Chapter 6 Cluster Analysis
Previously, we have investigated two consecutive words using ngrams, associations of any two words, and associations rules for word relationship. Here, we are interested in what words come together as a group or cluster. In doing so, we need to measure the closeness of words. One way to measure it is to first consider the frequency of each word for all professors. Then we get a matrix of data with words on the columns and frequency on the rows. Then we can calculate the distance of a pair of words. Note that pairwise correlation can be used as a closeness or distance matrix.
We first get the frequency of words for each professor using tidytext. Then we convert the data to a document-term matrix using the cast_dtm function in the tidytext package. After that, we change the document term matrix to a regular matrix using as.matrix.
prof.tm <- unnest_tokens(prof1000, word, comments)
stopwords <- read_csv("data/stopwords.evaluation.csv")
prof.tm <- prof.tm %>% anti_join(stopwords)
prof.dtm <- prof.tm %>% count(profid, word) %>% cast_dtm(profid, word, n)
prof.dtm
## <<DocumentTermMatrix (documents: 1000, terms: 8155)>>
## Non-/sparse entries: 229773/7925227
## Sparsity : 97%
## Maximal term length: 19
## Weighting : term frequency (tf)Note that most of the elements of the matrix are 0. The full matrix would have a total of \(1000\times 8,155\) elements and only 229,773 of them are not 0 and 97% of elements are 0.
If the words are very rarely used, then their association would be low. Therefore, here we focus on frequently used words. Particularly, we remove words that have at lease a sparsity of 0.4, meaning for 40% of professors, the words were not used. After this, there are 57 words left in the data. The sparsity is only 22%.
prof.subset <- removeSparseTerms(prof.dtm, 0.4)
tm::inspect(prof.subset)
## <<DocumentTermMatrix (documents: 1000, terms: 57)>>
## Non-/sparse entries: 44327/12673
## Sparsity : 22%
## Maximal term length: 11
## Weighting : term frequency (tf)
## Sample :
## Terms
## Docs easy good great hard help learn lecture test time work
## 105 20 25 14 43 27 18 27 105 19 10
## 133 58 47 62 62 25 28 54 121 28 23
## 3 36 30 52 45 19 16 100 62 16 12
## 562 35 35 29 35 30 12 32 19 52 13
## 641 38 38 45 47 55 10 47 65 30 40
## 658 113 28 45 23 14 17 39 90 11 1
## 721 119 25 40 18 8 30 67 46 25 11
## 782 65 23 36 10 18 45 27 58 15 1
## 960 123 144 172 96 152 114 419 64 77 67
## 973 50 35 69 16 46 28 2 7 31 586.1 Hierarchical cluster analysis
We first cluster the words into groups through hierarchical cluster analysis based on the dissimilarity of the words. One way to measure the dissimilarity is to use the Euclidean distance. Imagine that each word is a point in a 1000-dimensional space formed by the 1,000 professors. The number of times of the word used by the professors is the coordinates in the space. Then for any two words, the distance in the space can be calculated as a measure of dissimilarity. A larger distance indicates more dissimilarity of the two words. Overall, a 57 by 57 distance matrix can be formed. Part of the matrix is shown below.
d <- dist(t(prof.subset), method = "euclidian")
as.matrix(round(d))[1:8, 1:8]
## actual answer assignment awesome bad best better boring
## actual 0 85 139 83 71 191 64 113
## answer 85 0 149 107 86 203 88 122
## assignment 139 149 0 151 150 213 145 170
## awesome 83 107 151 0 103 166 93 132
## bad 71 86 150 103 0 213 68 106
## best 191 203 213 166 213 0 207 224
## better 64 88 145 93 68 207 0 115
## boring 113 122 170 132 106 224 115 0A Euclidean distance is a straight-line distance between two points in a Euclidean space. For example, consider a 3-dimensional space shown in the figure below (the figure was adapted from Wikipedia: https://en.wikipedia.org/wiki/Euclidean_distance). For two points \(p\) and \(q\) with the coordinates \((p_1, p_2, p_3)\) and \((q_1, q_2, q_3)\), respectively. The Euclidean distance between them is
\[ d(p,q) = \sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + (p_3-q_3)^2}. \]
With the distance matrix, we can then cluster the words. The R function hclust provides several methods for cluster analysis. The different methods generally follow the same procedure. Initially, each word is assumed to its own cluster. Then the algorithm proceeds iteratively, at each stage joining the two most similar clusters and continuing until there is just a single cluster. At each stage distances between clusters are recomputed by the Lance–Williams algorithm according to the particular clustering method being used.
The default method used by hclust is called complete-linkage clustering, also known as farthest neighbor clustering. At the beginning of the clustering process, each element is in a cluster of its own. Then the two clusters separated by the shortest distance are combined. For two clusters, the distance of them is the maximum distance among any pair of elements from the two clusters.
Mathematically, the distance between clusters \(X\) and \(Y\) is defined as
\[
D(X,Y)= \max_{x\in X, y\in Y} d(x,y)
\]
where \(d(x,y)\) is the distance between an element \(x\) in the cluster \(X\) and an element \(y\) in the cluster \(Y\).
The following code conducts the cluster analysis based on the distance matrix for the words.
fit <- hclust(d = d, method = "complete")
fit
##
## Call:
## hclust(d = d, method = "complete")
##
## Cluster method : complete
## Distance : euclidean
## Number of objects: 57It is often easier to visualize the clusters through a plot. Such a plot is called a “dendrogram”, which has a tree structure. For example, the following code draws the cluster dendrogram and then adds a rectangular box around each cluster. From it, we can easily see which words belong to which groups. For example, if we decide to keep 6 clusters, we can see that (1) “lecture”, “easy”, and “test” form their own clusters; (2) “great”, “good”, “help”, “hard”, and “work” belong to one cluster; (3) “study”, “exam”, and “note” belong to one cluster; and (4) the rest of words belong to one cluster. We will discuss how to choose the optimal number of clusters later.
6.2 k-means
k-means clustering is a popular method for clustering. It aims to partition the observations into k clusters in which each observation belongs to the cluster with the nearest mean.
The idea of the k-means is straightforward. Suppose we want to cluster the data in the figure below. Our task is to group the data into two clusters.
In the first step, we can randomly initialize two points (the blue and red circles), called the cluster centroids. Then, we can cluster each data into the closest cluster centroid.
After that, we can then calculate the new cluster centroids based on the elements in them.
Then, we can re-assign each element to the closest centroid again.
The above procedure can be repeated so the cluster centroid will not move any further.
6.2.1 k-means in R
The R function kmeans can be used for k-means clustering. To do it, we need to specify the number of clusters. For example, a k-means clustering with two clusters is conducted using the following code. Note that the function kmeans needs the input of raw data other than a distance matrix.
From the output, we can see that the two clusters have 50 and 7 words, respectively. The cluster membership can be identified using the clustering vector. The means of each cluster are also provided.
kmeans.data <- as.matrix(t(prof.subset))
kfit <- kmeans(kmeans.data, 2)
kfit
## K-means clustering with 2 clusters of sizes 7, 50
##
## Cluster means:
## 1 2 3 4 5 6 7 8 9
## 1 13.42857 1.857143 49.14286 7.00 28.42857 2.571429 2.857143 1.428571 8.857143
## 2 7.06000 0.920000 17.90000 3.04 10.16000 1.060000 1.320000 0.760000 3.160000
## 10 11 12 13 14 15 16 17
## 1 32.28571 5.571429 4.571429 6.571429 15.28571 2.714286 11.42857 11.85714
## 2 11.58000 2.780000 1.440000 3.400000 4.40000 1.180000 3.70000 5.78000
## 18 19 20 21 22 23 24 25
## 1 6.571429 6.857143 2.285714 3.142857 4.857143 15.28571 6.714286 7.428571
## 2 5.640000 2.700000 0.880000 1.580000 2.560000 5.38000 2.740000 2.100000
## 26 27 28 29 30 31 32 33 34
## 1 15.14286 3.714286 11.0 3.00 3.428571 7.428571 7.571429 5.714286 7.714286
## 2 6.92000 1.660000 4.1 0.84 0.960000 2.300000 3.120000 2.000000 2.640000
## 35 36 37 38 39 40 41 42
## 1 15.57143 4.714286 14.85714 2.857143 0.4285714 13.57143 5.714286 10.14286
## 2 5.82000 1.520000 6.30000 1.140000 0.1400000 4.00000 1.880000 3.58000
## 43 44 45 46 47 48 49 50 51
## 1 1.857143 7.00 3.714286 4.714286 6.142857 3.571429 5.428571 20.00 5.714286
## 2 0.700000 2.56 1.540000 2.900000 2.680000 1.920000 2.720000 6.52 3.240000
## 52 53 54 55 56 57 58 59
## 1 11.57143 3.285714 5.571429 5.142857 16.71429 9.142857 6.285714 19.71429
## 2 3.70000 1.500000 1.540000 1.580000 5.48000 3.180000 2.240000 7.64000
## 60 61 62 63 64 65 66 67 68
## 1 8.857143 11.57143 8.857143 7.571429 7.285714 2.428571 19.28571 8.00 8.285714
## 2 5.220000 2.62000 2.620000 2.360000 2.000000 2.040000 6.86000 3.36 4.020000
## 69 70 71 72 73 74 75 76 77
## 1 17.00 8.571429 1.142857 8.142857 10.28571 14.42857 9.857143 10.0 16.28571
## 2 6.02 4.000000 0.660000 2.040000 2.90000 6.04000 4.780000 3.8 5.02000
## 78 79 80 81 82 83 84 85 86
## 1 17.71429 20.42857 27.42857 4.857143 29.14286 4.285714 19.0 3.714286 15.00
## 2 5.72000 7.24000 10.22000 1.180000 8.16000 2.140000 5.2 1.540000 5.06
## 87 88 89 90 91 92 93 94 95
## 1 2.857143 2.285714 8.285714 22.28571 2.285714 15.57143 2.00 3.428571 22.85714
## 2 1.040000 1.020000 2.440000 7.22000 0.760000 7.08000 1.56 0.760000 7.62000
## 96 97 98 99 100 101 102 103 104
## 1 3.0 7.142857 7.714286 11.42857 3.428571 18.14286 9.285714 6.00 5.571429
## 2 1.2 2.920000 2.860000 3.42000 1.460000 5.94000 3.760000 1.52 1.780000
## 105 106 107 108 109 110 111 112 113
## 1 37.28571 4.857143 13.14286 4.428571 3.0 5.714286 4.142857 3.857143 2.428571
## 2 12.76000 3.860000 4.46000 0.800000 1.1 2.380000 1.960000 1.780000 0.500000
## 114 115 116 117 118 119 120 121 122
## 1 0.5714286 6.428571 9.285714 3.714286 2.428571 23.42857 14.0 22.28571 14.71429
## 2 0.8800000 3.480000 3.440000 1.580000 1.680000 9.08000 4.2 10.28000 5.06000
## 123 124 125 126 127 128 129 130 131 132
## 1 5.428571 3.00 8.285714 15.85714 12.28571 8.00 9.571429 3.0 6.571429 7.0
## 2 1.880000 0.88 4.180000 5.40000 5.62000 3.34 4.620000 1.1 1.400000 2.2
## 133 134 135 136 137 138 139 140 141
## 1 61.28571 2.00 3.142857 2.857143 4.285714 9.142857 12.57143 18.28571 4.714286
## 2 22.60000 0.88 1.580000 1.420000 1.940000 5.800000 4.92000 8.20000 2.000000
## 142 143 144 145 146 147 148 149
## 1 1.714286 29.28571 2.285714 11.14286 18.42857 1.571429 9.285714 2.571429
## 2 1.860000 8.16000 0.820000 3.36000 4.98000 1.100000 2.420000 1.260000
## 150 151 152 153 154 155 156 157 158 159 160
## 1 0.1428571 5.00 2.285714 4.0 5.714286 4.142857 2.571429 5.142857 19.0 3.00 6.0
## 2 0.6200000 1.78 0.880000 1.2 2.500000 1.860000 1.140000 2.220000 8.1 1.76 2.4
## 161 162 163 164 165 166 167 168
## 1 5.142857 27.28571 1.428571 5.285714 24.00 5.714286 0.7142857 7.142857
## 2 1.840000 10.14000 0.500000 2.720000 6.32 1.320000 0.5200000 3.760000
## 169 170 171 172 173 174 175 176 177
## 1 9.857143 25.42857 7.571429 25.28571 5.285714 3.285714 7.00 7.857143 6.142857
## 2 5.640000 8.56000 2.420000 6.86000 1.580000 1.260000 3.82 3.900000 2.580000
## 178 179 180 181 182 183 184 185 186
## 1 11.85714 2.857143 2.00 6.571429 7.285714 3.285714 8.285714 9.714286 5.142857
## 2 2.48000 1.260000 2.08 2.340000 3.760000 1.740000 2.680000 3.520000 1.280000
## 187 188 189 190 191 192 193 194
## 1 3.714286 3.714286 5.571429 11.28571 9.142857 3.571429 3.428571 32.57143
## 2 1.460000 1.380000 1.520000 3.46000 3.060000 1.500000 1.360000 8.84000
## 195 196 197 198 199 200 201 202
## 1 3.857143 6.857143 4.714286 9.857143 3.714286 2.142857 10.28571 8.714286
## 2 1.220000 3.260000 2.720000 4.220000 1.340000 1.460000 4.46000 4.180000
## 203 204 205 206 207 208 209 210 211
## 1 4.571429 15.57143 2.714286 8.00 2.714286 13.14286 7.00 20.85714 8.857143
## 2 2.120000 7.04000 1.560000 2.38 1.300000 8.22000 2.32 5.42000 5.740000
## 212 213 214 215 216 217 218 219 220
## 1 4.857143 16.00 6.857143 4.428571 1.571429 13.85714 2.857143 4.142857 6.428571
## 2 2.320000 3.84 2.360000 2.680000 1.400000 3.76000 1.480000 1.440000 4.340000
## 221 222 223 224 225 226 227 228 229
## 1 3.857143 4.428571 5.428571 5.857143 16.85714 7.285714 2.571429 3.0 5.142857
## 2 2.360000 1.540000 1.160000 2.480000 4.94000 4.200000 1.660000 1.5 2.160000
## 230 231 232 233 234 235 236 237 238
## 1 5.428571 6.142857 4.285714 9.571429 9.00 13.71429 4.428571 3.857143 14.57143
## 2 1.160000 3.080000 2.040000 2.460000 4.82 6.46000 1.760000 1.480000 3.92000
## 239 240 241 242 243 244 245 246 247
## 1 8.00 10.14286 4.571429 5.142857 2.428571 2.714286 4.571429 4.714286 4.00
## 2 2.08 4.00000 1.860000 1.900000 1.340000 1.140000 2.520000 2.100000 1.26
## 248 249 250 251 252 253 254 255 256
## 1 3.285714 2.142857 7.857143 2.285714 4.714286 3.857143 5.571429 5.571429 20.00
## 2 2.160000 0.840000 4.000000 1.860000 2.240000 1.220000 2.640000 2.220000 7.46
## 257 258 259 260 261 262 263 264 265
## 1 7.0 6.571429 8.714286 5.714286 5.571429 0.8571429 7.857143 17.00 13.42857
## 2 2.3 2.400000 3.700000 3.400000 2.700000 0.6000000 2.600000 4.64 6.40000
## 266 267 268 269 270 271 272 273
## 1 4.285714 1.857143 12.71429 2.285714 7.857143 4.285714 12.57143 3.714286
## 2 2.160000 1.060000 4.20000 1.040000 3.540000 2.200000 5.62000 1.120000
## 274 275 276 277 278 279 280 281 282
## 1 7.428571 8.857143 4.857143 5.571429 17.71429 3.857143 3.00 4.714286 4.714286
## 2 2.380000 4.160000 2.880000 2.200000 6.96000 1.900000 1.12 1.820000 1.260000
## 283 284 285 286 287 288 289 290 291
## 1 18.28571 5.571429 12.85714 6.571429 17.00 8.142857 5.285714 18.57143 4.00
## 2 6.42000 1.540000 4.74000 2.440000 6.36 3.220000 1.600000 6.16000 1.06
## 292 293 294 295 296 297 298 299 300
## 1 8.857143 21.57143 6.428571 12.85714 6.00 29.28571 5.571429 7.285714 5.142857
## 2 2.240000 9.62000 3.220000 4.30000 1.88 9.50000 3.460000 2.520000 1.820000
## 301 302 303 304 305 306 307 308 309
## 1 5.571429 2.285714 5.857143 17.28571 4.714286 7.428571 5.428571 12.00 4.00
## 2 2.140000 1.380000 3.460000 5.80000 2.300000 4.020000 2.180000 2.74 1.38
## 310 311 312 313 314 315 316 317 318 319
## 1 10.00 12.71429 8.571429 4.0 7.714286 7.00 6.142857 20.14286 26.71429 5.857143
## 2 3.84 4.86000 2.480000 0.8 5.800000 3.52 2.140000 7.78000 7.20000 2.380000
## 320 321 322 323 324 325 326 327 328
## 1 7.285714 1.142857 8.285714 10.71429 10.0 8.0 4.857143 5.142857 1.571429
## 2 3.260000 0.260000 4.100000 3.92000 4.3 2.1 1.840000 1.600000 1.220000
## 329 330 331 332 333 334 335 336
## 1 4.857143 4.714286 2.714286 5.571429 4.571429 4.857143 13.42857 15.14286
## 2 3.320000 1.620000 1.140000 2.280000 1.720000 1.080000 3.88000 7.16000
## 337 338 339 340 341 342 343 344
## 1 4.142857 10.71429 14.57143 7.571429 7.857143 8.142857 3.571429 2.857143
## 2 2.580000 3.72000 6.78000 2.780000 2.740000 2.680000 1.200000 1.160000
## 345 346 347 348 349 350 351 352 353
## 1 4.285714 17.71429 2.714286 2.714286 8.00 8.571429 18.42857 8.428571 4.857143
## 2 1.480000 7.54000 0.860000 1.420000 2.42 2.000000 6.44000 3.200000 1.760000
## 354 355 356 357 358 359 360 361 362 363
## 1 6.714286 6.571429 4.285714 3.428571 3.00 5.428571 5.00 1.714286 7.714286 7.00
## 2 2.300000 3.100000 1.260000 3.860000 0.64 1.580000 2.66 0.840000 2.560000 3.12
## 364 365 366 367 368 369 370 371
## 1 12.85714 12.71429 9.142857 4.857143 6.571429 4.428571 4.714286 11.14286
## 2 4.86000 6.66000 2.940000 1.360000 2.180000 2.000000 2.120000 3.88000
## 372 373 374 375 376 377 378 379 380
## 1 9.714286 9.857143 11.71429 9.142857 4.142857 3.00 5.285714 6.00 5.428571
## 2 4.340000 3.600000 4.52000 3.240000 1.280000 1.68 2.220000 4.08 2.920000
## 381 382 383 384 385 386 387 388
## 1 10.00 0.8571429 14.57143 9.571429 5.142857 7.571429 2.857143 2.142857
## 2 4.42 0.9600000 4.98000 3.420000 2.180000 2.500000 1.060000 0.880000
## 389 390 391 392 393 394 395 396 397
## 1 2.142857 10.14286 3.857143 8.714286 2.285714 11.71429 11.00 3.428571 13.00
## 2 1.260000 3.20000 2.160000 2.140000 1.060000 3.70000 3.06 1.380000 5.02
## 398 399 400 401 402 403 404 405 406
## 1 6.571429 2.142857 6.428571 4.142857 6.142857 2.428571 5.571429 6.285714 9.00
## 2 2.000000 0.760000 2.620000 1.020000 3.080000 1.660000 2.480000 1.300000 3.86
## 407 408 409 410 411 412 413 414 415
## 1 9.285714 7.142857 13.14286 13.57143 9.714286 2.571429 9.571429 5.00 6.00
## 2 3.720000 4.800000 4.74000 5.20000 2.420000 2.040000 4.180000 1.16 1.88
## 416 417 418 419 420 421 422 423
## 1 4.428571 3.285714 2.428571 9.142857 1.428571 4.857143 14.85714 6.428571
## 2 2.220000 1.500000 2.280000 6.100000 1.100000 2.320000 7.70000 2.260000
## 424 425 426 427 428 429 430 431 432
## 1 2.142857 1.857143 4.142857 16.14286 7.285714 11.71429 7.142857 3.00 6.142857
## 2 1.580000 1.420000 2.020000 6.12000 2.160000 4.82000 2.540000 1.74 3.620000
## 433 434 435 436 437 438 439 440 441
## 1 10.42857 1.857143 15.14286 7.714286 2.285714 4.714286 4.142857 3.714286 3.00
## 2 4.72000 1.160000 6.04000 2.420000 0.620000 2.100000 2.220000 1.320000 1.28
## 442 443 444 445 446 447 448 449 450
## 1 7.714286 3.857143 6.714286 3.428571 14.42857 2.857143 5.00 5.714286 10.14286
## 2 4.240000 1.120000 2.440000 1.520000 5.96000 1.160000 2.62 1.640000 3.66000
## 451 452 453 454 455 456 457 458 459
## 1 7.00 5.428571 3.714286 2.571429 15.14286 12.28571 6.142857 16.28571 10.14286
## 2 2.54 3.440000 2.380000 0.920000 3.60000 4.54000 2.560000 5.02000 3.36000
## 460 461 462 463 464 465 466 467
## 1 5.857143 3.714286 3.857143 9.428571 12.14286 11.57143 2.571429 4.857143
## 2 2.260000 2.140000 2.600000 5.100000 5.34000 3.86000 1.700000 1.660000
## 468 469 470 471 472 473 474 475 476
## 1 5.142857 6.714286 8.428571 9.285714 4.571429 4.571429 2.285714 7.285714 5.0
## 2 2.300000 1.760000 3.060000 3.440000 1.580000 2.260000 1.020000 3.220000 2.1
## 477 478 479 480 481 482 483 484
## 1 5.571429 5.857143 2.714286 5.285714 3.285714 5.285714 5.571429 5.428571
## 2 2.300000 3.340000 1.680000 3.380000 1.140000 1.600000 1.620000 3.380000
## 485 486 487 488 489 490 491 492 493
## 1 5.428571 5.571429 3.285714 5.285714 6.285714 1.142857 5.571429 4.00 4.714286
## 2 1.340000 1.480000 2.880000 1.880000 2.540000 0.860000 1.400000 1.44 1.220000
## 494 495 496 497 498 499 500 501 502
## 1 17.71429 3.857143 3.428571 18.00 1.714286 7.428571 10.14286 7.00 13.28571
## 2 6.54000 2.100000 2.080000 5.72 0.860000 1.740000 4.02000 1.86 3.52000
## 503 504 505 506 507 508 509 510 511
## 1 8.285714 3.714286 3.00 17.85714 8.428571 3.428571 5.857143 7.714286 4.714286
## 2 2.920000 2.420000 1.32 4.20000 1.740000 2.020000 1.820000 2.580000 1.300000
## 512 513 514 515 516 517 518 519 520
## 1 6.285714 4.571429 7.714286 9.0 3.428571 5.857143 4.857143 6.571429 3.571429
## 2 4.560000 1.740000 3.100000 2.5 1.260000 1.500000 2.180000 1.380000 1.560000
## 521 522 523 524 525 526 527 528 529
## 1 7.142857 3.857143 3.00 9.00 6.428571 15.28571 5.285714 2.714286 7.857143
## 2 3.140000 1.240000 0.96 4.84 1.540000 8.22000 2.120000 1.300000 3.000000
## 530 531 532 533 534 535 536 537
## 1 14.28571 9.857143 4.428571 9.142857 9.571429 14.57143 8.857143 6.857143
## 2 5.06000 3.920000 1.660000 3.540000 5.740000 3.98000 2.140000 1.880000
## 538 539 540 541 542 543 544 545 546
## 1 2.857143 8.00 4.428571 9.857143 3.142857 18.71429 5.142857 5.285714 1.428571
## 2 1.520000 3.56 2.260000 2.720000 1.540000 5.82000 1.620000 1.780000 0.900000
## 547 548 549 550 551 552 553 554 555
## 1 7.714286 4.714286 6.285714 3.857143 8.0 11.28571 15.14286 8.857143 3.00
## 2 2.900000 1.400000 4.040000 3.000000 3.3 4.46000 4.76000 2.580000 1.66
## 556 557 558 559 560 561 562 563 564
## 1 5.857143 2.714286 10.42857 9.428571 10.00 4.285714 30.71429 6.714286 29.85714
## 2 1.760000 0.840000 2.80000 4.320000 3.62 2.160000 13.70000 2.580000 9.02000
## 565 566 567 568 569 570 571 572 573
## 1 4.285714 6.285714 3.285714 12.57143 8.428571 3.142857 5.00 12.14286 6.285714
## 2 1.560000 2.000000 1.500000 3.52000 1.960000 2.160000 3.24 4.68000 2.660000
## 574 575 576 577 578 579 580 581 582
## 1 5.285714 7.142857 5.285714 14.57143 2.0 3.428571 4.714286 7.428571 23.14286
## 2 2.120000 2.360000 3.080000 6.44000 1.5 2.100000 1.720000 2.600000 9.08000
## 583 584 585 586 587 588 589 590 591
## 1 13.28571 6.714286 9.428571 9.714286 4.00 8.00 11.00 5.285714 2.285714
## 2 3.00000 1.960000 2.660000 3.920000 2.06 2.84 4.52 1.860000 1.200000
## 592 593 594 595 596 597 598 599 600
## 1 6.571429 10.42857 3.00 9.285714 6.00 3.857143 6.285714 2.571429 5.571429
## 2 2.820000 4.94000 1.28 7.760000 2.62 1.900000 2.320000 1.200000 1.900000
## 601 602 603 604 605 606 607 608 609
## 1 1.428571 3.428571 10.71429 11.42857 3.428571 2.0 13.00 6.714286 15.85714
## 2 1.020000 1.680000 4.56000 3.92000 1.000000 0.7 3.46 3.400000 6.42000
## 610 611 612 613 614 615 616 617 618
## 1 3.857143 7.142857 5.714286 8.571429 6.142857 9.285714 6.571429 7.00 12.85714
## 2 1.600000 2.100000 1.880000 2.400000 2.560000 3.760000 2.960000 2.56 3.52000
## 619 620 621 622 623 624 625 626 627
## 1 0.8571429 3.00 1.571429 2.571429 6.285714 3.571429 18.85714 4.571429 19.28571
## 2 1.3000000 1.56 0.960000 1.400000 2.660000 2.640000 4.56000 1.100000 7.24000
## 628 629 630 631 632 633 634 635
## 1 5.857143 3.285714 5.571429 8.428571 2.428571 3.285714 17.57143 2.857143
## 2 1.780000 1.000000 2.060000 5.260000 2.300000 2.020000 4.74000 1.700000
## 636 637 638 639 640 641 642 643
## 1 4.285714 3.714286 3.142857 3.714286 16.42857 47.85714 6.142857 12.85714
## 2 1.720000 2.160000 2.100000 1.380000 6.00000 17.74000 2.880000 4.32000
## 644 645 646 647 648 649 650 651 652
## 1 11.85714 8.142857 5.714286 2.00 3.142857 4.571429 4.428571 7.428571 5.142857
## 2 5.54000 3.040000 2.880000 1.92 1.780000 1.940000 2.380000 2.300000 2.240000
## 653 654 655 656 657 658 659 660
## 1 4.142857 5.714286 5.142857 2.857143 9.428571 50.28571 15.14286 2.714286
## 2 1.900000 2.500000 2.100000 1.020000 3.880000 13.98000 3.74000 2.780000
## 661 662 663 664 665 666 667 668 669
## 1 4.714286 10.28571 9.285714 7.285714 9.428571 3.00 4.285714 3.142857 16.57143
## 2 1.720000 2.88000 3.840000 2.240000 2.200000 1.48 2.200000 1.880000 6.44000
## 670 671 672 673 674 675 676 677 678
## 1 6.571429 3.714286 17.00 10.14286 11.85714 23.14286 15.57143 10.14286 6.428571
## 2 2.740000 2.000000 6.46 3.10000 3.68000 8.16000 4.32000 7.22000 2.900000
## 679 680 681 682 683 684 685 686
## 1 5.714286 4.857143 7.285714 6.571429 3.142857 9.857143 13.42857 4.428571
## 2 2.260000 2.120000 2.340000 2.240000 2.000000 4.200000 6.78000 1.640000
## 687 688 689 690 691 692 693 694 695
## 1 2.714286 2.00 4.285714 13.14286 6.285714 12.71429 3.714286 2.428571 16.85714
## 2 1.560000 1.74 2.480000 4.64000 3.560000 5.66000 1.760000 2.080000 4.82000
## 696 697 698 699 700 701 702 703 704
## 1 5.857143 7.857143 8.142857 5.142857 6.00 6.285714 8.857143 3.285714 18.85714
## 2 2.400000 2.080000 3.920000 2.100000 2.22 2.080000 3.260000 2.220000 5.68000
## 705 706 707 708 709 710 711 712 713
## 1 3.857143 3.142857 9.142857 3.0 7.857143 6.428571 13.28571 15.85714 4.714286
## 2 1.980000 1.780000 2.640000 1.6 5.220000 1.940000 4.64000 5.30000 1.340000
## 714 715 716 717 718 719 720 721 722
## 1 28.42857 3.285714 4.714286 3.142857 7.714286 2.857143 5.285714 46.14286 5.00
## 2 7.16000 1.820000 1.280000 1.160000 3.500000 1.240000 2.020000 15.84000 2.34
## 723 724 725 726 727 728 729 730 731
## 1 4.714286 12.57143 3.142857 8.00 9.285714 5.714286 24.71429 4.571429 12.28571
## 2 1.980000 3.54000 2.160000 3.82 2.840000 2.320000 5.98000 1.920000 2.90000
## 732 733 734 735 736 737 738 739
## 1 3.285714 4.714286 8.428571 4.571429 11.14286 8.857143 7.285714 4.571429
## 2 1.600000 1.960000 3.960000 1.640000 3.90000 3.980000 2.340000 1.520000
## 740 741 742 743 744 745 746 747 748
## 1 21.14286 6.00 2.428571 6.142857 8.285714 4.571429 15.28571 21.85714 5.00
## 2 8.48000 2.26 1.480000 2.320000 3.500000 1.500000 6.22000 5.84000 2.12
## 749 750 751 752 753 754 755 756 757
## 1 7.857143 9.571429 5.142857 1.714286 5.285714 3.714286 2.00 4.142857 6.714286
## 2 2.240000 3.520000 2.560000 1.120000 2.320000 1.880000 1.38 1.100000 2.400000
## 758 759 760 761 762 763 764 765 766 767
## 1 4.571429 1.0 3.714286 5.142857 9.0 3.142857 8.857143 6.285714 4.857143 5.0
## 2 1.460000 0.7 1.520000 2.100000 3.6 1.220000 3.080000 3.360000 2.240000 1.8
## 768 769 770 771 772 773 774 775 776
## 1 17.85714 12.42857 6.714286 3.571429 4.285714 4.714286 18.85714 9.571429 12.00
## 2 7.30000 3.64000 1.600000 2.120000 1.280000 2.280000 6.62000 3.300000 3.82
## 777 778 779 780 781 782 783 784 785 786
## 1 13.57143 3.714286 8.428571 9.00 4.142857 33.85714 4.285714 2.285714 4.0 5.00
## 2 5.46000 2.040000 3.680000 4.54 1.380000 14.00000 1.600000 1.180000 1.6 2.06
## 787 788 789 790 791 792 793 794 795
## 1 8.714286 7.142857 4.714286 6.00 6.857143 4.285714 28.28571 19.14286 4.285714
## 2 2.920000 1.840000 1.560000 1.62 2.400000 1.400000 11.74000 5.24000 1.840000
## 796 797 798 799 800 801 802 803 804
## 1 5.571429 11.42857 4.00 10.85714 7.571429 8.285714 6.142857 12.85714 5.0
## 2 2.640000 4.64000 2.42 4.20000 2.640000 3.040000 3.200000 5.84000 1.8
## 805 806 807 808 809 810 811 812 813
## 1 5.857143 4.00 3.857143 12.57143 9.857143 8.285714 6.857143 8.571429 3.714286
## 2 2.680000 1.74 1.500000 4.32000 3.600000 3.060000 2.900000 1.740000 1.420000
## 814 815 816 817 818 819 820 821 822
## 1 10.57143 17.85714 3.857143 4.857143 3.428571 14.00 3.571429 3.714286 2.285714
## 2 2.96000 6.50000 1.420000 1.600000 1.980000 4.92 2.100000 1.400000 1.920000
## 823 824 825 826 827 828 829 830 831
## 1 12.28571 6.00 4.00 21.85714 4.142857 9.714286 2.428571 6.714286 4.285714
## 2 3.28000 2.62 1.38 7.74000 1.280000 4.520000 1.340000 1.800000 2.560000
## 832 833 834 835 836 837 838 839
## 1 5.857143 5.714286 16.28571 3.857143 7.285714 8.285714 4.571429 4.142857
## 2 2.100000 2.320000 7.92000 1.620000 2.700000 2.140000 1.960000 2.200000
## 840 841 842 843 844 845 846 847 848
## 1 7.142857 2.142857 9.0 4.142857 9.285714 6.428571 2.714286 4.142857 6.428571
## 2 2.440000 2.020000 3.6 2.260000 3.880000 2.100000 1.440000 2.280000 2.200000
## 849 850 851 852 853 854 855 856 857
## 1 2.571429 6.00 5.714286 7.714286 5.571429 2.571429 3.285714 5.714286 30.14286
## 2 2.260000 2.44 2.580000 1.980000 2.380000 1.820000 2.300000 2.860000 12.38000
## 858 859 860 861 862 863 864 865
## 1 11.57143 4.714286 8.428571 6.571429 6.571429 13.57143 6.857143 2.571429
## 2 4.82000 1.940000 3.500000 2.440000 2.780000 4.90000 1.760000 1.540000
## 866 867 868 869 870 871 872 873
## 1 18.42857 8.142857 6.571429 4.285714 3.428571 4.428571 5.857143 8.428571
## 2 5.94000 3.060000 3.000000 2.360000 1.740000 2.000000 1.940000 2.880000
## 874 875 876 877 878 879 880 881
## 1 6.428571 6.714286 10.57143 4.142857 18.28571 7.714286 16.28571 1.428571
## 2 2.200000 2.540000 2.82000 1.900000 6.52000 2.700000 8.02000 1.460000
## 882 883 884 885 886 887 888 889 890
## 1 4.142857 4.428571 4.571429 8.285714 3.00 5.142857 6.714286 5.857143 8.285714
## 2 1.740000 1.460000 1.900000 3.940000 2.18 2.040000 2.700000 2.100000 2.980000
## 891 892 893 894 895 896 897 898 899
## 1 3.857143 9.571429 9.00 1.428571 4.714286 6.571429 4.857143 13.14286 6.571429
## 2 1.880000 3.520000 3.68 1.120000 1.800000 2.140000 2.300000 3.86000 2.700000
## 900 901 902 903 904 905 906 907
## 1 3.714286 11.42857 13.14286 4.571429 29.71429 5.142857 4.714286 6.857143
## 2 1.520000 2.02000 4.14000 1.980000 8.60000 2.040000 2.480000 2.560000
## 908 909 910 911 912 913 914 915
## 1 2.142857 2.285714 14.57143 5.142857 12.14286 4.142857 3.142857 4.142857
## 2 0.920000 2.600000 4.08000 2.560000 3.98000 2.720000 1.560000 1.700000
## 916 917 918 919 920 921 922 923 924 925
## 1 11.71429 10.0 8.00 4.285714 4.714286 2.285714 25.85714 5.857143 6.00 6.00
## 2 4.54000 3.9 3.26 1.480000 2.700000 1.440000 12.64000 1.720000 1.98 2.02
## 926 927 928 929 930 931 932 933 934
## 1 8.00 4.285714 5.285714 3.285714 12.28571 6.714286 7.428571 3.571429 2.142857
## 2 2.48 1.520000 1.680000 1.600000 4.94000 3.860000 2.680000 2.500000 1.940000
## 935 936 937 938 939 940 941 942 943
## 1 3.857143 8.142857 9.857143 9.714286 7.857143 13.85714 10.28571 3.714286 8.00
## 2 1.800000 2.480000 3.000000 3.580000 2.640000 6.28000 3.92000 1.820000 2.38
## 944 945 946 947 948 949 950 951 952
## 1 10.42857 2.714286 7.571429 10.57143 7.00 9.857143 4.285714 6.00 8.142857
## 2 3.04000 1.180000 2.800000 3.74000 1.98 3.820000 2.040000 1.92 2.120000
## 953 954 955 956 957 958 959 960
## 1 5.714286 4.714286 6.142857 28.71429 7.142857 2.285714 17.42857 167.1429
## 2 2.360000 1.940000 2.240000 10.20000 2.220000 1.580000 5.54000 45.7800
## 961 962 963 964 965 966 967 968 969
## 1 4.571429 6.714286 17.00 5.714286 6.714286 6.428571 3.428571 7.00 4.428571
## 2 2.100000 2.280000 5.52 3.280000 1.900000 2.340000 1.920000 3.22 2.120000
## 970 971 972 973 974 975 976 977 978
## 1 14.71429 4.142857 8.857143 32.14286 3.00 2.571429 3.00 4.571429 11.85714
## 2 3.70000 2.280000 2.860000 16.52000 2.18 1.880000 2.02 2.020000 3.80000
## 979 980 981 982 983 984 985 986 987
## 1 5.285714 4.428571 7.571429 8.00 6.142857 3.142857 8.285714 7.571429 5.428571
## 2 2.480000 1.600000 1.520000 2.42 2.200000 1.780000 2.180000 2.120000 2.340000
## 988 989 990 991 992 993 994 995 996
## 1 4.857143 5.142857 4.571429 5.428571 3.142857 5.285714 6.857143 10.00 3.571429
## 2 1.600000 2.860000 2.560000 2.220000 1.920000 2.780000 2.200000 3.12 2.260000
## 997 998 999 1000
## 1 7.571429 8.571429 7.285714 8.142857
## 2 2.400000 3.100000 1.780000 2.600000
##
## Clustering vector:
## actual answer assignment awesome bad best
## 2 2 2 2 2 2
## better boring care clear definite difficult
## 2 2 2 2 2 2
## don't easy exam expect explain extreme
## 2 1 2 2 2 2
## final fine fun good grade grading
## 2 2 2 1 2 2
## great hard help helpful interesting know
## 1 1 1 2 2 2
## learn lecture like little love material
## 2 1 2 2 2 2
## most nice note pass pretty question
## 2 2 2 2 2 2
## recommend semester study subject talk test
## 2 2 2 2 2 1
## time understand well willing work book
## 2 2 2 2 2 2
## enjoy funny read
## 2 2 2
##
## Within cluster sum of squares by cluster:
## [1] 357551.1 696885.9
## (between_SS / total_SS = 24.9 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss" "tot.withinss"
## [6] "betweenss" "size" "iter" "ifault"The output also includes the information on individual (withinss) and total within-cluster (tot.withinss) sum of squares (SS). The individual SS is calculated as the sum of squares within each cluster. The total is the sum of them. In addition, the total sum of squares (totss) and between-cluster sum of squares (betweenss) can also be obtained from the output. For this example, the ratio of between SS and total SS is 24.9%. This means that 24.9% variation in the data can be explained by the two clusters.
6.2.2 Determine the number of clusters
Many methods can be used to determine the best number of clusters. Here we use the “eblow” method. For each given number of clusters, we can calculate how much variance in the data can be explained by the clustering. Typically, this will increase with the number of clusters. However, the increase would slow down at a certain point and that’s where we choose the number of clusters.
For example, the following code obtains the explained variances for 1 to 7 clusters.
k <- 7
varper <- NULL
for (i in 1:k) {
kfit <- kmeans(kmeans.data, i)
varper <- c(varper, kfit$betweenss/kfit$totss)
}
varper
## [1] -2.537936e-14 2.487755e-01 3.640712e-01 4.618219e-01 5.120916e-01
## [6] 5.485253e-01 5.867760e-01
plot(1:k, varper, xlab = "# of clusters", ylab = "explained variance")
From the plot, after 3 clusters, the increase in the explained variance becomes slower - there is an elbow here. Therefore, we might use 3 clusters here.
kfit <- kmeans(kmeans.data, 3)
kfit$cluster
## actual answer assignment awesome bad best
## 3 3 3 3 3 1
## better boring care clear definite difficult
## 3 3 3 3 3 3
## don't easy exam expect explain extreme
## 1 2 1 3 3 3
## final fine fun good grade grading
## 3 3 3 1 1 3
## great hard help helpful interesting know
## 1 1 1 1 3 1
## learn lecture like little love material
## 1 2 1 3 3 1
## most nice note pass pretty question
## 3 3 3 3 3 1
## recommend semester study subject talk test
## 1 3 1 3 3 2
## time understand well willing work book
## 1 1 1 3 1 3
## enjoy funny read
## 3 3 36.3 k-medoids
k-medoids is another clustering method. Both k-means and k-medoids attempt to minimize the distance between points within a cluster. In contrast to k-means, k-medoids chooses datapoints in the data as centers (called medoids). Another difference is the use of the distance. k-medoids typically use the Manhattan distance to define the dissimilarity. For example, for the example used in k-means, the distance is defined as below for k-medoids.
\[ d(p,q) = |p_1-q_1| + |p_2-q_2| + |p_3-q_3|. \]
A common implementation of k-medoid clustering is the Partitioning Around Medoids (PAM) algorithm. In R, the function pam of the R package cluster can be used. Different from the kmeans function, the pam function can take both a data matrix or a dissimilarity matrix.
For example,
kmedoid.res <- pam(kmeans.data, 2, metric = "manhantan")
kmedoid.res
## Medoids:
## ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## most 37 4 2 15 1 12 3 0 2 8 15 2 0 4 2 2 1 9 5 3 2 0 2 4 3 1
## good 22 15 3 30 2 24 4 3 3 14 18 7 9 11 18 4 10 20 8 13 7 3 5 14 11 8
## 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
## most 9 3 3 0 1 0 2 0 3 4 0 5 2 0 6 1 1 0 2 3 0 1 2 2 0
## good 12 6 11 4 4 13 6 12 9 6 5 15 2 1 15 4 16 2 15 7 5 5 3 7 9
## 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
## most 6 1 0 0 2 8 0 1 7 8 3 2 2 1 0 5 0 2 1 2 5 0 0 7 3
## good 8 7 3 0 5 15 4 4 24 18 9 10 9 5 5 20 5 8 17 13 2 7 12 13 17
## 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
## most 4 7 8 4 6 0 2 1 2 0 1 0 1 1 11 0 7 1 1 10 1 1 4 4
## good 8 7 18 16 41 4 30 4 22 6 9 0 3 4 7 6 34 6 0 8 2 4 7 7
## 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117
## most 2 7 0 0 0 10 3 4 0 2 4 4 2 0 1 4 4 1
## good 3 7 18 6 11 25 8 16 2 4 1 6 3 2 1 10 7 4
## 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135
## most 2 7 4 11 2 1 2 7 2 1 2 1 2 1 2 20 1 0
## good 1 13 12 27 12 4 2 13 14 12 5 7 0 5 8 47 0 2
## 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153
## most 1 1 1 5 3 2 2 5 0 4 3 1 2 2 1 3 0 1
## good 2 6 16 5 34 4 1 23 2 3 13 2 7 1 0 1 1 5
## 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171
## most 4 4 0 1 4 4 1 1 8 0 5 5 0 2 3 7 7 2
## good 3 7 2 5 10 2 6 7 29 4 4 26 2 1 13 6 12 4
## 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
## most 6 2 1 3 2 1 1 0 6 1 2 2 0 3 1 1 2 2
## good 9 1 2 9 12 8 11 1 2 4 11 0 7 7 3 3 5 4
## 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207
## most 2 5 5 1 6 0 1 1 4 3 1 3 6 3 5 0 2 0
## good 12 8 7 1 19 1 11 1 11 5 0 5 4 7 13 4 4 1
## 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225
## most 6 3 3 4 2 4 3 6 3 1 4 3 1 4 1 1 3 4
## good 22 7 17 13 2 9 4 3 3 9 0 1 17 1 1 5 5 12
## 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243
## most 2 3 1 1 0 3 4 1 3 5 2 0 4 2 3 4 0 1
## good 11 1 5 9 2 6 4 13 8 9 6 3 10 9 7 1 1 3
## 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261
## most 0 4 2 0 2 0 7 0 4 1 3 1 3 0 2 5 6 3
## good 6 6 7 7 4 2 10 2 6 4 11 11 19 4 4 11 9 5
## 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279
## most 1 1 4 6 2 0 5 0 3 0 6 1 1 6 2 3 2 2
## good 2 9 11 11 4 3 15 3 10 4 11 6 13 4 5 7 10 5
## 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297
## most 1 1 1 12 0 5 3 3 4 0 5 1 3 7 5 2 0 12
## good 5 2 3 13 3 16 5 7 10 0 17 2 9 23 7 13 5 29
## 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315
## most 3 1 0 1 1 4 6 1 2 2 2 2 6 1 0 1 4 2
## good 5 5 4 2 4 5 15 3 11 9 7 8 12 10 8 6 8 4
## 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333
## most 1 19 2 2 0 0 4 2 6 2 1 3 0 4 1 1 1 1
## good 11 21 20 3 8 0 17 15 6 5 5 4 3 4 5 3 4 5
## 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351
## most 0 2 10 3 3 7 1 0 2 2 3 2 7 0 3 2 2 5
## good 9 6 12 6 11 14 2 5 8 5 0 4 32 3 4 9 8 16
## 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369
## most 8 3 3 2 4 5 0 2 4 1 1 3 5 7 4 1 0 0
## good 5 5 7 8 4 3 4 6 4 1 6 5 18 13 9 9 5 5
## 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387
## most 4 2 1 1 4 4 0 1 1 2 2 3 0 5 3 6 5 0
## good 7 8 6 13 18 7 1 2 7 4 4 12 1 19 6 3 9 1
## 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405
## most 1 0 3 3 5 3 5 1 1 1 1 1 3 2 1 2 0 3
## good 1 3 17 11 2 5 7 13 7 15 4 5 3 3 4 2 3 0
## 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423
## most 6 4 3 4 2 1 2 5 6 2 0 2 3 6 2 2 3 3
## good 9 10 14 18 10 8 4 12 5 3 4 3 2 12 1 6 20 3
## 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441
## most 4 2 3 3 2 8 2 0 2 5 0 2 3 0 1 3 1 0
## good 1 3 2 19 6 13 5 10 5 12 3 16 4 3 2 9 6 3
## 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459
## most 2 1 1 1 3 0 0 0 4 3 5 0 0 3 2 0 8 4
## good 7 5 11 4 15 2 4 10 11 3 4 3 7 10 12 8 26 12
## 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477
## most 3 0 4 4 1 1 1 3 3 1 1 1 1 5 0 1 3 2
## good 3 6 9 9 23 5 4 6 10 10 8 15 7 4 5 8 2 7
## 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495
## most 6 0 3 0 4 2 4 0 0 2 1 1 3 0 1 0 2 3
## good 6 5 4 3 4 5 2 7 4 5 7 10 0 3 5 2 26 3
## 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513
## most 3 2 2 1 4 1 3 5 1 1 4 2 0 1 4 0 3 2
## good 6 12 7 8 4 3 8 13 4 3 19 0 4 4 4 6 14 3
## 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
## most 4 1 1 1 1 1 2 3 2 1 2 1 5 4 2 4 4 3
## good 4 8 4 2 5 9 3 7 2 2 4 6 18 2 3 8 14 12
## 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549
## most 1 1 8 2 2 0 2 3 0 0 6 8 0 1 2 1 1 2
## good 6 2 13 5 8 5 4 5 5 6 4 21 8 2 4 2 3 9
## 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567
## most 0 0 6 3 2 1 0 0 1 2 2 2 13 1 12 0 0 0
## good 11 9 8 13 8 6 3 3 8 11 9 6 35 8 23 7 4 4
## 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585
## most 1 1 2 3 7 3 2 1 3 1 1 6 1 1 8 2 0 0
## good 9 13 1 10 9 2 0 7 3 10 2 3 0 7 19 11 5 4
## 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603
## most 2 1 3 5 2 0 0 3 3 15 2 2 1 0 1 4 3 1
## good 3 3 7 6 7 2 6 9 1 15 7 2 8 1 0 1 2 9
## 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621
## most 2 1 0 0 2 6 1 1 1 0 1 2 2 3 7 1 2 2
## good 7 3 0 5 13 15 7 11 6 7 4 8 6 7 10 1 3 3
## 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639
## most 0 2 3 5 0 5 0 0 0 3 0 1 3 2 0 3 4 1
## good 4 5 4 9 5 10 2 6 7 4 0 5 8 2 8 1 3 4
## 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657
## most 5 12 4 3 3 11 3 3 3 0 1 0 3 0 7 1 2 2
## good 17 38 6 7 11 1 7 3 2 2 1 3 7 4 3 2 2 11
## 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675
## most 10 3 3 1 1 3 0 1 3 11 2 4 2 0 8 1 2 4
## good 28 10 3 4 11 9 7 8 3 0 3 12 4 0 14 8 4 16
## 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693
## most 3 8 2 2 0 4 3 3 3 8 0 4 2 7 3 3 4 0
## good 18 7 4 5 0 5 3 4 11 9 4 3 4 4 13 2 22 3
## 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711
## most 3 1 2 0 4 1 2 1 1 1 4 2 0 1 1 3 4 4
## good 4 9 6 4 14 4 5 7 5 5 8 3 0 8 8 6 9 10
## 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729
## most 10 0 6 0 0 1 9 0 1 11 5 1 1 1 2 1 2 2
## good 16 1 16 4 2 2 3 4 5 25 1 2 6 1 13 11 2 21
## 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747
## most 0 2 2 1 3 0 3 5 2 0 5 1 0 1 1 1 8 2
## good 5 7 6 3 10 4 6 10 10 3 20 5 2 8 2 3 10 9
## 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765
## most 1 2 1 3 0 5 2 1 0 2 3 0 0 3 3 1 1 2
## good 2 9 8 2 2 10 3 4 3 4 2 1 5 6 8 2 9 6
## 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783
## most 0 1 6 3 1 4 0 5 6 8 0 2 1 1 3 2 6 2
## good 3 2 15 11 2 7 4 4 21 4 14 7 1 3 12 4 23 1
## 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801
## most 0 0 1 2 1 0 2 2 2 4 2 1 2 4 3 2 3 0
## good 3 1 2 7 5 7 8 8 1 13 6 1 9 8 2 13 5 9
## 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819
## most 1 5 1 1 1 2 2 2 3 1 1 2 3 5 1 0 5 6
## good 9 11 1 7 3 3 10 5 7 4 1 3 5 14 7 7 2 6
## 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837
## most 4 2 0 1 3 1 4 2 4 1 1 2 2 0 13 2 3 3
## good 5 7 3 1 1 6 11 4 9 1 3 7 2 4 20 4 3 7
## 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855
## most 1 1 2 2 2 1 2 1 0 1 1 4 1 4 1 2 0 1
## good 1 3 4 4 7 6 7 7 5 3 5 3 7 4 2 2 0 6
## 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873
## most 1 7 6 1 2 0 2 4 1 0 11 1 0 0 2 4 5 1
## good 4 34 5 0 7 8 0 3 3 2 16 4 5 6 1 3 11 5
## 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891
## most 3 1 0 3 6 7 2 1 1 1 2 9 0 2 0 0 2 1
## good 7 6 2 1 10 5 9 1 1 2 7 8 7 3 4 5 2 3
## 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909
## most 0 3 2 3 1 0 2 1 0 4 5 3 1 6 1 2 1 1
## good 14 5 0 4 5 4 5 5 4 5 4 5 25 3 2 9 0 1
## 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927
## most 2 1 5 2 2 1 4 6 4 0 2 4 8 0 2 0 1 1
## good 7 1 3 1 4 2 6 10 3 2 7 4 21 4 2 9 7 2
## 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945
## most 0 0 4 3 2 3 3 1 1 2 4 1 5 3 2 4 3 0
## good 5 3 11 3 5 6 3 4 2 7 5 3 10 7 3 7 4 3
## 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963
## most 1 0 2 8 3 3 0 1 1 2 9 2 0 3 44 3 1 7
## good 6 13 4 8 6 7 4 2 4 5 14 7 1 20 144 0 12 9
## 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981
## most 1 2 0 2 1 1 2 2 3 9 4 5 4 0 1 1 0 2
## good 4 2 8 3 11 2 8 2 7 35 1 1 0 3 19 3 1 4
## 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999
## most 1 1 4 2 2 1 1 3 5 1 2 2 1 0 2 0 1 0
## good 2 5 5 2 1 8 4 5 2 4 2 4 9 2 3 4 4 5
## 1000
## most 4
## good 9
## Clustering vector:
## actual answer assignment awesome bad best
## 1 1 1 1 1 1
## better boring care clear definite difficult
## 1 1 1 1 1 1
## don't easy exam expect explain extreme
## 1 2 1 1 1 1
## final fine fun good grade grading
## 1 1 1 2 1 1
## great hard help helpful interesting know
## 2 2 2 1 1 1
## learn lecture like little love material
## 1 2 1 1 1 1
## most nice note pass pretty question
## 1 1 1 1 1 1
## recommend semester study subject talk test
## 1 1 1 1 1 2
## time understand well willing work book
## 1 1 1 1 1 1
## enjoy funny read
## 1 1 1
## Objective function:
## build swap
## 138.8718 138.8718
##
## Available components:
## [1] "medoids" "id.med" "clustering" "objective" "isolation"
## [6] "clusinfo" "silinfo" "diss" "call" "data"6.3.1 Visualization
Many times, it is useful to visualize the results. When there are more than 2 clustering variables such as in our case with 1,000 clustering variables, principal component analysis (PCA) is often conducted to find 2 principal components and the visualize the data in the two-dimensional space. However, PCA cannot directly be conducted when the sample size is small. In this case, we use the dissimilarity/distance matrix for analysis instead.
The R function clusplot from the package cluster is used. Note that kmedoid.res$diss is the distance matrix from the previous k-medoids analysis.
