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Type 'q()' to quit R. > library(robustbase) > > source(system.file("xtraR/test_MCD.R", package = "robustbase"))#-> doMCDdata > ## ../inst/xtraR/test_MCD.R > ## instead of relying on system.file("test-tools-1.R", package="Matrix"): > source(system.file("xtraR/test-tools.R", package = "robustbase")) # showProc.time(), relErr() > showProc.time() Time (user system elapsed): 0 0 0 > > ## -- now do it: > options(digits = 5) > set.seed(101) # <<-- sub-sampling algorithm now based on R's RNG and seed > doMCDdata() Call: doMCDdata() Data Set n p h(alf) LOG(obj) ============================================= bushfire 38 5 22 18.135810 Best subsample: [1] 1 2 3 4 5 6 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Outliers: 16 : [1] 7 8 9 10 11 12 29 30 31 32 33 34 35 36 37 38 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=22); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 18.1 Robust Estimate of Location: V1 V2 V3 V4 V5 105 147 274 218 279 Robust Estimate of Covariance: V1 V2 V3 V4 V5 V1 346 268 -1692 -381 -311 V2 268 236 -1125 -230 -194 V3 -1692 -1125 9993 2455 1951 V4 -381 -230 2455 647 505 V5 -311 -194 1951 505 398 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= heart 12 2 7 5.678742 Best subsample: [1] 1 3 4 5 7 9 11 Outliers: 0 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=7); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 5.68 Robust Estimate of Location: height weight 38.3 33.1 Robust Estimate of Covariance: height weight height 135 259 weight 259 564 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= starsCYG 47 2 25 -8.031215 Best subsample: [1] 1 2 4 6 8 10 12 13 16 24 25 26 28 32 33 37 38 39 40 41 42 43 44 45 46 Outliers: 7 : [1] 7 9 11 14 20 30 34 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=25); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -8.03 Robust Estimate of Location: log.Te log.light 4.41 4.95 Robust Estimate of Covariance: log.Te log.light log.Te 0.0132 0.0394 log.light 0.0394 0.2743 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= stackloss 21 3 12 5.472581 Best subsample: [1] 4 5 6 7 8 9 10 11 12 13 14 20 Outliers: 9 : [1] 1 2 3 15 16 17 18 19 21 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=12); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 5.47 Robust Estimate of Location: Air.Flow Water.Temp Acid.Conc. 59.5 20.8 87.3 Robust Estimate of Covariance: Air.Flow Water.Temp Acid.Conc. Air.Flow 6.29 5.85 5.74 Water.Temp 5.85 9.23 6.14 Acid.Conc. 5.74 6.14 23.25 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= phosphor 18 2 10 6.878847 Best subsample: [1] 3 5 8 9 11 12 13 14 15 17 Outliers: 3 : [1] 1 6 10 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=10); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 6.88 Robust Estimate of Location: inorg organic 13.4 38.8 Robust Estimate of Covariance: inorg organic inorg 129 130 organic 130 182 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= coleman 20 5 13 1.286808 Best subsample: [1] 2 3 4 5 7 8 12 13 14 16 17 19 20 Outliers: 7 : [1] 1 6 9 10 11 15 18 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=13); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 1.29 Robust Estimate of Location: salaryP fatherWc sstatus teacherSc motherLev 2.76 48.38 6.12 25.00 6.40 Robust Estimate of Covariance: salaryP fatherWc sstatus teacherSc motherLev salaryP 0.253 1.79 -0.266 0.151 0.075 fatherWc 1.786 1303.38 330.496 12.604 34.503 sstatus -0.266 330.50 119.888 3.833 10.131 teacherSc 0.151 12.60 3.833 0.785 0.555 motherLev 0.075 34.50 10.131 0.555 1.043 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= salinity 28 3 16 1.326364 Best subsample: [1] 1 2 6 7 8 12 13 14 18 20 21 22 25 26 27 28 Outliers: 4 : [1] 5 16 23 24 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=16); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 1.33 Robust Estimate of Location: X1 X2 X3 10.08 2.78 22.78 Robust Estimate of Covariance: X1 X2 X3 X1 10.44 1.01 -3.19 X2 1.01 3.83 -1.44 X3 -3.19 -1.44 2.39 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wood 20 5 13 -36.270094 Best subsample: [1] 1 2 3 5 9 10 12 13 14 15 17 18 20 Outliers: 7 : [1] 4 6 7 8 11 16 19 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=13); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -36.3 Robust Estimate of Location: x1 x2 x3 x4 x5 0.587 0.122 0.531 0.538 0.892 Robust Estimate of Covariance: x1 x2 x3 x4 x5 x1 0.010025 1.88e-03 0.003153 -0.000586 -1.63e-03 x2 0.001881 4.85e-04 0.001269 -0.000052 2.36e-05 x3 0.003153 1.27e-03 0.006632 -0.000871 3.52e-04 x4 -0.000586 -5.20e-05 -0.000871 0.002846 1.83e-03 x5 -0.001630 2.36e-05 0.000352 0.001828 2.77e-03 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= hbk 75 3 39 -1.047858 Best subsample: [1] 15 16 17 18 19 20 21 22 23 24 26 27 31 32 33 35 36 37 38 40 43 49 50 51 54 [26] 55 56 58 59 61 63 64 66 67 70 71 72 73 74 Outliers: 14 : [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=39); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -1.05 Robust Estimate of Location: X1 X2 X3 1.54 1.78 1.69 Robust Estimate of Covariance: X1 X2 X3 X1 1.227 0.055 0.127 X2 0.055 1.249 0.153 X3 0.127 0.153 1.160 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= Animals 28 2 15 14.555543 Best subsample: [1] 1 3 4 5 10 11 17 18 19 20 21 22 23 26 27 Outliers: 14 : [1] 2 6 7 8 9 12 13 14 15 16 23 24 25 28 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=15); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 14.6 Robust Estimate of Location: body brain 18.7 64.9 Robust Estimate of Covariance: body brain body 929 1576 brain 1576 5646 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= milk 86 8 47 -28.837261 Best subsample: [1] 5 7 8 9 10 21 22 24 30 31 32 33 34 35 38 39 45 46 51 53 54 55 56 57 58 [26] 59 60 61 62 63 64 65 66 67 68 69 71 72 76 78 79 80 81 82 83 84 86 Outliers: 20 : [1] 1 2 3 11 12 13 14 15 16 17 18 20 27 41 44 47 70 74 75 77 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=47); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -28.8 Robust Estimate of Location: X1 X2 X3 X4 X5 X6 X7 X8 1.03 35.87 33.09 26.15 25.13 25.06 123.14 14.39 Robust Estimate of Covariance: X1 X2 X3 X4 X5 X6 X7 X1 3.86e-07 0.000115 0.000135 0.000132 0.000118 0.000101 0.000538 X2 1.15e-04 1.901695 0.321524 0.228041 0.164447 0.261330 1.804532 X3 1.35e-04 0.321524 1.189750 0.869795 0.851445 0.857952 0.777883 X4 1.32e-04 0.228041 0.869795 0.684723 0.651039 0.652613 0.603585 X5 1.18e-04 0.164447 0.851445 0.651039 0.680275 0.655047 0.608406 X6 1.01e-04 0.261330 0.857952 0.652613 0.655047 0.680328 0.601059 X7 5.38e-04 1.804532 0.777883 0.603585 0.608406 0.601059 4.022100 X8 1.29e-05 0.238712 0.201708 0.132675 0.115217 0.125472 0.389816 X8 X1 1.29e-05 X2 2.39e-01 X3 2.02e-01 X4 1.33e-01 X5 1.15e-01 X6 1.25e-01 X7 3.90e-01 X8 1.54e-01 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= lactic 20 2 11 0.359580 Best subsample: [1] 1 2 3 4 5 7 8 9 10 11 12 Outliers: 4 : [1] 17 18 19 20 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=11); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 0.36 Robust Estimate of Location: X Y 3.86 5.01 Robust Estimate of Covariance: X Y X 10.6 14.6 Y 14.6 21.3 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pension 18 2 10 16.675508 Best subsample: [1] 1 2 3 4 5 6 8 9 11 12 Outliers: 5 : [1] 14 15 16 17 18 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=10); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 16.7 Robust Estimate of Location: Income Reserves 52.3 560.9 Robust Estimate of Covariance: Income Reserves Income 1420 11932 Reserves 11932 208643 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pilot 20 2 11 6.487287 Best subsample: [1] 2 3 6 7 9 12 15 16 17 18 20 Outliers: 0 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=11); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 6.49 Robust Estimate of Location: X Y 101.1 67.7 Robust Estimate of Covariance: X Y X 3344 1070 Y 1070 343 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= vaso 39 2 21 -3.972244 Best subsample: [1] 3 4 8 14 18 19 20 21 22 23 24 25 26 27 28 33 34 35 37 38 39 Outliers: 4 : [1] 1 2 17 31 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=21); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -3.97 Robust Estimate of Location: Volume Rate 1.16 1.72 Robust Estimate of Covariance: Volume Rate Volume 0.313 -0.167 Rate -0.167 0.728 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wagnerGrowth 63 6 35 6.569262 Best subsample: [1] 2 3 4 5 6 7 9 10 11 12 14 16 17 18 20 23 25 27 31 32 35 36 38 41 44 [26] 48 52 53 54 55 56 57 58 60 62 Outliers: 17 : [1] 1 8 15 21 22 24 26 28 29 33 39 42 43 46 50 61 63 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=35); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 6.57 Robust Estimate of Location: Region PA GPA HS GHS y 11.318 34.050 -2.049 2.498 0.289 6.650 Robust Estimate of Covariance: Region PA GPA HS GHS y Region 32.797 14.685 -1.650 -1.0301 -0.2907 -10.601 PA 14.685 25.961 -6.038 -1.5554 0.1318 -25.877 GPA -1.650 -6.038 5.352 0.3838 -0.1690 4.583 HS -1.030 -1.555 0.384 0.9156 -0.0486 3.194 GHS -0.291 0.132 -0.169 -0.0486 0.1205 -0.209 y -10.601 -25.877 4.583 3.1936 -0.2085 70.718 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= fish 158 6 82 8.878062 Best subsample: [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 [20] 20 21 22 23 24 25 26 27 28 30 31 32 35 36 37 42 43 44 45 [39] 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 107 109 110 111 [58] 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 [77] 131 134 135 136 137 139 Outliers: 69 : [1] 29 38 39 40 41 61 62 63 64 65 66 67 68 69 70 71 72 73 74 [20] 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 [39] 94 95 96 97 98 99 100 101 102 103 104 133 140 141 142 143 144 145 146 [58] 147 148 149 150 151 152 153 154 155 156 157 158 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=82); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 8.88 Robust Estimate of Location: Weight Length1 Length2 Length3 Height Width 331.9 24.4 26.6 29.7 31.2 14.7 Robust Estimate of Covariance: Weight Length1 Length2 Length3 Height Width Weight 75099.7 1549.5 1699.2 2119.44 1638.37 -69.924 Length1 1549.5 35.3 38.4 47.11 32.40 -1.697 Length2 1699.2 38.4 41.9 51.43 35.78 -1.797 Length3 2119.4 47.1 51.4 64.10 47.40 -2.549 Height 1638.4 32.4 35.8 47.40 48.97 -2.586 Width -69.9 -1.7 -1.8 -2.55 -2.59 0.833 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pottery 27 6 17 -10.586933 Best subsample: [1] 1 2 4 5 6 9 10 11 13 14 15 19 20 21 22 26 27 Outliers: 9 : [1] 3 8 12 16 17 18 23 24 25 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=17); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -10.6 Robust Estimate of Location: SI AL FE MG CA TI 54.983 15.206 9.700 3.817 5.211 0.859 Robust Estimate of Covariance: SI AL FE MG CA TI SI 20.5823 2.2874 -0.0204 2.1265 -1.8023 0.08821 AL 2.2874 4.0361 -0.6302 -2.4997 0.2084 -0.02038 FE -0.0204 -0.6302 0.2780 0.5338 -0.3512 0.01427 MG 2.1265 -2.4997 0.5338 2.7956 -0.1579 0.02847 CA -1.8023 0.2084 -0.3512 -0.1579 1.2324 -0.03465 TI 0.0882 -0.0204 0.0143 0.0285 -0.0347 0.00175 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= rice 105 6 56 -14.438945 Best subsample: [1] 2 4 6 8 10 12 15 17 18 21 24 27 29 30 31 32 33 34 36 [20] 37 38 41 44 45 47 51 52 53 55 59 60 61 65 67 70 72 76 78 [39] 79 80 81 82 83 84 85 86 90 92 93 94 95 97 98 99 102 105 Outliers: 11 : [1] 9 28 40 42 49 58 62 64 71 75 77 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=56); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -14.4 Robust Estimate of Location: Favor Appearance Taste Stickiness -0.2844 0.0786 -0.1453 0.0378 Toughness Overall_evaluation 0.0833 -0.2363 Robust Estimate of Covariance: Favor Appearance Taste Stickiness Toughness Favor 0.453 0.373 0.449 0.425 -0.198 Appearance 0.373 0.598 0.576 0.568 -0.318 Taste 0.449 0.576 0.723 0.695 -0.379 Stickiness 0.425 0.568 0.695 0.834 -0.470 Toughness -0.198 -0.318 -0.379 -0.470 0.439 Overall_evaluation 0.534 0.661 0.815 0.843 -0.465 Overall_evaluation Favor 0.534 Appearance 0.661 Taste 0.815 Stickiness 0.843 Toughness -0.465 Overall_evaluation 0.986 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= un86 73 7 40 17.017767 Best subsample: [1] 1 9 10 12 14 16 17 18 20 23 24 26 27 31 33 37 39 41 42 45 47 48 49 50 51 [26] 52 55 56 57 60 61 62 63 64 65 67 70 71 72 73 Outliers: 29 : [1] 3 4 5 6 7 8 11 13 15 19 21 22 28 29 30 35 36 38 40 43 44 46 53 54 58 [26] 59 66 68 69 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=40); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 17 Robust Estimate of Location: POP MOR CAR DR GNP DEN TB 20.364 69.750 6.463 0.859 1.133 59.998 0.439 Robust Estimate of Covariance: POP MOR CAR DR GNP DEN TB POP 575.827 243.29 -12.910 -2.4098 -3.0456 160.82 0.4208 MOR 243.291 2376.56 -282.081 -33.9548 -33.9168 -718.68 -1.0522 CAR -12.910 -282.08 56.808 5.6651 6.4636 86.27 0.2616 DR -2.410 -33.95 5.665 0.9009 0.5568 18.60 0.0154 GNP -3.046 -33.92 6.464 0.5568 1.3929 10.67 0.0067 DEN 160.825 -718.68 86.269 18.6034 10.6747 2512.64 -1.1705 TB 0.421 -1.05 0.262 0.0154 0.0067 -1.17 0.0181 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wages 36 9 23 25.658041 Best subsample: [1] 1 2 3 6 7 8 10 11 12 14 15 17 20 21 22 23 25 26 27 33 34 35 36 Outliers: 13 : [1] 4 5 9 13 16 18 19 24 28 29 30 31 32 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=23); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 25.7 Robust Estimate of Location: HRS RATE ERSP ERNO NEIN ASSET AGE DEP 2140.17 2.85 1133.30 307.48 343.26 6539.43 39.57 2.44 SCHOOL 10.07 Robust Estimate of Covariance: HRS RATE ERSP ERNO NEIN ASSET HRS 4433.91 19.7358 -3585.03 -990.563 8227.4 184546 RATE 19.74 0.2393 8.06 1.048 59.2 1373 ERSP -3585.03 8.0565 12399.96 995.108 -4363.3 -78026 ERNO -990.56 1.0481 995.11 2190.581 -426.0 -9925 NEIN 8227.37 59.1712 -4363.27 -425.985 19585.3 441574 ASSET 184546.39 1373.0630 -78025.61 -9925.182 441574.2 10017473 AGE -46.58 -0.2052 18.34 19.517 -83.0 -1898 DEP -6.57 -0.0985 -2.85 0.499 -20.6 -471 SCHOOL 59.89 0.5677 7.54 -4.821 153.0 3541 AGE DEP SCHOOL HRS -4.66e+01 -6.5659 59.885 RATE -2.05e-01 -0.0985 0.568 ERSP 1.83e+01 -2.8522 7.540 ERNO 1.95e+01 0.4986 -4.821 NEIN -8.30e+01 -20.6329 153.022 ASSET -1.90e+03 -471.1344 3540.557 AGE 7.72e-01 0.0412 -0.684 DEP 4.12e-02 0.0873 -0.240 SCHOOL -6.84e-01 -0.2402 1.453 -------------------------------------------------------- ======================================================== > doMCDdata(method="DetMCD"); warnings() Call: doMCDdata(method = "DetMCD") Data Set n p h(alf) LOG(obj) ============================================= bushfire 38 5 22 18.135810 Best subsample: [1] 1 2 3 4 5 6 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Outliers: 16 : [1] 7 8 9 10 11 12 29 30 31 32 33 34 35 36 37 38 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=22) Call: covMcd(x = x, nsamp = "deterministic") iBest: 2, 3, 6; C-step iterations: 2, 3, 3, 2, 2, 3 Log(Det.): 18.1 Robust Estimate of Location: V1 V2 V3 V4 V5 105 147 274 218 279 Robust Estimate of Covariance: V1 V2 V3 V4 V5 V1 346 268 -1692 -381 -311 V2 268 236 -1125 -230 -194 V3 -1692 -1125 9993 2455 1951 V4 -381 -230 2455 647 505 V5 -311 -194 1951 505 398 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= heart 12 2 7 5.678742 Best subsample: [1] 1 3 4 5 7 9 11 Outliers: 0 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=7) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 6; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): 5.68 Robust Estimate of Location: height weight 38.3 33.1 Robust Estimate of Covariance: height weight height 135 259 weight 259 564 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= starsCYG 47 2 25 -8.028718 Best subsample: [1] 1 6 10 12 13 16 23 24 25 26 28 31 32 33 37 38 39 40 41 42 43 44 45 46 47 Outliers: 7 : [1] 7 9 11 14 20 30 34 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=25) Call: covMcd(x = x, nsamp = "deterministic") iBest: 5; C-step iterations: 4, 4, 4, 4, 4, 4 Log(Det.): -8.03 Robust Estimate of Location: log.Te log.light 4.41 4.95 Robust Estimate of Covariance: log.Te log.light log.Te 0.0132 0.0394 log.light 0.0394 0.2743 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= stackloss 21 3 12 6.577286 Best subsample: [1] 4 5 6 7 8 9 11 13 16 18 19 20 Outliers: 2 : [1] 1 2 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=12) Call: covMcd(x = x, nsamp = "deterministic") iBest: 6; C-step iterations: 3, 3, 3, 3, 2, 2 Log(Det.): 6.58 Robust Estimate of Location: Air.Flow Water.Temp Acid.Conc. 58.4 20.5 86.1 Robust Estimate of Covariance: Air.Flow Water.Temp Acid.Conc. Air.Flow 56.3 13.33 26.68 Water.Temp 13.3 8.28 6.98 Acid.Conc. 26.7 6.98 37.97 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= phosphor 18 2 10 7.732906 Best subsample: [1] 2 4 5 7 8 9 11 12 14 16 Outliers: 1 : [1] 6 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=10) Call: covMcd(x = x, nsamp = "deterministic") iBest: 4; C-step iterations: 3, 3, 3, 3, 3, 3 Log(Det.): 7.73 Robust Estimate of Location: inorg organic 12.5 40.8 Robust Estimate of Covariance: inorg organic inorg 124 101 organic 101 197 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= coleman 20 5 13 2.149184 Best subsample: [1] 3 4 5 7 8 12 13 14 16 17 18 19 20 Outliers: 2 : [1] 6 10 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=13) Call: covMcd(x = x, nsamp = "deterministic") iBest: 5; C-step iterations: 2, 2, 2, 2, 2, 3 Log(Det.): 2.15 Robust Estimate of Location: salaryP fatherWc sstatus teacherSc motherLev 2.76 41.08 2.76 25.01 6.27 Robust Estimate of Covariance: salaryP fatherWc sstatus teacherSc motherLev salaryP 0.391 2.96 2.15 0.447 0.110 fatherWc 2.956 1358.64 442.72 12.235 32.842 sstatus 2.146 442.72 205.59 6.464 11.382 teacherSc 0.447 12.23 6.46 1.179 0.510 motherLev 0.110 32.84 11.38 0.510 0.919 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= salinity 28 3 16 1.940763 Best subsample: [1] 1 8 10 12 13 14 15 17 18 20 21 22 25 26 27 28 Outliers: 2 : [1] 5 16 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=16) Call: covMcd(x = x, nsamp = "deterministic") iBest: 4, 5; C-step iterations: 2, 2, 2, 3, 2, 2 Log(Det.): 1.94 Robust Estimate of Location: X1 X2 X3 10.50 2.58 23.12 Robust Estimate of Covariance: X1 X2 X3 X1 10.90243 -0.00457 -1.46 X2 -0.00457 3.85051 -1.95 X3 -1.46156 -1.94604 3.21 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wood 20 5 13 -35.240819 Best subsample: [1] 1 2 3 5 9 11 12 13 14 15 17 18 20 Outliers: 4 : [1] 4 6 8 19 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=13) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): -35.2 Robust Estimate of Location: x1 x2 x3 x4 x5 0.582 0.125 0.530 0.534 0.888 Robust Estimate of Covariance: x1 x2 x3 x4 x5 x1 0.010502 0.001810 2.08e-03 -0.000641 -9.61e-04 x2 0.001810 0.000555 8.76e-04 -0.000203 -4.70e-05 x3 0.002081 0.000876 5.60e-03 -0.001106 -1.26e-05 x4 -0.000641 -0.000203 -1.11e-03 0.004266 2.60e-03 x5 -0.000961 -0.000047 -1.26e-05 0.002602 2.95e-03 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= hbk 75 3 39 -1.045501 Best subsample: [1] 15 17 18 19 20 21 22 23 24 26 27 28 29 32 33 35 36 38 40 41 43 48 49 50 51 [26] 54 55 56 58 59 63 64 66 67 70 71 72 73 74 Outliers: 14 : [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=39) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 6; C-step iterations: 5, 5, 5, 5, 4, 5 Log(Det.): -1.05 Robust Estimate of Location: X1 X2 X3 1.54 1.78 1.69 Robust Estimate of Covariance: X1 X2 X3 X1 1.227 0.055 0.127 X2 0.055 1.249 0.153 X3 0.127 0.153 1.160 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= Animals 28 2 15 14.555543 Best subsample: [1] 1 3 4 5 10 11 17 18 19 20 21 22 23 26 27 Outliers: 14 : [1] 2 6 7 8 9 12 13 14 15 16 23 24 25 28 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=15) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 5, 6; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): 14.6 Robust Estimate of Location: body brain 18.7 64.9 Robust Estimate of Covariance: body brain body 929 1576 brain 1576 5646 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= milk 86 8 47 -28.844954 Best subsample: [1] 5 8 9 10 21 22 23 24 26 30 31 32 33 34 35 36 37 38 39 46 51 53 54 55 56 [26] 57 58 59 60 61 62 63 64 65 66 67 68 69 71 72 76 78 79 81 83 84 86 Outliers: 20 : [1] 1 2 3 11 12 13 14 15 16 17 18 20 27 41 44 47 70 74 75 77 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=47) Call: covMcd(x = x, nsamp = "deterministic") iBest: 5; C-step iterations: 3, 3, 4, 3, 3, 4 Log(Det.): -28.8 Robust Estimate of Location: X1 X2 X3 X4 X5 X6 X7 X8 1.03 35.90 33.10 26.16 25.13 25.07 123.13 14.39 Robust Estimate of Covariance: X1 X2 X3 X4 X5 X6 X7 X1 4.62e-07 8.16e-05 0.000162 0.000159 0.000141 0.000132 0.000604 X2 8.16e-05 1.73e+00 0.199526 0.156489 0.081711 0.201994 1.499551 X3 1.62e-04 2.00e-01 1.148093 0.849962 0.824682 0.847271 0.650728 X4 1.59e-04 1.56e-01 0.849962 0.676140 0.638434 0.649539 0.547968 X5 1.41e-04 8.17e-02 0.824682 0.638434 0.663934 0.648983 0.531229 X6 1.32e-04 2.02e-01 0.847271 0.649539 0.648983 0.683114 0.569006 X7 6.04e-04 1.50e+00 0.650728 0.547968 0.531229 0.569006 3.702975 X8 3.53e-06 2.01e-01 0.182321 0.124043 0.103206 0.118964 0.321120 X8 X1 3.53e-06 X2 2.01e-01 X3 1.82e-01 X4 1.24e-01 X5 1.03e-01 X6 1.19e-01 X7 3.21e-01 X8 1.44e-01 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= lactic 20 2 11 0.359580 Best subsample: [1] 1 2 3 4 5 7 8 9 10 11 12 Outliers: 4 : [1] 17 18 19 20 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=11) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 5, 6; C-step iterations: 3, 3, 3, 2, 3, 3 Log(Det.): 0.36 Robust Estimate of Location: X Y 3.86 5.01 Robust Estimate of Covariance: X Y X 10.6 14.6 Y 14.6 21.3 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pension 18 2 10 16.675508 Best subsample: [1] 1 2 3 4 5 6 8 9 11 12 Outliers: 5 : [1] 14 15 16 17 18 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=10) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 5, 6; C-step iterations: 2, 2, 2, 3, 3, 2 Log(Det.): 16.7 Robust Estimate of Location: Income Reserves 52.3 560.9 Robust Estimate of Covariance: Income Reserves Income 1420 11932 Reserves 11932 208643 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pilot 20 2 11 7.023173 Best subsample: [1] 1 2 3 4 8 11 12 13 14 15 19 Outliers: 0 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=11) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 5, 6; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): 7.02 Robust Estimate of Location: X Y 103.0 68.6 Robust Estimate of Covariance: X Y X 2581 830 Y 830 268 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= vaso 39 2 21 -3.972244 Best subsample: [1] 3 4 8 14 18 19 20 21 22 23 24 25 26 27 28 33 34 35 37 38 39 Outliers: 4 : [1] 1 2 17 31 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=21) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 6; C-step iterations: 3, 3, 3, 2, 2, 3 Log(Det.): -3.97 Robust Estimate of Location: Volume Rate 1.16 1.72 Robust Estimate of Covariance: Volume Rate Volume 0.313 -0.167 Rate -0.167 0.728 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wagnerGrowth 63 6 35 6.511864 Best subsample: [1] 2 3 4 5 6 7 9 10 11 12 13 16 17 18 20 23 25 27 31 32 35 36 38 41 44 [26] 48 51 52 53 54 55 56 57 60 62 Outliers: 15 : [1] 1 8 15 21 22 28 29 33 39 42 43 46 49 50 63 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=35) Call: covMcd(x = x, nsamp = "deterministic") iBest: 3; C-step iterations: 3, 3, 3, 3, 3, 3 Log(Det.): 6.51 Robust Estimate of Location: Region PA GPA HS GHS y 10.91 33.65 -2.05 2.43 0.31 6.98 Robust Estimate of Covariance: Region PA GPA HS GHS y Region 35.136 17.7291 -1.400 -0.6554 -0.4728 -14.930 PA 17.729 28.4297 -5.525 -1.2444 -0.0452 -29.618 GPA -1.400 -5.5245 5.217 0.3954 -0.2152 3.825 HS -0.655 -1.2444 0.395 0.7273 -0.0107 2.151 GHS -0.473 -0.0452 -0.215 -0.0107 0.1728 0.844 y -14.930 -29.6181 3.825 2.1514 0.8440 79.051 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= fish 158 6 82 8.880459 Best subsample: [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 [20] 20 21 22 23 24 25 26 27 35 36 37 42 43 44 45 46 47 48 49 [39] 50 51 52 53 54 55 56 57 58 59 60 106 107 108 109 110 111 112 113 [58] 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 [77] 134 135 136 137 138 139 Outliers: 64 : [1] 29 38 39 40 61 62 63 64 65 67 68 69 72 73 74 75 76 77 78 [20] 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 [39] 98 99 100 101 102 103 104 140 141 142 143 144 145 146 147 148 149 150 151 [58] 152 153 154 155 156 157 158 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=82) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 6; C-step iterations: 7, 7, 7, 5, 6, 6 Log(Det.): 8.88 Robust Estimate of Location: Weight Length1 Length2 Length3 Height Width 316.3 24.1 26.3 29.3 31.0 14.7 Robust Estimate of Covariance: Weight Length1 Length2 Length3 Height Width Weight 64662.2 1412.34 1541.95 1917.21 1420.83 -61.15 Length1 1412.3 34.14 37.04 45.07 29.25 -1.26 Length2 1541.9 37.04 40.26 49.04 32.21 -1.34 Length3 1917.2 45.07 49.04 60.82 43.03 -2.15 Height 1420.8 29.25 32.21 43.03 46.50 -2.66 Width -61.1 -1.26 -1.34 -2.15 -2.66 1.02 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pottery 27 6 17 -10.586933 Best subsample: [1] 1 2 4 5 6 9 10 11 13 14 15 19 20 21 22 26 27 Outliers: 9 : [1] 3 8 12 16 17 18 23 24 25 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=17) Call: covMcd(x = x, nsamp = "deterministic") iBest: 3; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): -10.6 Robust Estimate of Location: SI AL FE MG CA TI 54.983 15.206 9.700 3.817 5.211 0.859 Robust Estimate of Covariance: SI AL FE MG CA TI SI 20.5823 2.2874 -0.0204 2.1265 -1.8023 0.08821 AL 2.2874 4.0361 -0.6302 -2.4997 0.2084 -0.02038 FE -0.0204 -0.6302 0.2780 0.5338 -0.3512 0.01427 MG 2.1265 -2.4997 0.5338 2.7956 -0.1579 0.02847 CA -1.8023 0.2084 -0.3512 -0.1579 1.2324 -0.03465 TI 0.0882 -0.0204 0.0143 0.0285 -0.0347 0.00175 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= rice 105 6 56 -14.423048 Best subsample: [1] 4 6 8 10 13 15 16 17 18 25 27 29 30 31 32 33 34 36 37 [20] 38 44 45 47 51 52 53 55 59 60 65 66 67 70 72 74 76 78 79 [39] 80 81 82 83 84 85 86 90 92 93 94 95 97 98 99 100 101 105 Outliers: 13 : [1] 9 19 28 40 42 43 49 58 62 64 71 75 77 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=56) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1; C-step iterations: 3, 3, 3, 3, 5, 3 Log(Det.): -14.4 Robust Estimate of Location: Favor Appearance Taste Stickiness -0.2950 0.0799 -0.1555 0.0363 Toughness Overall_evaluation 0.0530 -0.2284 Robust Estimate of Covariance: Favor Appearance Taste Stickiness Toughness Favor 0.466 0.389 0.471 0.447 -0.198 Appearance 0.389 0.610 0.592 0.570 -0.293 Taste 0.471 0.592 0.760 0.718 -0.356 Stickiness 0.447 0.570 0.718 0.820 -0.419 Toughness -0.198 -0.293 -0.356 -0.419 0.400 Overall_evaluation 0.557 0.669 0.838 0.846 -0.425 Overall_evaluation Favor 0.557 Appearance 0.669 Taste 0.838 Stickiness 0.846 Toughness -0.425 Overall_evaluation 0.987 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= un86 73 7 40 17.117142 Best subsample: [1] 2 9 10 12 14 16 17 18 19 20 23 24 25 26 27 31 32 33 37 39 42 48 49 50 51 [26] 52 55 56 57 60 61 62 63 64 65 67 70 71 72 73 Outliers: 30 : [1] 3 4 5 6 7 8 11 13 15 21 22 28 29 30 35 36 38 40 41 43 44 45 46 53 54 [26] 58 59 66 68 69 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=40) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 6; C-step iterations: 3, 2, 3, 3, 2, 3 Log(Det.): 17.1 Robust Estimate of Location: POP MOR CAR DR GNP DEN TB 17.036 68.512 6.444 0.877 1.134 64.140 0.433 Robust Estimate of Covariance: POP MOR CAR DR GNP DEN TB POP 361.0402 195.296 -6.28 -0.0191 -2.06758 57.896 -0.06089 MOR 195.2957 2389.391 -279.44 -33.7257 -33.85782 -920.991 -0.99323 CAR -6.2818 -279.436 57.58 5.7749 6.58636 78.132 0.24976 DR -0.0191 -33.726 5.77 0.9066 0.56604 16.926 0.01980 GNP -2.0676 -33.858 6.59 0.5660 1.42442 9.285 0.00682 DEN 57.8963 -920.991 78.13 16.9262 9.28454 3530.116 -0.97487 TB -0.0609 -0.993 0.25 0.0198 0.00682 -0.975 0.01636 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wages 36 9 23 25.722758 Best subsample: [1] 1 2 3 6 7 8 10 11 14 15 17 20 21 22 23 25 27 29 31 33 34 35 36 Outliers: 13 : [1] 4 5 9 12 13 16 18 19 24 26 28 30 32 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=23) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): 25.7 Robust Estimate of Location: HRS RATE ERSP ERNO NEIN ASSET AGE DEP 2150.30 2.90 1117.87 306.83 356.52 6861.39 39.32 2.45 SCHOOL 10.24 Robust Estimate of Covariance: HRS RATE ERSP ERNO NEIN ASSET AGE HRS 3933.64 15.729 -4649.70 -630.546 6647.3 149167 -3.94e+01 RATE 15.73 0.210 5.82 3.237 49.4 1151 -1.35e-01 ERSP -4649.70 5.816 15177.00 464.307 -4624.8 -90372 5.56e+01 ERNO -630.55 3.237 464.31 2297.265 171.8 3479 1.32e+01 NEIN 6647.32 49.379 -4624.80 171.845 16303.6 365111 -4.96e+01 ASSET 149166.83 1151.179 -90372.14 3478.864 365110.6 8242103 -1.16e+03 AGE -39.36 -0.135 55.58 13.223 -49.6 -1164 7.22e-01 DEP -1.61 -0.071 -17.61 -0.431 -16.4 -366 -7.67e-02 SCHOOL 46.30 0.470 -0.21 2.524 119.8 2788 -4.46e-01 DEP SCHOOL HRS -1.6112 46.295 RATE -0.0710 0.470 ERSP -17.6121 -0.210 ERNO -0.4309 2.524 NEIN -16.4023 119.836 ASSET -366.3185 2788.094 AGE -0.0767 -0.446 DEP 0.0849 -0.149 SCHOOL -0.1485 1.122 -------------------------------------------------------- ======================================================== > ## vvvv no timing for 'R CMD Rdiff' outputs > doMCDdata(nrep = 12, time=FALSE) Call: doMCDdata(nrep = 12, time = FALSE) Data Set n p h(alf) LOG(obj) ============================================= bushfire 38 5 22 18.135810 Best subsample: [1] 1 2 3 4 5 6 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Outliers: 16 : [1] 7 8 9 10 11 12 29 30 31 32 33 34 35 36 37 38 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=22); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 18.1 Robust Estimate of Location: V1 V2 V3 V4 V5 105 147 274 218 279 Robust Estimate of Covariance: V1 V2 V3 V4 V5 V1 346 268 -1692 -381 -311 V2 268 236 -1125 -230 -194 V3 -1692 -1125 9993 2455 1951 V4 -381 -230 2455 647 505 V5 -311 -194 1951 505 398 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= heart 12 2 7 5.678742 Best subsample: [1] 1 3 4 5 7 9 11 Outliers: 0 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=7); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 5.68 Robust Estimate of Location: height weight 38.3 33.1 Robust Estimate of Covariance: height weight height 135 259 weight 259 564 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= starsCYG 47 2 25 -8.031215 Best subsample: [1] 1 2 4 6 8 10 12 13 16 24 25 26 28 32 33 37 38 39 40 41 42 43 44 45 46 Outliers: 7 : [1] 7 9 11 14 20 30 34 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=25); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -8.03 Robust Estimate of Location: log.Te log.light 4.41 4.95 Robust Estimate of Covariance: log.Te log.light log.Te 0.0132 0.0394 log.light 0.0394 0.2743 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= stackloss 21 3 12 5.472581 Best subsample: [1] 4 5 6 7 8 9 10 11 12 13 14 20 Outliers: 9 : [1] 1 2 3 15 16 17 18 19 21 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=12); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 5.47 Robust Estimate of Location: Air.Flow Water.Temp Acid.Conc. 59.5 20.8 87.3 Robust Estimate of Covariance: Air.Flow Water.Temp Acid.Conc. Air.Flow 6.29 5.85 5.74 Water.Temp 5.85 9.23 6.14 Acid.Conc. 5.74 6.14 23.25 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= phosphor 18 2 10 6.878847 Best subsample: [1] 3 5 8 9 11 12 13 14 15 17 Outliers: 3 : [1] 1 6 10 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=10); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 6.88 Robust Estimate of Location: inorg organic 13.4 38.8 Robust Estimate of Covariance: inorg organic inorg 129 130 organic 130 182 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= coleman 20 5 13 1.286808 Best subsample: [1] 2 3 4 5 7 8 12 13 14 16 17 19 20 Outliers: 7 : [1] 1 6 9 10 11 15 18 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=13); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 1.29 Robust Estimate of Location: salaryP fatherWc sstatus teacherSc motherLev 2.76 48.38 6.12 25.00 6.40 Robust Estimate of Covariance: salaryP fatherWc sstatus teacherSc motherLev salaryP 0.253 1.79 -0.266 0.151 0.075 fatherWc 1.786 1303.38 330.496 12.604 34.503 sstatus -0.266 330.50 119.888 3.833 10.131 teacherSc 0.151 12.60 3.833 0.785 0.555 motherLev 0.075 34.50 10.131 0.555 1.043 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= salinity 28 3 16 1.326364 Best subsample: [1] 1 2 6 7 8 12 13 14 18 20 21 22 25 26 27 28 Outliers: 4 : [1] 5 16 23 24 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=16); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 1.33 Robust Estimate of Location: X1 X2 X3 10.08 2.78 22.78 Robust Estimate of Covariance: X1 X2 X3 X1 10.44 1.01 -3.19 X2 1.01 3.83 -1.44 X3 -3.19 -1.44 2.39 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wood 20 5 13 -36.270094 Best subsample: [1] 1 2 3 5 9 10 12 13 14 15 17 18 20 Outliers: 7 : [1] 4 6 7 8 11 16 19 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=13); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -36.3 Robust Estimate of Location: x1 x2 x3 x4 x5 0.587 0.122 0.531 0.538 0.892 Robust Estimate of Covariance: x1 x2 x3 x4 x5 x1 0.010025 1.88e-03 0.003153 -0.000586 -1.63e-03 x2 0.001881 4.85e-04 0.001269 -0.000052 2.36e-05 x3 0.003153 1.27e-03 0.006632 -0.000871 3.52e-04 x4 -0.000586 -5.20e-05 -0.000871 0.002846 1.83e-03 x5 -0.001630 2.36e-05 0.000352 0.001828 2.77e-03 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= hbk 75 3 39 -1.047858 Best subsample: [1] 15 16 17 18 19 20 21 22 23 24 26 27 31 32 33 35 36 37 38 40 43 49 50 51 54 [26] 55 56 58 59 61 63 64 66 67 70 71 72 73 74 Outliers: 14 : [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=39); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -1.05 Robust Estimate of Location: X1 X2 X3 1.54 1.78 1.69 Robust Estimate of Covariance: X1 X2 X3 X1 1.227 0.055 0.127 X2 0.055 1.249 0.153 X3 0.127 0.153 1.160 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= Animals 28 2 15 14.555543 Best subsample: [1] 1 3 4 5 10 11 17 18 19 20 21 22 23 26 27 Outliers: 14 : [1] 2 6 7 8 9 12 13 14 15 16 23 24 25 28 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=15); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 14.6 Robust Estimate of Location: body brain 18.7 64.9 Robust Estimate of Covariance: body brain body 929 1576 brain 1576 5646 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= milk 86 8 47 -28.893938 Best subsample: [1] 5 7 8 9 10 21 22 24 26 30 31 32 33 34 35 37 38 39 45 46 51 53 54 55 56 [26] 57 58 59 60 61 63 64 65 66 67 68 69 71 72 76 78 79 80 81 83 84 86 Outliers: 21 : [1] 1 2 3 11 12 13 14 15 16 17 18 20 27 41 44 47 50 70 74 75 77 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=47); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -28.9 Robust Estimate of Location: X1 X2 X3 X4 X5 X6 X7 X8 1.03 35.76 33.05 26.12 25.10 25.04 122.94 14.36 Robust Estimate of Covariance: X1 X2 X3 X4 X5 X6 X7 X1 4.68e-07 8.92e-05 0.00018 0.000172 0.000152 0.000143 0.000625 X2 8.92e-05 1.56e+00 0.21893 0.161497 0.101095 0.197334 1.278580 X3 1.80e-04 2.19e-01 1.17765 0.868701 0.855642 0.864872 0.690044 X4 1.72e-04 1.61e-01 0.86870 0.688578 0.659177 0.660833 0.565031 X5 1.52e-04 1.01e-01 0.85564 0.659177 0.692458 0.667944 0.570607 X6 1.43e-04 1.97e-01 0.86487 0.660833 0.667944 0.693997 0.572373 X7 6.25e-04 1.28e+00 0.69004 0.565031 0.570607 0.572373 3.468208 X8 3.52e-06 1.15e-01 0.17236 0.114700 0.097445 0.107939 0.211966 X8 X1 3.52e-06 X2 1.15e-01 X3 1.72e-01 X4 1.15e-01 X5 9.74e-02 X6 1.08e-01 X7 2.12e-01 X8 1.16e-01 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= lactic 20 2 11 0.359580 Best subsample: [1] 1 2 3 4 5 7 8 9 10 11 12 Outliers: 4 : [1] 17 18 19 20 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=11); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 0.36 Robust Estimate of Location: X Y 3.86 5.01 Robust Estimate of Covariance: X Y X 10.6 14.6 Y 14.6 21.3 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pension 18 2 10 16.675508 Best subsample: [1] 1 2 3 4 5 6 8 9 11 12 Outliers: 5 : [1] 14 15 16 17 18 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=10); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 16.7 Robust Estimate of Location: Income Reserves 52.3 560.9 Robust Estimate of Covariance: Income Reserves Income 1420 11932 Reserves 11932 208643 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pilot 20 2 11 6.487287 Best subsample: [1] 2 3 6 7 9 12 15 16 17 18 20 Outliers: 0 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=11); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 6.49 Robust Estimate of Location: X Y 101.1 67.7 Robust Estimate of Covariance: X Y X 3344 1070 Y 1070 343 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= vaso 39 2 21 -3.972244 Best subsample: [1] 3 4 8 14 18 19 20 21 22 23 24 25 26 27 28 33 34 35 37 38 39 Outliers: 4 : [1] 1 2 17 31 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=21); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -3.97 Robust Estimate of Location: Volume Rate 1.16 1.72 Robust Estimate of Covariance: Volume Rate Volume 0.313 -0.167 Rate -0.167 0.728 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wagnerGrowth 63 6 35 6.569262 Best subsample: [1] 2 3 4 5 6 7 9 10 11 12 14 16 17 18 20 23 25 27 31 32 35 36 38 41 44 [26] 48 52 53 54 55 56 57 58 60 62 Outliers: 17 : [1] 1 8 15 21 22 24 26 28 29 33 39 42 43 46 50 61 63 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=35); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 6.57 Robust Estimate of Location: Region PA GPA HS GHS y 11.318 34.050 -2.049 2.498 0.289 6.650 Robust Estimate of Covariance: Region PA GPA HS GHS y Region 32.797 14.685 -1.650 -1.0301 -0.2907 -10.601 PA 14.685 25.961 -6.038 -1.5554 0.1318 -25.877 GPA -1.650 -6.038 5.352 0.3838 -0.1690 4.583 HS -1.030 -1.555 0.384 0.9156 -0.0486 3.194 GHS -0.291 0.132 -0.169 -0.0486 0.1205 -0.209 y -10.601 -25.877 4.583 3.1936 -0.2085 70.718 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= fish 158 6 82 8.859084 Best subsample: [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 [20] 20 21 22 23 24 25 26 27 28 32 35 36 37 42 43 44 45 46 47 [39] 48 49 50 51 52 53 54 55 56 57 58 59 60 107 109 110 111 112 113 [58] 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 [77] 134 135 136 137 138 139 Outliers: 63 : [1] 29 38 39 40 41 61 62 63 64 65 67 68 69 72 73 74 75 76 77 [20] 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 [39] 97 98 99 100 101 102 103 104 140 142 143 144 146 147 148 149 150 151 152 [58] 153 154 155 156 157 158 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=82); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 8.86 Robust Estimate of Location: Weight Length1 Length2 Length3 Height Width 329.9 24.5 26.6 29.7 31.1 14.7 Robust Estimate of Covariance: Weight Length1 Length2 Length3 Height Width Weight 69083.0 1477.81 1613.6 1992.62 1439.32 -62.12 Length1 1477.8 34.68 37.6 45.51 28.82 -1.31 Length2 1613.6 37.61 40.9 49.52 31.81 -1.40 Length3 1992.6 45.51 49.5 61.16 42.65 -2.25 Height 1439.3 28.82 31.8 42.65 46.74 -2.82 Width -62.1 -1.31 -1.4 -2.25 -2.82 1.01 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pottery 27 6 17 -10.586933 Best subsample: [1] 1 2 4 5 6 9 10 11 13 14 15 19 20 21 22 26 27 Outliers: 9 : [1] 3 8 12 16 17 18 23 24 25 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=17); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -10.6 Robust Estimate of Location: SI AL FE MG CA TI 54.983 15.206 9.700 3.817 5.211 0.859 Robust Estimate of Covariance: SI AL FE MG CA TI SI 20.5823 2.2874 -0.0204 2.1265 -1.8023 0.08821 AL 2.2874 4.0361 -0.6302 -2.4997 0.2084 -0.02038 FE -0.0204 -0.6302 0.2780 0.5338 -0.3512 0.01427 MG 2.1265 -2.4997 0.5338 2.7956 -0.1579 0.02847 CA -1.8023 0.2084 -0.3512 -0.1579 1.2324 -0.03465 TI 0.0882 -0.0204 0.0143 0.0285 -0.0347 0.00175 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= rice 105 6 56 -14.463986 Best subsample: [1] 2 4 6 8 10 12 15 18 21 22 24 29 30 31 32 33 34 36 37 [20] 38 41 44 45 47 51 52 53 54 55 59 61 65 67 68 69 70 72 76 [39] 78 79 80 81 82 83 84 85 86 92 93 94 95 97 98 99 102 105 Outliers: 13 : [1] 9 14 19 28 40 42 49 58 62 71 75 77 89 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=56); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): -14.5 Robust Estimate of Location: Favor Appearance Taste Stickiness -0.2731 0.0600 -0.1468 0.0646 Toughness Overall_evaluation 0.0894 -0.2192 Robust Estimate of Covariance: Favor Appearance Taste Stickiness Toughness Favor 0.388 0.323 0.393 0.389 -0.195 Appearance 0.323 0.503 0.494 0.494 -0.270 Taste 0.393 0.494 0.640 0.629 -0.361 Stickiness 0.389 0.494 0.629 0.815 -0.486 Toughness -0.195 -0.270 -0.361 -0.486 0.451 Overall_evaluation 0.471 0.575 0.723 0.772 -0.457 Overall_evaluation Favor 0.471 Appearance 0.575 Taste 0.723 Stickiness 0.772 Toughness -0.457 Overall_evaluation 0.882 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= un86 73 7 40 16.965868 Best subsample: [1] 1 9 10 12 14 16 17 18 20 23 24 26 27 31 32 33 37 39 42 45 47 48 49 50 51 [26] 52 55 56 57 60 61 62 63 64 65 67 70 71 72 73 Outliers: 31 : [1] 3 4 5 6 7 8 11 13 15 19 21 22 25 28 29 30 34 35 36 38 40 43 44 46 53 [26] 54 58 59 66 68 69 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=40); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 17 Robust Estimate of Location: POP MOR CAR DR GNP DEN TB 21.110 70.143 6.581 0.834 1.169 53.692 0.444 Robust Estimate of Covariance: POP MOR CAR DR GNP DEN TB POP 589.850 246.49 -15.648 -2.0167 -3.86409 287.611 0.34871 MOR 246.492 2370.52 -287.608 -32.1359 -35.44415 -673.259 -1.06385 CAR -15.648 -287.61 58.485 5.7597 6.63441 107.711 0.25254 DR -2.017 -32.14 5.760 0.8406 0.59370 14.998 0.01685 GNP -3.864 -35.44 6.634 0.5937 1.42649 16.827 0.00217 DEN 287.611 -673.26 107.711 14.9975 16.82656 1629.056 -0.45076 TB 0.349 -1.06 0.253 0.0169 0.00217 -0.451 0.01829 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wages 36 9 23 25.781100 Best subsample: [1] 1 2 3 6 7 8 9 10 11 14 17 18 20 21 22 23 25 26 27 33 34 35 36 Outliers: 13 : [1] 4 5 12 13 15 16 19 24 28 29 30 31 32 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=23); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x) Log(Det.): 25.8 Robust Estimate of Location: HRS RATE ERSP ERNO NEIN ASSET AGE DEP 2150.65 2.82 1135.17 293.00 345.61 6552.57 39.44 2.39 SCHOOL 10.09 Robust Estimate of Covariance: HRS RATE ERSP ERNO NEIN ASSET HRS 6092.92 15.1803 -2410.41 -2.40e+03 8807.1 193759 RATE 15.18 0.2017 10.24 -9.14e-01 49.0 1142 ERSP -2410.41 10.2403 13932.45 5.36e+02 -3271.0 -55432 ERNO -2397.41 -0.9137 535.75 2.36e+03 -2020.8 -44920 NEIN 8807.14 48.9680 -3271.05 -2.02e+03 18314.6 409392 ASSET 193758.66 1142.4445 -55432.35 -4.49e+04 409392.2 9215780 AGE -63.71 -0.1916 15.01 3.02e+01 -95.6 -2149 DEP -9.89 -0.0562 -9.65 4.99e+00 -16.7 -374 SCHOOL 62.36 0.4612 23.49 -1.68e+01 136.8 3150 AGE DEP SCHOOL HRS -6.37e+01 -9.8883 62.357 RATE -1.92e-01 -0.0562 0.461 ERSP 1.50e+01 -9.6481 23.487 ERNO 3.02e+01 4.9941 -16.814 NEIN -9.56e+01 -16.6945 136.752 ASSET -2.15e+03 -373.6149 3150.225 AGE 9.46e-01 0.0972 -0.765 DEP 9.72e-02 0.0580 -0.169 SCHOOL -7.65e-01 -0.1688 1.256 -------------------------------------------------------- ======================================================== > doMCDdata(nrep = 12, time=FALSE, method="DetMCD"); warnings() Call: doMCDdata(nrep = 12, time = FALSE, method = "DetMCD") Data Set n p h(alf) LOG(obj) ============================================= bushfire 38 5 22 18.135810 Best subsample: [1] 1 2 3 4 5 6 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Outliers: 16 : [1] 7 8 9 10 11 12 29 30 31 32 33 34 35 36 37 38 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=22) Call: covMcd(x = x, nsamp = "deterministic") iBest: 2, 3, 6; C-step iterations: 2, 3, 3, 2, 2, 3 Log(Det.): 18.1 Robust Estimate of Location: V1 V2 V3 V4 V5 105 147 274 218 279 Robust Estimate of Covariance: V1 V2 V3 V4 V5 V1 346 268 -1692 -381 -311 V2 268 236 -1125 -230 -194 V3 -1692 -1125 9993 2455 1951 V4 -381 -230 2455 647 505 V5 -311 -194 1951 505 398 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= heart 12 2 7 5.678742 Best subsample: [1] 1 3 4 5 7 9 11 Outliers: 0 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=7) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 6; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): 5.68 Robust Estimate of Location: height weight 38.3 33.1 Robust Estimate of Covariance: height weight height 135 259 weight 259 564 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= starsCYG 47 2 25 -8.028718 Best subsample: [1] 1 6 10 12 13 16 23 24 25 26 28 31 32 33 37 38 39 40 41 42 43 44 45 46 47 Outliers: 7 : [1] 7 9 11 14 20 30 34 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=25) Call: covMcd(x = x, nsamp = "deterministic") iBest: 5; C-step iterations: 4, 4, 4, 4, 4, 4 Log(Det.): -8.03 Robust Estimate of Location: log.Te log.light 4.41 4.95 Robust Estimate of Covariance: log.Te log.light log.Te 0.0132 0.0394 log.light 0.0394 0.2743 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= stackloss 21 3 12 6.577286 Best subsample: [1] 4 5 6 7 8 9 11 13 16 18 19 20 Outliers: 2 : [1] 1 2 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=12) Call: covMcd(x = x, nsamp = "deterministic") iBest: 6; C-step iterations: 3, 3, 3, 3, 2, 2 Log(Det.): 6.58 Robust Estimate of Location: Air.Flow Water.Temp Acid.Conc. 58.4 20.5 86.1 Robust Estimate of Covariance: Air.Flow Water.Temp Acid.Conc. Air.Flow 56.3 13.33 26.68 Water.Temp 13.3 8.28 6.98 Acid.Conc. 26.7 6.98 37.97 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= phosphor 18 2 10 7.732906 Best subsample: [1] 2 4 5 7 8 9 11 12 14 16 Outliers: 1 : [1] 6 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=10) Call: covMcd(x = x, nsamp = "deterministic") iBest: 4; C-step iterations: 3, 3, 3, 3, 3, 3 Log(Det.): 7.73 Robust Estimate of Location: inorg organic 12.5 40.8 Robust Estimate of Covariance: inorg organic inorg 124 101 organic 101 197 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= coleman 20 5 13 2.149184 Best subsample: [1] 3 4 5 7 8 12 13 14 16 17 18 19 20 Outliers: 2 : [1] 6 10 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=13) Call: covMcd(x = x, nsamp = "deterministic") iBest: 5; C-step iterations: 2, 2, 2, 2, 2, 3 Log(Det.): 2.15 Robust Estimate of Location: salaryP fatherWc sstatus teacherSc motherLev 2.76 41.08 2.76 25.01 6.27 Robust Estimate of Covariance: salaryP fatherWc sstatus teacherSc motherLev salaryP 0.391 2.96 2.15 0.447 0.110 fatherWc 2.956 1358.64 442.72 12.235 32.842 sstatus 2.146 442.72 205.59 6.464 11.382 teacherSc 0.447 12.23 6.46 1.179 0.510 motherLev 0.110 32.84 11.38 0.510 0.919 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= salinity 28 3 16 1.940763 Best subsample: [1] 1 8 10 12 13 14 15 17 18 20 21 22 25 26 27 28 Outliers: 2 : [1] 5 16 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=16) Call: covMcd(x = x, nsamp = "deterministic") iBest: 4, 5; C-step iterations: 2, 2, 2, 3, 2, 2 Log(Det.): 1.94 Robust Estimate of Location: X1 X2 X3 10.50 2.58 23.12 Robust Estimate of Covariance: X1 X2 X3 X1 10.90243 -0.00457 -1.46 X2 -0.00457 3.85051 -1.95 X3 -1.46156 -1.94604 3.21 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wood 20 5 13 -35.240819 Best subsample: [1] 1 2 3 5 9 11 12 13 14 15 17 18 20 Outliers: 4 : [1] 4 6 8 19 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=13) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): -35.2 Robust Estimate of Location: x1 x2 x3 x4 x5 0.582 0.125 0.530 0.534 0.888 Robust Estimate of Covariance: x1 x2 x3 x4 x5 x1 0.010502 0.001810 2.08e-03 -0.000641 -9.61e-04 x2 0.001810 0.000555 8.76e-04 -0.000203 -4.70e-05 x3 0.002081 0.000876 5.60e-03 -0.001106 -1.26e-05 x4 -0.000641 -0.000203 -1.11e-03 0.004266 2.60e-03 x5 -0.000961 -0.000047 -1.26e-05 0.002602 2.95e-03 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= hbk 75 3 39 -1.045501 Best subsample: [1] 15 17 18 19 20 21 22 23 24 26 27 28 29 32 33 35 36 38 40 41 43 48 49 50 51 [26] 54 55 56 58 59 63 64 66 67 70 71 72 73 74 Outliers: 14 : [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=39) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 6; C-step iterations: 5, 5, 5, 5, 4, 5 Log(Det.): -1.05 Robust Estimate of Location: X1 X2 X3 1.54 1.78 1.69 Robust Estimate of Covariance: X1 X2 X3 X1 1.227 0.055 0.127 X2 0.055 1.249 0.153 X3 0.127 0.153 1.160 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= Animals 28 2 15 14.555543 Best subsample: [1] 1 3 4 5 10 11 17 18 19 20 21 22 23 26 27 Outliers: 14 : [1] 2 6 7 8 9 12 13 14 15 16 23 24 25 28 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=15) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 5, 6; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): 14.6 Robust Estimate of Location: body brain 18.7 64.9 Robust Estimate of Covariance: body brain body 929 1576 brain 1576 5646 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= milk 86 8 47 -28.844954 Best subsample: [1] 5 8 9 10 21 22 23 24 26 30 31 32 33 34 35 36 37 38 39 46 51 53 54 55 56 [26] 57 58 59 60 61 62 63 64 65 66 67 68 69 71 72 76 78 79 81 83 84 86 Outliers: 20 : [1] 1 2 3 11 12 13 14 15 16 17 18 20 27 41 44 47 70 74 75 77 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=47) Call: covMcd(x = x, nsamp = "deterministic") iBest: 5; C-step iterations: 3, 3, 4, 3, 3, 4 Log(Det.): -28.8 Robust Estimate of Location: X1 X2 X3 X4 X5 X6 X7 X8 1.03 35.90 33.10 26.16 25.13 25.07 123.13 14.39 Robust Estimate of Covariance: X1 X2 X3 X4 X5 X6 X7 X1 4.62e-07 8.16e-05 0.000162 0.000159 0.000141 0.000132 0.000604 X2 8.16e-05 1.73e+00 0.199526 0.156489 0.081711 0.201994 1.499551 X3 1.62e-04 2.00e-01 1.148093 0.849962 0.824682 0.847271 0.650728 X4 1.59e-04 1.56e-01 0.849962 0.676140 0.638434 0.649539 0.547968 X5 1.41e-04 8.17e-02 0.824682 0.638434 0.663934 0.648983 0.531229 X6 1.32e-04 2.02e-01 0.847271 0.649539 0.648983 0.683114 0.569006 X7 6.04e-04 1.50e+00 0.650728 0.547968 0.531229 0.569006 3.702975 X8 3.53e-06 2.01e-01 0.182321 0.124043 0.103206 0.118964 0.321120 X8 X1 3.53e-06 X2 2.01e-01 X3 1.82e-01 X4 1.24e-01 X5 1.03e-01 X6 1.19e-01 X7 3.21e-01 X8 1.44e-01 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= lactic 20 2 11 0.359580 Best subsample: [1] 1 2 3 4 5 7 8 9 10 11 12 Outliers: 4 : [1] 17 18 19 20 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=11) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 5, 6; C-step iterations: 3, 3, 3, 2, 3, 3 Log(Det.): 0.36 Robust Estimate of Location: X Y 3.86 5.01 Robust Estimate of Covariance: X Y X 10.6 14.6 Y 14.6 21.3 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pension 18 2 10 16.675508 Best subsample: [1] 1 2 3 4 5 6 8 9 11 12 Outliers: 5 : [1] 14 15 16 17 18 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=10) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 5, 6; C-step iterations: 2, 2, 2, 3, 3, 2 Log(Det.): 16.7 Robust Estimate of Location: Income Reserves 52.3 560.9 Robust Estimate of Covariance: Income Reserves Income 1420 11932 Reserves 11932 208643 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pilot 20 2 11 7.023173 Best subsample: [1] 1 2 3 4 8 11 12 13 14 15 19 Outliers: 0 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=11) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 5, 6; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): 7.02 Robust Estimate of Location: X Y 103.0 68.6 Robust Estimate of Covariance: X Y X 2581 830 Y 830 268 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= vaso 39 2 21 -3.972244 Best subsample: [1] 3 4 8 14 18 19 20 21 22 23 24 25 26 27 28 33 34 35 37 38 39 Outliers: 4 : [1] 1 2 17 31 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=21) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 6; C-step iterations: 3, 3, 3, 2, 2, 3 Log(Det.): -3.97 Robust Estimate of Location: Volume Rate 1.16 1.72 Robust Estimate of Covariance: Volume Rate Volume 0.313 -0.167 Rate -0.167 0.728 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wagnerGrowth 63 6 35 6.511864 Best subsample: [1] 2 3 4 5 6 7 9 10 11 12 13 16 17 18 20 23 25 27 31 32 35 36 38 41 44 [26] 48 51 52 53 54 55 56 57 60 62 Outliers: 15 : [1] 1 8 15 21 22 28 29 33 39 42 43 46 49 50 63 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=35) Call: covMcd(x = x, nsamp = "deterministic") iBest: 3; C-step iterations: 3, 3, 3, 3, 3, 3 Log(Det.): 6.51 Robust Estimate of Location: Region PA GPA HS GHS y 10.91 33.65 -2.05 2.43 0.31 6.98 Robust Estimate of Covariance: Region PA GPA HS GHS y Region 35.136 17.7291 -1.400 -0.6554 -0.4728 -14.930 PA 17.729 28.4297 -5.525 -1.2444 -0.0452 -29.618 GPA -1.400 -5.5245 5.217 0.3954 -0.2152 3.825 HS -0.655 -1.2444 0.395 0.7273 -0.0107 2.151 GHS -0.473 -0.0452 -0.215 -0.0107 0.1728 0.844 y -14.930 -29.6181 3.825 2.1514 0.8440 79.051 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= fish 158 6 82 8.880459 Best subsample: [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 [20] 20 21 22 23 24 25 26 27 35 36 37 42 43 44 45 46 47 48 49 [39] 50 51 52 53 54 55 56 57 58 59 60 106 107 108 109 110 111 112 113 [58] 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 [77] 134 135 136 137 138 139 Outliers: 64 : [1] 29 38 39 40 61 62 63 64 65 67 68 69 72 73 74 75 76 77 78 [20] 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 [39] 98 99 100 101 102 103 104 140 141 142 143 144 145 146 147 148 149 150 151 [58] 152 153 154 155 156 157 158 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=82) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 6; C-step iterations: 7, 7, 7, 5, 6, 6 Log(Det.): 8.88 Robust Estimate of Location: Weight Length1 Length2 Length3 Height Width 316.3 24.1 26.3 29.3 31.0 14.7 Robust Estimate of Covariance: Weight Length1 Length2 Length3 Height Width Weight 64662.2 1412.34 1541.95 1917.21 1420.83 -61.15 Length1 1412.3 34.14 37.04 45.07 29.25 -1.26 Length2 1541.9 37.04 40.26 49.04 32.21 -1.34 Length3 1917.2 45.07 49.04 60.82 43.03 -2.15 Height 1420.8 29.25 32.21 43.03 46.50 -2.66 Width -61.1 -1.26 -1.34 -2.15 -2.66 1.02 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pottery 27 6 17 -10.586933 Best subsample: [1] 1 2 4 5 6 9 10 11 13 14 15 19 20 21 22 26 27 Outliers: 9 : [1] 3 8 12 16 17 18 23 24 25 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=17) Call: covMcd(x = x, nsamp = "deterministic") iBest: 3; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): -10.6 Robust Estimate of Location: SI AL FE MG CA TI 54.983 15.206 9.700 3.817 5.211 0.859 Robust Estimate of Covariance: SI AL FE MG CA TI SI 20.5823 2.2874 -0.0204 2.1265 -1.8023 0.08821 AL 2.2874 4.0361 -0.6302 -2.4997 0.2084 -0.02038 FE -0.0204 -0.6302 0.2780 0.5338 -0.3512 0.01427 MG 2.1265 -2.4997 0.5338 2.7956 -0.1579 0.02847 CA -1.8023 0.2084 -0.3512 -0.1579 1.2324 -0.03465 TI 0.0882 -0.0204 0.0143 0.0285 -0.0347 0.00175 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= rice 105 6 56 -14.423048 Best subsample: [1] 4 6 8 10 13 15 16 17 18 25 27 29 30 31 32 33 34 36 37 [20] 38 44 45 47 51 52 53 55 59 60 65 66 67 70 72 74 76 78 79 [39] 80 81 82 83 84 85 86 90 92 93 94 95 97 98 99 100 101 105 Outliers: 13 : [1] 9 19 28 40 42 43 49 58 62 64 71 75 77 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=56) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1; C-step iterations: 3, 3, 3, 3, 5, 3 Log(Det.): -14.4 Robust Estimate of Location: Favor Appearance Taste Stickiness -0.2950 0.0799 -0.1555 0.0363 Toughness Overall_evaluation 0.0530 -0.2284 Robust Estimate of Covariance: Favor Appearance Taste Stickiness Toughness Favor 0.466 0.389 0.471 0.447 -0.198 Appearance 0.389 0.610 0.592 0.570 -0.293 Taste 0.471 0.592 0.760 0.718 -0.356 Stickiness 0.447 0.570 0.718 0.820 -0.419 Toughness -0.198 -0.293 -0.356 -0.419 0.400 Overall_evaluation 0.557 0.669 0.838 0.846 -0.425 Overall_evaluation Favor 0.557 Appearance 0.669 Taste 0.838 Stickiness 0.846 Toughness -0.425 Overall_evaluation 0.987 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= un86 73 7 40 17.117142 Best subsample: [1] 2 9 10 12 14 16 17 18 19 20 23 24 25 26 27 31 32 33 37 39 42 48 49 50 51 [26] 52 55 56 57 60 61 62 63 64 65 67 70 71 72 73 Outliers: 30 : [1] 3 4 5 6 7 8 11 13 15 21 22 28 29 30 35 36 38 40 41 43 44 45 46 53 54 [26] 58 59 66 68 69 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=40) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1, 2, 3, 4, 6; C-step iterations: 3, 2, 3, 3, 2, 3 Log(Det.): 17.1 Robust Estimate of Location: POP MOR CAR DR GNP DEN TB 17.036 68.512 6.444 0.877 1.134 64.140 0.433 Robust Estimate of Covariance: POP MOR CAR DR GNP DEN TB POP 361.0402 195.296 -6.28 -0.0191 -2.06758 57.896 -0.06089 MOR 195.2957 2389.391 -279.44 -33.7257 -33.85782 -920.991 -0.99323 CAR -6.2818 -279.436 57.58 5.7749 6.58636 78.132 0.24976 DR -0.0191 -33.726 5.77 0.9066 0.56604 16.926 0.01980 GNP -2.0676 -33.858 6.59 0.5660 1.42442 9.285 0.00682 DEN 57.8963 -920.991 78.13 16.9262 9.28454 3530.116 -0.97487 TB -0.0609 -0.993 0.25 0.0198 0.00682 -0.975 0.01636 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wages 36 9 23 25.722758 Best subsample: [1] 1 2 3 6 7 8 10 11 14 15 17 20 21 22 23 25 27 29 31 33 34 35 36 Outliers: 13 : [1] 4 5 9 12 13 16 18 19 24 26 28 30 32 ------------- *MCD() result: -------------------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=23) Call: covMcd(x = x, nsamp = "deterministic") iBest: 1; C-step iterations: 2, 2, 2, 2, 2, 2 Log(Det.): 25.7 Robust Estimate of Location: HRS RATE ERSP ERNO NEIN ASSET AGE DEP 2150.30 2.90 1117.87 306.83 356.52 6861.39 39.32 2.45 SCHOOL 10.24 Robust Estimate of Covariance: HRS RATE ERSP ERNO NEIN ASSET AGE HRS 3933.64 15.729 -4649.70 -630.546 6647.3 149167 -3.94e+01 RATE 15.73 0.210 5.82 3.237 49.4 1151 -1.35e-01 ERSP -4649.70 5.816 15177.00 464.307 -4624.8 -90372 5.56e+01 ERNO -630.55 3.237 464.31 2297.265 171.8 3479 1.32e+01 NEIN 6647.32 49.379 -4624.80 171.845 16303.6 365111 -4.96e+01 ASSET 149166.83 1151.179 -90372.14 3478.864 365110.6 8242103 -1.16e+03 AGE -39.36 -0.135 55.58 13.223 -49.6 -1164 7.22e-01 DEP -1.61 -0.071 -17.61 -0.431 -16.4 -366 -7.67e-02 SCHOOL 46.30 0.470 -0.21 2.524 119.8 2788 -4.46e-01 DEP SCHOOL HRS -1.6112 46.295 RATE -0.0710 0.470 ERSP -17.6121 -0.210 ERNO -0.4309 2.524 NEIN -16.4023 119.836 ASSET -366.3185 2788.094 AGE -0.0767 -0.446 DEP 0.0849 -0.149 SCHOOL -0.1485 1.122 -------------------------------------------------------- ======================================================== > doMCDdata(nrep = 12, time=FALSE, method = "MASS") Call: doMCDdata(nrep = 12, time = FALSE, method = "MASS") Data Set n p h(alf) LOG(obj) ============================================= bushfire 38 5 22 18.135810 Best subsample: [1] 1 2 3 4 5 6 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Outliers: 0 ------------- *MCD() result: -------------------------- $center V1 V2 V3 V4 V5 109.44 149.56 260.32 215.12 276.88 $cov V1 V2 V3 V4 V5 V1 376.67 280.03 -1628.02 -342.01 -285.07 V2 280.03 236.42 -987.14 -188.82 -162.89 V3 -1628.02 -987.14 10203.64 2369.63 1917.58 V4 -342.01 -188.82 2369.63 589.94 468.56 V5 -285.07 -162.89 1917.58 468.56 375.19 $msg [1] "0 singular samples of size 6 out of 3000" $crit [1] 18.136 $best [1] 1 2 3 4 5 6 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 $n.obs [1] 38 $quan [1] 22 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= heart 12 2 7 5.678742 Best subsample: [1] 1 3 4 5 7 9 11 Outliers: 0 ------------- *MCD() result: -------------------------- $center height weight 40.358 38.125 $cov height weight height 142.46 298.91 weight 298.91 679.01 $msg [1] "0 singular samples of size 3 out of 220" $crit [1] 5.6787 $best [1] 1 3 4 5 7 9 11 $n.obs [1] 12 $quan [1] 7 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= starsCYG 47 2 25 -8.031215 Best subsample: [1] 1 2 4 6 8 10 12 13 16 24 25 26 28 32 33 37 38 39 40 41 42 43 44 45 46 Outliers: 0 ------------- *MCD() result: -------------------------- $center log.Te log.light 4.409 4.949 $cov log.Te log.light log.Te 0.011789 0.035179 log.light 0.035179 0.244869 $msg [1] "12 singular samples of size 3 out of 1500" $crit [1] -8.0312 $best [1] 1 2 4 6 8 10 12 13 16 24 25 26 28 32 33 37 38 39 40 41 42 43 44 45 46 $n.obs [1] 47 $quan [1] 25 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= stackloss 21 3 12 5.472581 Best subsample: [1] 4 5 6 7 8 9 10 11 12 13 14 20 Outliers: 0 ------------- *MCD() result: -------------------------- $center Air.Flow Water.Temp Acid.Conc. 56.706 20.235 85.529 $cov Air.Flow Water.Temp Acid.Conc. Air.Flow 23.4706 7.5735 16.1029 Water.Temp 7.5735 6.3162 5.3676 Acid.Conc. 16.1029 5.3676 32.3897 $msg [1] "88 singular samples of size 4 out of 2000" $crit [1] 5.4726 $best [1] 4 5 6 7 8 9 10 11 12 13 14 20 $n.obs [1] 21 $quan [1] 12 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= phosphor 18 2 10 6.878847 Best subsample: [1] 3 5 8 9 11 12 13 14 15 17 Outliers: 0 ------------- *MCD() result: -------------------------- $center inorg organic 15.215 39.385 $cov inorg organic inorg 95.47 116.49 organic 116.49 171.76 $msg [1] "1 singular samples of size 3 out of 816" $crit [1] 6.8788 $best [1] 3 5 8 9 11 12 13 14 15 17 $n.obs [1] 18 $quan [1] 10 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= coleman 20 5 13 1.286808 Best subsample: [1] 2 3 4 5 7 8 12 13 14 16 17 19 20 Outliers: 0 ------------- *MCD() result: -------------------------- $center salaryP fatherWc sstatus teacherSc motherLev 2.8253 44.6267 4.7133 25.1240 6.3320 $cov salaryP fatherWc sstatus teacherSc motherLev salaryP 0.18916 -0.30888 0.14262 0.17971 0.02461 fatherWc -0.30888 683.87325 196.89588 3.30523 17.29381 sstatus 0.14262 196.89588 85.94311 1.68507 5.58631 teacherSc 0.17971 3.30523 1.68507 0.51571 0.21891 motherLev 0.02461 17.29381 5.58631 0.21891 0.50172 $msg [1] "0 singular samples of size 6 out of 3000" $crit [1] 1.2868 $best [1] 2 3 4 5 7 8 12 13 14 16 17 19 20 $n.obs [1] 20 $quan [1] 13 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= salinity 28 3 16 1.326364 Best subsample: [1] 1 2 6 7 8 12 13 14 18 20 21 22 25 26 27 28 Outliers: 0 ------------- *MCD() result: -------------------------- $center X1 X2 X3 10.0826 2.7826 22.7777 $cov X1 X2 X3 X1 9.14332 0.88241 -2.7916 X2 0.88241 3.35968 -1.2622 X3 -2.79160 -1.26222 2.0924 $msg [1] "9 singular samples of size 4 out of 2000" $crit [1] 1.3264 $best [1] 1 2 6 7 8 12 13 14 18 20 21 22 25 26 27 28 $n.obs [1] 28 $quan [1] 16 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wood 20 5 13 -36.270094 Best subsample: [1] 1 2 3 5 9 10 12 13 14 15 17 18 20 Outliers: 0 ------------- *MCD() result: -------------------------- $center x1 x2 x3 x4 x5 0.57613 0.12294 0.53127 0.53760 0.88913 $cov x1 x2 x3 x4 x5 x1 5.2757e-03 7.8749e-04 1.2965e-03 -2.0514e-05 -4.0002e-04 x2 7.8749e-04 2.2023e-04 5.4362e-04 2.3846e-05 2.7230e-05 x3 1.2965e-03 5.4362e-04 3.0435e-03 -7.0560e-04 -4.4395e-05 x4 -2.0514e-05 2.3846e-05 -7.0560e-04 2.1388e-03 1.3511e-03 x5 -4.0002e-04 2.7230e-05 -4.4395e-05 1.3511e-03 1.5946e-03 $msg [1] "0 singular samples of size 6 out of 3000" $crit [1] -36.27 $best [1] 1 2 3 5 9 10 12 13 14 15 17 18 20 $n.obs [1] 20 $quan [1] 13 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= hbk 75 3 39 -1.047858 Best subsample: [1] 15 16 17 18 19 20 21 22 23 24 26 27 31 32 33 35 36 37 38 40 43 49 50 51 54 [26] 55 56 58 59 61 63 64 66 67 70 71 72 73 74 Outliers: 0 ------------- *MCD() result: -------------------------- $center X1 X2 X3 1.5583 1.8033 1.6600 $cov X1 X2 X3 X1 1.124845 0.022175 0.15373 X2 0.022175 1.138972 0.18149 X3 0.153729 0.181492 1.04346 $msg [1] "0 singular samples of size 4 out of 2000" $crit [1] -1.0479 $best [1] 15 16 17 18 19 20 21 22 23 24 26 27 31 32 33 35 36 37 38 40 43 49 50 51 54 [26] 55 56 58 59 61 63 64 66 67 70 71 72 73 74 $n.obs [1] 75 $quan [1] 39 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= Animals 28 2 15 14.555543 Best subsample: [1] 1 3 4 5 10 11 17 18 19 20 21 22 23 26 27 Outliers: 0 ------------- *MCD() result: -------------------------- $center body brain 48.331 127.321 $cov body brain body 4978.6 7801.4 brain 7801.4 21693.7 $msg [1] "0 singular samples of size 3 out of 3276" $crit [1] 14.556 $best [1] 1 3 4 5 10 11 17 18 19 20 21 22 23 26 27 $n.obs [1] 28 $quan [1] 15 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= milk 86 8 47 -28.931843 Best subsample: [1] 5 8 9 10 22 23 24 26 30 31 32 33 34 35 37 38 39 45 46 51 53 54 55 56 57 [26] 58 59 60 61 63 64 65 66 67 68 69 71 72 76 78 79 81 83 84 86 Outliers: 0 ------------- *MCD() result: -------------------------- $center X1 X2 X3 X4 X5 X6 X7 X8 1.0302 35.7571 33.0540 26.1206 25.1000 25.0365 122.9397 14.3559 $cov X1 X2 X3 X4 X5 X6 X7 X1 4.2168e-07 8.0438e-05 0.00016232 0.00015533 0.00013742 0.00012898 0.00056354 X2 8.0438e-05 1.4057e+00 0.19735023 0.14557604 0.09112903 0.17788018 1.15253456 X3 1.6232e-04 1.9735e-01 1.06155658 0.78306196 0.77129032 0.77961086 0.62201741 X4 1.5533e-04 1.4558e-01 0.78306196 0.62069636 0.59419355 0.59568612 0.50932924 X5 1.3742e-04 9.1129e-02 0.77129032 0.59419355 0.62419355 0.60209677 0.51435484 X6 1.2898e-04 1.7788e-01 0.77961086 0.59568612 0.60209677 0.62558116 0.51594726 X7 5.6354e-04 1.1525e+00 0.62201741 0.50932924 0.51435484 0.51594726 3.12630312 X8 3.1754e-06 1.0393e-01 0.15537148 0.10339299 0.08783871 0.09729826 0.19106964 X8 X1 3.1754e-06 X2 1.0393e-01 X3 1.5537e-01 X4 1.0339e-01 X5 8.7839e-02 X6 9.7298e-02 X7 1.9107e-01 X8 1.0417e-01 $msg [1] "30 singular samples of size 9 out of 3000" $crit [1] -28.932 $best [1] 5 8 9 10 22 23 24 26 30 31 32 33 34 35 37 38 39 45 46 51 53 54 55 56 57 [26] 58 59 60 61 63 64 65 66 67 68 69 71 72 76 78 79 81 83 84 86 $n.obs [1] 86 $quan [1] 47 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= lactic 20 2 11 0.359580 Best subsample: [1] 1 2 3 4 5 7 8 9 10 11 12 Outliers: 0 ------------- *MCD() result: -------------------------- $center X Y 4.625 5.925 $cov X Y X 12.117 15.843 Y 15.843 21.705 $msg [1] "23 singular samples of size 3 out of 1140" $crit [1] 0.35958 $best [1] 1 2 3 4 5 7 8 9 10 11 12 $n.obs [1] 20 $quan [1] 11 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pension 18 2 10 16.675508 Best subsample: [1] 1 2 3 4 5 6 8 9 11 12 Outliers: 0 ------------- *MCD() result: -------------------------- $center Income Reserves 59.371 671.279 $cov Income Reserves Income 1787.4 20057 Reserves 20056.9 329315 $msg [1] "0 singular samples of size 3 out of 816" $crit [1] 16.676 $best [1] 1 2 3 4 5 6 8 9 11 12 $n.obs [1] 18 $quan [1] 10 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pilot 20 2 11 6.487287 Best subsample: [1] 2 3 6 7 9 12 15 16 17 18 20 Outliers: 0 ------------- *MCD() result: -------------------------- $center X Y 97.647 66.529 $cov X Y X 2761.49 876.45 Y 876.45 279.01 $msg [1] "21 singular samples of size 3 out of 1140" $crit [1] 6.4873 $best [1] 2 3 6 7 9 12 15 16 17 18 20 $n.obs [1] 20 $quan [1] 11 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= vaso 39 2 21 -3.972244 Best subsample: [1] 3 4 8 14 18 19 20 21 22 23 24 25 26 27 28 33 34 35 37 38 39 Outliers: 0 ------------- *MCD() result: -------------------------- $center Volume Rate 1.2528 1.6717 $cov Volume Rate Volume 0.41213 -0.19935 Rate -0.19935 0.68865 $msg [1] "12 singular samples of size 3 out of 1500" $crit [1] -3.9722 $best [1] 3 4 8 14 18 19 20 21 22 23 24 25 26 27 28 33 34 35 37 38 39 $n.obs [1] 39 $quan [1] 21 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wagnerGrowth 63 6 35 6.511864 Best subsample: [1] 2 3 4 5 6 7 9 10 11 12 13 16 17 18 20 23 25 27 31 32 35 36 38 41 44 [26] 48 51 52 53 54 55 56 57 60 62 Outliers: 0 ------------- *MCD() result: -------------------------- $center Region PA GPA HS GHS y 10.91837 33.63327 -1.99857 2.49347 0.34224 7.59102 $cov Region PA GPA HS GHS y Region 31.32653 15.7286054 -0.47426 -0.891794 -0.3977296 -12.67116 PA 15.72861 24.2037016 -4.27558 -1.253049 -0.0075054 -25.15602 GPA -0.47426 -4.2755818 4.65617 0.236160 -0.1258220 3.50016 HS -0.89179 -1.2530491 0.23616 0.766040 -0.0161350 2.61401 GHS -0.39773 -0.0075054 -0.12582 -0.016135 0.2047386 0.83022 y -12.67116 -25.1560159 3.50016 2.614013 0.8302206 72.72663 $msg [1] "0 singular samples of size 7 out of 3000" $crit [1] 6.5119 $best [1] 2 3 4 5 6 7 9 10 11 12 13 16 17 18 20 23 25 27 31 32 35 36 38 41 44 [26] 48 51 52 53 54 55 56 57 60 62 $n.obs [1] 63 $quan [1] 35 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= fish 158 6 82 8.859084 Best subsample: [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 [20] 20 21 22 23 24 25 26 27 28 32 35 36 37 42 43 44 45 46 47 [39] 48 49 50 51 52 53 54 55 56 57 58 59 60 107 109 110 111 112 113 [58] 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 [77] 134 135 136 137 138 139 Outliers: 0 ------------- *MCD() result: -------------------------- $center Weight Length1 Length2 Length3 Height Width 348.230 24.800 26.995 30.037 30.954 14.774 $cov Weight Length1 Length2 Length3 Height Width Weight 73259.775 1601.93442 1742.15601 2089.13982 1305.5681 -15.32084 Length1 1601.934 38.83231 41.92827 49.17163 25.9685 -0.15279 Length2 1742.156 41.92827 45.35661 53.31114 28.8038 -0.17589 Length3 2089.140 49.17163 53.31114 64.01178 38.9279 -0.86682 Height 1305.568 25.96846 28.80382 38.92787 45.3144 -2.35869 Width -15.321 -0.15279 -0.17589 -0.86682 -2.3587 1.04154 $msg [1] "0 singular samples of size 7 out of 3000" $crit [1] 8.8591 $best [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 [20] 20 21 22 23 24 25 26 27 28 32 35 36 37 42 43 44 45 46 47 [39] 48 49 50 51 52 53 54 55 56 57 58 59 60 107 109 110 111 112 113 [58] 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 [77] 134 135 136 137 138 139 $n.obs [1] 158 $quan [1] 82 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pottery 27 6 17 -10.586933 Best subsample: [1] 1 2 4 5 6 9 10 11 13 14 15 19 20 21 22 26 27 Outliers: 0 ------------- *MCD() result: -------------------------- $center SI AL FE MG CA TI 54.98333 15.20556 9.70000 3.81667 5.21111 0.85944 $cov SI AL FE MG CA TI SI 13.063824 1.451863 -0.0129412 1.349706 -1.143922 0.0559902 AL 1.451863 2.561732 -0.4000000 -1.586569 0.132288 -0.0129379 FE -0.012941 -0.400000 0.1764706 0.338824 -0.222941 0.0090588 MG 1.349706 -1.586569 0.3388235 1.774412 -0.100196 0.0180686 CA -1.143922 0.132288 -0.2229412 -0.100196 0.782222 -0.0219935 TI 0.055990 -0.012938 0.0090588 0.018069 -0.021993 0.0011114 $msg [1] "0 singular samples of size 7 out of 3000" $crit [1] -10.587 $best [1] 1 2 4 5 6 9 10 11 13 14 15 19 20 21 22 26 27 $n.obs [1] 27 $quan [1] 17 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= rice 105 6 56 -14.463986 Best subsample: [1] 2 4 6 8 10 12 15 18 21 22 24 29 30 31 32 33 34 36 37 [20] 38 41 44 45 47 51 52 53 54 55 59 61 65 67 68 69 70 72 76 [39] 78 79 80 81 82 83 84 85 86 92 93 94 95 97 98 99 102 105 Outliers: 0 ------------- *MCD() result: -------------------------- $center Favor Appearance Taste Stickiness -0.261143 0.091901 -0.131516 0.074637 Toughness Overall_evaluation 0.068473 -0.209516 $cov Favor Appearance Taste Stickiness Toughness Favor 0.37972 0.32416 0.37997 0.37180 -0.18998 Appearance 0.32416 0.54625 0.51414 0.51737 -0.29632 Taste 0.37997 0.51414 0.62865 0.62208 -0.35794 Stickiness 0.37180 0.51737 0.62208 0.79643 -0.46309 Toughness -0.18998 -0.29632 -0.35794 -0.46309 0.42723 Overall_evaluation 0.45786 0.59799 0.71700 0.77311 -0.44109 Overall_evaluation Favor 0.45786 Appearance 0.59799 Taste 0.71700 Stickiness 0.77311 Toughness -0.44109 Overall_evaluation 0.88794 $msg [1] "0 singular samples of size 7 out of 3000" $crit [1] -14.464 $best [1] 2 4 6 8 10 12 15 18 21 22 24 29 30 31 32 33 34 36 37 [20] 38 41 44 45 47 51 52 53 54 55 59 61 65 67 68 69 70 72 76 [39] 78 79 80 81 82 83 84 85 86 92 93 94 95 97 98 99 102 105 $n.obs [1] 105 $quan [1] 56 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= un86 73 7 40 16.891076 Best subsample: [1] 9 10 12 14 16 17 18 20 23 24 26 27 31 32 33 37 39 41 42 45 47 48 49 50 51 [26] 52 55 56 57 60 61 62 63 64 65 67 70 71 72 73 Outliers: 0 ------------- *MCD() result: -------------------------- $center POP MOR CAR DR GNP DEN TB 22.52457 68.52174 7.12370 0.90130 1.24739 59.68239 0.44217 $cov POP MOR CAR DR GNP DEN TB POP 770.25647 229.95690 -14.63580 -2.645204 -6.6936678 18.7093 0.6785543 MOR 229.95690 2101.32174 -275.88086 -31.396696 -36.6697198 -653.4731 -0.8918261 CAR -14.63580 -275.88086 65.63702 6.005357 9.2014676 90.1669 0.2285518 DR -2.64520 -31.39670 6.00536 0.849207 0.7307013 17.1491 0.0130815 GNP -6.69367 -36.66972 9.20147 0.730701 2.0202730 14.0366 0.0030325 DEN 18.70930 -653.47305 90.16686 17.149066 14.0366419 2208.4689 -1.1267831 TB 0.67855 -0.89183 0.22855 0.013082 0.0030325 -1.1268 0.0158174 $msg [1] "0 singular samples of size 8 out of 3000" $crit [1] 16.891 $best [1] 9 10 12 14 16 17 18 20 23 24 26 27 31 32 33 37 39 41 42 45 47 48 49 50 51 [26] 52 55 56 57 60 61 62 63 64 65 67 70 71 72 73 $n.obs [1] 73 $quan [1] 40 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wages 36 9 23 25.139287 Best subsample: [1] 1 2 3 6 7 8 10 11 14 17 20 21 22 23 25 26 27 29 31 33 34 35 36 Outliers: 0 ------------- *MCD() result: -------------------------- $center HRS RATE ERSP ERNO NEIN ASSET AGE DEP 2144.5652 2.8484 1115.4348 303.0870 340.9565 6491.8261 39.4000 2.4663 SCHOOL 10.0913 $cov HRS RATE ERSP ERNO NEIN ASSET HRS 3.1775e+03 12.703652 -3.0743e+03 -675.5968 5390.0711 121911.56 RATE 1.2704e+01 0.136870 4.0405e+00 -1.8722 34.5884 805.56 ERSP -3.0743e+03 4.040484 1.0453e+04 165.3696 -3044.3439 -58968.01 ERNO -6.7560e+02 -1.872176 1.6537e+02 1062.9921 -840.3142 -20437.21 NEIN 5.3901e+03 34.588383 -3.0443e+03 -840.3142 12176.3162 273686.95 ASSET 1.2191e+05 805.557943 -5.8968e+04 -20437.2115 273686.9466 6197644.24 AGE -3.9518e+01 -0.177027 3.3995e+01 8.4773 -61.5682 -1451.17 DEP -5.6454e-01 -0.035799 -1.1652e+01 1.6422 -8.4696 -184.88 SCHOOL 3.9405e+01 0.325095 9.9032e-01 -7.9219 90.6542 2110.15 AGE DEP SCHOOL HRS -3.9518e+01 -0.564543 39.405138 RATE -1.7703e-01 -0.035799 0.325095 ERSP 3.3995e+01 -11.651729 0.990316 ERNO 8.4773e+00 1.642154 -7.921937 NEIN -6.1568e+01 -8.469623 90.654150 ASSET -1.4512e+03 -184.879808 2110.152964 AGE 7.4818e-01 -0.042855 -0.569545 DEP -4.2855e-02 0.051342 -0.073393 SCHOOL -5.6955e-01 -0.073393 0.840830 $msg [1] "0 singular samples of size 10 out of 3000" $crit [1] 25.139 $best [1] 1 2 3 6 7 8 10 11 14 17 20 21 22 23 25 26 27 29 31 33 34 35 36 $n.obs [1] 36 $quan [1] 23 -------------------------------------------------------- ======================================================== > > ###--- now the "close to singular" mahalanobis case: > set.seed(6) > (c3 <- covMcd(mort3)) Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=34); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = mort3) Log(Det.): 37.5 Robust Estimate of Location: MO70 MAGE CI68 MDOC DENS NONW EDUC IN69 113.06 287.91 164.84 155.76 19.56 2.23 558.23 106.91 Robust Estimate of Covariance: MO70 MAGE CI68 MDOC DENS NONW EDUC IN69 MO70 540.1 497.3190 62.46 107.37 -20.17 10.6107 -766.8 -107.88 MAGE 497.3 747.7620 4.57 222.17 -75.53 -0.0169 -443.7 3.43 CI68 62.5 4.5663 588.93 -253.26 -65.58 5.1532 -659.0 19.76 MDOC 107.4 222.1690 -253.26 1629.27 -28.72 1.9300 296.0 100.86 DENS -20.2 -75.5255 -65.58 -28.72 124.97 8.2075 -325.0 -42.70 NONW 10.6 -0.0169 5.15 1.93 8.21 2.1620 -85.5 -7.87 EDUC -766.8 -443.7455 -659.00 295.96 -324.95 -85.4782 5850.0 446.59 IN69 -107.9 3.4259 19.76 100.86 -42.70 -7.8714 446.6 111.20 > (c3. <- covMcd(mort3, nsamp="deterministic")) Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=34) Call: covMcd(x = mort3, nsamp = "deterministic") iBest: 2; C-step iterations: 3, 3, 3, 2, 2, 3 Log(Det.): 38 Robust Estimate of Location: MO70 MAGE CI68 MDOC DENS NONW EDUC IN69 111.41 286.52 164.44 155.62 19.50 2.24 556.46 107.68 Robust Estimate of Covariance: MO70 MAGE CI68 MDOC DENS NONW EDUC IN69 MO70 599.72 511.165 84.86 128.44 -5.08 14.133 -643 -113.52 MAGE 511.16 736.618 12.89 184.81 -78.70 0.249 -353 12.26 CI68 84.86 12.891 630.10 -235.70 -58.52 4.483 -657 37.68 MDOC 128.44 184.814 -235.70 1520.48 11.12 6.648 287 61.60 DENS -5.08 -78.700 -58.52 11.12 134.09 9.919 -282 -50.87 NONW 14.13 0.249 4.48 6.65 9.92 2.407 -75 -8.91 EDUC -643.45 -352.533 -657.02 286.66 -282.25 -75.004 5333 375.55 IN69 -113.52 12.257 37.68 61.60 -50.87 -8.912 376 136.08 > stopifnot(log(c3$crit) <= log(c3.$crit), + print(log(c3.$crit / c3$crit)) <= 0.8) [1] 0.013676 > ## see 0.516 / 0.291 {with seed 7} > ## > ## rescale variables: > scaleV <- c(0.1, 0.1, 1, 1, .001, 0.1, 0.1, 100) > mm <- data.matrix(mort3) * rep(scaleV, each = nrow(mort3)) > C3 <- covMcd(mm) > C3. <- covMcd(mm, nsamp="deterministic") > stopifnot(C3$mcd.wt == c3$mcd.wt)# here, not for all seeds! > > ## error ("computationally singular") with old (too high) default tolerance: > try( covMcd(mm, control= rrcov.control(tol = 1e-10)) ) Error in solve.default(cov, ...) : system is computationally singular: reciprocal condition number = 2.4435e-11 > try( covMcd(mm, control= rrcov.control(tol = 1e-10), nsamp="deterministic") ) Error in solve.default(cov, ...) : system is computationally singular: reciprocal condition number = 2.71945e-11 > > showProc.time() Time (user system elapsed): 2.48 0.23 2.72 > > ## "large" examples using different algo branches {seg.fault in version 0.4-4}: > > n <- 600 ## - partitioning will be triggered > set.seed(1) > X <- matrix(round(100*rnorm(n * 3)), n, 3) > (cX <- covMcd(X)) Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=302); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = X) Log(Det.): 25.1 Robust Estimate of Location: [1] 0.141 -6.083 -0.703 Robust Estimate of Covariance: [,1] [,2] [,3] [1,] 9321 -111 376 [2,] -111 10842 242 [3,] 376 242 11330 > cX. <- covMcd(X, nsamp="deterministic", scalefn = scaleTau2) > i <- names(cX); i <- i[!(i %in% c("call", "nsamp", "method", "raw.weights"))] > stopifnot(sum(cX.$raw.weights != cX$raw.weights) <= 2, + all.equal(cX[i], cX.[i], tol= 1/9)) > > n <- 2000 ## - nesting will be triggered > set.seed(4) > X <- matrix(round(100*rnorm(n * 3)), n, 3) > set.seed(1) > summary(cX <- covMcd(X)) # <- show newly activated print.summary.mcd(.) Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=1002); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = X) Log(Det.): 24.9 Robust Estimate of Location: [1] 0.422 -0.422 -0.252 Robust Estimate of Covariance: [,1] [,2] [,3] [1,] 9861.1 -60.6 74 [2,] -60.6 9472.4 -273 [3,] 74.0 -273.0 10034 Eigenvalues: [1] 10172 9836 9360 Robust Distances: Min. 1st Qu. Median Mean 3rd Qu. Max. 0.0171 1.1700 2.4300 3.0100 4.1400 15.8000 Robustness weights: 52 observations c(118,122,130,152,155,159,181,246,266,376,394,432,483,517,608,666,730,774,792,840,913,971,986,1067,1095,1099,1106,1129,1167,1179,1222,1261,1276,1293,1412,1443,1499,1534,1570,1573,1589,1596,1622,1633,1712,1714,1726,1784,1795,1805,1845,1891) are outliers with |weight| = 0 ( < 5e-05); 1948 weights are ~= 1. > cX. <- covMcd(X, nsamp="deterministic", scalefn = scaleTau2) > i2 <- i[i != "mcd.wt"] > stopifnot(print(sum(cX.$raw.weights != cX$raw.weights)) <= 3, # 2 + all.equal(cX[i2], cX.[i2], tol= 1/10))# 1/16 [1] 2 > > set.seed(1) ## testing of 'raw.only' : > cXo <- covMcd(X, raw.only=TRUE) > i <- paste0("raw.", c("cov", "center", "cnp2")) > stopifnot(cXo$raw.only, all.equal(cX[i], cXo[i], tol = 1e-15), + c("best", "mah") %in% setdiff(names(cX), names(cXo))) > showProc.time() Time (user system elapsed): 0.24 0.03 0.27 > > ## Now, some small sample cases: > > ## maximal values: > n. <- 10 > p. <- 8 > set.seed(44) > (X. <- cbind(1:n., round(10*rt(n.,3)), round(10*rt(n.,2)), + matrix(round(10*rnorm(n. * (p.-3)), 1), nrow = n., ncol = p.-3))) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [1,] 1 8 0 -3.6 4.7 3.0 -7.7 -3.3 [2,] 2 -24 3 5.7 -15.6 13.5 -8.9 -10.0 [3,] 3 -1 0 17.0 -1.9 19.0 17.4 -5.8 [4,] 4 -9 2 0.1 -6.0 -11.5 18.6 25.8 [5,] 5 -6 -31 2.4 10.0 9.6 5.4 -4.8 [6,] 6 6 -3 -12.3 -4.6 17.2 -4.6 15.2 [7,] 7 22 16 -2.8 -2.2 -5.2 -2.2 5.6 [8,] 8 23 5 -9.0 -10.4 -2.6 -5.7 2.0 [9,] 9 1 -9 2.1 -5.6 4.1 2.8 -3.0 [10,] 10 -17 -2 -8.8 -7.8 6.5 4.2 17.7 > > ## 2 x 1 ---> Error > r <- tryCatch(covMcd(X.[1:2, 2, drop=FALSE]), error=function(e)e) > stopifnot(inherits(r, "error"), + grepl("too small sample size", r$message)) > > ## 3 x 2 --- ditto > r <- tryCatch(covMcd(X.[1:3, 2:3]), error=function(e)e) > stopifnot(inherits(r, "error"), + grepl("too small sample size", r$message)) > > ## 5 x 3 [ n < 2 p ! ] --- also works for MASS > X <- X.[1:5, 1:3] > set.seed(101) > ## the finite-sample correction is definitely doubtful: > summary(cc <- covMcd(X, use.correction = FALSE)) Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=4); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = X, use.correction = FALSE) Log(Det.): 4.3 Robust Estimate of Location: [1] 2.50 -6.50 1.25 Robust Estimate of Covariance: [,1] [,2] [,3] [1,] 1.797 -5.03 0.539 [2,] -5.033 198.80 -20.671 [3,] 0.539 -20.67 2.427 Eigenvalues: [1] 201.080 1.669 0.274 Robust Distances: Min. 1st Qu. Median Mean 3rd Qu. Max. 2.1 2.1 2.1 752.0 2.1 3750.0 Robustness weights: [1] 1 1 1 1 0 Warning message: In covMcd(X, use.correction = FALSE) : n < 2 * p, i.e., possibly too small sample size > str(cc) ## best = 2 3 4 5 List of 19 $ call : language covMcd(x = X, use.correction = FALSE) $ nsamp : num 500 $ method : chr "Fast MCD(alpha=0.5 ==> h=4); nsamp = 500; (n,k)mini = (300,5)" $ cov : num [1:3, 1:3] 1.797 -5.033 0.539 -5.033 198.8 ... $ center : num [1:3] 2.5 -6.5 1.25 $ n.obs : int 5 $ best : int [1:4] 1 2 3 4 $ alpha : num 0.5 $ quan : num 4 $ raw.cov : num [1:3, 1:3] 2.474 -6.928 0.742 -6.928 273.675 ... $ raw.center : num [1:3] 2.5 -6.5 1.25 $ raw.weights: num [1:5] 1 1 1 1 0 $ crit : num 4.3 $ raw.mah : num [1:5] 1.52 1.52 1.52 1.52 2724.44 $ mah : num [1:5] 2.09 2.09 2.09 2.09 3750.56 $ mcd.wt : num [1:5] 1 1 1 1 0 $ X : num [1:5, 1:3] 1 2 3 4 5 8 -24 -1 -9 -6 ... ..- attr(*, "dimnames")=List of 2 .. ..$ : chr [1:5] "1" "2" "3" "4" ... .. ..$ : NULL $ raw.cnp2 : num [1:2] 1.48 1 $ cnp2 : num [1:2] 1.08 1 - attr(*, "class")= chr "mcd" > if(hasMASS <- requireNamespace("MASS", quietly=TRUE)) { + mcc <- MASS::cov.mcd(X) + stopifnot(cc$best == mcc$best, + all.equal(cc$center, mcc$center, tolerance = 1e-10), + all.equal(c(mcc$cov / cc$raw.cov), rep(0.673549282206, 3*3))) + } > ## p = 4 -- 6 x 4 & 7 x 4 [ n < 2 p ! ] > p <- 4 > n <- 7 > X <- X.[1:n, 1+(1:p)] > stopifnot(dim(X) == c(n,p)) > (cc <- covMcd(X, use.correction = FALSE)) Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=6); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = X, use.correction = FALSE) Log(Det.): 15.6 Robust Estimate of Location: [1] 0.333 3.000 0.683 -4.267 Robust Estimate of Covariance: [,1] [,2] [,3] [,4] [1,] 264.3 50.669 -68.183 85.250 [2,] 50.7 47.688 -0.617 -0.724 [3,] -68.2 -0.617 104.485 -12.456 [4,] 85.2 -0.724 -12.456 47.227 Warning message: In covMcd(X, use.correction = FALSE) : n < 2 * p, i.e., possibly too small sample size > str(cc) ## best = 1 2 4 5 6 7 List of 19 $ call : language covMcd(x = X, use.correction = FALSE) $ nsamp : num 500 $ method : chr "Fast MCD(alpha=0.5 ==> h=6); nsamp = 500; (n,k)mini = (300,5)" $ cov : num [1:4, 1:4] 264.3 50.7 -68.2 85.2 50.7 ... $ center : num [1:4] 0.333 3 0.683 -4.267 $ n.obs : int 7 $ best : int [1:6] 1 2 3 4 6 7 $ alpha : num 0.5 $ quan : num 6 $ raw.cov : num [1:4, 1:4] 319 61.2 -82.3 102.9 61.2 ... $ raw.center : num [1:4] 0.333 3 0.683 -4.267 $ raw.weights: num [1:7] 1 1 1 1 0 1 1 $ crit : num 15.6 $ raw.mah : num [1:7] 2.546 2.477 3.224 0.835 24.765 ... $ mah : num [1:7] 3.07 2.99 3.89 1.01 29.89 ... $ mcd.wt : num [1:7] 1 1 1 1 0 1 1 $ X : num [1:7, 1:4] 8 -24 -1 -9 -6 6 22 0 3 0 ... ..- attr(*, "dimnames")=List of 2 .. ..$ : chr [1:7] "1" "2" "3" "4" ... .. ..$ : NULL $ raw.cnp2 : num [1:2] 1.28 1 $ cnp2 : num [1:2] 1.06 1 - attr(*, "class")= chr "mcd" > if(hasMASS) { + mcc <- MASS::cov.mcd(X) + stopifnot(cc$best == mcc$best, + all.equal(cc$center, mcc$center, tolerance = 1e-10), + all.equal(c(mcc$cov / cc$raw.cov), rep(0.7782486992881, p*p))) + } > > n <- 6 > X <- X[1:n,] > (cc <- covMcd(X, use.correction = FALSE)) Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=5); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = X, use.correction = FALSE) Log(Det.): 7.67 Robust Estimate of Location: [1] -4.00 0.40 1.38 -4.68 Robust Estimate of Covariance: [,1] [,2] [,3] [,4] [1,] 180.4 -26.61 -61.1 92.26 [2,] -26.6 5.64 13.7 -9.48 [3,] -61.1 13.69 126.7 -13.27 [4,] 92.3 -9.48 -13.3 57.67 Warning message: In covMcd(X, use.correction = FALSE) : n < 2 * p, i.e., possibly too small sample size > if(hasMASS) { + mcc <- MASS::cov.mcd(X) + stopifnot(cc$best == mcc$best, + all.equal(cc$center, mcc$center, tolerance = 1e-10), + all.equal(c(mcc$cov / cc$raw.cov), rep(0.7528695976179, p*p))) + } > > showProc.time() Time (user system elapsed): 0.04 0 0.05 > > ## nsamp = "exact" -- here for p=7 > coleman.x <- data.matrix(coleman[, 1:6]) > showSys.time(CcX <- covMcd(coleman.x, nsamp= "exact")) Time user system elapsed Time 0.52 0.00 0.51 > showSys.time(Ccd <- covMcd(coleman.x, nsamp= "deterministic")) Time user system elapsed Time 0 0 0 > stopifnot(all.equal(CcX$best, + c(2, 5:9, 11,13, 14:16, 19:20), tolerance=0), + intersect(CcX$best, Ccd$best) == c(2,5,7,8,13,14,16,19,20), + relErr(CcX$crit, Ccd$crit) < 0.35 # see ~ 0.34 + ) > summary(Ccd) Minimum Covariance Determinant (MCD) estimator approximation. Method: Deterministic MCD(alpha=0.5 ==> h=13) Call: covMcd(x = coleman.x, nsamp = "deterministic") iBest: 1; C-step iterations: 3, 2, 2, 2, 2, 2 Log(Det.): 1.85 Robust Estimate of Location: salaryP fatherWc sstatus teacherSc motherLev Y 2.76 48.38 6.12 25.00 6.40 36.53 Robust Estimate of Covariance: salaryP fatherWc sstatus teacherSc motherLev Y salaryP 0.3367 2.37 -0.354 0.201 0.0997 -0.863 fatherWc 2.3742 1732.58 439.328 16.754 45.8650 254.718 sstatus -0.3541 439.33 159.367 5.095 13.4665 90.241 teacherSc 0.2013 16.75 5.095 1.044 0.7381 2.075 motherLev 0.0997 45.86 13.466 0.738 1.3859 7.403 Y -0.8630 254.72 90.241 2.075 7.4030 57.565 Eigenvalues: [1] 1.89e+03 5.81e+01 5.10e+00 8.07e-01 2.16e-01 1.93e-02 Robust Distances: Min. 1st Qu. Median Mean 3rd Qu. Max. 0.859 1.920 2.850 15.100 21.500 71.900 Robustness weights: 7 observations c(1,6,9,10,11,15,18) are outliers with |weight| = 0 ( < 0.005); 13 weights are ~= 1. > > > demo(determinMCD)## ../demo/determinMCD.R demo(determinMCD) ---- ~~~~~~~~~~~ > library(robustbase) > source(system.file("xtraR/test_MCD.R", package = "robustbase"))#-> doMCDdata() > ##' This version of domcd() runs *both* "Fast" and "deterministic" MCD > ##' @title covMcd() "workhorse" function -- *passed* to and from doMCDdata() > ##' @param x data set: n x p numeric matrix > ##' @param xname "promise" which will be substituted() and printed > ##' @param nrep number of repetition: only sensible for *timing* > ##' @param time > ##' @param short > ##' @param full > ##' @param lname optional: > ##' @param seed optional: > ##' @param trace optional: > domcd.2 <- function(x, xname, nrep=1, + do.exact = NULL, # <- smart default, globally customizable + time = get("time", parent.frame()), # compromise + short = get("short", parent.frame()), # compromise + full = get("full", parent.frame()), # compromise + lname=20, seed=123, trace=FALSE) + { + if(short && full) + stop("you should not set both 'full' and 'short' to TRUE") + force(xname)# => evaluate when it is a data(<>, ..) call + n <- dim(x)[1] + p <- dim(x)[2] + metha <- "FastMCD" + methb <- "detMCD" + if(is.null(do.exact)) { + nLarge <- if(exists("nLarge", mode="numeric")) + get("nLarge", mode="numeric") else 5000 + do.exact <- choose(n, p+1L) < nLarge + } + set.seed(seed); mcda <- covMcd(x, trace=trace) + set.seed(seed); mcdb <- covMcd(x, nsamp="deterministic", trace=trace) + if(do.exact) { + methX <- "exactMCD" + set.seed(seed); mcdX <- covMcd(x, nsamp="exact", trace=trace) + } + mkRes <- function(mcd) + sprintf("%3d %3d %3d %12.6f\n", n,p, mcd$quan, mcd$crit) + xresa <- mkRes(mcda) + xresb <- mkRes(mcdb) + if(do.exact) xresX <- mkRes(mcdX) + if(time) { + tim1 <- function(meth) + sprintf("%10.3f\n", system.time(repMCD(x, nrep, meth))[1]/nrep) + xresa <- paste(xresa, tim1(metha)) + xresb <- paste(xresb, tim1(methb)) + if(do.exact) xresX <- paste(xresX, tim1(methX)) + } + if(full) { + header <- get("header", parent.frame()) + header(time) + } + ## lname: must fit to header(): + x.meth <- paste(xname, format(c(metha, methb, if(do.exact) methX))) + cat(sprintf("%*s", lname, x.meth[1]), xresa) + cat(sprintf("%*s", lname, x.meth[2]), xresb) + if(do.exact) cat(sprintf("%*s", lname, x.meth[3]), xresX) + cat("Best subsamples: \n") + cat(sprintf(" %10s: ", metha)); print(mcda$best) + if(identical(mcdb$best, mcda$best)) + cat(sprintf(" %s is the same as %s\n", methb, metha)) + else { + cat(sprintf(" %10s: ", methb)); print(mcdb$best) + cat(sprintf(" Difference %s - %s:", methb, metha)) + print(setdiff(mcdb$best, mcda$best)) + } + if(do.exact) { + if(identical(mcda$best, mcdX$best)) + cat(sprintf(" %s is the same as %s\n", methX, metha)) + else if(identical(mcdb$best, mcdX$best)) + cat(sprintf(" %s is the same as %s\n", methX, methb)) + else { + cat(sprintf(" %10s: ", methX)); print(mcdX$best) + } + } + if(!short) { + cat("Details about", metha,": ") + ibad <- which(mcda$wt==0) + names(ibad) <- NULL + nbad <- length(ibad) + cat("Outliers: ",nbad,"\n") + if(nbad > 0) + print(ibad) + if(full){ + cat("-------------\n") + print(mcda) + } + cat("--------------------------------------------------------\n") + } + } > doMCDdata(domcd = domcd.2) Call: doMCDdata(domcd = domcd.2) Data Set n p h(alf) LOG(obj) ============================================= bushfire FastMCD 38 5 22 18.135810 bushfire detMCD 38 5 22 18.135810 Best subsamples: FastMCD: [1] 1 2 3 4 5 6 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 detMCD is the same as FastMCD Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=22); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 18.1 Robust Estimate of Location: V1 V2 V3 V4 V5 105 147 274 218 279 Robust Estimate of Covariance: V1 V2 V3 V4 V5 V1 346 268 -1692 -381 -311 V2 268 236 -1125 -230 -194 V3 -1692 -1125 9993 2455 1951 V4 -381 -230 2455 647 505 V5 -311 -194 1951 505 398 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= heart FastMCD 12 2 7 5.678742 heart detMCD 12 2 7 5.678742 heart exactMCD 12 2 7 5.678742 Best subsamples: FastMCD: [1] 1 3 4 5 7 9 11 detMCD is the same as FastMCD exactMCD is the same as FastMCD Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=7); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 5.68 Robust Estimate of Location: height weight 38.3 33.1 Robust Estimate of Covariance: height weight height 135 259 weight 259 564 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= starsCYG FastMCD 47 2 25 -8.031215 starsCYG detMCD 47 2 25 -8.028718 Best subsamples: FastMCD: [1] 1 2 4 6 8 10 12 13 16 24 25 26 28 32 33 37 38 39 40 41 42 43 44 45 46 detMCD: [1] 1 6 10 12 13 16 23 24 25 26 28 31 32 33 37 38 39 40 41 42 43 44 45 46 47 Difference detMCD - FastMCD:[1] 23 31 47 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=25); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): -8.03 Robust Estimate of Location: log.Te log.light 4.41 4.95 Robust Estimate of Covariance: log.Te log.light log.Te 0.0132 0.0394 log.light 0.0394 0.2743 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= stackloss FastMCD 21 3 12 5.472581 stackloss detMCD 21 3 12 6.577286 Best subsamples: FastMCD: [1] 4 5 6 7 8 9 10 11 12 13 14 20 detMCD: [1] 4 5 6 7 8 9 11 13 16 18 19 20 Difference detMCD - FastMCD:[1] 16 18 19 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=12); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 5.47 Robust Estimate of Location: Air.Flow Water.Temp Acid.Conc. 59.5 20.8 87.3 Robust Estimate of Covariance: Air.Flow Water.Temp Acid.Conc. Air.Flow 6.29 5.85 5.74 Water.Temp 5.85 9.23 6.14 Acid.Conc. 5.74 6.14 23.25 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= phosphor FastMCD 18 2 10 6.878847 phosphor detMCD 18 2 10 7.732906 phosphor exactMCD 18 2 10 6.878847 Best subsamples: FastMCD: [1] 3 5 8 9 11 12 13 14 15 17 detMCD: [1] 2 4 5 7 8 9 11 12 14 16 Difference detMCD - FastMCD:[1] 2 4 7 16 exactMCD is the same as FastMCD Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=10); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 6.88 Robust Estimate of Location: inorg organic 13.4 38.8 Robust Estimate of Covariance: inorg organic inorg 129 130 organic 130 182 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= coleman FastMCD 20 5 13 1.286808 coleman detMCD 20 5 13 2.149184 Best subsamples: FastMCD: [1] 2 3 4 5 7 8 12 13 14 16 17 19 20 detMCD: [1] 3 4 5 7 8 12 13 14 16 17 18 19 20 Difference detMCD - FastMCD:[1] 18 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=13); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 1.29 Robust Estimate of Location: salaryP fatherWc sstatus teacherSc motherLev 2.76 48.38 6.12 25.00 6.40 Robust Estimate of Covariance: salaryP fatherWc sstatus teacherSc motherLev salaryP 0.253 1.79 -0.266 0.151 0.075 fatherWc 1.786 1303.38 330.496 12.604 34.503 sstatus -0.266 330.50 119.888 3.833 10.131 teacherSc 0.151 12.60 3.833 0.785 0.555 motherLev 0.075 34.50 10.131 0.555 1.043 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= salinity FastMCD 28 3 16 1.326364 salinity detMCD 28 3 16 1.940763 Best subsamples: FastMCD: [1] 1 2 6 7 8 12 13 14 18 20 21 22 25 26 27 28 detMCD: [1] 1 8 10 12 13 14 15 17 18 20 21 22 25 26 27 28 Difference detMCD - FastMCD:[1] 10 15 17 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=16); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 1.33 Robust Estimate of Location: X1 X2 X3 10.08 2.78 22.78 Robust Estimate of Covariance: X1 X2 X3 X1 10.44 1.01 -3.19 X2 1.01 3.83 -1.44 X3 -3.19 -1.44 2.39 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wood FastMCD 20 5 13 -36.270094 wood detMCD 20 5 13 -35.240819 Best subsamples: FastMCD: [1] 1 2 3 5 9 10 12 13 14 15 17 18 20 detMCD: [1] 1 2 3 5 9 11 12 13 14 15 17 18 20 Difference detMCD - FastMCD:[1] 11 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=13); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): -36.3 Robust Estimate of Location: x1 x2 x3 x4 x5 0.587 0.122 0.531 0.538 0.892 Robust Estimate of Covariance: x1 x2 x3 x4 x5 x1 0.010025 1.88e-03 0.003153 -0.000586 -1.63e-03 x2 0.001881 4.85e-04 0.001269 -0.000052 2.36e-05 x3 0.003153 1.27e-03 0.006632 -0.000871 3.52e-04 x4 -0.000586 -5.20e-05 -0.000871 0.002846 1.83e-03 x5 -0.001630 2.36e-05 0.000352 0.001828 2.77e-03 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= hbk FastMCD 75 3 39 -1.047858 hbk detMCD 75 3 39 -1.045501 Best subsamples: FastMCD: [1] 15 16 17 18 19 20 21 22 23 24 26 27 31 32 33 35 36 37 38 40 43 49 50 51 54 [26] 55 56 58 59 61 63 64 66 67 70 71 72 73 74 detMCD: [1] 15 17 18 19 20 21 22 23 24 26 27 28 29 32 33 35 36 38 40 41 43 48 49 50 51 [26] 54 55 56 58 59 63 64 66 67 70 71 72 73 74 Difference detMCD - FastMCD:[1] 28 29 41 48 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=39); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): -1.05 Robust Estimate of Location: X1 X2 X3 1.54 1.78 1.69 Robust Estimate of Covariance: X1 X2 X3 X1 1.227 0.055 0.127 X2 0.055 1.249 0.153 X3 0.127 0.153 1.160 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= Animals FastMCD 28 2 15 14.555543 Animals detMCD 28 2 15 14.555543 Animals exactMCD 28 2 15 14.555543 Best subsamples: FastMCD: [1] 1 3 4 5 10 11 17 18 19 20 21 22 23 26 27 detMCD is the same as FastMCD exactMCD is the same as FastMCD Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=15); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 14.6 Robust Estimate of Location: body brain 18.7 64.9 Robust Estimate of Covariance: body brain body 929 1576 brain 1576 5646 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= milk FastMCD 86 8 47 -28.931843 milk detMCD 86 8 47 -28.844954 Best subsamples: FastMCD: [1] 5 7 8 9 10 22 23 24 26 30 31 32 33 34 35 37 38 39 45 46 49 51 53 54 55 [26] 56 57 58 59 60 61 63 64 65 66 67 68 69 71 72 76 78 79 81 83 84 86 detMCD: [1] 5 8 9 10 21 22 23 24 26 30 31 32 33 34 35 36 37 38 39 46 51 53 54 55 56 [26] 57 58 59 60 61 62 63 64 65 66 67 68 69 71 72 76 78 79 81 83 84 86 Difference detMCD - FastMCD:[1] 21 36 62 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=47); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): -28.9 Robust Estimate of Location: X1 X2 X3 X4 X5 X6 X7 X8 1.03 35.76 33.05 26.12 25.10 25.04 122.94 14.36 Robust Estimate of Covariance: X1 X2 X3 X4 X5 X6 X7 X1 4.68e-07 8.92e-05 0.00018 0.000172 0.000152 0.000143 0.000625 X2 8.92e-05 1.56e+00 0.21893 0.161497 0.101095 0.197334 1.278580 X3 1.80e-04 2.19e-01 1.17765 0.868701 0.855642 0.864872 0.690044 X4 1.72e-04 1.61e-01 0.86870 0.688578 0.659177 0.660833 0.565031 X5 1.52e-04 1.01e-01 0.85564 0.659177 0.692458 0.667944 0.570607 X6 1.43e-04 1.97e-01 0.86487 0.660833 0.667944 0.693997 0.572373 X7 6.25e-04 1.28e+00 0.69004 0.565031 0.570607 0.572373 3.468208 X8 3.52e-06 1.15e-01 0.17236 0.114700 0.097445 0.107939 0.211966 X8 X1 3.52e-06 X2 1.15e-01 X3 1.72e-01 X4 1.15e-01 X5 9.74e-02 X6 1.08e-01 X7 2.12e-01 X8 1.16e-01 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= lactic FastMCD 20 2 11 0.359580 lactic detMCD 20 2 11 0.359580 lactic exactMCD 20 2 11 0.359580 Best subsamples: FastMCD: [1] 1 2 3 4 5 7 8 9 10 11 12 detMCD is the same as FastMCD exactMCD is the same as FastMCD Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=11); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 0.36 Robust Estimate of Location: X Y 3.86 5.01 Robust Estimate of Covariance: X Y X 10.6 14.6 Y 14.6 21.3 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pension FastMCD 18 2 10 16.675508 pension detMCD 18 2 10 16.675508 pension exactMCD 18 2 10 16.675508 Best subsamples: FastMCD: [1] 1 2 3 4 5 6 8 9 11 12 detMCD is the same as FastMCD exactMCD is the same as FastMCD Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=10); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 16.7 Robust Estimate of Location: Income Reserves 52.3 560.9 Robust Estimate of Covariance: Income Reserves Income 1420 11932 Reserves 11932 208643 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pilot FastMCD 20 2 11 6.487287 pilot detMCD 20 2 11 7.023173 pilot exactMCD 20 2 11 6.487287 Best subsamples: FastMCD: [1] 2 3 6 7 9 12 15 16 17 18 20 detMCD: [1] 1 2 3 4 8 11 12 13 14 15 19 Difference detMCD - FastMCD:[1] 1 4 8 11 13 14 19 exactMCD is the same as FastMCD Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=11); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 6.49 Robust Estimate of Location: X Y 101.1 67.7 Robust Estimate of Covariance: X Y X 3344 1070 Y 1070 343 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= vaso FastMCD 39 2 21 -3.972244 vaso detMCD 39 2 21 -3.972244 Best subsamples: FastMCD: [1] 3 4 8 14 18 19 20 21 22 23 24 25 26 27 28 33 34 35 37 38 39 detMCD is the same as FastMCD Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=21); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): -3.97 Robust Estimate of Location: Volume Rate 1.16 1.72 Robust Estimate of Covariance: Volume Rate Volume 0.313 -0.167 Rate -0.167 0.728 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wagnerGrowth FastMCD 63 6 35 6.610368 wagnerGrowth detMCD 63 6 35 6.511864 Best subsamples: FastMCD: [1] 2 3 4 5 6 7 9 10 11 12 13 16 18 20 23 25 26 27 31 32 35 36 38 41 44 [26] 48 51 52 53 54 55 56 57 60 62 detMCD: [1] 2 3 4 5 6 7 9 10 11 12 13 16 17 18 20 23 25 27 31 32 35 36 38 41 44 [26] 48 51 52 53 54 55 56 57 60 62 Difference detMCD - FastMCD:[1] 17 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=35); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 6.61 Robust Estimate of Location: Region PA GPA HS GHS y 10.872 33.408 -1.919 2.461 0.369 7.879 Robust Estimate of Covariance: Region PA GPA HS GHS y Region 35.3392 15.3047 0.0232 -0.55317 -0.35789 -8.77 PA 15.3047 21.7332 -3.7447 -0.73355 0.09486 -24.54 GPA 0.0232 -3.7447 5.1610 0.23781 -0.14571 5.34 HS -0.5532 -0.7336 0.2378 0.68319 0.00711 2.02 GHS -0.3579 0.0949 -0.1457 0.00711 0.24753 1.40 y -8.7720 -24.5388 5.3353 2.01779 1.39654 78.82 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= fish FastMCD 158 6 82 8.879005 fish detMCD 158 6 82 8.880459 Best subsamples: FastMCD: [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 [20] 20 21 22 23 24 25 26 27 28 30 32 35 36 37 42 43 44 45 46 [39] 47 48 49 50 51 52 53 54 55 56 57 58 59 60 107 109 110 111 113 [58] 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 [77] 134 135 136 137 138 139 detMCD: [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 [20] 20 21 22 23 24 25 26 27 35 36 37 42 43 44 45 46 47 48 49 [39] 50 51 52 53 54 55 56 57 58 59 60 106 107 108 109 110 111 112 113 [58] 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 [77] 134 135 136 137 138 139 Difference detMCD - FastMCD:[1] 106 108 112 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=82); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 8.88 Robust Estimate of Location: Weight Length1 Length2 Length3 Height Width 329.9 24.5 26.6 29.7 31.1 14.7 Robust Estimate of Covariance: Weight Length1 Length2 Length3 Height Width Weight 69083.0 1477.81 1613.6 1992.62 1439.32 -62.12 Length1 1477.8 34.68 37.6 45.51 28.82 -1.31 Length2 1613.6 37.61 40.9 49.52 31.81 -1.40 Length3 1992.6 45.51 49.5 61.16 42.65 -2.25 Height 1439.3 28.82 31.8 42.65 46.74 -2.82 Width -62.1 -1.31 -1.4 -2.25 -2.82 1.01 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= pottery FastMCD 27 6 17 -10.586933 pottery detMCD 27 6 17 -10.586933 Best subsamples: FastMCD: [1] 1 2 4 5 6 9 10 11 13 14 15 19 20 21 22 26 27 detMCD is the same as FastMCD Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=17); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): -10.6 Robust Estimate of Location: SI AL FE MG CA TI 54.983 15.206 9.700 3.817 5.211 0.859 Robust Estimate of Covariance: SI AL FE MG CA TI SI 20.5823 2.2874 -0.0204 2.1265 -1.8023 0.08821 AL 2.2874 4.0361 -0.6302 -2.4997 0.2084 -0.02038 FE -0.0204 -0.6302 0.2780 0.5338 -0.3512 0.01427 MG 2.1265 -2.4997 0.5338 2.7956 -0.1579 0.02847 CA -1.8023 0.2084 -0.3512 -0.1579 1.2324 -0.03465 TI 0.0882 -0.0204 0.0143 0.0285 -0.0347 0.00175 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= rice FastMCD 105 6 56 -14.463986 rice detMCD 105 6 56 -14.423048 Best subsamples: FastMCD: [1] 2 4 6 8 10 12 15 18 21 22 24 29 30 31 32 33 34 36 37 [20] 38 41 44 45 47 51 52 53 54 55 59 61 65 67 68 69 70 72 76 [39] 78 79 80 81 82 83 84 85 86 92 93 94 95 97 98 99 102 105 detMCD: [1] 4 6 8 10 13 15 16 17 18 25 27 29 30 31 32 33 34 36 37 [20] 38 44 45 47 51 52 53 55 59 60 65 66 67 70 72 74 76 78 79 [39] 80 81 82 83 84 85 86 90 92 93 94 95 97 98 99 100 101 105 Difference detMCD - FastMCD: [1] 13 16 17 25 27 60 66 74 90 100 101 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=56); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): -14.5 Robust Estimate of Location: Favor Appearance Taste Stickiness -0.2731 0.0600 -0.1468 0.0646 Toughness Overall_evaluation 0.0894 -0.2192 Robust Estimate of Covariance: Favor Appearance Taste Stickiness Toughness Favor 0.388 0.323 0.393 0.389 -0.195 Appearance 0.323 0.503 0.494 0.494 -0.270 Taste 0.393 0.494 0.640 0.629 -0.361 Stickiness 0.389 0.494 0.629 0.815 -0.486 Toughness -0.195 -0.270 -0.361 -0.486 0.451 Overall_evaluation 0.471 0.575 0.723 0.772 -0.457 Overall_evaluation Favor 0.471 Appearance 0.575 Taste 0.723 Stickiness 0.772 Toughness -0.457 Overall_evaluation 0.882 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= un86 FastMCD 73 7 40 16.891076 un86 detMCD 73 7 40 17.117142 Best subsamples: FastMCD: [1] 9 10 12 14 16 17 18 20 23 24 26 27 31 32 33 37 39 41 42 45 47 48 49 50 51 [26] 52 55 56 57 60 61 62 63 64 65 67 70 71 72 73 detMCD: [1] 2 9 10 12 14 16 17 18 19 20 23 24 25 26 27 31 32 33 37 39 42 48 49 50 51 [26] 52 55 56 57 60 61 62 63 64 65 67 70 71 72 73 Difference detMCD - FastMCD:[1] 2 19 25 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=40); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 16.9 Robust Estimate of Location: POP MOR CAR DR GNP DEN TB 20.364 69.750 6.463 0.859 1.133 59.998 0.439 Robust Estimate of Covariance: POP MOR CAR DR GNP DEN TB POP 575.827 243.29 -12.910 -2.4098 -3.0456 160.82 0.4208 MOR 243.291 2376.56 -282.081 -33.9548 -33.9168 -718.68 -1.0522 CAR -12.910 -282.08 56.808 5.6651 6.4636 86.27 0.2616 DR -2.410 -33.95 5.665 0.9009 0.5568 18.60 0.0154 GNP -3.046 -33.92 6.464 0.5568 1.3929 10.67 0.0067 DEN 160.825 -718.68 86.269 18.6034 10.6747 2512.64 -1.1705 TB 0.421 -1.05 0.262 0.0154 0.0067 -1.17 0.0181 -------------------------------------------------------- Data Set n p h(alf) LOG(obj) ============================================= wages FastMCD 36 9 23 25.658041 wages detMCD 36 9 23 25.722758 Best subsamples: FastMCD: [1] 1 2 3 6 7 8 10 11 12 14 15 17 20 21 22 23 25 26 27 33 34 35 36 detMCD: [1] 1 2 3 6 7 8 10 11 14 15 17 20 21 22 23 25 27 29 31 33 34 35 36 Difference detMCD - FastMCD:[1] 29 31 Details about FastMCD : Outliers: 0 ------------- Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=23); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = x, trace = trace) Log(Det.): 25.7 Robust Estimate of Location: HRS RATE ERSP ERNO NEIN ASSET AGE DEP 2140.17 2.85 1133.30 307.48 343.26 6539.43 39.57 2.44 SCHOOL 10.07 Robust Estimate of Covariance: HRS RATE ERSP ERNO NEIN ASSET HRS 4433.91 19.7358 -3585.03 -990.563 8227.4 184546 RATE 19.74 0.2393 8.06 1.048 59.2 1373 ERSP -3585.03 8.0565 12399.96 995.108 -4363.3 -78026 ERNO -990.56 1.0481 995.11 2190.581 -426.0 -9925 NEIN 8227.37 59.1712 -4363.27 -425.985 19585.3 441574 ASSET 184546.39 1373.0630 -78025.61 -9925.182 441574.2 10017473 AGE -46.58 -0.2052 18.34 19.517 -83.0 -1898 DEP -6.57 -0.0985 -2.85 0.499 -20.6 -471 SCHOOL 59.89 0.5677 7.54 -4.821 153.0 3541 AGE DEP SCHOOL HRS -4.66e+01 -6.5659 59.885 RATE -2.05e-01 -0.0985 0.568 ERSP 1.83e+01 -2.8522 7.540 ERNO 1.95e+01 0.4986 -4.821 NEIN -8.30e+01 -20.6329 153.022 ASSET -1.90e+03 -471.1344 3540.557 AGE 7.72e-01 0.0412 -0.684 DEP 4.12e-02 0.0873 -0.240 SCHOOL -6.84e-01 -0.2402 1.453 -------------------------------------------------------- ======================================================== > warnings() ## in one example n < 2 * p .. Warning message: In covMcd(X, use.correction = FALSE) : n < 2 * p, i.e., possibly too small sample size > ###' Test the exact fit property of CovMcd -------------------------------- > > ##' generate "exact fit" data > d.exact <- function(seed=seed, p=2) { + stopifnot(p >= 1) + set.seed(seed) + n1 <- 45 + x1 <- matrix(rnorm(p*n1), nrow=n1, ncol=p) + x1[,p] <- x1[,p] + 3 + n2 <- 55 + m2 <- 3 + x <- rbind(x1, cbind(matrix(rnorm((p-1)*n2), n2, p-1), rep(m2,n2))) + colnames(x) <- paste0("X", 1:p) + x + } > plot(d.exact(18, p=2)) > pairs(d.exact(1234, p=3), gap=0.1) > for(p in c(2,4)) + for(sid in c(2, 4, 18, 1234)) { + cat("\nseed = ",sid,"; p = ",p,":\n") + d.x <- d.exact(sid, p=p) + d2 <- covMcd(d.x) + ## Gave error {for p=2, seeds 2, 4, 18 also on 64-bit}: + ## At line 729 of file rffastmcd.f + ## Fortran runtime error: Index '6' of dimension 1 of array 'z' above upper bound of 4 + print(d2) + if(FALSE) ## FIXME fails when calling eigen() in "r6pack()" + d2. <- covMcd(d.x, nsamp = "deterministic", scalefn = Qn) + stopifnot(d2$singularity$kind == "on.hyperplane") + } seed = 2 ; p = 2 : Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=51); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = d.x) The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. Log(Det.): -Inf Robust Estimate of Location: X1 X2 0.158 3.000 Robust Estimate of Covariance: X1 X2 X1 4.26 0 X2 0.00 0 seed = 4 ; p = 2 : Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=51); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = d.x) The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. Log(Det.): -Inf Robust Estimate of Location: X1 X2 0.0133 3.0000 Robust Estimate of Covariance: X1 X2 X1 3.59e+00 -8.53e-17 X2 -8.53e-17 0.00e+00 seed = 18 ; p = 2 : Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=51); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = d.x) The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. Log(Det.): -Inf Robust Estimate of Location: X1 X2 0.0277 3.0000 Robust Estimate of Covariance: X1 X2 X1 3.01e+00 -1.14e-16 X2 -5.69e-17 0.00e+00 seed = 1234 ; p = 2 : Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=51); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = d.x) The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. Log(Det.): -Inf Robust Estimate of Location: X1 X2 0.0621 3.0000 Robust Estimate of Covariance: X1 X2 X1 2.86e+00 -1.14e-16 X2 -1.14e-16 0.00e+00 seed = 2 ; p = 4 : Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=52); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = d.x) The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) Log(Det.): -Inf Robust Estimate of Location: X1 X2 X3 X4 0.033 0.197 0.160 3.000 Robust Estimate of Covariance: X1 X2 X3 X4 X1 2.52e+00 0.185 -0.113 -2.61e-16 X2 1.85e-01 2.236 -0.141 0.00e+00 X3 -1.13e-01 -0.141 1.789 0.00e+00 X4 -2.61e-16 0.000 0.000 0.00e+00 seed = 4 ; p = 4 : Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=52); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = d.x) The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) Log(Det.): -Inf Robust Estimate of Location: X1 X2 X3 X4 -0.00568 -0.08741 -0.08413 3.00000 Robust Estimate of Covariance: X1 X2 X3 X4 X1 2.24e+00 1.70e-01 -8.48e-02 1.07e-16 X2 1.70e-01 1.95e+00 -1.09e-01 1.49e-16 X3 -8.48e-02 -1.09e-01 2.20e+00 1.49e-16 X4 1.07e-16 7.45e-17 1.49e-16 0.00e+00 seed = 18 ; p = 4 : Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=52); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = d.x) The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) Log(Det.): -Inf Robust Estimate of Location: X1 X2 X3 X4 -0.1567 0.0415 0.0109 3.0000 Robust Estimate of Covariance: X1 X2 X3 X4 X1 2.59e+00 3.97e-02 -4.40e-01 -4.47e-16 X2 3.97e-02 2.21e+00 -3.44e-01 -7.45e-17 X3 -4.40e-01 -3.44e-01 2.55e+00 2.61e-16 X4 -4.47e-16 -7.45e-17 2.52e-16 0.00e+00 seed = 1234 ; p = 4 : Minimum Covariance Determinant (MCD) estimator approximation. Method: Fast MCD(alpha=0.5 ==> h=52); nsamp = 500; (n,k)mini = (300,5) Call: covMcd(x = d.x) The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) Log(Det.): -Inf Robust Estimate of Location: X1 X2 X3 X4 0.1901 0.0538 0.0847 3.0000 Robust Estimate of Covariance: X1 X2 X3 X4 X1 2.79e+00 2.21e-01 0.1668 5.96e-16 X2 2.21e-01 2.02e+00 -0.0216 1.49e-16 X3 1.67e-01 -2.16e-02 1.7319 0.00e+00 X4 5.96e-16 1.49e-16 0.0000 0.00e+00 > ## TODO: also get examples of other singularity$kind's > Warning messages: 1: In covMcd(d.x) : The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. 2: In covMcd(d.x) : The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. 3: In covMcd(d.x) : The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. 4: In covMcd(d.x) : The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. 5: In covMcd(d.x) : The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) 6: In covMcd(d.x) : The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) 7: In covMcd(d.x) : The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) 8: In covMcd(d.x) : The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) > ## ----------- including simple "exactfit" (code = 3) > warnings() Warning messages: 1: In covMcd(d.x) : The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. 2: In covMcd(d.x) : The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. 3: In covMcd(d.x) : The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. 4: In covMcd(d.x) : The 51-th order statistic of the absolute deviation of variable 2 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the line with equation 0 (x_i1-m_1) + 1 (x_i2-m_2) = 0 with (m_1,m_2) the mean of these observations. 5: In covMcd(d.x) : The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) 6: In covMcd(d.x) : The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) 7: In covMcd(d.x) : The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) 8: In covMcd(d.x) : The 52-th order statistic of the absolute deviation of variable 4 is zero. There are 55 observations (in the entire dataset of 100 obs.) lying on the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p) = 0 with (m_1, ..., m_p) the mean of these observations and coefficients a_i from the vector a <- c(0, 0, 0, 1) > > showProc.time() Time (user system elapsed): 1.35 0.05 1.39 > if(!robustbase:::doExtras()) quit() > proc.time() user system elapsed 4.28 0.37 4.64