Muthukumar Srinivasan
Muthukumar Srinivasan
December 22, 2015
This is an R Markdown document. Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. For more details on using R Markdown see http://rmarkdown.rstudio.com.
When you click the Knit button a document will be generated that includes both content as well as the output of any embedded R code chunks within the document. You can embed an R code chunk like this:
#########Part2-Question 2.1
######### using this function knitr is an engine for dynamic report generation with R. It is a #########package in the statistical programming language R that enables integration of R code into #########LaTeX, LyX, HTML, Markdown, AsciiDoc, and reStructuredText documents.
library("knitr")
binomialFunction<-function (x) {
dbinom(x,20,0.5)
}
#Let's setup an array x values
###### building array
x <- seq(0,20,by=1)
px <- sapply(x,binomialFunction)
######let's obtain f(x)
fx <- sample(x,1000,replace=TRUE,prob=px)
par(mfrow=c(1,2))
hist(fx, breaks=x)
#######Part2-Question 2.2
#######2.2. Principal Components Analysis. For the auto data set attached with the nal
#######exam, please perform a Principal Components Analysis by performing an SVD on the 4
#######independent variables (with mpg as the dependent variable) and select the top 2 directions.
#######Please scatter plot the data set after it has been projected to these two dimensions. Your
#######code should print out the two orthogonal vectors and also perform the scatter plot of the
#######data after it has been projected to these two dimensions.
auto_mpg_data = read.fwf(file='C:\\temp\\auto-mpg.data', widths = c(5,11,11,10,8))
auto_mpg_data## V1 V2 V3 V4 V5
## 1 18.0 8 307.0 130.0 3504 . 1
## 2 15.0 8 350.0 165.0 3693 . 1
## 3 18.0 8 318.0 150.0 3436 . 1
## 4 16.0 8 304.0 150.0 3433 . 1
## 5 17.0 8 302.0 140.0 3449 . 1
## 6 15.0 8 429.0 198.0 4341 . 1
## 7 14.0 8 454.0 220.0 4354 .
## 8 14.0 8 440.0 215.0 4312 .
## 9 14.0 8 455.0 225.0 4425 . 1
## 10 15.0 8 390.0 190.0 3850 .
## 11 15.0 8 383.0 170.0 3563 . 1
## 12 14.0 8 340.0 160.0 3609 .
## 13 15.0 8 400.0 150.0 3761 .
## 14 14.0 8 455.0 225.0 3086 . 1
## 15 24.0 4 113.0 95.00 2372 . 1
## 16 22.0 6 198.0 95.00 2833 . 1
## 17 18.0 6 199.0 97.00 2774 . 1
## 18 21.0 6 200.0 85.00 2587 . 1
## 19 27.0 4 97.00 88.00 2130 . 1
## 20 26.0 4 97.00 46.00 1835 . 2
## 21 25.0 4 110.0 87.00 2672 . 1
## 22 24.0 4 107.0 90.00 2430 . 1
## 23 25.0 4 104.0 95.00 2375 . 1
## 24 26.0 4 121.0 113.0 2234 . 1
## 25 21.0 6 199.0 90.00 2648 . 1
## 26 10.0 8 360.0 215.0 4615 . 1
## 27 10.0 8 307.0 200.0 4376 . 1
## 28 11.0 8 318.0 210.0 4382 . 1
## 29 9.0 8 304.0 193.0 4732 . 1
## 30 27.0 4 97.00 88.00 2130 . 1
## 31 28.0 4 140.0 90.00 2264 . 1
## 32 25.0 4 113.0 95.00 2228 . 1
## 33 25.0 4 98.00 ? 2046 . 1
## 34 19.0 6 232.0 100.0 2634 . 1
## 35 16.0 6 225.0 105.0 3439 . 1
## 36 17.0 6 250.0 100.0 3329 . 1
## 37 19.0 6 250.0 88.00 3302 . 1
## 38 18.0 6 232.0 100.0 3288 . 1
## 39 14.0 8 350.0 165.0 4209 . 1
## 40 14.0 8 400.0 175.0 4464 . 1
## 41 14.0 8 351.0 153.0 4154 . 1
## 42 14.0 8 318.0 150.0 4096 . 1
## 43 12.0 8 383.0 180.0 4955 . 1
## 44 13.0 8 400.0 170.0 4746 . 1
## 45 13.0 8 400.0 175.0 5140 . 1
## 46 18.0 6 258.0 110.0 2962 . 1
## 47 22.0 4 140.0 72.00 2408 . 1
## 48 19.0 6 250.0 100.0 3282 . 1
## 49 18.0 6 250.0 88.00 3139 . 1
## 50 23.0 4 122.0 86.00 2220 . 1
## 51 28.0 4 116.0 90.00 2123 . 1
## 52 30.0 4 79.00 70.00 2074 . 1
## 53 30.0 4 88.00 76.00 2065 . 1
## 54 31.0 4 71.00 65.00 1773 . 1
## 55 35.0 4 72.00 69.00 1613 . 1
## 56 27.0 4 97.00 60.00 1834 . 1
## 57 26.0 4 91.00 70.00 1955 . 2
## 58 24.0 4 113.0 95.00 2278 . 1
## 59 25.0 4 97.50 80.00 2126 . 1
## 60 23.0 4 97.00 54.00 2254 . 2
## 61 20.0 4 140.0 90.00 2408 . 1
## 62 21.0 4 122.0 86.00 2226 . 1
## 63 13.0 8 350.0 165.0 4274 . 1
## 64 14.0 8 400.0 175.0 4385 . 1
## 65 15.0 8 318.0 150.0 4135 . 1
## 66 14.0 8 351.0 153.0 4129 . 1
## 67 17.0 8 304.0 150.0 3672 . 1
## 68 11.0 8 429.0 208.0 4633 . 1
## 69 13.0 8 350.0 155.0 4502 . 1
## 70 12.0 8 350.0 160.0 4456 . 1
## 71 13.0 8 400.0 190.0 4422 . 1
## 72 19.0 3 70.00 97.00 2330 . 1
## 73 15.0 8 304.0 150.0 3892 . 1
## 74 13.0 8 307.0 130.0 4098 . 1
## 75 13.0 8 302.0 140.0 4294 . 1
## 76 14.0 8 318.0 150.0 4077 . 1
## 77 18.0 4 121.0 112.0 2933 . 1
## 78 22.0 4 121.0 76.00 2511 . 1
## 79 21.0 4 120.0 87.00 2979 . 1
## 80 26.0 4 96.00 69.00 2189 . 1
## 81 22.0 4 122.0 86.00 2395 . 1
## 82 28.0 4 97.00 92.00 2288 . 1
## 83 23.0 4 120.0 97.00 2506 . 1
## 84 28.0 4 98.00 80.00 2164 . 1
## 85 27.0 4 97.00 88.00 2100 . 1
## 86 13.0 8 350.0 175.0 4100 . 1
## 87 14.0 8 304.0 150.0 3672 . 1
## 88 13.0 8 350.0 145.0 3988 . 1
## 89 14.0 8 302.0 137.0 4042 . 1
## 90 15.0 8 318.0 150.0 3777 . 1
## 91 12.0 8 429.0 198.0 4952 . 1
## 92 13.0 8 400.0 150.0 4464 . 1
## 93 13.0 8 351.0 158.0 4363 . 1
## 94 14.0 8 318.0 150.0 4237 . 1
## 95 13.0 8 440.0 215.0 4735 . 1
## 96 12.0 8 455.0 225.0 4951 . 1
## 97 13.0 8 360.0 175.0 3821 . 1
## 98 18.0 6 225.0 105.0 3121 . 1
## 99 16.0 6 250.0 100.0 3278 . 1
## 100 18.0 6 232.0 100.0 2945 . 1
## 101 18.0 6 250.0 88.00 3021 . 1
## 102 23.0 6 198.0 95.00 2904 . 1
## 103 26.0 4 97.00 46.00 1950 . 2
## 104 11.0 8 400.0 150.0 4997 . 1
## 105 12.0 8 400.0 167.0 4906 . 1
## 106 13.0 8 360.0 170.0 4654 . 1
## 107 12.0 8 350.0 180.0 4499 . 1
## 108 18.0 6 232.0 100.0 2789 . 1
## 109 20.0 4 97.00 88.00 2279 . 1
## 110 21.0 4 140.0 72.00 2401 . 1
## 111 22.0 4 108.0 94.00 2379 . 1
## 112 18.0 3 70.00 90.00 2124 . 1
## 113 19.0 4 122.0 85.00 2310 . 1
## 114 21.0 6 155.0 107.0 2472 . 1
## 115 26.0 4 98.00 90.00 2265 . 1
## 116 15.0 8 350.0 145.0 4082 . 1
## 117 16.0 8 400.0 230.0 4278 . 9
## 118 29.0 4 68.00 49.00 1867 . 1
## 119 24.0 4 116.0 75.00 2158 . 1
## 120 20.0 4 114.0 91.00 2582 . 1
## 121 19.0 4 121.0 112.0 2868 . 1
## 122 15.0 8 318.0 150.0 3399 . 1
## 123 24.0 4 121.0 110.0 2660 . 1
## 124 20.0 6 156.0 122.0 2807 . 1
## 125 11.0 8 350.0 180.0 3664 . 1
## 126 20.0 6 198.0 95.00 3102 . 1
## 127 21.0 6 200.0 ? 2875 . 1
## 128 19.0 6 232.0 100.0 2901 . 1
## 129 15.0 6 250.0 100.0 3336 . 1
## 130 31.0 4 79.00 67.00 1950 . 1
## 131 26.0 4 122.0 80.00 2451 . 1
## 132 32.0 4 71.00 65.00 1836 . 2
## 133 25.0 4 140.0 75.00 2542 . 1
## 134 16.0 6 250.0 100.0 3781 . 1
## 135 16.0 6 258.0 110.0 3632 . 1
## 136 18.0 6 225.0 105.0 3613 . 1
## 137 16.0 8 302.0 140.0 4141 . 1
## 138 13.0 8 350.0 150.0 4699 . 1
## 139 14.0 8 318.0 150.0 4457 . 1
## 140 14.0 8 302.0 140.0 4638 . 1
## 141 14.0 8 304.0 150.0 4257 . 1
## 142 29.0 4 98.00 83.00 2219 . 1
## 143 26.0 4 79.00 67.00 1963 . 1
## 144 26.0 4 97.00 78.00 2300 . 1
## 145 31.0 4 76.00 52.00 1649 . 1
## 146 32.0 4 83.00 61.00 2003 . 1
## 147 28.0 4 90.00 75.00 2125 . 1
## 148 24.0 4 90.00 75.00 2108 . 1
## 149 26.0 4 116.0 75.00 2246 . 1
## 150 24.0 4 120.0 97.00 2489 . 1
## 151 26.0 4 108.0 93.00 2391 . 1
## 152 31.0 4 79.00 67.00 2000 . 1
## 153 19.0 6 225.0 95.00 3264 . 1
## 154 18.0 6 250.0 105.0 3459 . 1
## 155 15.0 6 250.0 72.00 3432 . 2
## 156 15.0 6 250.0 72.00 3158 . 1
## 157 16.0 8 400.0 170.0 4668 . 1
## 158 15.0 8 350.0 145.0 4440 . 1
## 159 16.0 8 318.0 150.0 4498 . 1
## 160 14.0 8 351.0 148.0 4657 . 1
## 161 17.0 6 231.0 110.0 3907 . 2
## 162 16.0 6 250.0 105.0 3897 . 1
## 163 15.0 6 258.0 110.0 3730 . 1
## 164 18.0 6 225.0 95.00 3785 . 1
## 165 21.0 6 231.0 110.0 3039 . 1
## 166 20.0 8 262.0 110.0 3221 . 1
## 167 13.0 8 302.0 129.0 3169 . 1
## 168 29.0 4 97.00 75.00 2171 . 1
## 169 23.0 4 140.0 83.00 2639 . 1
## 170 20.0 6 232.0 100.0 2914 . 1
## 171 23.0 4 140.0 78.00 2592 . 1
## 172 24.0 4 134.0 96.00 2702 . 1
## 173 25.0 4 90.00 71.00 2223 . 1
## 174 24.0 4 119.0 97.00 2545 . 1
## 175 18.0 6 171.0 97.00 2984 . 1
## 176 29.0 4 90.00 70.00 1937 . 1
## 177 19.0 6 232.0 90.00 3211 . 1
## 178 23.0 4 115.0 95.00 2694 . 1
## 179 23.0 4 120.0 88.00 2957 . 1
## 180 22.0 4 121.0 98.00 2945 . 1
## 181 25.0 4 121.0 115.0 2671 . 1
## 182 33.0 4 91.00 53.00 1795 . 1
## 183 28.0 4 107.0 86.00 2464 . 1
## 184 25.0 4 116.0 81.00 2220 . 1
## 185 25.0 4 140.0 92.00 2572 . 1
## 186 26.0 4 98.00 79.00 2255 . 1
## 187 27.0 4 101.0 83.00 2202 . 1
## 188 17.5 8 305.0 140.0 4215 . 1
## 189 16.0 8 318.0 150.0 4190 . 1
## 190 15.5 8 304.0 120.0 3962 . 1
## 191 14.5 8 351.0 152.0 4215 . 1
## 192 22.0 6 225.0 100.0 3233 . 1
## 193 22.0 6 250.0 105.0 3353 . 1
## 194 24.0 6 200.0 81.00 3012 . 1
## 195 22.5 6 232.0 90.00 3085 . 1
## 196 29.0 4 85.00 52.00 2035 . 2
## 197 24.5 4 98.00 60.00 2164 . 2
## 198 29.0 4 90.00 70.00 1937 . 1
## 199 33.0 4 91.00 53.00 1795 . 1
## 200 20.0 6 225.0 100.0 3651 . 1
## 201 18.0 6 250.0 78.00 3574 . 2
## 202 18.5 6 250.0 110.0 3645 . 1
## 203 17.5 6 258.0 95.00 3193 . 1
## 204 29.5 4 97.00 71.00 1825 . 1
## 205 32.0 4 85.00 70.00 1990 . 1
## 206 28.0 4 97.00 75.00 2155 . 1
## 207 26.5 4 140.0 72.00 2565 . 1
## 208 20.0 4 130.0 102.0 3150 . 1
## 209 13.0 8 318.0 150.0 3940 . 1
## 210 19.0 4 120.0 88.00 3270 . 2
## 211 19.0 6 156.0 108.0 2930 . 1
## 212 16.5 6 168.0 120.0 3820 . 1
## 213 16.5 8 350.0 180.0 4380 . 1
## 214 13.0 8 350.0 145.0 4055 . 1
## 215 13.0 8 302.0 130.0 3870 . 1
## 216 13.0 8 318.0 150.0 3755 . 1
## 217 31.5 4 98.00 68.00 2045 . 1
## 218 30.0 4 111.0 80.00 2155 . 1
## 219 36.0 4 79.00 58.00 1825 . 1
## 220 25.5 4 122.0 96.00 2300 . 1
## 221 33.5 4 85.00 70.00 1945 . 1
## 222 17.5 8 305.0 145.0 3880 . 1
## 223 17.0 8 260.0 110.0 4060 . 1
## 224 15.5 8 318.0 145.0 4140 . 1
## 225 15.0 8 302.0 130.0 4295 . 1
## 226 17.5 6 250.0 110.0 3520 . 1
## 227 20.5 6 231.0 105.0 3425 . 1
## 228 19.0 6 225.0 100.0 3630 . 1
## 229 18.5 6 250.0 98.00 3525 . 1
## 230 16.0 8 400.0 180.0 4220 . 1
## 231 15.5 8 350.0 170.0 4165 . 1
## 232 15.5 8 400.0 190.0 4325 . 1
## 233 16.0 8 351.0 149.0 4335 . 1
## 234 29.0 4 97.00 78.00 1940 . 1
## 235 24.5 4 151.0 88.00 2740 . 1
## 236 26.0 4 97.00 75.00 2265 . 1
## 237 25.5 4 140.0 89.00 2755 . 1
## 238 30.5 4 98.00 63.00 2051 . 1
## 239 33.5 4 98.00 83.00 2075 . 1
## 240 30.0 4 97.00 67.00 1985 . 1
## 241 30.5 4 97.00 78.00 2190 . 1
## 242 22.0 6 146.0 97.00 2815 . 1
## 243 21.5 4 121.0 110.0 2600 . 1
## 244 21.5 3 80.00 110.0 2720 . 1
## 245 43.1 4 90.00 48.00 1985 . 2
## 246 36.1 4 98.00 66.00 1800 . 1
## 247 32.8 4 78.00 52.00 1985 . 1
## 248 39.4 4 85.00 70.00 2070 . 1
## 249 36.1 4 91.00 60.00 1800 . 1
## 250 19.9 8 260.0 110.0 3365 . 1
## 251 19.4 8 318.0 140.0 3735 . 1
## 252 20.2 8 302.0 139.0 3570 . 1
## 253 19.2 6 231.0 105.0 3535 . 1
## 254 20.5 6 200.0 95.00 3155 . 1
## 255 20.2 6 200.0 85.00 2965 . 1
## 256 25.1 4 140.0 88.00 2720 . 1
## 257 20.5 6 225.0 100.0 3430 . 1
## 258 19.4 6 232.0 90.00 3210 . 1
## 259 20.6 6 231.0 105.0 3380 . 1
## 260 20.8 6 200.0 85.00 3070 . 1
## 261 18.6 6 225.0 110.0 3620 . 1
## 262 18.1 6 258.0 120.0 3410 . 1
## 263 19.2 8 305.0 145.0 3425 . 1
## 264 17.7 6 231.0 165.0 3445 . 1
## 265 18.1 8 302.0 139.0 3205 . 1
## 266 17.5 8 318.0 140.0 4080 . 1
## 267 30.0 4 98.00 68.00 2155 . 1
## 268 27.5 4 134.0 95.00 2560 . 1
## 269 27.2 4 119.0 97.00 2300 . 1
## 270 30.9 4 105.0 75.00 2230 . 1
## 271 21.1 4 134.0 95.00 2515 . 1
## 272 23.2 4 156.0 105.0 2745 . 1
## 273 23.8 4 151.0 85.00 2855 . 1
## 274 23.9 4 119.0 97.00 2405 . 1
## 275 20.3 5 131.0 103.0 2830 . 1
## 276 17.0 6 163.0 125.0 3140 . 1
## 277 21.6 4 121.0 115.0 2795 . 1
## 278 16.2 6 163.0 133.0 3410 . 1
## 279 31.5 4 89.00 71.00 1990 . 1
## 280 29.5 4 98.00 68.00 2135 . 1
## 281 21.5 6 231.0 115.0 3245 . 1
## 282 19.8 6 200.0 85.00 2990 . 1
## 283 22.3 4 140.0 88.00 2890 . 1
## 284 20.2 6 232.0 90.00 3265 . 1
## 285 20.6 6 225.0 110.0 3360 . 1
## 286 17.0 8 305.0 130.0 3840 . 1
## 287 17.6 8 302.0 129.0 3725 . 1
## 288 16.5 8 351.0 138.0 3955 . 1
## 289 18.2 8 318.0 135.0 3830 . 1
## 290 16.9 8 350.0 155.0 4360 . 1
## 291 15.5 8 351.0 142.0 4054 . 1
## 292 19.2 8 267.0 125.0 3605 . 1
## 293 18.5 8 360.0 150.0 3940 . 1
## 294 31.9 4 89.00 71.00 1925 . 1
## 295 34.1 4 86.00 65.00 1975 . 1
## 296 35.7 4 98.00 80.00 1915 . 1
## 297 27.4 4 121.0 80.00 2670 . 1
## 298 25.4 5 183.0 77.00 3530 . 2
## 299 23.0 8 350.0 125.0 3900 . 1
## 300 27.2 4 141.0 71.00 3190 . 2
## 301 23.9 8 260.0 90.00 3420 . 2
## 302 34.2 4 105.0 70.00 2200 . 1
## 303 34.5 4 105.0 70.00 2150 . 1
## 304 31.8 4 85.00 65.00 2020 . 1
## 305 37.3 4 91.00 69.00 2130 . 1
## 306 28.4 4 151.0 90.00 2670 . 1
## 307 28.8 6 173.0 115.0 2595 . 1
## 308 26.8 6 173.0 115.0 2700 . 1
## 309 33.5 4 151.0 90.00 2556 . 1
## 310 41.5 4 98.00 76.00 2144 . 1
## 311 38.1 4 89.00 60.00 1968 . 1
## 312 32.1 4 98.00 70.00 2120 . 1
## 313 37.2 4 86.00 65.00 2019 . 1
## 314 28.0 4 151.0 90.00 2678 . 1
## 315 26.4 4 140.0 88.00 2870 . 1
## 316 24.3 4 151.0 90.00 3003 . 2
## 317 19.1 6 225.0 90.00 3381 . 1
## 318 34.3 4 97.00 78.00 2188 . 1
## 319 29.8 4 134.0 90.00 2711 . 1
## 320 31.3 4 120.0 75.00 2542 . 1
## 321 37.0 4 119.0 92.00 2434 . 1
## 322 32.2 4 108.0 75.00 2265 . 1
## 323 46.6 4 86.00 65.00 2110 . 1
## 324 27.9 4 156.0 105.0 2800 . 1
## 325 40.8 4 85.00 65.00 2110 . 1
## 326 44.3 4 90.00 48.00 2085 . 2
## 327 43.4 4 90.00 48.00 2335 . 2
## 328 36.4 5 121.0 67.00 2950 . 1
## 329 30.0 4 146.0 67.00 3250 . 2
## 330 44.6 4 91.00 67.00 1850 . 1
## 331 40.9 4 85.00 ? 1835 . 1
## 332 33.8 4 97.00 67.00 2145 . 1
## 333 29.8 4 89.00 62.00 1845 . 1
## 334 32.7 6 168.0 132.0 2910 . 1
## 335 23.7 3 70.00 100.0 2420 . 1
## 336 35.0 4 122.0 88.00 2500 . 1
## 337 23.6 4 140.0 ? 2905 . 1
## 338 32.4 4 107.0 72.00 2290 . 1
## 339 27.2 4 135.0 84.00 2490 . 1
## 340 26.6 4 151.0 84.00 2635 . 1
## 341 25.8 4 156.0 92.00 2620 . 1
## 342 23.5 6 173.0 110.0 2725 . 1
## 343 30.0 4 135.0 84.00 2385 . 1
## 344 39.1 4 79.00 58.00 1755 . 1
## 345 39.0 4 86.00 64.00 1875 . 1
## 346 35.1 4 81.00 60.00 1760 . 1
## 347 32.3 4 97.00 67.00 2065 . 1
## 348 37.0 4 85.00 65.00 1975 . 1
## 349 37.7 4 89.00 62.00 2050 . 1
## 350 34.1 4 91.00 68.00 1985 . 1
## 351 34.7 4 105.0 63.00 2215 . 1
## 352 34.4 4 98.00 65.00 2045 . 1
## 353 29.9 4 98.00 65.00 2380 . 2
## 354 33.0 4 105.0 74.00 2190 . 1
## 355 34.5 4 100.0 ? 2320 . 1
## 356 33.7 4 107.0 75.00 2210 . 1
## 357 32.4 4 108.0 75.00 2350 . 1
## 358 32.9 4 119.0 100.0 2615 . 1
## 359 31.6 4 120.0 74.00 2635 . 1
## 360 28.1 4 141.0 80.00 3230 . 2
## 361 30.7 6 145.0 76.00 3160 . 1
## 362 25.4 6 168.0 116.0 2900 . 1
## 363 24.2 6 146.0 120.0 2930 . 1
## 364 22.4 6 231.0 110.0 3415 . 1
## 365 26.6 8 350.0 105.0 3725 . 1
## 366 20.2 6 200.0 88.00 3060 . 1
## 367 17.6 6 225.0 85.00 3465 . 1
## 368 28.0 4 112.0 88.00 2605 . 1
## 369 27.0 4 112.0 88.00 2640 . 1
## 370 34.0 4 112.0 88.00 2395 . 1
## 371 31.0 4 112.0 85.00 2575 . 1
## 372 29.0 4 135.0 84.00 2525 . 1
## 373 27.0 4 151.0 90.00 2735 . 1
## 374 24.0 4 140.0 92.00 2865 . 1
## 375 23.0 4 151.0 ? 3035 . 2
## 376 36.0 4 105.0 74.00 1980 . 1
## 377 37.0 4 91.00 68.00 2025 . 1
## 378 31.0 4 91.00 68.00 1970 . 1
## 379 38.0 4 105.0 63.00 2125 . 1
## 380 36.0 4 98.00 70.00 2125 . 1
## 381 36.0 4 120.0 88.00 2160 . 1
## 382 36.0 4 107.0 75.00 2205 . 1
## 383 34.0 4 108.0 70.00 2245 1
## 384 38.0 4 91.00 67.00 1965 . 1
## 385 32.0 4 91.00 67.00 1965 . 1
## 386 38.0 4 91.00 67.00 1995 . 1
## 387 25.0 6 181.0 110.0 2945 . 1
## 388 38.0 6 262.0 85.00 3015 . 1
## 389 26.0 4 156.0 92.00 2585 . 1
## 390 22.0 6 232.0 112.0 2835 1
## 391 32.0 4 144.0 96.00 2665 . 1
## 392 36.0 4 135.0 84.00 2370 . 1
## 393 27.0 4 151.0 90.00 2950 . 1
## 394 27.0 4 140.0 86.00 2790 . 1
## 395 44.0 4 97.00 52.00 2130 . 2
## 396 32.0 4 135.0 84.00 2295 . 1
## 397 28.0 4 120.0 79.00 2625 . 1
## 398 31.0 4 119.0 82.00 2720 . 1colnames(auto_mpg_data) <- c("displacement", "horsepower", "weight", "acceleration","mpg")
A <- scale(data.matrix(auto_mpg_data[1:4]))
mpg <- data.matrix(auto_mpg_data[5])
colnames(mpg)<- "mpg"
#######calculate the covariance matrix
cov <- cov(A)
print(cov)## displacement horsepower weight acceleration
## displacement 1.0000000 -0.6522363 0.4215846 -0.8317409
## horsepower -0.6522363 1.0000000 -0.5256948 0.7590080
## weight 0.4215846 -0.5256948 1.0000000 -0.4807430
## acceleration -0.8317409 0.7590080 -0.4807430 1.0000000####### Now, let's do SVD on the covariance Matrix
LSA <- svd(cov)
print(LSA)## $d
## [1] 2.8621693 0.6571824 0.3341016 0.1465468
##
## $u
## [,1] [,2] [,3] [,4]
## [1,] -0.5176258 -0.37976775 -0.52752464 0.55637911
## [2,] 0.5185625 0.01775182 -0.80995911 -0.27339360
## [3,] -0.4048118 0.88250074 -0.23941248 -0.00124319
## [4,] 0.5470685 0.27686418 0.09146395 0.78466342
##
## $v
## [,1] [,2] [,3] [,4]
## [1,] -0.5176258 -0.37976775 -0.52752464 0.55637911
## [2,] 0.5185625 0.01775182 -0.80995911 -0.27339360
## [3,] -0.4048118 0.88250074 -0.23941248 -0.00124319
## [4,] 0.5470685 0.27686418 0.09146395 0.78466342####### and the variance eigenvectors are
LSA$d## [1] 2.8621693 0.6571824 0.3341016 0.1465468LSA$u## [,1] [,2] [,3] [,4]
## [1,] -0.5176258 -0.37976775 -0.52752464 0.55637911
## [2,] 0.5185625 0.01775182 -0.80995911 -0.27339360
## [3,] -0.4048118 0.88250074 -0.23941248 -0.00124319
## [4,] 0.5470685 0.27686418 0.09146395 0.78466342LSA$v## [,1] [,2] [,3] [,4]
## [1,] -0.5176258 -0.37976775 -0.52752464 0.55637911
## [2,] 0.5185625 0.01775182 -0.80995911 -0.27339360
## [3,] -0.4048118 0.88250074 -0.23941248 -0.00124319
## [4,] 0.5470685 0.27686418 0.09146395 0.78466342#########Plot as
plot(LSA$d,type='b',pch=10,xlab='Singular value',ylab='magnitude')
#### Compare the result
my.ir.pca <- prcomp(A,center=FALSE,scale.=FALSE)
my.ir.pca## Standard deviations:
## [1] 1.6917947 0.8106679 0.5780152 0.3828143
##
## Rotation:
## PC1 PC2 PC3 PC4
## displacement 0.5176258 -0.37976775 0.52752464 0.55637911
## horsepower -0.5185625 0.01775182 0.80995911 -0.27339360
## weight 0.4048118 0.88250074 0.23941248 -0.00124319
## acceleration -0.5470685 0.27686418 -0.09146395 0.78466342my.ir.pca$sdev^2## [1] 2.8621693 0.6571824 0.3341016 0.1465468#################
LSA <- svd(A)
# Let's now project the data onto these two dimensions
depth <-2
us <- as.matrix(LSA$u[, 1:depth])
vs <- as.matrix(LSA$v[, 1:depth])
ds <- as.matrix(diag(LSA$d)[1:depth, 1:depth])
A_dn <- us %*% ds %*% t(vs)
colnames(A_dn) <- c("displacement", "horsepower", "weight", "acceleration")
pairs(~.,data=cbind(A_dn,mpg), main=paste("Scatterplot Matrix of auto-mpg Data onto the First",depth,"Dimensions."))
#colnames(A) <- c("displacement", "horsepower", "weight", "acceleration")
pairs(~.,data=cbind(A,mpg), main="Scatterplot Matrix of auto-mpg Data (Original)")
#######Part2-Question 2.3
infinity<-5000
n <- c(1:infinity)
q <-(1-1/n)^n
plot(n,q)
###identify maximum value of Q
lim <- max(q)
lim## [1] 0.3678427#####pofInfinity is equal to 1-lim
1-lim## [1] 0.6321573# In this program lines, we will do the experiment and boostrap n elements from a sequence of values from 1:n
bsExperiment <- function(n){
# Define the sequence of values to sample from
values <- seq(1:n)
# let's sample with replacement n-times
samples <- replicate(n,{sample(values,1,replace=TRUE)})
# return the percentage of values samples in this iteration
return(length(unique(samples))/n)
}
# The output of a single experiment with n=1000 is:
n <- 10000
bsExperiment(n)## [1] 0.6326### we can repeat for several values
m<-500
bsExperiment(m)## [1] 0.614You can also embed plots, for example: Note that the echo = FALSE parameter was added to the code chunk to prevent printing of the R code that generated the plot.