Multioutput least square SVR-based multivariate EWMA control chart: The performance evaluation and application

Figures & data

Table 1. The parameter of VAR (1) and VMA (1)

	$\mathbf{\ell }=2$	$\mathbf{\ell }=3$
VAR (1)	${\mathbf{\Phi }}_1=\left[\begin{array}{cc}{\varphi }_{11}& {\varphi }_{12}\\ {\varphi }_{21}& {\varphi }_{22}\end{array}\right]=\left[\begin{array}{cc}0.80& 0.05\\ 0.05& 0.90\end{array}\right]$	${\mathbf{\Phi }}_1=\left[\begin{array}{ccc}{\varphi }_{11}& {\varphi }_{12}& {\varphi }_{13}\\ {\varphi }_{21}& {\varphi }_{22}& {\varphi }_{23}\\ {\varphi }_{31}& {\varphi }_{32}& {\varphi }_{33}\end{array}\right]=\left[\begin{array}{ccc}0.80& 0.05& 0.05\\ 0.05& 0.90& 0.05\\ 0.05& 0.05& 0.80\end{array}\right]$
VMA (1)	${\mathbf{\Phi }}_1=\left[\begin{array}{cc}{\theta }_{11}& {\theta }_{12}\\ {\theta }_{21}& {\theta }_{22}\end{array}\right]=\left[\begin{array}{cc}0.80& 0.05\\ 0.05& 0.90\end{array}\right]$	${\mathbf{\Phi }}_1=\left[\begin{array}{ccc}{\theta }_{11}& {\theta }_{12}& {\theta }_{13}\\ {\theta }_{21}& {\theta }_{22}& {\theta }_{23}\\ {\theta }_{31}& {\theta }_{32}& {\theta }_{33}\end{array}\right]=\left[\begin{array}{ccc}0.80& 0.05& 0.05\\ 0.05& 0.90& 0.05\\ 0.05& 0.05& 0.80\end{array}\right]$

Table 2. The ARL of the proposed control chart for VAR (1) data with $\mathrm{\ell }=2$

Mean shift	Smoothing parameter $\lambda$ (UCL)
	0.05 (7.35)	0.10 (8.64)	0.20 (9.65)	0.30 (10.08)	0.40 (10.31)	0.50 (10.44)	0.60 (10.52)	0.80 (10.58)
0.0	195.432	194.534	196.955	197.716	193.038	194.770	193.628	201.382
0.2	62.111	101.494	63.865	85.563	93.990	123.780	124.384	134.702
0.4	18.879	27.976	22.734	29.649	31.788	52.657	52.641	63.193
0.6	9.419	12.873	11.092	13.688	14.605	22.459	25.516	29.789
0.8	5.725	7.287	6.946	7.878	8.285	10.944	13.396	15.608
1.0	4.218	4.929	4.598	5.065	5.357	7.031	7.875	9.796
1.2	2.943	3.643	3.609	3.586	3.891	4.612	5.380	6.081
1.4	2.410	2.803	2.887	2.875	2.968	3.398	3.709	4.400
1.6	1.987	2.302	2.300	2.401	2.476	2.592	2.925	3.092
1.8	1.649	1.965	1.936	1.975	2.054	2.194	2.329	2.478
2.0	1.485	1.649	1.697	1.736	1.709	1.885	1.919	2.057
2.2	1.368	1.495	1.518	1.509	1.521	1.593	1.669	1.744
2.4	1.218	1.367	1.375	1.420	1.389	1.433	1.474	1.480
2.6	1.173	1.245	1.228	1.277	1.264	1.285	1.339	1.313
2.8	1.094	1.144	1.181	1.185	1.168	1.185	1.203	1.199
3.0	1.044	1.098	1.103	1.099	1.087	1.134	1.161	1.115

Table 3. The ARL of the proposed control chart for VAR (1) data with $\mathrm{\ell }=3$

Mean shift	Smoothing parameter $\lambda$ (UCL)
	0.05 (9.41)	0.10 (10.81)	0.20 (11.9)	0.30 (12.35)	0.40 (12.58)	0.50 (12.71)	0.60 (12.79)	0.80 (12.85)
0.0	190.831	192.013	197.262	197.362	203.900	202.197	196.837	196.551
0.2	44.821	49.418	70.124	59.641	88.620	124.289	131.451	145.293
0.4	14.417	16.334	20.221	19.865	29.249	49.041	64.726	74.554
0.6	7.149	8.480	9.430	10.187	13.471	18.805	27.244	35.420
0.8	4.390	5.331	5.860	5.884	7.811	8.986	13.446	16.849
1.0	3.128	3.523	3.904	4.070	4.978	5.275	6.568	8.775
1.2	2.410	2.793	2.946	3.095	3.582	3.543	3.997	5.552
1.4	1.957	2.192	2.282	2.317	2.665	2.563	2.781	3.677
1.6	1.648	1.779	1.931	1.999	2.208	2.058	2.150	2.640
1.8	1.414	1.547	1.654	1.634	1.831	1.744	1.703	2.111
2.0	1.254	1.368	1.443	1.463	1.583	1.497	1.510	1.613
2.2	1.176	1.219	1.273	1.294	1.364	1.331	1.299	1.397
2.4	1.078	1.122	1.169	1.193	1.251	1.168	1.171	1.252
2.6	1.052	1.071	1.098	1.125	1.145	1.103	1.108	1.139
2.8	1.031	1.039	1.044	1.056	1.087	1.056	1.059	1.077
3.0	1.009	1.021	1.030	1.029	1.036	1.028	1.030	1.041

Table 4. The ARL of the proposed control chart for VMA (1) data with $\mathrm{\ell }=2$

Mean shift	Smoothing parameter $\lambda$ (UCL)
	0.05 (7.35)	0.10 (8.64)	0.20 (9.65)	0.30 (10.08)	0.40 (10.31)	0.50 (10.44)	0.60 (10.52)	0.80 (10.58)
0.0	191.528	209.450	200.855	197.182	192.157	207.018	201.489	196.645
0.2	102.481	99.689	150.308	142.579	149.902	154.908	168.190	176.479
0.4	36.124	39.184	63.490	79.445	97.133	103.503	127.560	133.873
0.6	18.967	20.904	28.430	45.911	52.547	57.546	88.078	94.690
0.8	12.351	13.128	16.808	27.673	30.440	35.718	58.271	68.000
1.0	8.811	9.057	11.192	17.812	19.439	21.353	36.545	45.659
1.2	6.631	7.080	7.927	12.675	12.706	14.073	25.118	33.433
1.4	5.154	5.530	6.615	9.752	9.167	10.402	16.800	21.784
1.6	4.155	4.346	4.863	7.062	7.159	7.269	11.822	15.606
1.8	3.564	3.796	4.121	6.087	5.528	5.634	8.453	11.410
2.0	2.930	3.174	3.492	4.912	4.680	4.577	6.961	9.186
2.2	2.450	2.780	2.798	4.277	3.921	3.562	5.284	7.104
2.4	2.145	2.426	2.479	3.738	3.172	3.076	4.598	5.683
2.6	1.895	2.146	2.189	3.258	2.847	2.579	3.626	4.699
2.8	1.776	1.988	1.921	2.931	2.534	2.257	3.042	3.676
3.0	1.606	1.766	1.747	2.644	2.157	1.953	2.742	3.177

Table 5. The ARL of the proposed control chart for VMA (1) data with $\mathrm{\ell }=3$

Mean shift	Smoothing parameter $\lambda$ (UCL)
	0.05 (9.41)	0.10 (10.81)	0.20 (11.9)	0.30 (12.35)	0.40 (12.58)	0.50 (12.71)	0.60 (12.79)	0.80 (12.85)
0.0	195.872	210.231	196.055	195.429	209.942	203.506	194.486	190.051
0.2	92.936	116.680	139.969	157.858	178.459	184.604	178.316	182.577
0.4	36.082	44.680	73.573	111.413	137.050	148.439	150.332	167.394
0.6	18.760	23.992	40.047	70.143	84.013	110.885	115.314	137.174
0.8	12.716	15.186	23.313	45.452	57.504	78.676	77.957	108.204
1.0	9.263	11.019	15.442	27.668	36.516	54.794	61.504	90.722
1.2	7.254	8.006	11.135	18.472	24.962	37.455	40.961	67.690
1.4	5.612	6.770	8.149	13.825	17.412	25.201	29.017	51.550
1.6	4.794	5.690	6.796	9.836	12.638	19.328	20.683	36.996
1.8	3.996	4.743	5.470	8.384	9.365	14.044	14.784	27.976
2.0	3.562	4.035	4.682	6.624	7.603	10.770	11.025	21.127
2.2	2.971	3.462	3.921	5.710	6.339	8.557	8.528	16.110
2.4	2.635	2.961	3.391	4.760	5.223	7.106	6.877	13.217
2.6	2.252	2.638	3.016	4.166	4.449	5.714	5.798	10.981
2.8	2.102	2.451	2.655	3.619	3.868	4.965	4.458	8.562
3.0	1.893	2.219	2.487	3.156	3.301	4.290	3.882	6.937

Figure 1. The ARLs comparison of the proposed control chart for (a) VAR (1) and (b) VMA (1) data.

Figure 2. Time series plots of (a) hourly water turbidity data and (b) hourly chlorine residual data in Phase I.

Figure 3. MLS-SVR-based MEWMA control chart for water quality data in Phase I.

Figure 4. Time series plots of (a) hourly water turbidity data and (b) hourly chlorine residual data in Phase II.

Figure 5. MLS-SVR-based MEWMA control chart for water quality data in Phase II.

Figure 6. Time series plots of (a) updated hourly water turbidity data and (b) updated hourly chlorine residual data in Phase II.

Figure 7. MLS-SVR-based MEWMA control chart for updated water quality data in phase II.