921	1.5353	1.1203	1.0291	1.0061	1.0010
6	2.1756	1.3936	1.0738	1.0140	1.0022	1.0002
7	1.9519	1.2966	1.0460	1.0068	1.0008	1.0001
8	1.7861	1.2274	1.0290	1.0033	1.0003	1.0000
9	1.6590	1.1766	1.0183	1.0016	1.0001
10	1.5588	1.1384	1.0117	1.0008	1.0000
11	1.4783	1.1092	1.0074	1.0004
12	1.4124	1.0867	1.0047	1.0002
13	1.3578	1.0691	1.0030	1.0001
14	1.3120	1.0552	1.0019	1.0000
15	1.2733	1.0443	1.0012
16	1.2403	1.0356	1.0008
17	1.2119	1.0286	1.0005
18	1.1873	1.0230	1.0003
19	1.1660	1.0186	1.0002
20	1.1475	1.0150	1.0001
21	1.1312	1.0121	1.0001
22	1.1169	1.0098	1.0001
23	1.1043	1.0079	1.0000
24	1.0932	1.0064
25	1.0834	1.0052
26	1.0746	1.0042
27	1.0669	1.0034
28	1.0600	1.0027
29	1.0538	1.0022
30	1.0483	1.0018
35	1.0284	1.0006
40	1.0168	1.0002
45	1.0100	1.0001
50	1.0060	1.0000

Table 3. ARL profile for the asymptotic MEWMA control chart schemes

			r
p	λ		0.05	0.10	0.20	0.30	0.40	0.50
		H=	7.37	8.64	9.64	10.08	10.33	10.44
2	0.0		200.2	200.2	200.4	199.8	200.5	199.5
	0.2		85.5	98.4	118.1	133.2	147.3	153.5
	0.5		26.8	28.0	34.6	43.7	53.6	63.7
	1.0		11.3	10.2	10.3	11.4	13.3	15.7
	1.5		7.2	6.1	5.5	5.5	5.8	6.4
	2.0		5.3	4.4	3.8	3.5	3.5	3.6
	3.0		3.6	2.9	2.4	2.2	2.0	1.9
	4.0		2.8	2.2	1.9	1.7	1.5	1.4
	5.0		2.2	2.0	1.5	1.3	1.1	1.1
		H=	11.26	12.79	13.89	14.35	14.59	14.71
4	0.0		199.8	200.1	200.2	199.9	200.0	199.9
	0.2		101.0	117.6	139.4	153.8	164.9	169.9
	0.5		32.1	35.1	46.2	59.0	72.9	85.9
	1.0		13.5	12.2	12.7	14.7	18.0	22.2
	1.5		8.6	7.3	6.6	6.7	7.3	8.4
	2.0		6.3	5.2	4.4	4.2	4.2	4.5
	3.0		4.2	3.4	2.8	2.5	2.3	2.3
	4.0		3.2	2.6	2.1	1.9	1.7	1.6
	5.0		2.7	2.1	1.8	1.5	1.3	1.2
		H=	20.72	22.68	24.05	24.57	24.87	25.03
10	0.0		200.5	200.3	199.7	199.8	199.7	200.2
	0.2		123.0	139.8	160.9	169.6	177.0	182.9
	0.5		42.6	48.7	67.5	85.2	101.5	114.3
	1.0		17.5	16.1	18.0	22.7	29.1	37.3
	1.5		11.1	9.3	8.6	9.3	10.9	13.5
	2.0		8.2	6.6	5.6	5.4	5.7	6.5
	3.0		5.4	4.3	3.4	3.1	2.9	2.9
	4.0		4.2	3.3	2.5	2.2	2.1	1.9
	5.0		3.4	2.7	2.1	1.9	1.7	1.5

Table 4. ARL profile for the exact MEWMA control chart schemes

			r
p	λ		0.05		0.10		0.20		0.30		0.40		0.50
		H =	7.37	7.72	8.64	8.79	9.64	9.70	10.08	10.11	10.33	10.34	10.44	10.46
2	0.0		169.3	199.5	186.5	199.9	194.0	199.5	196.6	199.5	198.7	199.5	198.6	200.1
	0.2		70.3	78.2	90.5	95.1	113.8	116.7	130.7	132.5	145.7	146.4	152.5	153.8
	0.5		19.8	21.1	24.4	25.0	32.9	33.3	42.2	42.6	52.6	52.8	62.9	63.4
	1.0		6.6	6.9	7.7	7.8	9.1	9.2	10.7	10.7	12.8	12.8	15.4	15.5
	1.5		3.5	3.6	4.0	4.0	4.4	4.5	4.8	4.9	5.4	5.4	6.1	6.1
	2.0		2.3	2.4	2.6	2.6	2.8	2.8	3.0	3.0	3.1	3.1	3.3	3.3
	3.0		1.4	1.4	1.5	1.5	1.6	1.6	1.6	1.6	1.7	1.7	1.7	1.7
	4.0		1.1	1.1	1.1	1.1	1.2	1.2	1.2	1.2	1.2	1.2	1.2	1.2
	5.0		1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0
		H =	11.26	11.70	12.79	12.98	13.89	13.95	14.35	14.39	14.59	14.61	14.71	14.73
4	0.0		169.1	200.1	186.9	199.8	194.1	199.5	196.1	199.7	197.6	199.7	198.5	200.0
	0.2		82.4	93.9	108.0	114.8	135.0	137.8	151.0	153.5	163.2	164.5	168.5	170.1
	0.5		24.0	25.6	30.7	31.9	44.0	44.6	57.5	58.1	71.6	72.0	85.0	85.8
	1.0		8.0	8.4	9.4	9.6	11.3	11.5	13.9	14.0	17.4	17.4	21.8	21.9
	1.5		4.2	4.4	4.8	4.9	5.4	5.4	6.0	6.0	6.9	6.9	8.0	8.0
	2.0		2.7	2.8	3.1	3.1	3.4	3.4	3.6	3.6	3.8	3.8	4.2	4.2
	3.0		1.5	1.6	1.7	1.7	1.8	1.8	1.9	1.9	1.9	1.9	2.0	2.0
	4.0		1.1	1.2	1.2	1.2	1.3	1.3	1.3	1.3	1.3	1.3	1.3	1.3
	5.0		1.0	1.0	1.0	1.0	1.0	1.1	1.1	1.1	1.1	1.1	1.1	1.1
		H =	20.72	21.37	22.68	22.94	24.05	24.15	24.57	24.64	24.87	24.92	25.03	25.06
10	0.0		165.6	199.9	185.1	200.2	193.2	199.6	195.8	199.9	197.1	200.2	198.3	200.4
	0.2		98.8	117.2	127.9	136.9	155.3	159.6	166.3	169.8	174.5	177.1	181.1	18.28
	0.5		31.7	34.7	42.9	44.7	64.0	65.4	82.7	84.4	99.8	101.3	112.9	114.1
	1.0		10.5	11.1	12.5	12.9	16.2	16.4	21.5	21.8	28.2	28.5	36.6	36.8
	1.5		5.4	5.7	6.2	6.4	7.3	7.3	8.5	8.5	10.3	10.4	13.0	13.1
	2.0		3.4	3.6	3.9	4.0	4.3	4.3	4.7	4.7	5.2	5.2	6.1	6.1
	3.0		1.9	2.0	2.1	2.1	2.2	2.3	2.3	2.3	2.4	2.4	2.5	2.5
	4.0		1.3	1.3	1.4	1.4	1.5	1.5	1.5	1.5	1.6	1.6	1.6	1.6
	5.0		1.1	1.1	1.1	1.1	1.2	1.2	1.2	1.2	1.2	1.2	1.2	1.2

Table 5. Q _t,3 and Q _t,4 statistics for r=0.1

Obs. No., t	X 1	X 2	Q t,3	Q t,4
1	2.404	−1.426	1.485	7.815
2	1.624	1.489	2.734	7.949
3	1.454	0.653	4.626	9.873
4	−0.768	−0.814	2.475	4.346
5	0.776	0.531	3.120	4.790
6	0.918	1.332	4.506	6.280
7	2.146	−0.721	7.522	9.753
8	2.816	−0.912	13.490	16.559
9	−0.245	−0.932	10.442	12.286
10	0.0245	0.735	8.357	9.514
11	0.379	−0.935	7.890	8.752
12	0.920	−1.076	9.183	9.979
13	1.742	−1.232	13.133	14.041
14	0.457	−0.639	12.639	13.337
15	−1.335	−1.555	9.488	9.908
16	0.152	−0.828	9.414	9.749
17	0.569	1.558	6.436	6.620
18	2.369	1.279	9.361	9.577
19	1.283	−0.049	10.939	11.143
20	1.850	−0.886	14.784	15.005

Figure 7. The asymptotic and exact MEWMA control charts.