Consider thep= 9 process variables in Table 11.5.(a) Perform a PCA on the first 30 observations. Be sure to work with the standardized variables.(b) How much variability is explained if only the firstr= 3 principal components are retained?(c) Construct an appropriate set of pairwise plots of the firstr= 3 principal component scores.(d) Now consider the last 10 observations. Obtain the principal component scores and plot them on the chart in part (c). Does the process seem to be in control? TABLE 1 1 .5 Cascade Process Data Observation x1 x2 x3 x4 x5 x6 x7 x8 x9 y1 Residuals y2 1 12.78 0.15 91 56 1.54 7.38 1.75 5.89 1.11 951.5 0.81498 87 2 14.97 0.1 90 49 1.54 7.14 1.71 5.91 1.109 952.2 −0.31685 88 3 15.43 0.07 90 41 1.47 7.33 1.64 5.92 1.104 952.3 −0.28369 86 4 14.95 0.12 89 43 1.54 7.21 1.93 5.71 1.103 951.8 −0.45924 89 5 16.17 0.1 83 42 1.67 7.23 1.86 5.63 1.103 952.3 −0.56512 86 6 17.25 0.07 84 54 1.49 7.15 1.68 5.8 1.099 952.2 −0.22592 91 7 16.57 0.12 89 61 1.64 7.23 1.82 5.88 1.096 950.2 −0.55431 99 8 19.31 0.08 99 60 1.46 7.74 1.69 6.13 1.092 950.5 −0.18874 100 9 18.75 0.04 99 52 1.89 7.57 2.02 6.27 1.084 950.6 0.15245 103 10 16.99 0.09 98 57 1.66 7.51 1.82 6.38 1.086 949.8 −0.33580 107 11 18.2 0.13 98 49 1.66 7.27 1.92 6.3 1.089 951.2 −0.85525 98 12 16.2 0.16 97 52 2.16 7.21 2.34 6.07 1.089 950.6 0.47027 96 13 14.72 0.12 82 61 1.49 7.33 1.72 6.01 1.092 948.9 −1.74107 93 14 14.42 0.13 81 63 1.16 7.5 1.5 6.11 1.094 951.7 0.62057 91 15 11.02 0.1 83 56 1.56 7.14 1.73 6.14 1.102 951.5 0.72583 91 16 9.82 0.1 86 53 1.26 7.32 1.54 6.15 1.112 951.3 −0.03421 93 17 11.41 0.12 87 49 1.29 7.22 1.57 6.13 1.114 952.9 0.28093 91 18 14.74 0.1 81 42 1.55 7.17 1.77 6.28 1.114 953.9 −1.87257 94 19 14.5 0.08 84 53 1.57 7.23 1.69 6.28 1.109 953.3 −0.20805 96 20 14.71 0.09 89 46 1.45 7.23 1.67 6.12 1.108 952.6 −0.66749 94 21 15.26 0.13 91 47 1.74 7.28 1.98 6.19 1.105 952.3 −0.75390 99 22 17.3 0.12 95 47 1.57 7.18 1.86 6.06 1.098 952.6 −0.03479 95 23 17.62 0.06 95 42 2.05 7.15 2.14 6.15 1.096 952.9 0.24439 92 24 18.21 0.06 93 41 1.46 7.28 1.61 6.11 1.096 953.9 0.67889 87 25 14.38 0.1 90 46 1.42 7.29 1.73 6.13 1.1 954.2 1.94313 89 26 12.13 0.14 87 50 1.76 7.21 1.9 6.31 1.112 951.9 −0.92344 98 27 12.72 0.1 90 47 1.52 7.25 1.79 6.25 1.112 952.3 −0.74707 95 28 17.42 0.1 89 51 1.33 7.38 1.51 6.01 1.111 953.7 −0.21053 88 29 17.63 0.11 87 45 1.51 7.42 1.68 6.11 1.103 954.7 0.66802 86 30 16.17 0.05 83 57 1.41 7.35 1.62 6.14 1.105 954.6 1.35076 84 31 16.88 0.16 86 58 2.1 7.15 2.28 6.42 1.105 954.8 0.61137 91 32 13.87 0.16 85 46 2.1 7.11 2.16 6.44 1.106 954.4 0.56960 92 33 14.56 0.05 84 41 1.34 7.14 1.51 6.24 1.113 955 −0.09131 88 34 15.35 0.12 83 40 1.52 7.08 1.81 6 1.114 956.5 1.03785 83 35 15.91 0.12 81 45 1.76 7.26 1.9 6.07 1.116 955.3 −0.07282 83 36 14.32 0.11 85 47 1.58 7.15 1.72 6.02 1.113 954.2 0.53440 86 37 15.43 0.13 86 43 1.46 7.15 1.73 6.11 1.115 955.4 0.16379 85 38 14.47 0.08 85 54 1.62 7.1 1.78 6.15 1.118 953.8 −0.37110 88 39 14.74 0.07 84 52 1.47 7.24 1.66 5.89 1.112 953.2 0.17177 83 40 16.28 0.13 86 49 1.72 7.05 1.89 5.91 1.109 954.2 0.47427 85

Consider the p = 9 process variables in Table 11.5. (a) Perform a PCA on the first 30 observations. Be sure to work with the standardized variables. (b) How much variability is explained if only the...

Consider thep= 9 process variables in Table 11.5.

(a) Perform a PCA on the first 30 observations. Be sure to work with the standardized variables.

(b) How much variability is explained if only the firstr= 3 principal components are retained?

(d) Now consider the last 10 observations. Obtain the principal component scores and plot them on the chart in part (c). Does the process seem to be in control?

TABLE 1 1 .5
Cascade Process Data
Observation	x₁	x₂	x₃	x₄	x₅	x₆	x₇	x₈	x₉	y₁	Residuals	y₂
1	12.78	0.15	91	56	1.54	7.38	1.75	5.89	1.11	951.5	0.81498	87
2	14.97	0.1	90	49	1.54	7.14	1.71	5.91	1.109	952.2	−0.31685	88
3	15.43	0.07	90	41	1.47	7.33	1.64	5.92	1.104	952.3	−0.28369	86
4	14.95	0.12	89	43	1.54	7.21	1.93	5.71	1.103	951.8	−0.45924	89
5	16.17	0.1	83	42	1.67	7.23	1.86	5.63	1.103	952.3	−0.56512	86
6	17.25	0.07	84	54	1.49	7.15	1.68	5.8	1.099	952.2	−0.22592	91
7	16.57	0.12	89	61	1.64	7.23	1.82	5.88	1.096	950.2	−0.55431	99
8	19.31	0.08	99	60	1.46	7.74	1.69	6.13	1.092	950.5	−0.18874	100
9	18.75	0.04	99	52	1.89	7.57	2.02	6.27	1.084	950.6	0.15245	103
10	16.99	0.09	98	57	1.66	7.51	1.82	6.38	1.086	949.8	−0.33580	107
11	18.2	0.13	98	49	1.66	7.27	1.92	6.3	1.089	951.2	−0.85525	98
12	16.2	0.16	97	52	2.16	7.21	2.34	6.07	1.089	950.6	0.47027	96
13	14.72	0.12	82	61	1.49	7.33	1.72	6.01	1.092	948.9	−1.74107	93
14	14.42	0.13	81	63	1.16	7.5	1.5	6.11	1.094	951.7	0.62057	91
15	11.02	0.1	83	56	1.56	7.14	1.73	6.14	1.102	951.5	0.72583	91
16	9.82	0.1	86	53	1.26	7.32	1.54	6.15	1.112	951.3	−0.03421	93
17	11.41	0.12	87	49	1.29	7.22	1.57	6.13	1.114	952.9	0.28093	91
18	14.74	0.1	81	42	1.55	7.17	1.77	6.28	1.114	953.9	−1.87257	94
19	14.5	0.08	84	53	1.57	7.23	1.69	6.28	1.109	953.3	−0.20805	96
20	14.71	0.09	89	46	1.45	7.23	1.67	6.12	1.108	952.6	−0.66749	94
21	15.26	0.13	91	47	1.74	7.28	1.98	6.19	1.105	952.3	−0.75390	99
22	17.3	0.12	95	47	1.57	7.18	1.86	6.06	1.098	952.6	−0.03479	95
23	17.62	0.06	95	42	2.05	7.15	2.14	6.15	1.096	952.9	0.24439	92
24	18.21	0.06	93	41	1.46	7.28	1.61	6.11	1.096	953.9	0.67889	87
25	14.38	0.1	90	46	1.42	7.29	1.73	6.13	1.1	954.2	1.94313	89
26	12.13	0.14	87	50	1.76	7.21	1.9	6.31	1.112	951.9	−0.92344	98
27	12.72	0.1	90	47	1.52	7.25	1.79	6.25	1.112	952.3	−0.74707	95
28	17.42	0.1	89	51	1.33	7.38	1.51	6.01	1.111	953.7	−0.21053	88
29	17.63	0.11	87	45	1.51	7.42	1.68	6.11	1.103	954.7	0.66802	86
30	16.17	0.05	83	57	1.41	7.35	1.62	6.14	1.105	954.6	1.35076	84
31	16.88	0.16	86	58	2.1	7.15	2.28	6.42	1.105	954.8	0.61137	91
32	13.87	0.16	85	46	2.1	7.11	2.16	6.44	1.106	954.4	0.56960	92
33	14.56	0.05	84	41	1.34	7.14	1.51	6.24	1.113	955	−0.09131	88
34	15.35	0.12	83	40	1.52	7.08	1.81	6	1.114	956.5	1.03785	83
35	15.91	0.12	81	45	1.76	7.26	1.9	6.07	1.116	955.3	−0.07282	83
36	14.32	0.11	85	47	1.58	7.15	1.72	6.02	1.113	954.2	0.53440	86
37	15.43	0.13	86	43	1.46	7.15	1.73	6.11	1.115	955.4	0.16379	85
38	14.47	0.08	85	54	1.62	7.1	1.78	6.15	1.118	953.8	−0.37110	88
39	14.74	0.07	84	52	1.47	7.24	1.66	5.89	1.112	953.2	0.17177	83
40	16.28	0.13	86	49	1.72	7.05	1.89	5.91	1.109	954.2	0.47427	85

May 25, 2022

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Consider the p = 9 process variables in Table 11.5. (a) Perform a PCA on the first 30 observations. Be sure to work with the standardized variables. (b) How much variability is explained if only the...

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