1. For this problem your inputs are a training data set and a validation set. Bo

Question

1. For this problem your inputs are a training data set and a validation set. Both data sets have coordinates in n-space and the corresponding classification for each observation in the data set. In practice, one has a set of points for which the classification is known. We divide this set into two sets: one is a training data set and one is a validation set. We then use the training set to devise an algorithm to predict which classification should be assigned to new observations. We can then run the algorithm on the data we have in the validation set. Since we know the correct classification for these points, we can determine the error rate of our algorithm had these data been new observations. In this problem you will write code to test the k-nearest-neighbors (knn) regression algorithm. The algorithm works as follows: we are given a new observation and an odd number, k. We find the k nearest neighbors from the training set for the new observation. We then assign the new observation to whichever classification the majority of the k-nearest-neighbors has. For example, if k = 5, and three of the nearest neighbors have a classification of 2, then the new observation is assigned to class 2. Write two functions. The first has prototype k.nn <-function(k,v.data,t.data) Here k is a positive integer to denote the numbers of nearest neighbors to find. The v.data parameter is a matrix. Each row contains the coordinates of an observation. If v.data has 100 observations in 2 dimensional space, then v.data is a matrix of dimension 100x2. The t.data is formed similarly. We must have ncol(t.data) equal to ncol(v.data). The output is a nrow(v.data) by k matrix. Each row of the output contains the k nearest neighbors for the corresponding observation in v.data. Store the observation number (or row number) from t.data in the output matrix. Example: Suppose the first observation of v.data has as its 3 nearest neighbors the 10th,23rd, and 45th observation from t.data. If k = 3, then the first row of the output matrix will be 10,23,45. The second function has prototype vote <- function(class.id,knn.out) Here knn.out is the output of your k.nn function, and class.id is a vector containing the class id of the training data set. The output is a vector of length nrow(knn.out).

validate.set

"id" "x" "y"
1 26.338 41.535
2 29.006 51.537
1 27.053 43.935
1 24.293 48.647
2 29.414 45.933
1 25.444 44.697
1 17.503 47.216
2 30.41 45.35
2 29.179 50.698
2 27.775 47.875
1 24.04 44.401
2 29.534 47.583
1 23.95 42.891
2 28.943 45.932
1 19.076 48.38
1 20.63 43.082
1 20.255 46.118
1 20.8 43.477
1 20.036 46.713
1 24.834 43.793
1 20.615 45.711
2 26.157 48.699
1 26.885 43.067
2 29.386 44.536
1 24.892 45.434
2 29.922 48.499
2 27.687 47.475
2 33.753 44.165
2 33.206 44.347
1 24.294 45.443
1 18.685 41.636
2 30.647 43.249
2 29.837 45.465
2 30.594 48.361
2 28.576 47.667
1 21.378 43.893
2 33.631 45.627
2 30.419 48.647
1 25.074 46.427
1 20.569 45.849
1 22.476 42.359
1 26.209 43.918
2 31.796 44.064
1 19.937 43.914
2 31.83 47.359
1 24.115 45.015
1 24.779 43.828
2 29.746 47.488
1 15.768 44.412
2 28.481 46.295
1 19.393 43.095
1 20.937 48.212
2 29.021 47.787
2 31.529 45.894
1 23.479 45.378
2 25.43 48.709
2 29.104 50.541
1 24.757 40.591
1 23.505 45.566
2 31.585 49.955
1 21.567 45.035
2 28.528 47.341
2 30.267 47.47
2 28.251 46.892
1 18.37 45.81
2 30.666 48.442
1 22.636 47.573
2 26.604 50.065
1 21.274 43.386
2 26.659 51.223
2 30.024 47.439
2 25.975 48.285
1 23.5 43.481
2 26.427 49.644
2 28.324 49.885
1 21.245 46.125
1 20.758 43.607
2 31.302 44.807
2 30.172 48.81
2 30.015 45.643
1 21.741 44.845
2 27.476 49.855
1 19.894 48.095
1 21.241 47.115
2 31.096 43.03
1 25.748 48.371
1 23.461 41.494
2 29.852 45.838
2 30.187 47.291
2 27.934 50.28
2 29.177 48.502
2 31.631 46.01
1 23.99 41.356
1 26.083 46.515
2 30.778 44.147
2 33.268 44.432
1 23.437 44.646
1 24.147 44.219
2 30.231 45.649
1 21.054 45.957
2 28.585 50.652
2 29.819 45.566
2 28.351 50.734
1 22.058 45.505
1 24.069 44.548
2 28.494 44.718
1 26.826 45.36
2 28.444 47.291
2 27.764 45.819
2 29.481 46.858
2 31.404 42.986
2 29.716 47.899
1 19.596 47.093
2 28.928 47.969
1 20.124 44.876
1 25.194 44.001
2 29.203 46.955
2 29.702 45.109
2 31.387 49.209
2 28.519 45.138
2 30.112 45.648
1 22.417 47.132
1 21.239 45.783
2 27.156 49.504
2 32.431 42.854
2 29.486 46.835
2 29.026 47.006
1 17.893 45.512
1 19.794 45.1
2 30.973 47.466
2 27.749 47.309
1 21.174 46.337
1 25.123 44.681
2 26.781 50.516
2 28.628 49.434
2 28.863 45.182
1 22.79 44.797
1 23.548 43.592
1 24.019 45.067
1 24.856 42.717
1 19.778 43.65
2 30.541 49.441
1 22.244 46.401
2 31.213 44.101
1 23.34 44.711
1 21.408 42.687
2 30.794 42.003
1 22.615 45.44
2 28.814 48.545
2 30.123 48.823
2 30.533 49.632
1 24.401 42.17
1 21.757 41.167
1 22.805 43.729
2 28.838 46.841
2 29.448 47.906
2 31.301 47.078
2 28.68 47.927
2 31.226 46.385
1 22.155 44.922
2 32.498 45.156
2 32.411 43.059
2 28.834 46.836
2 30.021 45.989
2 27.794 47.662
2 31.888 45.842
1 22 43.245
2 29.485 49.78
1 22.513 45.961
2 31.372 45.752
2 28.561 46.313
2 27.924 45.666
2 28.089 47.293
2 30.48 46.642
1 18.685 45.048
2 28.6 45.045
2 30.085 43.874
1 23.145 44.032
1 20.063 44.072
2 28.628 46.16
1 19.403 52.32
1 20.585 45.947
2 28.684 45.511
1 22.484 41.624
2 28.944 45.954
1 19.6 46.662
2 26.391 49.594
2 28.252 47.126
2 30.038 45.182
1 24.495 42.363
2 29.897 46.762
1 21 44.015
1 19.704 43.259
1 27.656 44.706
2 29.619 48.313
1 22.741 45.276
1 19.543 46.614
1 25.046 42.051
1 21.683 48.959
2 26.668 48.896
1 22.112 45.177
1 21.748 44.231
2 29.801 49.096
1 22.299 44.026
2 27.358 52.618
2 32.132 47.54
1 21.713 47.676
2 31.329 44.316
2 30.169 49.856
1 22.725 46.401
2 31.316 45.703
1 26.969 45.105
1 19.443 45.339
1 23.793 47.021
2 32.512 46.388
1 21.886 45.959
1 23.66 44.679
2 30.812 45.686
2 30.585 45.966
1 24.359 41.739
1 23.969 43.923
2 30.869 48.555
1 29.699 43.575
2 30.196 42.659
1 24.783 46.583
2 29.109 49.005
2 31.002 45.181
1 18.802 42.556
1 27.32 46.327
1 18.435 46.434
1 15.588 46.95
1 20.642 48.108
1 22.453 42.078
1 18.166 49.408
1 21.954 47.909
1 20.827 41.691
2 31.423 45.353
2 27.289 42.774
2 29.092 48.305
1 23.935 44.116
2 26.018 51.444
2 32.985 45.384
1 25.052 42.204
1 17.057 46.493
1 26.123 44.019
1 22.17 42.914
2 28.729 47.717
1 23.764 43.576
1 21.438 47.938
1 20.934 45.927
2 31.844 46.063
2 31.287 45.216
2 31.204 43.418
1 22.867 42.315
2 25.972 50.786
1 21.318 45.08
2 29.445 45.512
2 30.261 48.129
2 31.532 48.508
2 29.578 50.289
2 29.662 47.579
2 29.809 48.26
2 28.505 51.228
2 29.53 48.666
1 18.495 44.987
2 27.196 44.994
1 21.362 43.744
2 29.731 46.764
2 28.609 48.655
2 30.978 51.473
1 22.042 44.213
2 32.137 45.331
2 28.623 45.594
2 31.804 44.67
1 23.626 44.436
1 24.119 46.987
1 19.375 46.639
1 21.822 44.835
2 28.807 47.876
1 21.664 43.947
2 26.389 46.373
2 33.763 43.004
2 28.029 46.174
2 30.804 46.373
2 28.194 46.825
1 25.154 47.46
2 32.838 45.374
1 23.219 44.544
1 18.621 44.87
2 27.523 53.699
2 31.84 45.775
2 28.246 49.08
2 29.571 47.015
2 31.754 45.509
1 23.926 43.231
2 29.106 43.461
2 30.511 46.652
1 21.744 45.26
2 27.88 44.27
2 32.462 46.623
2 31.712 45.705
2 29.257 46.784
1 21.167 47.641
1 25.662 43.065
2 30.924 46.248
1 24.335 46.145
2 28.515 50.29
2 29.708 48.509
1 23.655 44.313
2 30.9 44.943
1 27.811 44.086
2 28.576 47.358
2 26.851 47.355
1 22.358 48.31
2 26.089 48.337
2 28.053 49.326
1 20.352 45.496
2 26.463 47.757
2 31.937 45.964
2 29.354 49.153
2 29.649 46.289
1 20.84 43.867
1 19.323 43.021
1 20.214 44.817
1 26.766 45.716
1 26.743 42.568
2 31.038 47.012
2 33.005 46.177
2 28.84 49.66
2 29.503 49.324
2 29.746 46.927
1 15.774 47.174
2 28.899 46.674
1 26.976 45.05
2 30.826 45.101
2 30.505 45.302
1 19.002 45.146
2 30.973 45.379
1 22.452 46.431
2 30.41 46.327
2 30.416 45.575
2 28.252 46.382
2 30.92 41.833
2 29.126 48.451
2 27.769 50.209
2 27.061 51.905
2 27.247 44.793
2 27.331 49.6
2 29.551 46.8
2 31.178 46.053
2 28.429 46.609
2 28.036 46.521
2 32.629 44.474
1 16.819 46.919
1 21.504 45.743
2 28.874 43.794
2 32.601 45.807
2 31.806 49.505
2 30.44 45.995
1 22.883 41.924
2 30.13 46.874
2 29.32 46.197
1 24.46 45.638
1 26.815 45.236
2 33.489 43.25
1 20.796 43.728
2 27.85 48.69
1 23.985 41.281
2 28.052 48.402
2 30.278 45.651
1 22.582 44.146
2 26.611 47.729
2 30.687 44.947
1 22.6 43.209
2 30.609 45.847
1 24.312 43.404
2 27.373 51.075
1 22.964 45.907
1 21.014 44.933
2 29.236 47.632
1 24.273 44.853
1 21.967 47.084
1 24.184 42.732
2 27.978 48.579
1 22.014 46.556
2 27.379 51.062
2 30.964 46.584
2 30.601 41.828
1 21.243 44.424
2 29.63 46.5
2 31.82 44.505
1 24.27 44.325
2 28.454 49.688
1 20.286 46.707
2 31.077 47.941
1 25.55 42.878
1 21.641 45.882
2 28.816 45.281
1 21.669 45.034
2 30.985 48.978
2 28.218 43.842
2 31.165 50.132
1 24.495 43.844
2 27.472 47.812
2 31.277 44.693
2 28.769 47.739
2 30.354 46.72
1 23.837 47.124
2 28.34 45.434
2 33.581 42
2 31.186 45.964
2 30.066 48.531
1 23.666 45.03
1 18.73 46.781
2 25.754 50
2 29.774 50.266
2 27.022 51.035
2 30.176 43.822
2 26.375 54.581
2 30.031 47.943
1 19.951 47.894
2 30.019 47.383
2 27.553 47.456
1 24.711 44.01
1 19.902 44.7
2 31.241 44.789
2 31.531 46.118
1 23.204 41.82
2 30.669 46.181
2 28.428 45.842
1 17.772 47.662
2 30.352 46.197
1 21.018 46.231
1 25.442 42.813
2 28.339 49.745
2 28.503 46.505
2 30.575 47.936
2 27.983 46.987
1 19.103 39.831
2 29.081 51.297
1 25.35 46.52
1 19.545 46.331
1 22.044 47.159
1 24.049 44.279
2 31.482 44.311
1 21.609 47.779
1 18.2 45.659
2 29.567 47.163
1 17.426 46.857
1 22.898 43.286
2 27.873 48.719
2 28.802 47.311
2 28.434 44.836
1 16.191 45.974
2 30.74 43.416
2 31.106 45.994
1 23.965 45.966
2 28.155 50.42
1 20.739 48.868

train.set:

"id" "x" "y"
2 29.168 47.706
1 24.227 46.126
2 31.738 44.98
2 29.779 48.096
2 26.772 47.925
1 22.833 43.714
1 20.681 44.786
2 32.536 49.213
2 29.269 46.725
1 14.104 44.953
1 20.997 47.428
1 24.537 44.216
2 27.309 51.78
2 32.891 42.698
1 26.93 43.427
2 31.384 46.975
1 19.223 45.099
2 29.606 44.794
2 30.353 47.284
2 30.97 46.273
2 31.506 47.333
2 32.031 46.119
1 20.299 44.756
1 20.888 44.543
2 28.252 49.814
2 29.57 46.655
2 25.938 52.129
2 29.675 47.645
1 23.721 43.713
2 32.38 46.668
1 24.664 43.804
1 21.552 43.663
2 29.304 49.269
1 21.951 43.405
2 31.724 46.84
2 29.619 46.244
2 33.173 43.185
1 26.734 40.747
2 30.531 50.806
2 28.938 49.537
1 19.037 44.7
1 18.444 48.143
1 25.156 42.403
2 30.741 45.136
2 33.754 43.297
2 29.837 43.159
1 23.567 46.419
2 26.88 47.982
1 22.321 43.198
2 28.399 49.702
1 18.784 43.807
2 30.785 44.229
1 19.707 46.784
2 28.78 47.185
2 28.928 47.058
2 28.666 49.492
2 28.586 47.329
2 31.417 42.582
2 29.605 46.034
2 28.907 49.657
2 28.183 48.34
1 15.456 44.392
2 31.358 48.937
2 31.497 46.478
2 30.528 50.102
1 24.563 43.026
2 33.122 45.642
1 19.749 42.831
1 21.64 47.677
2 31.793 43.435
1 22.837 46.195
2 30.77 43.799
1 19.642 41.915
1 19.861 44.159
2 26.858 48.388
2 27.731 49.362
2 29.086 46.085
2 28.008 45.798
2 30.021 47.216
1 26.688 41.874
1 20.006 48.605
2 30.399 43.672
2 28.943 47.151
1 20.282 43.912
1 19.002 46.064
2 29.618 49.596
1 21.303 44.869
1 27.551 44.38
1 23.761 43.959
1 23.632 44.869
2 31.809 42.896
2 29.678 47.183
2 31.491 48.526
1 24.173 42.339
2 28.054 47.976
1 24.082 44.268
2 27.251 47.359
1 24.017 44.536
2 29.493 46.894
1 22.398 43.889
1 19.144 45.053
2 32.444 38.846
1 21.781 44.255
2 31.486 46.501
2 29.621 48.052
2 30.82 47.622
2 26.581 50.059
2 30.117 47.213
2 28.812 47.713
1 21.481 45.113
2 29.2 43.626
2 28.044 46.329
2 30.235 48.059
2 34.898 45.464
2 27.914 46.763
2 32.878 43.954
2 31.456 47.035
1 25.753 44.108
1 22.994 42.722
2 26.984 47.678
2 28.764 46.202
1 21.976 46.549
2 29.659 47.693
2 29.374 46.033
2 30.692 47.404
1 19.777 44.579
1 24.757 45.536
2 32.109 46.384
2 26.317 49.322
1 20.667 47.344
2 29.516 49.081
2 29.916 46.653
2 30.606 50.151
2 31.874 40.44
2 26.783 46.032
1 25.08 44.602
1 22.952 46.748
2 28.813 44.558
1 16.19 46.175
2 28.147 47.198
1 18.213 44.018
2 29.52 48.857
2 30.355 46.744
2 26.84 49.2
1 21.573 45.706
1 23.007 44.814
2 30.314 44.852
2 29.384 46.513
1 22.324 42.891
2 32.885 45.22
1 18.124 46.125
1 21.632 45.788
2 30.153 47.156
2 28.823 48.94
2 27.872 47.17
2 30.86 45.263
2 31.498 48.011
1 20.945 45.088
1 17.921 45.478
1 19.39 46.744
2 31.394 49.269
2 27.018 45.388
2 29.68 47.418
1 25.118 45.182
2 27.321 46.544
2 25.923 47.938
2 29.695 46.524
2 25.782 46.888
2 30.787 47.968
2 31.089 47.289
2 29.181 50.953
1 21.026 44.378
1 22.988 45.695
2 29.907 47.524
2 30.154 45.789
1 20.704 45.272
1 22.79 41.46
2 27.306 51.685
2 28.734 49.009
2 31.711 43.859
2 26.978 49.466
2 29.034 46.61
1 18.252 47.344
2 30.003 47.037
2 30.544 44.355
1 13.564 45.195
2 31.936 48.603
1 29.134 40.904
1 21.681 44.309
1 25.569 43.098
2 28.926 46.637
1 25.177 44.827
1 24.733 45.865
2 26.594 47.979
2 30.413 46.727
1 22.711 44.171
2 31.17 49.651
1 20.359 47.642
2 30.103 42.816
2 30.683 46.298
2 26.178 50.409
2 33.57 44.64
2 28.604 46.782
2 29.533 49.286
1 23.841 45.391
1 21.776 44.364
2 29.802 46.101
2 28.082 47.235
2 30.725 45.264
1 19.85 47.957
2 28.269 45.47
2 30.106 45.798
1 25.368 43.431
2 30.613 47.087
2 27.852 49.083
2 33.942 44.574
1 24.08 43.215
1 22.398 41.625
2 31.783 47.216
1 26.025 43.408
2 28.092 50.711
2 29.754 47.681
1 19.714 45.836
2 32.266 42.198
1 23.296 41.019
2 30.423 46.76
1 21.058 47.359
2 32.54 42.725
1 20.734 47.484
2 29.266 47.714
2 31.166 45.769
2 27.958 50.453
2 29.312 49.231
1 18.714 47.325
1 22.187 45.196
2 30.273 50.945
2 29.402 50.474
1 22.023 43.761
1 25.832 44.116
1 21.85 46.473
2 29.811 47.619
2 30.053 43.814
2 30.276 46.458
2 30.149 48.792
1 24.002 45.809
1 19.983 46.288
2 27.352 50.098
2 29.068 44.422
1 21.112 48.236
1 24.328 42.297
2 30.436 44.964
2 28.674 44.957
1 16.256 44.568
2 32.346 45.78
2 29.388 47.279
1 26.688 41.93
2 27.356 50.98
1 25.324 44.841
1 22.986 43.24
2 28.254 46.672
2 27.758 49.585
2 32.893 43.96
1 21.431 44.623
2 30.955 45.742
2 27.896 48.859
2 28.095 49.986
2 30.065 47.018
2 27.178 47.941
1 16.592 45.267
2 29.496 46.995
2 28.935 46.967
2 27.79 50.837
2 32.2 43.558
2 28.769 47.923
2 30.123 44.147
2 31.764 48.026
1 17.767 46.943
1 19.55 48.289
1 23.965 43.964
1 21.41 44.772
2 31.487 47.213
2 28.529 48.017
2 30.88 46.559
1 19.147 44.024
1 24.519 44.183
2 29.309 48.853
1 21.829 47.304
2 30.428 50.013
1 16.649 48.225
1 17.614 47.697
2 27.66 47.061
2 28.783 45.637
2 30.665 46.978
2 25.887 52.774
2 29.025 48.342
1 22.873 46.439
2 29.693 47.001
1 16.604 45.446
1 18.465 46.061
2 27.73 47.589
2 29.603 44.853
1 19.467 46.438
2 32.069 44.993
1 22.579 47.815
2 26.914 47.979
1 27.822 45.376
2 29.583 47.277
1 19.587 46.724
1 20.631 44.168
1 18.109 45.684
2 32.231 44.284
2 30.342 48.373
1 22.228 45.252
2 27.995 46.316
2 31.282 45.81
2 28.698 48.369
1 22.171 45.916
2 28.953 47.029
1 24.303 44.442
1 22.588 43.497
1 22.948 44.356
2 30.75 45.71
2 26.676 48.884
2 32.346 44.358
1 17.68 47.724
2 33.347 44.792
2 28.452 48.497
1 19.77 43.169
1 21.268 44.342
2 28.841 42.495
1 27.462 44.121
2 31.168 46.925
2 31.346 45.88
2 27.922 48.551
2 30.184 45.467
2 26.595 49.568
1 20.105 47.075
2 29.896 48.372
1 18.981 44.81
1 22.256 43.689
1 23.797 40.445
1 23.505 45.004
1 17.238 45.572
2 27.113 50.228
1 21.373 46.869
2 30.373 44.739
1 21.658 47.049
1 26.279 43.201
1 24.908 41.853
1 24.913 44.634
1 18.728 46.916
2 29.171 45.436
2 30.287 47.338
2 26.385 50.755
1 18.809 46.413
2 31.342 46.914
2 28.044 47.32
1 23.699 42.681
2 28.077 47.787
1 24.642 43.776
1 23.208 42.198
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2 29.251 50.829
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1 21.53 44.843
1 22.816 45.154
2 27.844 51.36
2 28.804 46.105
2 27.76 49.484
2 31.087 45.396
1 23.526 42.893
2 29.404 47.729
1 22.151 44.512
1 15.549 49.441
2 29.156 47.128
1 20.127 46.594
2 31.882 49.409
2 29.295 45.948
2 27.836 48.761
1 24.074 39.594
2 28.935 49.421
1 19.121 45.052
2 29.501 50.134
2 29.928 45.016
2 30.326 49.404
1 24.754 42.226
1 16.899 48.228
2 30.292 43.343
2 29.96 46.064
2 29.933 44.767
1 24.065 44.677
2 30.403 47.467
2 27.638 46.678
1 29.173 43.708
1 18.465 46.378
1 25.563 46.983
2 33.255 44.733
2 27.49 48.396
2 27.029 50.446
1 22.769 44.025
1 22.905 42.544
1 23.724 44.546
2 32.506 47.76
1 22.579 47.986
1 21.161 47.506
2 29.252 47.354
2 31.983 43.328
1 21.896 41.646
1 23.794 44.794
2 34.713 44.961
2 29.396 47.808
2 29.604 44.633
1 20.536 46.252
1 25.561 43.297
1 19.742 43.287
1 24.275 40.967
1 21.087 46.224
2 28.287 48.186
2 29.402 49.887
2 31.365 46.436
2 29.304 47.08
2 32.05 44.553
1 26.144 45.625
2 27.703 45.814
2 28.998 48.261
2 25.775 49.232
1 22.662 42.399
2 28.878 48.708
2 27.981 47.737
1 21.749 46.158
2 28.385 45.884
2 29.53 45.849
1 22.566 45.404
2 30.414 46.688
2 27.897 49.526
2 27.064 50.828
1 20.455 44.434
2 30.7 48.066
1 18.321 45.455
1 21.31 40.661
2 28.995 47.878
2 30.582 44.308
1 20.531 45.761
2 32.131 44.307
2 32.177 43.367
2 30.165 47.052
2 29.115 46
2 30.677 46.923
1 21.021 47.801
1 19.115 46.961
2 29.03 46.142
1 23.313 42.338
1 22.28 42.953
2 31.204 43.582
1 21.323 45.867
1 22.69 44.554
2 27.881 47.139
2 29.523 47.69
1 18.487 47.007
1 22.914 43.912
1 23.825 42.434
2 30.873 44.626
2 30.455 47.123
1 24.122 43.359
2 28.471 50.858
2 29.351 48.046
2 30.888 47.301
1 21.607 45.734
2 30.238 44.64
1 21.722 43.337
2 30.83 42.556
2 30.934 45.283
1 21 44.408
2 31.951 45.697
1 23.111 44.835
2 29.792 48.596
1 23.62 44.345
2 31.118 46.873
2 28.455 47.213
1 25.062 43.46
1 19.042 44.522
2 28.795 49.519
2 30.65 50.062
2 30.497 48.882
2 29.841 46.751
2 31.481 48.283
2 30.357 45.903
1 26.637 40.361
2 32.756 48.412
1 24.303 44.492
2 32.673 48.934
1 20.861 42.712
1 20.894 41.956
1 20.619 43.085
1 23.939 43.848
2 30.736 44.54
2 30.062 49.87
2 31.593 47.193
2 27.723 46.372
1 27.865 44.551
2 25.746 48.229
1 20.565 45.219
1 22.035 46.102
1 21.968 42.87
1 23.244 40.248
2 31.038 45.378
1 22.374 42.763
1 21.361 47.786
2 29.362 46.259
2 29.888 47.895
2 28.59 46.42
1 16.236 46.837
2 29.631 46.855
2 31.581 46.484
2 31.268 44.362
1 16.917 46.496
1 19.898 45.68
1 22.22 48.772
1 17.959 47.029
2 29.363 45.602
2 26.622 49.844
2 30.083 44.275
1 23.976 42.49
2 28.853 50.315
2 31.689 44.917
1 23.043 42.814
2 31.041 46.116
2 29.878 44.035
2 30.32 47.334
2 27.034 45.309
2 32.085 45.108
2 29.849 42.967
1 23.332 44.912
2 27.798 48.695
2 28.156 46.531
1 23.83 43.942
1 20.489 47.554
1 21.229 42.47
1 22.744 47.074
2 28.592 47.329
2 30.127 46.074
1 22.953 46.138
2 26.718 44.006
2 28.682 47.557
2 26.405 51.407
2 31.672 49.627
1 21.149 46.107
2 27.158 48.891
2 28.126 45.774
2 29.924 45.592
1 23.104 42.619
2 30.716 47.291
2 28.979 48.972
2 28.986 46.532
2 28.029 47.701
2 30.183 46.251
2 29.137 49.192
2 25.215 50.673
2 27.969 50.041
2 29.312 48.221
2 26.065 48.903
2 28.984 48.6
2 28.913 46.763
1 26.046 44.771
2 29.047 51.7
1 22.98 42.784
2 29.834 46.296
2 30.597 47.365
1 17.815 46.941
1 21.581 45.02
2 26.24 47.7
1 20.841 44.3
1 24.444 45.005
1 27.15 41.519
2 27.265 47.252
2 30.555 48.446
1 23.176 44.212
2 28.841 51.005
1 22.888 43.193
2 31.051 43.203
1 20.157 47.345
1 21.632 46.249
1 20.912 45.36
2 30.051 43.378
2 28.593 50.617
1 21.272 46.456
2 28.399 46.681
1 24.654 44.289
2 30.363 50.097
2 32.633 43.481
1 24.059 46.265
2 27.013 49.302
1 16.905 45.431
2 29.883 49.085
1 23.374 43.719
2 28.851 49.869
1 22.731 47.39
1 20.544 43.166
1 24.76 42.369
2 31.423 41.171
1 26.183 44.409
1 22.081 44.842
2 30.658 46.624
1 22.879 45.714
2 29.109 47.892
1 24.399 44.121
1 18.645 41.677
2 30.844 45.236
2 33.395 46.928
2 31.219 46.631
2 29.815 49.515
2 29.029 44.656
2 32.819 45.703
2 27.761 47.956
2 31.952 45.017
2 30.117 47.071
2 29.761 46.743
1 22.494 46.739
1 23.475 44.492
2 30.028 47.101
1 27.332 40.923
1 18.153 49.028
2 26.974 45.135
2 30.779 43.732
1 26.57 40.646
1 20.244 45.423
2 28.626 47.921
2 32.523 46.939
2 32.363 45.732
2 29.125 48.027
2 29.047 47.168
2 31.414 43.167
2 32.553 45.609
1 19.161 49.823
1 23.088 43.296
2 26.307 48.394
2 27.604 44.517
2 26.523 53.335
2 27.999 46.129
1 22.093 49.361
1 20.169 44.307
1 26.703 41.717
1 24.019 45.709
1 20.522 44.107
2 29.006 50.409
2 30.536 47.951
2 28.726 47.958
2 26.459 49.676
2 26.696 53.116
1 21.452 45.321
1 22.258 46.473
2 28.405 44.691
1 19.444 44.943
1 21.098 45.171
2 29.024 45.819
2 26.801 52.736
2 29.148 45.738
1 24.389 43.454
1 20.954 44.784
2 28.304 49.012
1 16.446 47.03
1 26.134 41.979
2 30.542 45.839
1 25.811 41.873
1 22.896 43.752
1 20.056 48.773
2 29.361 46.491
2 27.412 48.927
2 32.84 43.968
2 31.626 45.602
1 20.451 44.045
1 18.816 46.869
1 19.251 47.442
2 29.112 47.281
1 23.741 46.835
2 31.378 45.73
2 30.374 48.973
1 29.27 41.735
2 30.638 46.833
1 18.104 44.652
2 27.523 51.776
1 18.754 46.16
2 28.907 47.685
2 27.203 49.192
2 29.53 46.6
2 30.146 43.983
2 28.821 49.53
2 30.707 42.432
1 24.118 43.98
2 27.115 51.496
2 30.65 44.414
1 16.723 49.376
1 21.158 46.201
2 30.107 47.218
2 29.959 45.446
2 28.683 50.717
2 26.808 48.178
2 30.941 47.694
2 30.473 49.016
1 22.331 47.248
2 30.438 42.937
2 29.938 44.615
2 29.945 47.523
2 27.148 49.386
1 24.662 44.727
2 32.108 45.915
1 23.278 42.561
1 22.374 45.202
2 30.237 43.727
2 28.794 44.011
1 24.677 46.168
1 20.385 46.836
2 27.116 49.811
2 28.141 43.965
1 24.451 43.515
2 30.76 47.617
2 30.381 47.092
2 31.937 43.128
2 31.227 44.912
1 24.99 44.449
1 24.28 49.371
2 29.194 46.18
2 30.676 46.603
2 31.036 45.727
1 23.243 43.882
2 26.996 48.448
2 28.791 48.154
2 29.868 45.156
2 30.689 48.885
2 33.026 46.778
2 31.746 46.293
1 20.476 45.207
2 27.534 45.665
2 31.953 48.699
1 26.352 45.503
1 27.096 41.8
1 24.692 44.877
2 29.183 45.833
1 19.889 47.679
1 24.021 46.398
1 17.78 45.725
2 27.173 48.494
2 27.588 46.23
2 30.372 49.174
2 31.957 49.073
1 22.991 42.56
1 19.278 46.901
2 29.376 46.435
2 32.093 46.052
1 20.794 46.466
2 26.519 49.904
2 26.787 49.668
1 24.485 43.926
1 24.236 44.411
2 31.592 45.151
1 23.927 41.822
1 23.753 47.74
1 20.597 45.797
2 28.872 49.758
1 18.509 47.962
1 22.09 45.965
2 29.082 44.854
1 24.233 45.125
1 21.219 42.726
2 30.991 48.435
1 24.349 44.158
1 23.203 44.219
2 32.019 42.115
1 22.215 41.98
1 23.409 44.271
1 21.066 47.787
2 30.473 44.141
1 20.576 43.948
2 29.131 45.865
2 27.238 52.382
2 30.345 44.92
1 21.93 42.947
2 29.549 47.957
2 29.836 47.508
2 29.051 48.496
1 19.803 42.813
2 26.989 47.482
2 28.697 46.597
2 28.974 44.681
2 27.695 47.107
2 31.938 46.708
2 28.476 48.215
1 23.608 43.095
2 25.064 49.823
1 19.308 49.886
1 20.388 44.663
2 27.509 49.369
2 30.375 48.978
2 29.326 46.467
2 27.297 50.314
1 23.793 45.23
2 31.099 47.824
1 25.955 43.878
2 31.116 41.839
2 28.102 47.222
1 20.493 46.661
2 27.613 51.354
2 27.651 44.916
1 24.105 42.835
2 29.934 47.49
2 28.22 46.908
1 22.152 43.865
1 20.214 46.993
2 27.651 48.978
2 28.195 46.208
1 17.638 44.998
2 32.267 43.85
2 31.365 46.974
2 29.05 46.852
2 29.808 48.575
2 28.934 50.279
1 19.696 45.442
2 29.352 46.032
1 25.494 43.046
2 27.824 49.164
1 23.937 41.242
1 20.735 42.809
2 27.601 49.333
2 29.42 46.644
1 23.597 46.604
2 30.036 46.77
1 17.63 47.445
1 20.776 45.403
2 29.17 47.551
1 24.155 43.74
2 27.397 45.247
2 28.069 48.764
1 26.615 43.468
1 24.965 43.922
1 27.838 42.435
2 31.135 46.382
1 23.251 43.918
1 22.569 47.083
2 26.104 53.247
1 24.785 40.611
1 19.311 43.124
1 25.093 47.868
2 30.622 43.954
1 20.208 45.852
1 22.584 44.021
2 31.242 43.574
2 30.342 43.745
1 20.138 48.293
2 27.849 49.57
2 26.81 51.834
1 21.407 46.1
1 22.23 46.022
1 21.554 46.288
1 19.765 46.3
2 29.174 48.431
2 29.916 45.529
2 31.909 49.889
2 28.481 51.626
2 30.831 45.304
1 20.328 45.047
1 23.736 41.216
1 17.125 46.896
2 29.156 46.924
2 31.325 47.149
2 28.118 48.647
1 18.417 46.135
2 28.46 46.553
1 23.335 46.735
2 29.726 46.227
1 21.918 43.508
2 29.276 49.067
1 19.658 44.316
2 30.856 46.417
1 18.44 45.978
1 26.222 46.294
2 29.042 46.471
2 31.937 47.269
2 30.074 47.056
1 23.441 43.135
2 29.312 48.655
1 22.704 46.857
2 29.744 48.258
1 18.1 46.514
1 20.394 47.694
1 24.054 39.235
1 19.732 47.67
2 27.832 47.522
2 30.163 44.179
1 25.342 44.375
2 30.662 49.521
2 31.367 45.414
1 23.111 45.696
2 27.178 46.876
2 29.571 47.51
2 26.693 48.34
2 27.916 48.802
2 32.739 47.848
1 26.306 41.162
2 32.031 47.726
2 28.23 48.751
1 20.997 45.491
2 29.659 49.329
1 19.446 47.112
1 21.797 46.958
1 20.129 43.629
1 19.598 44.429
2 28.134 47.337
2 31.663 47.301
2 31.71 43.896
2 31.738 43.783
2 28.095 47.371
1 21.648 46.012
2 28.552 48.292
1 27.632 44.668
1 23.108 46.685
1 17.152 48.084
2 30.014 46.873
1 18.831 45.456
2 29.23 46.223
2 29.077 46.324
2 30.883 42.538
1 21.407 45.329
1 18.106 43.943
1 23.492 44.235
1 24.745 45.757
2 27.4 51.865
2 31.461 48.519
2 28.713 45.013
2 29.603 47.289
2 30.916 45.956
1 24.635 44.192
2 30.678 45.111
2 29.941 46.94
1 20.059 41.131
2 32.054 48.184
1 20.7 47.037
1 24.561 46.971
1 20.741 47.952
1 24.184 44.024
2 29.604 46.16
1 26.63 40.027
2 29.872 45.128
2 28.336 46.566
2 29.411 44.643
2 29.691 47.104
1 18.124 46.974
1 21.966 46.882
1 19.304 45.36
1 18.088 43.898
2 29.267 48.438
2 29.847 43.6
2 31.07 46.093
2 29.661 47.733
2 29.822 45.464
2 24.455 49.373
1 27.591 45.107
2 29.161 47.982
1 17.713 48.031
2 31.688 45.346
1 15.306 45.156
2 26.466 50.354
1 20.13 46.473
1 21.634 47.556
2 30.939 44.735
1 21.934 47.01
2 31.684 46.28
2 29.577 48.098
1 24.326 42.505
2 28.11 48.43
2 35.405 44.837
2 30.201 47.97
1 16.749 42.357
2 29.064 46.798
1 22.97 45.49
2 31.847 46.11
2 28.362 48.057
1 26.319 43.005
2 27.235 51.441
2 30.669 46.805
2 26.04 52.697
1 21.933 43.953
1 26.821 39.822
2 28.187 48.376
2 29.003 49.925
2 28.46 47.269
2 30.878 44.643
2 29.526 51.26
1 25.26 46.24
2 31.385 45.915
1 19.637 46.95
2 30.692 44.593
1 23.115 44.308
1 25.9 43.662
2 32.733 44.256
2 29.353 47.411

Explanation / Answer

Refining a k-Nearest-Neighbor classification

Machine learning algorithms provide methods of classifying objects into one of several groups based on the values of several explanatory variables. Nearest neighbor methods are easily implmented and easy to understand. There is no model associated to them, so errors have to be estimated computationally, but it provides one simple solution to classifying a new object based on known results in a reference set. kNN is a generalization of “if it walks like a duck, looks like a duck, and talks like a duck, it is probably a duck.” That is, objects that a close together with respect to the explanatory variables are likely to have the same classification.

The kNN algorithm

In the simplest setting, like the example we will do here, objects can fall into one of two classes, ( A ) or ( B ). We have a set of ( n ) measurements, ( v_1, dots, v_n ), of any object in question, and assume these are all numeric variables. We use a distance measure, namely Euclidean distance, to manifest when two objects are close with respect to these variables. So, given objects in the domain ( s ), ( r ), which have ( v_i ) measurements ( x_1, dots, x_n ), ( y_1, dots, y_n ), respectively, define [ dist(s, r) = sqrt(sum_i (x_i-y_i)^2). ] Depending on characteristics of the variables, other distance measures may be more appropriate, but we'll stick with Euclidean distance.

The kNN algorithm begins with a training set of objects for which we know not only the values of the explanatory variables but also the classifications ( A ) and ( B ). To predict the classification of a new object ( q ), the ( k=1 ) version of kNN would proceed by finding the element of the training set with the minimum distance from ( q ), suppose it is ( p_1 ), and predict the classification of ( q ) to be the same as ( p_1 ). If ( p_1 ) is rather isolated and there are lots of points in the other class almost as close to ( q ) this could be a misclassification. So, we generally pick some larger odd number ( m ) for ( k ); find ( p_1, dots,p_m ) closest to ( q ) and vote on what the classification of ( q ) should be. A rule of thumb in machine learning is to pick ( k ) near the square root of the size of the training set. In practice this does a good job of telling signal from noise.

Classifying samples as tumor or normal based on measurements from images

Typically, biopsy tissue samples from breast cancer are preserved on paraffin slide and examined visually by a pathologist. Cancerous tissues have more misshapen cells, larger nuclei, rough surfaces and other abnormal characteristics. Today, automated image analysis can collect measurements of the nuclei of the cells in a picture of a sample without human intervention. From the UC-Irvine machine learning archive we have the Wisconsin Breast Cancer Dataset, with nuclei measurements of 569 samples, some benign and some tumor. This is saved to disk and cleaned up here.

Creating training and test (validation) datasets

A large training set helps to give us a good model, but a large validation set increases the significance of the result you report. You need to strike a balance. The choice of what fraction to use for training may depend on how messy the data is, or how complex the modeling method is. A reasonable balance is 2/3 training and 1/3 validation. Furthermore, in splitting the data we want the training set (and implicitly the validation set) to be representative of the general population as far as this particular problem goes. Here, that means they should contain roughly the same fraction of benign and malignant cases as the whole population.

We'll do the split by randomly permuting the rows of data, selecting 1/3 for validation and having the rest left for training. Our rows are determined by the id variable.

Subset the data we'll use in the knn analysis accordingly.

Check that training and validation sets have

Executing knn

The knn algorithm is implemented in the package class. Install this if necessary and then load it.

What should k be? Let's use an odd number near the square root of size of the training set.

Check the agreement with the predictor.

Not very good as classifications go.

Testing different values of k

Let's sample a few other values of k and see how the performance changes. Let's try k=3, 7, 11, 31.

The best was actually 3, by a little. Sometimes, too many points hides subclusters that are fairly small. This can also be very dependent on the choice of training set.

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