This tutorial will walk you through some basic matrix notation that will be useful. First I will go over how to make a basic matrix:
A = matrix(
c(2, 4, 3, 1, 5, 7), # the data elements
nrow=2, # number of rows
ncol=3, # number of columns
byrow = TRUE) # fill matrix by rows
A # print the matrix ## [,1] [,2] [,3]
## [1,] 2 4 3
## [2,] 1 5 7Now that we know how to create a basic matrix and the format, we can use this same logic with any dataset we have. For this tutorial I am going to use the Boston dataset. Below are some points that we will look at that we will later change.
data(Boston)
class(Boston)## [1] "data.frame"str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...We will change the dataset into a matrix, see how these functions below change now that our dataset is a matrix.
mat <- as.matrix(Boston)
class(mat)## [1] "matrix"str(mat)## num [1:506, 1:14] 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## - attr(*, "dimnames")=List of 2
## ..$ : chr [1:506] "1" "2" "3" "4" ...
## ..$ : chr [1:14] "crim" "zn" "indus" "chas" ...we will now look at some other properties that can be used for analysis later on. The nrow and ncol functions tell us how many rows and columns our matrix(dataset) has.
nrow(mat)## [1] 506ncol(mat)## [1] 14Now we will look at specific values in the matrix. Using the matrix name with brackets will reference a row and a column, an example is below
# first value in mat, first row and first column
mat[1, 1]## [1] 0.00632# a middle value in mat, 250th row and 5th column
mat[250, 5]## [1] 0.431We can also reference specific cells by referencing a certain row and column name like we did below:
mat["10","age"]## [1] 85.9We can also refernce whole sections as well. If you wanna refernce several rows and columns add : between the range you wish to select, some examples are below:
mat[1:4, 1:2]## crim zn
## 1 0.00632 18
## 2 0.02731 0
## 3 0.02729 0
## 4 0.03237 0mat[5:8, 1:2]## crim zn
## 5 0.06905 0.0
## 6 0.02985 0.0
## 7 0.08829 12.5
## 8 0.14455 12.5If we don't want to reference a range, but want to reference more than one cell, we can use the c() function to create matrixes within our matrix.
mat[c(1,3,5), c(1,3)]## crim indus
## 1 0.00632 2.31
## 3 0.02729 7.07
## 5 0.06905 2.18If we only want to reference a certain row or column we can, we don't need to reference both:
# All columns from row 5
mat[5, ]## crim zn indus chas nox rm age dis
## 0.06905 0.00000 2.18000 0.00000 0.45800 7.14700 54.20000 6.06220
## rad tax ptratio black lstat medv
## 3.00000 222.00000 18.70000 396.90000 5.33000 36.20000# All rows from column 2
mat[, 2]## 1 2 3 4 5 6 7 8 9 10 11 12 13
## 18.0 0.0 0.0 0.0 0.0 0.0 12.5 12.5 12.5 12.5 12.5 12.5 12.5
## 14 15 16 17 18 19 20 21 22 23 24 25 26
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 27 28 29 30 31 32 33 34 35 36 37 38 39
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 40 41 42 43 44 45 46 47 48 49 50 51 52
## 75.0 75.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 21.0 21.0
## 53 54 55 56 57 58 59 60 61 62 63 64 65
## 21.0 21.0 75.0 90.0 85.0 100.0 25.0 25.0 25.0 25.0 25.0 25.0 17.5
## 66 67 68 69 70 71 72 73 74 75 76 77 78
## 80.0 80.0 12.5 12.5 12.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 79 80 81 82 83 84 85 86 87 88 89 90 91
## 0.0 0.0 25.0 25.0 25.0 25.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 92 93 94 95 96 97 98 99 100 101 102 103 104
## 0.0 28.0 28.0 28.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 105 106 107 108 109 110 111 112 113 114 115 116 117
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 118 119 120 121 122 123 124 125 126 127 128 129 130
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 131 132 133 134 135 136 137 138 139 140 141 142 143
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 144 145 146 147 148 149 150 151 152 153 154 155 156
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 157 158 159 160 161 162 163 164 165 166 167 168 169
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 170 171 172 173 174 175 176 177 178 179 180 181 182
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 183 184 185 186 187 188 189 190 191 192 193 194 195
## 0.0 0.0 0.0 0.0 0.0 45.0 45.0 45.0 45.0 45.0 45.0 60.0 60.0
## 196 197 198 199 200 201 202 203 204 205 206 207 208
## 80.0 80.0 80.0 80.0 95.0 95.0 82.5 82.5 95.0 95.0 0.0 0.0 0.0
## 209 210 211 212 213 214 215 216 217 218 219 220 221
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 222 223 224 225 226 227 228 229 230 231 232 233 234
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 235 236 237 238 239 240 241 242 243 244 245 246 247
## 0.0 0.0 0.0 0.0 30.0 30.0 30.0 30.0 30.0 30.0 22.0 22.0 22.0
## 248 249 250 251 252 253 254 255 256 257 258 259 260
## 22.0 22.0 22.0 22.0 22.0 22.0 22.0 80.0 80.0 90.0 20.0 20.0 20.0
## 261 262 263 264 265 266 267 268 269 270 271 272 273
## 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0
## 274 275 276 277 278 279 280 281 282 283 284 285 286
## 20.0 40.0 40.0 40.0 40.0 40.0 20.0 20.0 20.0 20.0 90.0 90.0 55.0
## 287 288 289 290 291 292 293 294 295 296 297 298 299
## 80.0 52.5 52.5 52.5 80.0 80.0 80.0 0.0 0.0 0.0 0.0 0.0 70.0
## 300 301 302 303 304 305 306 307 308 309 310 311 312
## 70.0 70.0 34.0 34.0 34.0 33.0 33.0 33.0 33.0 0.0 0.0 0.0 0.0
## 313 314 315 316 317 318 319 320 321 322 323 324 325
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 326 327 328 329 330 331 332 333 334 335 336 337 338
## 0.0 0.0 0.0 0.0 0.0 0.0 35.0 35.0 0.0 0.0 0.0 0.0 0.0
## 339 340 341 342 343 344 345 346 347 348 349 350 351
## 0.0 0.0 0.0 35.0 0.0 55.0 55.0 0.0 0.0 85.0 80.0 40.0 40.0
## 352 353 354 355 356 357 358 359 360 361 362 363 364
## 60.0 60.0 90.0 80.0 80.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 365 366 367 368 369 370 371 372 373 374 375 376 377
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 378 379 380 381 382 383 384 385 386 387 388 389 390
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 391 392 393 394 395 396 397 398 399 400 401 402 403
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 404 405 406 407 408 409 410 411 412 413 414 415 416
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 417 418 419 420 421 422 423 424 425 426 427 428 429
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 430 431 432 433 434 435 436 437 438 439 440 441 442
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 443 444 445 446 447 448 449 450 451 452 453 454 455
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 456 457 458 459 460 461 462 463 464 465 466 467 468
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 469 470 471 472 473 474 475 476 477 478 479 480 481
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 482 483 484 485 486 487 488 489 490 491 492 493 494
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## 495 496 497 498 499 500 501 502 503 504 505 506
## 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0Now we will look at some functions that are a bit more complicated that you can do with a matrix We will look at some summary functions:
# first row, all of the columns
column_1 <- mat[1, ]
# max particle size for resin 1
max(column_1)## [1] 396.9# max particle size for resin 2
max(mat[2, ])## [1] 396.9# minimum particle size for operator 3
min(mat[, 3])## [1] 0.46# mean for operator 3
mean(mat[, 3])## [1] 11.13678# median for operator 3
median(mat[, 3])## [1] 9.69# standard deviation for operator 3
sd(mat[, 3])## [1] 6.860353Another useful function is the apply function. This allows us the collasp the matrix down to the dimension specified by the margin. For example to find the average particle size of each column, we will calculate the mean of all the rows of the matrix:
avg_column <- apply(mat, 1, mean)I will now use the matrix notation we learned to run a two sample t-test:
t.test(mat[5,], mat[10,], paired=TRUE)##
## Paired t-test
##
## data: mat[5, ] and mat[10, ]
## t = -1.2579, df = 13, p-value = 0.2306
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -23.530384 6.212686
## sample estimates:
## mean of the differences
## -8.658849Specifically, do the means of the two vectors differ significantly?
result <- t.test(mat[5,], mat[10,], paired=TRUE)
names(result)## [1] "statistic" "parameter" "p.value" "conf.int" "estimate"
## [6] "null.value" "stderr" "alternative" "method" "data.name"result$p.value## [1] 0.2305678We can see from this analysis that our p-value is high, so we would conclude that the means of the two vectors don't differ significantly.
Now that we have went ove some basic notation of matrices, this will help you further in many other analysis and functions in R. Any dataset can be turned into a matrix and the notation is easier to use for many more functions such as t-test and many more analysis!