Assignment 2
Mason Schweder
6/18/2020
2
KNN classifiers is useful when you have classification data. Given a point KNN can find the k’th nearest neighbor and see which class those neighbors belong too. Then we determine that point falls into the class with the most neighbors in a single class. KNN regression is usefull in a linear or sometimes non-linear regression technique. Example we have a plot of linear data points and want to predict where a point will fall at a given x value. We measure the k’th closest points to a vertical line drawn at that x value and average those points to give an estimate what the value will be at that x.
9
Auto = read.csv("~/Auto.csv")
pairs(Auto)
Auto$cylinders = as.double(Auto$cylinders)
Auto$origin = as.double(Auto$origin)
Auto$horsepower = as.double(Auto$horsepower)
Auto$weight = as.double(Auto$weight)
Auto$year = as.double(Auto$year)
Auto## mpg cylinders displacement horsepower weight acceleration year origin
## 1 18.0 8 307.0 17 3504 12.0 70 1
## 2 15.0 8 350.0 35 3693 11.5 70 1
## 3 18.0 8 318.0 29 3436 11.0 70 1
## 4 16.0 8 304.0 29 3433 12.0 70 1
## 5 17.0 8 302.0 24 3449 10.5 70 1
## 6 15.0 8 429.0 42 4341 10.0 70 1
## 7 14.0 8 454.0 47 4354 9.0 70 1
## 8 14.0 8 440.0 46 4312 8.5 70 1
## 9 14.0 8 455.0 48 4425 10.0 70 1
## 10 15.0 8 390.0 40 3850 8.5 70 1
## 11 15.0 8 383.0 37 3563 10.0 70 1
## 12 14.0 8 340.0 34 3609 8.0 70 1
## 13 15.0 8 400.0 29 3761 9.5 70 1
## 14 14.0 8 455.0 48 3086 10.0 70 1
## 15 24.0 4 113.0 91 2372 15.0 70 3
## 16 22.0 6 198.0 91 2833 15.5 70 1
## 17 18.0 6 199.0 93 2774 15.5 70 1
## 18 21.0 6 200.0 81 2587 16.0 70 1
## 19 27.0 4 97.0 84 2130 14.5 70 3
## 20 26.0 4 97.0 50 1835 20.5 70 2
## 21 25.0 4 110.0 83 2672 17.5 70 2
## 22 24.0 4 107.0 86 2430 14.5 70 2
## 23 25.0 4 104.0 91 2375 17.5 70 2
## 24 26.0 4 121.0 10 2234 12.5 70 2
## 25 21.0 6 199.0 86 2648 15.0 70 1
## 26 10.0 8 360.0 46 4615 14.0 70 1
## 27 10.0 8 307.0 43 4376 15.0 70 1
## 28 11.0 8 318.0 45 4382 13.5 70 1
## 29 9.0 8 304.0 41 4732 18.5 70 1
## 30 27.0 4 97.0 84 2130 14.5 71 3
## 31 28.0 4 140.0 86 2264 15.5 71 1
## 32 25.0 4 113.0 91 2228 14.0 71 3
## 33 25.0 4 98.0 1 2046 19.0 71 1
## 34 19.0 6 232.0 2 2634 13.0 71 1
## 35 16.0 6 225.0 5 3439 15.5 71 1
## 36 17.0 6 250.0 2 3329 15.5 71 1
## 37 19.0 6 250.0 84 3302 15.5 71 1
## 38 18.0 6 232.0 2 3288 15.5 71 1
## 39 14.0 8 350.0 35 4209 12.0 71 1
## 40 14.0 8 400.0 38 4464 11.5 71 1
## 41 14.0 8 351.0 31 4154 13.5 71 1
## 42 14.0 8 318.0 29 4096 13.0 71 1
## 43 12.0 8 383.0 39 4955 11.5 71 1
## 44 13.0 8 400.0 37 4746 12.0 71 1
## 45 13.0 8 400.0 38 5140 12.0 71 1
## 46 18.0 6 258.0 8 2962 13.5 71 1
## 47 22.0 4 140.0 69 2408 19.0 71 1
## 48 19.0 6 250.0 2 3282 15.0 71 1
## 49 18.0 6 250.0 84 3139 14.5 71 1
## 50 23.0 4 122.0 82 2220 14.0 71 1
## 51 28.0 4 116.0 86 2123 14.0 71 2
## 52 30.0 4 79.0 67 2074 19.5 71 2
## 53 30.0 4 88.0 72 2065 14.5 71 2
## 54 31.0 4 71.0 62 1773 19.0 71 3
## 55 35.0 4 72.0 66 1613 18.0 71 3
## 56 27.0 4 97.0 57 1834 19.0 71 2
## 57 26.0 4 91.0 67 1955 20.5 71 1
## 58 24.0 4 113.0 91 2278 15.5 72 3
## 59 25.0 4 97.5 76 2126 17.0 72 1
## 60 23.0 4 97.0 55 2254 23.5 72 2
## 61 20.0 4 140.0 86 2408 19.5 72 1
## 62 21.0 4 122.0 82 2226 16.5 72 1
## 63 13.0 8 350.0 35 4274 12.0 72 1
## 64 14.0 8 400.0 38 4385 12.0 72 1
## 65 15.0 8 318.0 29 4135 13.5 72 1
## 66 14.0 8 351.0 31 4129 13.0 72 1
## 67 17.0 8 304.0 29 3672 11.5 72 1
## 68 11.0 8 429.0 44 4633 11.0 72 1
## 69 13.0 8 350.0 32 4502 13.5 72 1
## 70 12.0 8 350.0 34 4456 13.5 72 1
## 71 13.0 8 400.0 40 4422 12.5 72 1
## 72 19.0 3 70.0 93 2330 13.5 72 3
## 73 15.0 8 304.0 29 3892 12.5 72 1
## 74 13.0 8 307.0 17 4098 14.0 72 1
## 75 13.0 8 302.0 24 4294 16.0 72 1
## 76 14.0 8 318.0 29 4077 14.0 72 1
## 77 18.0 4 121.0 9 2933 14.5 72 2
## 78 22.0 4 121.0 72 2511 18.0 72 2
## 79 21.0 4 120.0 83 2979 19.5 72 2
## 80 26.0 4 96.0 66 2189 18.0 72 2
## 81 22.0 4 122.0 82 2395 16.0 72 1
## 82 28.0 4 97.0 88 2288 17.0 72 3
## 83 23.0 4 120.0 93 2506 14.5 72 3
## 84 28.0 4 98.0 76 2164 15.0 72 1
## 85 27.0 4 97.0 84 2100 16.5 72 3
## 86 13.0 8 350.0 38 4100 13.0 73 1
## 87 14.0 8 304.0 29 3672 11.5 73 1
## 88 13.0 8 350.0 26 3988 13.0 73 1
## 89 14.0 8 302.0 21 4042 14.5 73 1
## 90 15.0 8 318.0 29 3777 12.5 73 1
## 91 12.0 8 429.0 42 4952 11.5 73 1
## 92 13.0 8 400.0 29 4464 12.0 73 1
## 93 13.0 8 351.0 33 4363 13.0 73 1
## 94 14.0 8 318.0 29 4237 14.5 73 1
## 95 13.0 8 440.0 46 4735 11.0 73 1
## 96 12.0 8 455.0 48 4951 11.0 73 1
## 97 13.0 8 360.0 38 3821 11.0 73 1
## 98 18.0 6 225.0 5 3121 16.5 73 1
## 99 16.0 6 250.0 2 3278 18.0 73 1
## 100 18.0 6 232.0 2 2945 16.0 73 1
## 101 18.0 6 250.0 84 3021 16.5 73 1
## 102 23.0 6 198.0 91 2904 16.0 73 1
## 103 26.0 4 97.0 50 1950 21.0 73 2
## 104 11.0 8 400.0 29 4997 14.0 73 1
## 105 12.0 8 400.0 36 4906 12.5 73 1
## 106 13.0 8 360.0 37 4654 13.0 73 1
## 107 12.0 8 350.0 39 4499 12.5 73 1
## 108 18.0 6 232.0 2 2789 15.0 73 1
## 109 20.0 4 97.0 84 2279 19.0 73 3
## 110 21.0 4 140.0 69 2401 19.5 73 1
## 111 22.0 4 108.0 90 2379 16.5 73 3
## 112 18.0 3 70.0 86 2124 13.5 73 3
## 113 19.0 4 122.0 81 2310 18.5 73 1
## 114 21.0 6 155.0 6 2472 14.0 73 1
## 115 26.0 4 98.0 86 2265 15.5 73 2
## 116 15.0 8 350.0 26 4082 13.0 73 1
## 117 16.0 8 400.0 49 4278 9.5 73 1
## 118 29.0 4 68.0 52 1867 19.5 73 2
## 119 24.0 4 116.0 71 2158 15.5 73 2
## 120 20.0 4 114.0 87 2582 14.0 73 2
## 121 19.0 4 121.0 9 2868 15.5 73 2
## 122 15.0 8 318.0 29 3399 11.0 73 1
## 123 24.0 4 121.0 8 2660 14.0 73 2
## 124 20.0 6 156.0 14 2807 13.5 73 3
## 125 11.0 8 350.0 39 3664 11.0 73 1
## 126 20.0 6 198.0 91 3102 16.5 74 1
## 127 21.0 6 200.0 1 2875 17.0 74 1
## 128 19.0 6 232.0 2 2901 16.0 74 1
## 129 15.0 6 250.0 2 3336 17.0 74 1
## 130 31.0 4 79.0 64 1950 19.0 74 3
## 131 26.0 4 122.0 76 2451 16.5 74 1
## 132 32.0 4 71.0 62 1836 21.0 74 3
## 133 25.0 4 140.0 71 2542 17.0 74 1
## 134 16.0 6 250.0 2 3781 17.0 74 1
## 135 16.0 6 258.0 8 3632 18.0 74 1
## 136 18.0 6 225.0 5 3613 16.5 74 1
## 137 16.0 8 302.0 24 4141 14.0 74 1
## 138 13.0 8 350.0 29 4699 14.5 74 1
## 139 14.0 8 318.0 29 4457 13.5 74 1
## 140 14.0 8 302.0 24 4638 16.0 74 1
## 141 14.0 8 304.0 29 4257 15.5 74 1
## 142 29.0 4 98.0 79 2219 16.5 74 2
## 143 26.0 4 79.0 64 1963 15.5 74 2
## 144 26.0 4 97.0 74 2300 14.5 74 2
## 145 31.0 4 76.0 53 1649 16.5 74 3
## 146 32.0 4 83.0 58 2003 19.0 74 3
## 147 28.0 4 90.0 71 2125 14.5 74 1
## 148 24.0 4 90.0 71 2108 15.5 74 2
## 149 26.0 4 116.0 71 2246 14.0 74 2
## 150 24.0 4 120.0 93 2489 15.0 74 3
## 151 26.0 4 108.0 89 2391 15.5 74 3
## 152 31.0 4 79.0 64 2000 16.0 74 2
## 153 19.0 6 225.0 91 3264 16.0 75 1
## 154 18.0 6 250.0 5 3459 16.0 75 1
## 155 15.0 6 250.0 69 3432 21.0 75 1
## 156 15.0 6 250.0 69 3158 19.5 75 1
## 157 16.0 8 400.0 37 4668 11.5 75 1
## 158 15.0 8 350.0 26 4440 14.0 75 1
## 159 16.0 8 318.0 29 4498 14.5 75 1
## 160 14.0 8 351.0 27 4657 13.5 75 1
## 161 17.0 6 231.0 8 3907 21.0 75 1
## 162 16.0 6 250.0 5 3897 18.5 75 1
## 163 15.0 6 258.0 8 3730 19.0 75 1
## 164 18.0 6 225.0 91 3785 19.0 75 1
## 165 21.0 6 231.0 8 3039 15.0 75 1
## 166 20.0 8 262.0 8 3221 13.5 75 1
## 167 13.0 8 302.0 16 3169 12.0 75 1
## 168 29.0 4 97.0 71 2171 16.0 75 3
## 169 23.0 4 140.0 79 2639 17.0 75 1
## 170 20.0 6 232.0 2 2914 16.0 75 1
## 171 23.0 4 140.0 74 2592 18.5 75 1
## 172 24.0 4 134.0 92 2702 13.5 75 3
## 173 25.0 4 90.0 68 2223 16.5 75 2
## 174 24.0 4 119.0 93 2545 17.0 75 3
## 175 18.0 6 171.0 93 2984 14.5 75 1
## 176 29.0 4 90.0 67 1937 14.0 75 2
## 177 19.0 6 232.0 86 3211 17.0 75 1
## 178 23.0 4 115.0 91 2694 15.0 75 2
## 179 23.0 4 120.0 84 2957 17.0 75 2
## 180 22.0 4 121.0 94 2945 14.5 75 2
## 181 25.0 4 121.0 11 2671 13.5 75 2
## 182 33.0 4 91.0 54 1795 17.5 75 3
## 183 28.0 4 107.0 82 2464 15.5 76 2
## 184 25.0 4 116.0 77 2220 16.9 76 2
## 185 25.0 4 140.0 88 2572 14.9 76 1
## 186 26.0 4 98.0 75 2255 17.7 76 1
## 187 27.0 4 101.0 79 2202 15.3 76 2
## 188 17.5 8 305.0 24 4215 13.0 76 1
## 189 16.0 8 318.0 29 4190 13.0 76 1
## 190 15.5 8 304.0 13 3962 13.9 76 1
## 191 14.5 8 351.0 30 4215 12.8 76 1
## 192 22.0 6 225.0 2 3233 15.4 76 1
## 193 22.0 6 250.0 5 3353 14.5 76 1
## 194 24.0 6 200.0 77 3012 17.6 76 1
## 195 22.5 6 232.0 86 3085 17.6 76 1
## 196 29.0 4 85.0 53 2035 22.2 76 1
## 197 24.5 4 98.0 57 2164 22.1 76 1
## 198 29.0 4 90.0 67 1937 14.2 76 2
## 199 33.0 4 91.0 54 1795 17.4 76 3
## 200 20.0 6 225.0 2 3651 17.7 76 1
## 201 18.0 6 250.0 74 3574 21.0 76 1
## 202 18.5 6 250.0 8 3645 16.2 76 1
## 203 17.5 6 258.0 91 3193 17.8 76 1
## 204 29.5 4 97.0 68 1825 12.2 76 2
## 205 32.0 4 85.0 67 1990 17.0 76 3
## 206 28.0 4 97.0 71 2155 16.4 76 3
## 207 26.5 4 140.0 69 2565 13.6 76 1
## 208 20.0 4 130.0 3 3150 15.7 76 2
## 209 13.0 8 318.0 29 3940 13.2 76 1
## 210 19.0 4 120.0 84 3270 21.9 76 2
## 211 19.0 6 156.0 7 2930 15.5 76 3
## 212 16.5 6 168.0 13 3820 16.7 76 2
## 213 16.5 8 350.0 39 4380 12.1 76 1
## 214 13.0 8 350.0 26 4055 12.0 76 1
## 215 13.0 8 302.0 17 3870 15.0 76 1
## 216 13.0 8 318.0 29 3755 14.0 76 1
## 217 31.5 4 98.0 65 2045 18.5 77 3
## 218 30.0 4 111.0 76 2155 14.8 77 1
## 219 36.0 4 79.0 56 1825 18.6 77 2
## 220 25.5 4 122.0 92 2300 15.5 77 1
## 221 33.5 4 85.0 67 1945 16.8 77 3
## 222 17.5 8 305.0 26 3880 12.5 77 1
## 223 17.0 8 260.0 8 4060 19.0 77 1
## 224 15.5 8 318.0 26 4140 13.7 77 1
## 225 15.0 8 302.0 17 4295 14.9 77 1
## 226 17.5 6 250.0 8 3520 16.4 77 1
## 227 20.5 6 231.0 5 3425 16.9 77 1
## 228 19.0 6 225.0 2 3630 17.7 77 1
## 229 18.5 6 250.0 94 3525 19.0 77 1
## 230 16.0 8 400.0 39 4220 11.1 77 1
## 231 15.5 8 350.0 37 4165 11.4 77 1
## 232 15.5 8 400.0 40 4325 12.2 77 1
## 233 16.0 8 351.0 28 4335 14.5 77 1
## 234 29.0 4 97.0 74 1940 14.5 77 2
## 235 24.5 4 151.0 84 2740 16.0 77 1
## 236 26.0 4 97.0 71 2265 18.2 77 3
## 237 25.5 4 140.0 85 2755 15.8 77 1
## 238 30.5 4 98.0 60 2051 17.0 77 1
## 239 33.5 4 98.0 79 2075 15.9 77 1
## 240 30.0 4 97.0 64 1985 16.4 77 3
## 241 30.5 4 97.0 74 2190 14.1 77 2
## 242 22.0 6 146.0 93 2815 14.5 77 3
## 243 21.5 4 121.0 8 2600 12.8 77 2
## 244 21.5 3 80.0 8 2720 13.5 77 3
## 245 43.1 4 90.0 51 1985 21.5 78 2
## 246 36.1 4 98.0 63 1800 14.4 78 1
## 247 32.8 4 78.0 53 1985 19.4 78 3
## 248 39.4 4 85.0 67 2070 18.6 78 3
## 249 36.1 4 91.0 57 1800 16.4 78 3
## 250 19.9 8 260.0 8 3365 15.5 78 1
## 251 19.4 8 318.0 24 3735 13.2 78 1
## 252 20.2 8 302.0 23 3570 12.8 78 1
## 253 19.2 6 231.0 5 3535 19.2 78 1
## 254 20.5 6 200.0 91 3155 18.2 78 1
## 255 20.2 6 200.0 81 2965 15.8 78 1
## 256 25.1 4 140.0 84 2720 15.4 78 1
## 257 20.5 6 225.0 2 3430 17.2 78 1
## 258 19.4 6 232.0 86 3210 17.2 78 1
## 259 20.6 6 231.0 5 3380 15.8 78 1
## 260 20.8 6 200.0 81 3070 16.7 78 1
## 261 18.6 6 225.0 8 3620 18.7 78 1
## 262 18.1 6 258.0 13 3410 15.1 78 1
## 263 19.2 8 305.0 26 3425 13.2 78 1
## 264 17.7 6 231.0 35 3445 13.4 78 1
## 265 18.1 8 302.0 23 3205 11.2 78 1
## 266 17.5 8 318.0 24 4080 13.7 78 1
## 267 30.0 4 98.0 65 2155 16.5 78 1
## 268 27.5 4 134.0 91 2560 14.2 78 3
## 269 27.2 4 119.0 93 2300 14.7 78 3
## 270 30.9 4 105.0 71 2230 14.5 78 1
## 271 21.1 4 134.0 91 2515 14.8 78 3
## 272 23.2 4 156.0 5 2745 16.7 78 1
## 273 23.8 4 151.0 81 2855 17.6 78 1
## 274 23.9 4 119.0 93 2405 14.9 78 3
## 275 20.3 5 131.0 4 2830 15.9 78 2
## 276 17.0 6 163.0 15 3140 13.6 78 2
## 277 21.6 4 121.0 11 2795 15.7 78 2
## 278 16.2 6 163.0 19 3410 15.8 78 2
## 279 31.5 4 89.0 68 1990 14.9 78 2
## 280 29.5 4 98.0 65 2135 16.6 78 3
## 281 21.5 6 231.0 11 3245 15.4 79 1
## 282 19.8 6 200.0 81 2990 18.2 79 1
## 283 22.3 4 140.0 84 2890 17.3 79 1
## 284 20.2 6 232.0 86 3265 18.2 79 1
## 285 20.6 6 225.0 8 3360 16.6 79 1
## 286 17.0 8 305.0 17 3840 15.4 79 1
## 287 17.6 8 302.0 16 3725 13.4 79 1
## 288 16.5 8 351.0 22 3955 13.2 79 1
## 289 18.2 8 318.0 20 3830 15.2 79 1
## 290 16.9 8 350.0 32 4360 14.9 79 1
## 291 15.5 8 351.0 25 4054 14.3 79 1
## 292 19.2 8 267.0 15 3605 15.0 79 1
## 293 18.5 8 360.0 29 3940 13.0 79 1
## 294 31.9 4 89.0 68 1925 14.0 79 2
## 295 34.1 4 86.0 62 1975 15.2 79 3
## 296 35.7 4 98.0 76 1915 14.4 79 1
## 297 27.4 4 121.0 76 2670 15.0 79 1
## 298 25.4 5 183.0 73 3530 20.1 79 2
## 299 23.0 8 350.0 15 3900 17.4 79 1
## 300 27.2 4 141.0 68 3190 24.8 79 2
## 301 23.9 8 260.0 86 3420 22.2 79 1
## 302 34.2 4 105.0 67 2200 13.2 79 1
## 303 34.5 4 105.0 67 2150 14.9 79 1
## 304 31.8 4 85.0 62 2020 19.2 79 3
## 305 37.3 4 91.0 66 2130 14.7 79 2
## 306 28.4 4 151.0 86 2670 16.0 79 1
## 307 28.8 6 173.0 11 2595 11.3 79 1
## 308 26.8 6 173.0 11 2700 12.9 79 1
## 309 33.5 4 151.0 86 2556 13.2 79 1
## 310 41.5 4 98.0 72 2144 14.7 80 2
## 311 38.1 4 89.0 57 1968 18.8 80 3
## 312 32.1 4 98.0 67 2120 15.5 80 1
## 313 37.2 4 86.0 62 2019 16.4 80 3
## 314 28.0 4 151.0 86 2678 16.5 80 1
## 315 26.4 4 140.0 84 2870 18.1 80 1
## 316 24.3 4 151.0 86 3003 20.1 80 1
## 317 19.1 6 225.0 86 3381 18.7 80 1
## 318 34.3 4 97.0 74 2188 15.8 80 2
## 319 29.8 4 134.0 86 2711 15.5 80 3
## 320 31.3 4 120.0 71 2542 17.5 80 3
## 321 37.0 4 119.0 88 2434 15.0 80 3
## 322 32.2 4 108.0 71 2265 15.2 80 3
## 323 46.6 4 86.0 62 2110 17.9 80 3
## 324 27.9 4 156.0 5 2800 14.4 80 1
## 325 40.8 4 85.0 62 2110 19.2 80 3
## 326 44.3 4 90.0 51 2085 21.7 80 2
## 327 43.4 4 90.0 51 2335 23.7 80 2
## 328 36.4 5 121.0 64 2950 19.9 80 2
## 329 30.0 4 146.0 64 3250 21.8 80 2
## 330 44.6 4 91.0 64 1850 13.8 80 3
## 331 40.9 4 85.0 1 1835 17.3 80 2
## 332 33.8 4 97.0 64 2145 18.0 80 3
## 333 29.8 4 89.0 59 1845 15.3 80 2
## 334 32.7 6 168.0 18 2910 11.4 80 3
## 335 23.7 3 70.0 2 2420 12.5 80 3
## 336 35.0 4 122.0 84 2500 15.1 80 2
## 337 23.6 4 140.0 1 2905 14.3 80 1
## 338 32.4 4 107.0 69 2290 17.0 80 3
## 339 27.2 4 135.0 80 2490 15.7 81 1
## 340 26.6 4 151.0 80 2635 16.4 81 1
## 341 25.8 4 156.0 88 2620 14.4 81 1
## 342 23.5 6 173.0 8 2725 12.6 81 1
## 343 30.0 4 135.0 80 2385 12.9 81 1
## 344 39.1 4 79.0 56 1755 16.9 81 3
## 345 39.0 4 86.0 61 1875 16.4 81 1
## 346 35.1 4 81.0 57 1760 16.1 81 3
## 347 32.3 4 97.0 64 2065 17.8 81 3
## 348 37.0 4 85.0 62 1975 19.4 81 3
## 349 37.7 4 89.0 59 2050 17.3 81 3
## 350 34.1 4 91.0 65 1985 16.0 81 3
## 351 34.7 4 105.0 60 2215 14.9 81 1
## 352 34.4 4 98.0 62 2045 16.2 81 1
## 353 29.9 4 98.0 62 2380 20.7 81 1
## 354 33.0 4 105.0 70 2190 14.2 81 2
## 355 34.5 4 100.0 1 2320 15.8 81 2
## 356 33.7 4 107.0 71 2210 14.4 81 3
## 357 32.4 4 108.0 71 2350 16.8 81 3
## 358 32.9 4 119.0 2 2615 14.8 81 3
## 359 31.6 4 120.0 70 2635 18.3 81 3
## 360 28.1 4 141.0 76 3230 20.4 81 2
## 361 30.7 6 145.0 72 3160 19.6 81 2
## 362 25.4 6 168.0 12 2900 12.6 81 3
## 363 24.2 6 146.0 13 2930 13.8 81 3
## 364 22.4 6 231.0 8 3415 15.8 81 1
## 365 26.6 8 350.0 5 3725 19.0 81 1
## 366 20.2 6 200.0 84 3060 17.1 81 1
## 367 17.6 6 225.0 81 3465 16.6 81 1
## 368 28.0 4 112.0 84 2605 19.6 82 1
## 369 27.0 4 112.0 84 2640 18.6 82 1
## 370 34.0 4 112.0 84 2395 18.0 82 1
## 371 31.0 4 112.0 81 2575 16.2 82 1
## 372 29.0 4 135.0 80 2525 16.0 82 1
## 373 27.0 4 151.0 86 2735 18.0 82 1
## 374 24.0 4 140.0 88 2865 16.4 82 1
## 375 36.0 4 105.0 70 1980 15.3 82 2
## 376 37.0 4 91.0 65 2025 18.2 82 3
## 377 31.0 4 91.0 65 1970 17.6 82 3
## 378 38.0 4 105.0 60 2125 14.7 82 1
## 379 36.0 4 98.0 67 2125 17.3 82 1
## 380 36.0 4 120.0 84 2160 14.5 82 3
## 381 36.0 4 107.0 71 2205 14.5 82 3
## 382 34.0 4 108.0 67 2245 16.9 82 3
## 383 38.0 4 91.0 64 1965 15.0 82 3
## 384 32.0 4 91.0 64 1965 15.7 82 3
## 385 38.0 4 91.0 64 1995 16.2 82 3
## 386 25.0 6 181.0 8 2945 16.4 82 1
## 387 38.0 6 262.0 81 3015 17.0 82 1
## 388 26.0 4 156.0 88 2585 14.5 82 1
## 389 22.0 6 232.0 9 2835 14.7 82 1
## 390 32.0 4 144.0 92 2665 13.9 82 3
## 391 36.0 4 135.0 80 2370 13.0 82 1
## 392 27.0 4 151.0 86 2950 17.3 82 1
## 393 27.0 4 140.0 82 2790 15.6 82 1
## 394 44.0 4 97.0 53 2130 24.6 82 2
## 395 32.0 4 135.0 80 2295 11.6 82 1
## 396 28.0 4 120.0 75 2625 18.6 82 1
## 397 31.0 4 119.0 78 2720 19.4 82 1
## name
## 1 chevrolet chevelle malibu
## 2 buick skylark 320
## 3 plymouth satellite
## 4 amc rebel sst
## 5 ford torino
## 6 ford galaxie 500
## 7 chevrolet impala
## 8 plymouth fury iii
## 9 pontiac catalina
## 10 amc ambassador dpl
## 11 dodge challenger se
## 12 plymouth 'cuda 340
## 13 chevrolet monte carlo
## 14 buick estate wagon (sw)
## 15 toyota corona mark ii
## 16 plymouth duster
## 17 amc hornet
## 18 ford maverick
## 19 datsun pl510
## 20 volkswagen 1131 deluxe sedan
## 21 peugeot 504
## 22 audi 100 ls
## 23 saab 99e
## 24 bmw 2002
## 25 amc gremlin
## 26 ford f250
## 27 chevy c20
## 28 dodge d200
## 29 hi 1200d
## 30 datsun pl510
## 31 chevrolet vega 2300
## 32 toyota corona
## 33 ford pinto
## 34 amc gremlin
## 35 plymouth satellite custom
## 36 chevrolet chevelle malibu
## 37 ford torino 500
## 38 amc matador
## 39 chevrolet impala
## 40 pontiac catalina brougham
## 41 ford galaxie 500
## 42 plymouth fury iii
## 43 dodge monaco (sw)
## 44 ford country squire (sw)
## 45 pontiac safari (sw)
## 46 amc hornet sportabout (sw)
## 47 chevrolet vega (sw)
## 48 pontiac firebird
## 49 ford mustang
## 50 mercury capri 2000
## 51 opel 1900
## 52 peugeot 304
## 53 fiat 124b
## 54 toyota corolla 1200
## 55 datsun 1200
## 56 volkswagen model 111
## 57 plymouth cricket
## 58 toyota corona hardtop
## 59 dodge colt hardtop
## 60 volkswagen type 3
## 61 chevrolet vega
## 62 ford pinto runabout
## 63 chevrolet impala
## 64 pontiac catalina
## 65 plymouth fury iii
## 66 ford galaxie 500
## 67 amc ambassador sst
## 68 mercury marquis
## 69 buick lesabre custom
## 70 oldsmobile delta 88 royale
## 71 chrysler newport royal
## 72 mazda rx2 coupe
## 73 amc matador (sw)
## 74 chevrolet chevelle concours (sw)
## 75 ford gran torino (sw)
## 76 plymouth satellite custom (sw)
## 77 volvo 145e (sw)
## 78 volkswagen 411 (sw)
## 79 peugeot 504 (sw)
## 80 renault 12 (sw)
## 81 ford pinto (sw)
## 82 datsun 510 (sw)
## 83 toyouta corona mark ii (sw)
## 84 dodge colt (sw)
## 85 toyota corolla 1600 (sw)
## 86 buick century 350
## 87 amc matador
## 88 chevrolet malibu
## 89 ford gran torino
## 90 dodge coronet custom
## 91 mercury marquis brougham
## 92 chevrolet caprice classic
## 93 ford ltd
## 94 plymouth fury gran sedan
## 95 chrysler new yorker brougham
## 96 buick electra 225 custom
## 97 amc ambassador brougham
## 98 plymouth valiant
## 99 chevrolet nova custom
## 100 amc hornet
## 101 ford maverick
## 102 plymouth duster
## 103 volkswagen super beetle
## 104 chevrolet impala
## 105 ford country
## 106 plymouth custom suburb
## 107 oldsmobile vista cruiser
## 108 amc gremlin
## 109 toyota carina
## 110 chevrolet vega
## 111 datsun 610
## 112 maxda rx3
## 113 ford pinto
## 114 mercury capri v6
## 115 fiat 124 sport coupe
## 116 chevrolet monte carlo s
## 117 pontiac grand prix
## 118 fiat 128
## 119 opel manta
## 120 audi 100ls
## 121 volvo 144ea
## 122 dodge dart custom
## 123 saab 99le
## 124 toyota mark ii
## 125 oldsmobile omega
## 126 plymouth duster
## 127 ford maverick
## 128 amc hornet
## 129 chevrolet nova
## 130 datsun b210
## 131 ford pinto
## 132 toyota corolla 1200
## 133 chevrolet vega
## 134 chevrolet chevelle malibu classic
## 135 amc matador
## 136 plymouth satellite sebring
## 137 ford gran torino
## 138 buick century luxus (sw)
## 139 dodge coronet custom (sw)
## 140 ford gran torino (sw)
## 141 amc matador (sw)
## 142 audi fox
## 143 volkswagen dasher
## 144 opel manta
## 145 toyota corona
## 146 datsun 710
## 147 dodge colt
## 148 fiat 128
## 149 fiat 124 tc
## 150 honda civic
## 151 subaru
## 152 fiat x1.9
## 153 plymouth valiant custom
## 154 chevrolet nova
## 155 mercury monarch
## 156 ford maverick
## 157 pontiac catalina
## 158 chevrolet bel air
## 159 plymouth grand fury
## 160 ford ltd
## 161 buick century
## 162 chevroelt chevelle malibu
## 163 amc matador
## 164 plymouth fury
## 165 buick skyhawk
## 166 chevrolet monza 2+2
## 167 ford mustang ii
## 168 toyota corolla
## 169 ford pinto
## 170 amc gremlin
## 171 pontiac astro
## 172 toyota corona
## 173 volkswagen dasher
## 174 datsun 710
## 175 ford pinto
## 176 volkswagen rabbit
## 177 amc pacer
## 178 audi 100ls
## 179 peugeot 504
## 180 volvo 244dl
## 181 saab 99le
## 182 honda civic cvcc
## 183 fiat 131
## 184 opel 1900
## 185 capri ii
## 186 dodge colt
## 187 renault 12tl
## 188 chevrolet chevelle malibu classic
## 189 dodge coronet brougham
## 190 amc matador
## 191 ford gran torino
## 192 plymouth valiant
## 193 chevrolet nova
## 194 ford maverick
## 195 amc hornet
## 196 chevrolet chevette
## 197 chevrolet woody
## 198 vw rabbit
## 199 honda civic
## 200 dodge aspen se
## 201 ford granada ghia
## 202 pontiac ventura sj
## 203 amc pacer d/l
## 204 volkswagen rabbit
## 205 datsun b-210
## 206 toyota corolla
## 207 ford pinto
## 208 volvo 245
## 209 plymouth volare premier v8
## 210 peugeot 504
## 211 toyota mark ii
## 212 mercedes-benz 280s
## 213 cadillac seville
## 214 chevy c10
## 215 ford f108
## 216 dodge d100
## 217 honda accord cvcc
## 218 buick opel isuzu deluxe
## 219 renault 5 gtl
## 220 plymouth arrow gs
## 221 datsun f-10 hatchback
## 222 chevrolet caprice classic
## 223 oldsmobile cutlass supreme
## 224 dodge monaco brougham
## 225 mercury cougar brougham
## 226 chevrolet concours
## 227 buick skylark
## 228 plymouth volare custom
## 229 ford granada
## 230 pontiac grand prix lj
## 231 chevrolet monte carlo landau
## 232 chrysler cordoba
## 233 ford thunderbird
## 234 volkswagen rabbit custom
## 235 pontiac sunbird coupe
## 236 toyota corolla liftback
## 237 ford mustang ii 2+2
## 238 chevrolet chevette
## 239 dodge colt m/m
## 240 subaru dl
## 241 volkswagen dasher
## 242 datsun 810
## 243 bmw 320i
## 244 mazda rx-4
## 245 volkswagen rabbit custom diesel
## 246 ford fiesta
## 247 mazda glc deluxe
## 248 datsun b210 gx
## 249 honda civic cvcc
## 250 oldsmobile cutlass salon brougham
## 251 dodge diplomat
## 252 mercury monarch ghia
## 253 pontiac phoenix lj
## 254 chevrolet malibu
## 255 ford fairmont (auto)
## 256 ford fairmont (man)
## 257 plymouth volare
## 258 amc concord
## 259 buick century special
## 260 mercury zephyr
## 261 dodge aspen
## 262 amc concord d/l
## 263 chevrolet monte carlo landau
## 264 buick regal sport coupe (turbo)
## 265 ford futura
## 266 dodge magnum xe
## 267 chevrolet chevette
## 268 toyota corona
## 269 datsun 510
## 270 dodge omni
## 271 toyota celica gt liftback
## 272 plymouth sapporo
## 273 oldsmobile starfire sx
## 274 datsun 200-sx
## 275 audi 5000
## 276 volvo 264gl
## 277 saab 99gle
## 278 peugeot 604sl
## 279 volkswagen scirocco
## 280 honda accord lx
## 281 pontiac lemans v6
## 282 mercury zephyr 6
## 283 ford fairmont 4
## 284 amc concord dl 6
## 285 dodge aspen 6
## 286 chevrolet caprice classic
## 287 ford ltd landau
## 288 mercury grand marquis
## 289 dodge st. regis
## 290 buick estate wagon (sw)
## 291 ford country squire (sw)
## 292 chevrolet malibu classic (sw)
## 293 chrysler lebaron town @ country (sw)
## 294 vw rabbit custom
## 295 maxda glc deluxe
## 296 dodge colt hatchback custom
## 297 amc spirit dl
## 298 mercedes benz 300d
## 299 cadillac eldorado
## 300 peugeot 504
## 301 oldsmobile cutlass salon brougham
## 302 plymouth horizon
## 303 plymouth horizon tc3
## 304 datsun 210
## 305 fiat strada custom
## 306 buick skylark limited
## 307 chevrolet citation
## 308 oldsmobile omega brougham
## 309 pontiac phoenix
## 310 vw rabbit
## 311 toyota corolla tercel
## 312 chevrolet chevette
## 313 datsun 310
## 314 chevrolet citation
## 315 ford fairmont
## 316 amc concord
## 317 dodge aspen
## 318 audi 4000
## 319 toyota corona liftback
## 320 mazda 626
## 321 datsun 510 hatchback
## 322 toyota corolla
## 323 mazda glc
## 324 dodge colt
## 325 datsun 210
## 326 vw rabbit c (diesel)
## 327 vw dasher (diesel)
## 328 audi 5000s (diesel)
## 329 mercedes-benz 240d
## 330 honda civic 1500 gl
## 331 renault lecar deluxe
## 332 subaru dl
## 333 vokswagen rabbit
## 334 datsun 280-zx
## 335 mazda rx-7 gs
## 336 triumph tr7 coupe
## 337 ford mustang cobra
## 338 honda accord
## 339 plymouth reliant
## 340 buick skylark
## 341 dodge aries wagon (sw)
## 342 chevrolet citation
## 343 plymouth reliant
## 344 toyota starlet
## 345 plymouth champ
## 346 honda civic 1300
## 347 subaru
## 348 datsun 210 mpg
## 349 toyota tercel
## 350 mazda glc 4
## 351 plymouth horizon 4
## 352 ford escort 4w
## 353 ford escort 2h
## 354 volkswagen jetta
## 355 renault 18i
## 356 honda prelude
## 357 toyota corolla
## 358 datsun 200sx
## 359 mazda 626
## 360 peugeot 505s turbo diesel
## 361 volvo diesel
## 362 toyota cressida
## 363 datsun 810 maxima
## 364 buick century
## 365 oldsmobile cutlass ls
## 366 ford granada gl
## 367 chrysler lebaron salon
## 368 chevrolet cavalier
## 369 chevrolet cavalier wagon
## 370 chevrolet cavalier 2-door
## 371 pontiac j2000 se hatchback
## 372 dodge aries se
## 373 pontiac phoenix
## 374 ford fairmont futura
## 375 volkswagen rabbit l
## 376 mazda glc custom l
## 377 mazda glc custom
## 378 plymouth horizon miser
## 379 mercury lynx l
## 380 nissan stanza xe
## 381 honda accord
## 382 toyota corolla
## 383 honda civic
## 384 honda civic (auto)
## 385 datsun 310 gx
## 386 buick century limited
## 387 oldsmobile cutlass ciera (diesel)
## 388 chrysler lebaron medallion
## 389 ford granada l
## 390 toyota celica gt
## 391 dodge charger 2.2
## 392 chevrolet camaro
## 393 ford mustang gl
## 394 vw pickup
## 395 dodge rampage
## 396 ford ranger
## 397 chevy s-10auto = Auto[,-9]
cor(auto)## mpg cylinders displacement horsepower weight
## mpg 1.0000000 -0.7762599 -0.8044430 0.4228227 -0.8317389
## cylinders -0.7762599 1.0000000 0.9509199 -0.5466585 0.8970169
## displacement -0.8044430 0.9509199 1.0000000 -0.4820705 0.9331044
## horsepower 0.4228227 -0.5466585 -0.4820705 1.0000000 -0.4821507
## weight -0.8317389 0.8970169 0.9331044 -0.4821507 1.0000000
## acceleration 0.4222974 -0.5040606 -0.5441618 0.2662877 -0.4195023
## year 0.5814695 -0.3467172 -0.3698041 0.1274167 -0.3079004
## origin 0.5636979 -0.5649716 -0.6106643 0.2973734 -0.5812652
## acceleration year origin
## mpg 0.4222974 0.5814695 0.5636979
## cylinders -0.5040606 -0.3467172 -0.5649716
## displacement -0.5441618 -0.3698041 -0.6106643
## horsepower 0.2662877 0.1274167 0.2973734
## weight -0.4195023 -0.3079004 -0.5812652
## acceleration 1.0000000 0.2829009 0.2100836
## year 0.2829009 1.0000000 0.1843141
## origin 0.2100836 0.1843141 1.0000000auto$origin = factor(auto$origin, labels = c("American", "European", "Japanese"))
mpg_lm = lm(mpg ~ . , data = auto)
summary(mpg_lm)##
## Call:
## lm(formula = mpg ~ ., data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.9699 -2.0621 0.0693 1.9776 13.3807
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.278e+01 4.284e+00 -5.318 1.77e-07 ***
## cylinders -2.569e-01 3.353e-01 -0.766 0.44415
## displacement 2.018e-02 7.361e-03 2.741 0.00641 **
## horsepower 9.850e-03 6.780e-03 1.453 0.14710
## weight -7.152e-03 5.831e-04 -12.267 < 2e-16 ***
## acceleration 1.579e-01 7.681e-02 2.056 0.04048 *
## year 8.036e-01 5.006e-02 16.053 < 2e-16 ***
## originEuropean 2.728e+00 5.514e-01 4.947 1.13e-06 ***
## originJapanese 2.665e+00 5.333e-01 4.997 8.82e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.301 on 388 degrees of freedom
## Multiple R-squared: 0.8257, Adjusted R-squared: 0.8221
## F-statistic: 229.8 on 8 and 388 DF, p-value: < 2.2e-16There is a relationship between predictors and response. The p-vlaue is very small 2.2e-16. When looking at each predictor we see cylinders and horsepower however do not have significant p-values.
Displacement, Weight, Acceleration, Year, and Origin do have a significant relationship to mpg
Year coef = 0.8036. We interpret this as a 1 year increase in the car will result in an increase of 0.8036 in mpg. As cars become more modern their mpg goes up!
par(mfrow = c(2,2))
plot(mpg_lm)
The residual plot has a slight curve to it. This might suggest there is non-linearity among the relationships. Observation 323 and 327 appear in the residual plot and the qq plot. Observation 14 shows up with a very high leverage compared to other points.
summary(lm(formula = mpg ~ . * ., data = auto))##
## Call:
## lm(formula = mpg ~ . * ., data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.8783 -1.3913 -0.0053 1.2860 12.2457
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.016e+02 4.285e+01 2.370 0.018296 *
## cylinders 3.141e+00 7.716e+00 0.407 0.684219
## displacement -2.563e-01 1.743e-01 -1.471 0.142193
## horsepower 6.841e-02 1.432e-01 0.478 0.633173
## weight 1.026e-03 1.544e-02 0.066 0.947081
## acceleration -8.780e+00 1.636e+00 -5.366 1.44e-07 ***
## year -2.623e-01 5.289e-01 -0.496 0.620219
## originEuropean -4.071e+01 1.148e+01 -3.546 0.000442 ***
## originJapanese -3.126e+01 1.396e+01 -2.239 0.025782 *
## cylinders:displacement -2.737e-03 5.918e-03 -0.462 0.644018
## cylinders:horsepower 4.230e-03 1.108e-02 0.382 0.702852
## cylinders:weight 6.569e-04 7.700e-04 0.853 0.394141
## cylinders:acceleration 2.857e-01 1.339e-01 2.134 0.033476 *
## cylinders:year -1.235e-01 1.019e-01 -1.212 0.226276
## cylinders:originEuropean -6.225e-01 1.119e+00 -0.556 0.578395
## cylinders:originJapanese 1.049e+00 1.261e+00 0.833 0.405656
## displacement:horsepower 1.374e-04 3.020e-04 0.455 0.649468
## displacement:weight 1.466e-05 7.742e-06 1.893 0.059145 .
## displacement:acceleration -1.435e-03 2.714e-03 -0.529 0.597347
## displacement:year 3.155e-03 2.307e-03 1.368 0.172201
## displacement:originEuropean -3.426e-02 4.151e-02 -0.825 0.409762
## displacement:originJapanese 7.092e-02 4.782e-02 1.483 0.138945
## horsepower:weight -2.426e-05 2.318e-05 -1.047 0.295924
## horsepower:acceleration -6.044e-04 3.431e-03 -0.176 0.860246
## horsepower:year -5.648e-04 1.791e-03 -0.315 0.752652
## horsepower:originEuropean 2.916e-02 2.124e-02 1.373 0.170702
## horsepower:originJapanese 1.375e-03 2.663e-02 0.052 0.958846
## weight:acceleration -4.947e-05 2.003e-04 -0.247 0.805034
## weight:year -1.691e-04 1.796e-04 -0.942 0.347028
## weight:originEuropean 2.306e-03 2.341e-03 0.985 0.325219
## weight:originJapanese -4.380e-03 2.829e-03 -1.548 0.122513
## acceleration:year 9.784e-02 1.939e-02 5.046 7.18e-07 ***
## acceleration:originEuropean 1.043e+00 1.930e-01 5.403 1.19e-07 ***
## acceleration:originJapanese 7.133e-01 2.655e-01 2.686 0.007558 **
## year:originEuropean 3.081e-01 1.391e-01 2.214 0.027422 *
## year:originJapanese 2.353e-01 1.412e-01 1.666 0.096518 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.649 on 361 degrees of freedom
## Multiple R-squared: 0.8955, Adjusted R-squared: 0.8854
## F-statistic: 88.41 on 35 and 361 DF, p-value: < 2.2e-16I used the * to test all interactions since we have a relatively low number of predictors. Some of the interactions do appear statistically significant. Cylinders:Acceleration acceleration:year acceleration:origionEuropean acceleration:originJapanese year:originEurope all have some significance.
lm.fit2 = lm(formula = log(mpg) ~ ., data = auto)
lm.fit3 = lm(formula = mpg^2 ~., data = auto)
summary(lm.fit2)##
## Call:
## lm(formula = log(mpg) ~ ., data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.39718 -0.06971 0.00706 0.06908 0.34177
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.4171502 0.1542685 9.186 < 2e-16 ***
## cylinders -0.0170756 0.0120760 -1.414 0.158161
## displacement 0.0004969 0.0002651 1.875 0.061609 .
## horsepower 0.0003565 0.0002442 1.460 0.145112
## weight -0.0002967 0.0000210 -14.132 < 2e-16 ***
## acceleration 0.0052528 0.0027659 1.899 0.058290 .
## year 0.0320410 0.0018025 17.775 < 2e-16 ***
## originEuropean 0.0810993 0.0198565 4.084 5.37e-05 ***
## originJapanese 0.0662720 0.0192057 3.451 0.000621 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1189 on 388 degrees of freedom
## Multiple R-squared: 0.8803, Adjusted R-squared: 0.8778
## F-statistic: 356.7 on 8 and 388 DF, p-value: < 2.2e-16summary(lm.fit3)##
## Call:
## lm(formula = mpg^2 ~ ., data = auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -473.56 -138.76 -22.94 103.34 1061.09
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.093e+03 2.708e+02 -7.727 9.51e-14 ***
## cylinders -4.248e+00 2.120e+01 -0.200 0.84131
## displacement 1.490e+00 4.654e-01 3.202 0.00148 **
## horsepower 4.435e-01 4.287e-01 1.035 0.30154
## weight -3.816e-01 3.686e-02 -10.351 < 2e-16 ***
## acceleration 1.111e+01 4.856e+00 2.287 0.02274 *
## year 4.357e+01 3.165e+00 13.766 < 2e-16 ***
## originEuropean 1.798e+02 3.486e+01 5.158 4.00e-07 ***
## originJapanese 1.857e+02 3.372e+01 5.507 6.66e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 208.7 on 388 degrees of freedom
## Multiple R-squared: 0.7357, Adjusted R-squared: 0.7302
## F-statistic: 135 on 8 and 388 DF, p-value: < 2.2e-16When doing some transformations I found no significant changes to which predictors are significant to mpg for squaring or taking the log.
10
library(ISLR)## Warning: package 'ISLR' was built under R version 3.6.3##
## Attaching package: 'ISLR'## The following object is masked _by_ '.GlobalEnv':
##
## Autosales_lm = lm(Sales ~ Price + Urban + US, data = Carseats)
summary(sales_lm)##
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9206 -1.6220 -0.0564 1.5786 7.0581
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.043469 0.651012 20.036 < 2e-16 ***
## Price -0.054459 0.005242 -10.389 < 2e-16 ***
## UrbanYes -0.021916 0.271650 -0.081 0.936
## USYes 1.200573 0.259042 4.635 4.86e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335
## F-statistic: 41.52 on 3 and 396 DF, p-value: < 2.2e-16Urban is not significant to the model. Price and US are significant predictors to Sales. A 1 unit increase in price results in a -0.054 drop in sales. Being built in the US increases the likelyhood/number of sales.
Sales = 13.04 - 0.054 * Price + 1.20 * USYes if we want to include Urban Sales = 13.04 - 0.054 * Price - 0.022 * UrbanYes + 1.20 * USYES
We can reject the null for Price and US predictors as they have a very low p-value. We should keep them in the model and remove UrbanYes
sales2 = lm(Sales ~ Price + US, data = Carseats)
summary(sales2)##
## Call:
## lm(formula = Sales ~ Price + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9269 -1.6286 -0.0574 1.5766 7.0515
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.03079 0.63098 20.652 < 2e-16 ***
## Price -0.05448 0.00523 -10.416 < 2e-16 ***
## USYes 1.19964 0.25846 4.641 4.71e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2354
## F-statistic: 62.43 on 2 and 397 DF, p-value: < 2.2e-16The model in a) Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335 The model in e) Multiple R-squared: 0.2393, Adjusted R-squared: 0.2354 Both models with and without the UrbanYes explain about 23-24% of the variance.
confint(sales2, level = 0.95)## 2.5 % 97.5 %
## (Intercept) 11.79032020 14.27126531
## Price -0.06475984 -0.04419543
## USYes 0.69151957 1.70776632par(mfrow = c(2,2))
plot(sales2)
- Residual graph is a very nice looking straight line with no evident urvature. QQ plot is virtually a straight line. The leverage plot does seem to have one point farther than the others.
12
This will happen if the sum of the x^2 and y^2 values are the same.
set.seed(1)
x = 1:100
sum(x^2)## [1] 338350y = 2 * x + rnorm(100, sd = 0.1)
sum(y^2)## [1] 1353606regX = lm(x ~ y)
regY = lm(y ~ x)
summary(regX)##
## Call:
## lm(formula = x ~ y)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.114850 -0.029264 -0.000719 0.030278 0.116917
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.462e-03 9.096e-03 -0.71 0.479
## y 5.000e-01 7.818e-05 6395.53 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04513 on 98 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 4.09e+07 on 1 and 98 DF, p-value: < 2.2e-16summary(regY)##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.234005 -0.060584 0.001551 0.058514 0.229747
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0131666 0.0181897 0.724 0.471
## x 1.9999549 0.0003127 6395.532 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09027 on 98 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 4.09e+07 on 1 and 98 DF, p-value: < 2.2e-16x = 1:100
sum(x^2)## [1] 338350y = 100:1
sum(y^2)## [1] 338350xval = lm(x ~ y)
yval = lm(y ~ x)
summary(xval)## Warning in summary.lm(xval): essentially perfect fit: summary may be
## unreliable##
## Call:
## lm(formula = x ~ y)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.680e-13 -4.300e-16 2.850e-15 5.302e-15 3.575e-14
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.010e+02 5.598e-15 1.804e+16 <2e-16 ***
## y -1.000e+00 9.624e-17 -1.039e+16 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.778e-14 on 98 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.08e+32 on 1 and 98 DF, p-value: < 2.2e-16summary(yval)## Warning in summary.lm(yval): essentially perfect fit: summary may be
## unreliable##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.575e-14 -5.302e-15 -2.850e-15 4.300e-16 2.680e-13
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.010e+02 5.598e-15 1.804e+16 <2e-16 ***
## x -1.000e+00 9.624e-17 -1.039e+16 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.778e-14 on 98 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.08e+32 on 1 and 98 DF, p-value: < 2.2e-16