Digit Recognition with PCA + KNN
Sample images from the MNIST dataset
The dataset contains gray-scale images of hand-drawn digits, from zero through nine.
Each image is 28 pixels in height and 28 pixels in width, for a total of 784 pixels in total. Each pixel has a single pixel-value associated with it, indicating the lightness or darkness of that pixel, with higher numbers meaning darker. This pixel-value is an integer between 0 and 255, inclusive
digits_df = read.csv("./data/train.csv",stringsAsFactors = F)
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## 3 0 0 0 0 0 0 0 0
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## 5 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0
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## 2 0 0 0 0 0 0 0 0
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## 5 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0
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## 2 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0
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## 1 0 0 0 0 0 0 0 0
## 2 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0
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## 2 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0
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## 2 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0
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## 2 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0
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## 2 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0
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## 2 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0
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## 1 0 0 0 0 0
## 2 0 0 0 0 0
## 3 0 0 0 0 0
## 4 0 0 0 0 0
## 5 0 0 0 0 0
## 6 0 0 0 0 0Labels are stored in a separate vector.
The pixel values are scaled by the largest value and then centred.
digits_labels = digits_df[,1]
digits_X = digits_df[,-1]/255
digits_scaled_x = as.data.frame(scale(digits_X,scale = FALSE,center = TRUE))
head(digits_scaled_x)## pixel0 pixel1 pixel2 pixel3 pixel4 pixel5 pixel6 pixel7 pixel8 pixel9
## 1 0 0 0 0 0 0 0 0 0 0
## 2 0 0 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0 0 0
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## 5 0 0 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0 0 0
## pixel10 pixel11 pixel12 pixel13 pixel14 pixel15
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## 2 0 0 -1.176471e-05 -4.388422e-05 -2.016807e-05 -8.403361e-07
## 3 0 0 -1.176471e-05 -4.388422e-05 -2.016807e-05 -8.403361e-07
## 4 0 0 -1.176471e-05 -4.388422e-05 -2.016807e-05 -8.403361e-07
## 5 0 0 -1.176471e-05 -4.388422e-05 -2.016807e-05 -8.403361e-07
## 6 0 0 -1.176471e-05 -4.388422e-05 -2.016807e-05 -8.403361e-07
## pixel16 pixel17 pixel18 pixel19 pixel20 pixel21 pixel22 pixel23 pixel24
## 1 0 0 0 0 0 0 0 0 0
## 2 0 0 0 0 0 0 0 0 0
## 3 0 0 0 0 0 0 0 0 0
## 4 0 0 0 0 0 0 0 0 0
## 5 0 0 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0 0 0
## pixel25 pixel26 pixel27 pixel28 pixel29 pixel30 pixel31 pixel32
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## 2 0 0 0 0 0 0 0 -1.493931e-06
## 3 0 0 0 0 0 0 0 -1.493931e-06
## 4 0 0 0 0 0 0 0 -1.493931e-06
## 5 0 0 0 0 0 0 0 -1.493931e-06
## 6 0 0 0 0 0 0 0 -1.493931e-06
## pixel33 pixel34 pixel35 pixel36 pixel37
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## 2 -5.135387e-06 -4.136321e-05 -0.0001069094 -0.0001996265 -0.0002604108
## 3 -5.135387e-06 -4.136321e-05 -0.0001069094 -0.0001996265 -0.0002604108
## 4 -5.135387e-06 -4.136321e-05 -0.0001069094 -0.0001996265 -0.0002604108
## 5 -5.135387e-06 -4.136321e-05 -0.0001069094 -0.0001996265 -0.0002604108
## 6 -5.135387e-06 -4.136321e-05 -0.0001069094 -0.0001996265 -0.0002604108
## pixel38 pixel39 pixel40 pixel41 pixel42
## 1 -0.0005081232 -0.0006828198 -0.0007502334 -0.0007474323 -0.0007688142
## 2 -0.0005081232 -0.0006828198 -0.0007502334 -0.0007474323 -0.0007688142
## 3 -0.0005081232 -0.0006828198 -0.0007502334 -0.0007474323 -0.0007688142
## 4 -0.0005081232 -0.0006828198 -0.0007502334 -0.0007474323 -0.0007688142
## 5 -0.0005081232 -0.0006828198 -0.0007502334 -0.0007474323 -0.0007688142
## 6 -0.0005081232 -0.0006828198 -0.0007502334 -0.0007474323 -0.0007688142
## pixel43 pixel44 pixel45 pixel46 pixel47
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## 2 -0.0006719888 -0.0006450047 -0.000594958 -0.0004129785 -0.0002383754
## 3 -0.0006719888 -0.0006450047 -0.000594958 -0.0004129785 -0.0002383754
## 4 -0.0006719888 -0.0006450047 -0.000594958 -0.0004129785 -0.0002383754
## 5 -0.0006719888 -0.0006450047 -0.000594958 -0.0004129785 -0.0002383754
## 6 -0.0006719888 -0.0006450047 -0.000594958 -0.0004129785 -0.0002383754
## pixel48 pixel49 pixel50 pixel51 pixel52 pixel53
## 1 -0.0001767507 -6.041083e-05 -4.126984e-05 -1.979458e-05 0 0
## 2 -0.0001767507 -6.041083e-05 -4.126984e-05 -1.979458e-05 0 0
## 3 -0.0001767507 -6.041083e-05 -4.126984e-05 -1.979458e-05 0 0
## 4 -0.0001767507 -6.041083e-05 -4.126984e-05 -1.979458e-05 0 0
## 5 -0.0001767507 -6.041083e-05 -4.126984e-05 -1.979458e-05 0 0
## 6 -0.0001767507 -6.041083e-05 -4.126984e-05 -1.979458e-05 0 0
## pixel54 pixel55 pixel56 pixel57 pixel58 pixel59 pixel60
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## 2 0 0 0 0 -5.975724e-06 -2.70775e-06 -2.875817e-05
## 3 0 0 0 0 -5.975724e-06 -2.70775e-06 -2.875817e-05
## 4 0 0 0 0 -5.975724e-06 -2.70775e-06 -2.875817e-05
## 5 0 0 0 0 -5.975724e-06 -2.70775e-06 -2.875817e-05
## 6 0 0 0 0 -5.975724e-06 -2.70775e-06 -2.875817e-05
## pixel61 pixel62 pixel63 pixel64 pixel65
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## 2 -3.538749e-05 -0.0002396825 -0.000592437 -0.001160224 -0.00209281
## 3 -3.538749e-05 -0.0002396825 -0.000592437 -0.001160224 -0.00209281
## 4 -3.538749e-05 -0.0002396825 -0.000592437 -0.001160224 -0.00209281
## 5 -3.538749e-05 -0.0002396825 -0.000592437 -0.001160224 -0.00209281
## 6 -3.538749e-05 -0.0002396825 -0.000592437 -0.001160224 -0.00209281
## pixel66 pixel67 pixel68 pixel69 pixel70
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## 2 -0.003409337 -0.005279272 -0.007770308 -0.01058273 -0.01329458
## 3 -0.003409337 -0.005279272 -0.007770308 -0.01058273 -0.01329458
## 4 -0.003409337 -0.005279272 -0.007770308 -0.01058273 -0.01329458
## 5 -0.003409337 -0.005279272 -0.007770308 -0.01058273 -0.01329458
## 6 -0.003409337 -0.005279272 -0.007770308 -0.01058273 -0.01329458
## pixel71 pixel72 pixel73 pixel74 pixel75 pixel76
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## 2 -0.0149085 -0.01466695 -0.01306984 -0.01052745 -0.007814192 -0.004689636
## 3 -0.0149085 -0.01466695 -0.01306984 -0.01052745 -0.007814192 -0.004689636
## 4 -0.0149085 -0.01466695 -0.01306984 -0.01052745 -0.007814192 -0.004689636
## 5 -0.0149085 -0.01466695 -0.01306984 -0.01052745 -0.007814192 -0.004689636
## 6 -0.0149085 -0.01466695 -0.01306984 -0.01052745 -0.007814192 -0.004689636
## pixel77 pixel78 pixel79 pixel80 pixel81
## 1 -0.00235761 -0.001152288 -0.0003861811 -0.0001370682 -3.295985e-05
## 2 -0.00235761 -0.001152288 -0.0003861811 -0.0001370682 -3.295985e-05
## 3 -0.00235761 -0.001152288 -0.0003861811 -0.0001370682 -3.295985e-05
## 4 -0.00235761 -0.001152288 -0.0003861811 -0.0001370682 -3.295985e-05
## 5 -0.00235761 -0.001152288 -0.0003861811 -0.0001370682 -3.295985e-05
## 6 -0.00235761 -0.001152288 -0.0003861811 -0.0001370682 -3.295985e-05
## pixel82 pixel83 pixel84 pixel85 pixel86 pixel87 pixel88
## 1 0 0 0 0 -1.419234e-05 -1.633987e-05 -6.43324e-05
## 2 0 0 0 0 -1.419234e-05 -1.633987e-05 -6.43324e-05
## 3 0 0 0 0 -1.419234e-05 -1.633987e-05 -6.43324e-05
## 4 0 0 0 0 -1.419234e-05 -1.633987e-05 -6.43324e-05
## 5 0 0 0 0 -1.419234e-05 -1.633987e-05 -6.43324e-05
## 6 0 0 0 0 -1.419234e-05 -1.633987e-05 -6.43324e-05
## pixel89 pixel90 pixel91 pixel92 pixel93
## 1 -0.0003520075 -0.0009248366 -0.002120542 -0.004677778 -0.008991597
## 2 -0.0003520075 -0.0009248366 -0.002120542 -0.004677778 -0.008991597
## 3 -0.0003520075 -0.0009248366 -0.002120542 -0.004677778 -0.008991597
## 4 -0.0003520075 -0.0009248366 -0.002120542 -0.004677778 -0.008991597
## 5 -0.0003520075 -0.0009248366 -0.002120542 -0.004677778 -0.008991597
## 6 -0.0003520075 -0.0009248366 -0.002120542 -0.004677778 -0.008991597
## pixel94 pixel95 pixel96 pixel97 pixel98 pixel99
## 1 -0.01477796 -0.02240738 -0.03039701 -0.03940728 -0.04732446 -0.05256844
## 2 -0.01477796 -0.02240738 -0.03039701 -0.03940728 -0.04732446 -0.05256844
## 3 -0.01477796 -0.02240738 -0.03039701 -0.03940728 -0.04732446 -0.05256844
## 4 -0.01477796 -0.02240738 -0.03039701 -0.03940728 -0.04732446 -0.05256844
## 5 -0.01477796 -0.02240738 -0.03039701 -0.03940728 -0.04732446 -0.05256844
## 6 -0.01477796 -0.02240738 -0.03039701 -0.03940728 -0.04732446 -0.05256844
## pixel100 pixel101 pixel102 pixel103 pixel104 pixel105
## 1 -0.05126555 -0.04538749 -0.03645294 -0.02630719 -0.01623539 -0.00890028
## 2 -0.05126555 -0.04538749 -0.03645294 -0.02630719 -0.01623539 -0.00890028
## 3 -0.05126555 -0.04538749 -0.03645294 -0.02630719 -0.01623539 -0.00890028
## 4 -0.05126555 -0.04538749 -0.03645294 -0.02630719 -0.01623539 -0.00890028
## 5 -0.05126555 -0.04538749 -0.03645294 -0.02630719 -0.01623539 -0.00890028
## 6 -0.05126555 -0.04538749 -0.03645294 -0.02630719 -0.01623539 -0.00890028
## pixel106 pixel107 pixel108 pixel109 pixel110
## 1 -0.004281979 -0.001665079 -0.0006587302 -9.309057e-05 -1.129785e-05
## 2 -0.004281979 -0.001665079 -0.0006587302 -9.309057e-05 -1.129785e-05
## 3 -0.004281979 -0.001665079 -0.0006587302 -9.309057e-05 -1.129785e-05
## 4 -0.004281979 -0.001665079 -0.0006587302 -9.309057e-05 -1.129785e-05
## 5 -0.004281979 -0.001665079 -0.0006587302 -9.309057e-05 -1.129785e-05
## 6 -0.004281979 -0.001665079 -0.0006587302 -9.309057e-05 -1.129785e-05
## pixel111 pixel112 pixel113 pixel114 pixel115
## 1 0 0 -3.548086e-06 -1.073763e-05 -6.302521e-05
## 2 0 0 -3.548086e-06 -1.073763e-05 -6.302521e-05
## 3 0 0 -3.548086e-06 -1.073763e-05 -6.302521e-05
## 4 0 0 -3.548086e-06 -1.073763e-05 -6.302521e-05
## 5 0 0 -3.548086e-06 -1.073763e-05 -6.302521e-05
## 6 0 0 -3.548086e-06 -1.073763e-05 -6.302521e-05
## pixel116 pixel117 pixel118 pixel119 pixel120
## 1 -0.0003461251 -0.001524276 -0.004039402 -0.009627077 -0.01942437
## 2 -0.0003461251 -0.001524276 -0.004039402 -0.009627077 -0.01942437
## 3 -0.0003461251 -0.001524276 -0.004039402 -0.009627077 -0.01942437
## 4 -0.0003461251 -0.001524276 -0.004039402 -0.009627077 -0.01942437
## 5 -0.0003461251 -0.001524276 -0.004039402 -0.009627077 -0.01942437
## 6 -0.0003461251 -0.001524276 -0.004039402 -0.009627077 -0.01942437
## pixel121 pixel122 pixel123 pixel124 pixel125 pixel126
## 1 -0.03402726 -0.05411569 -0.07972848 -0.10997628 -0.1415097 -0.1675057
## 2 -0.03402726 0.01647255 0.03791858 0.42727862 0.3957452 0.5854355
## 3 -0.03402726 -0.05411569 -0.07972848 -0.09821158 0.4114315 0.3775923
## 4 -0.03402726 -0.05411569 -0.07972848 -0.10997628 -0.1415097 -0.1675057
## 5 -0.03010570 0.04392353 0.43007544 0.49786685 0.8545687 0.8285727
## 6 -0.03402726 -0.05411569 -0.07972848 -0.09821158 0.4114315 0.6246512
## pixel127 pixel128 pixel129 pixel130 pixel131 pixel132
## 1 -0.1807542 -0.1746763 -0.15273931 -0.1214304 -0.08984136 0.67894566
## 2 0.1565007 0.1076767 -0.14881774 -0.1214304 -0.08984136 -0.05830924
## 3 -0.1689894 -0.1746763 -0.15273931 -0.1214304 -0.08984136 -0.05830924
## 4 -0.1807542 -0.1746763 -0.15273931 -0.1214304 -0.08984136 -0.05830924
## 5 0.8153243 0.4410100 -0.03509225 -0.1135873 -0.08984136 -0.05830924
## 6 0.8153243 0.5821865 0.01980971 -0.1214304 -0.08984136 -0.05830924
## pixel133 pixel134 pixel135 pixel136 pixel137
## 1 0.96591522 0.35078123 -0.008234827 -0.003289262 -0.0007951447
## 2 -0.03408478 -0.01784622 -0.008234827 -0.003289262 -0.0007951447
## 3 -0.03408478 -0.01784622 -0.008234827 -0.003289262 -0.0007951447
## 4 -0.03408478 -0.01784622 -0.008234827 -0.003289262 -0.0007951447
## 5 -0.03408478 -0.01784622 -0.008234827 -0.003289262 -0.0007951447
## 6 -0.03408478 -0.01784622 -0.008234827 -0.003289262 -0.0007951447
## pixel138 pixel139 pixel140 pixel141 pixel142 pixel143
## 1 -0.0001391223 0 0 0 -3.697479e-05 -0.0001882353
## 2 -0.0001391223 0 0 0 -3.697479e-05 -0.0001882353
## 3 -0.0001391223 0 0 0 -3.697479e-05 -0.0001882353
## 4 -0.0001391223 0 0 0 -3.697479e-05 -0.0001882353
## 5 -0.0001391223 0 0 0 -3.697479e-05 -0.0001882353
## 6 -0.0001391223 0 0 0 -3.697479e-05 -0.0001882353
## pixel144 pixel145 pixel146 pixel147 pixel148
## 1 -0.001614099 -0.005640523 -0.01395173 -0.02803894 -0.0506985994
## 2 -0.001614099 -0.005640523 -0.01395173 -0.02803894 0.0002817927
## 3 -0.001614099 -0.005640523 -0.01395173 -0.02803894 -0.0506985994
## 4 -0.001614099 -0.005640523 0.84879337 0.67392185 -0.0271691877
## 5 -0.001614099 -0.005640523 -0.01395173 -0.02803894 -0.0193260504
## 6 -0.001614099 -0.005640523 -0.01395173 -0.02803894 -0.0506985994
## pixel149 pixel150 pixel151 pixel152 pixel153 pixel154
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## 2 0.25325537 0.8540319 0.8181824 0.7600944 0.7019193 0.6576612
## 3 -0.08399953 -0.1263602 -0.1778961 -0.2006899 0.7019193 0.6576612
## 4 -0.08399953 -0.1263602 -0.1778961 -0.2359840 -0.2941591 -0.3384173
## 5 0.31992204 0.8657966 0.8142608 0.7561728 0.6979978 0.6537396
## 6 -0.08399953 -0.1263602 -0.1582882 0.4110748 0.7019193 0.3635435
## pixel155 pixel156 pixel157 pixel158 pixel159 pixel160
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## 2 0.4918112 0.6139204 0.2772447 -0.13239991 -0.1946972 -0.1348647
## 3 -0.3277966 -0.3507854 -0.3149121 -0.25789010 -0.1946972 -0.1348647
## 4 -0.3591692 -0.3507854 -0.2796180 0.04407068 -0.1946972 -0.1348647
## 5 0.6329877 0.6413714 0.6772447 0.18916872 -0.1868541 -0.1348647
## 6 0.2800465 0.6256852 0.6419506 0.02446284 -0.1946972 -0.1348647
## pixel161 pixel162 pixel163 pixel164 pixel165
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## 2 -0.08414725 -0.0479535 -0.02499832 -0.01139645 -0.003111204
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## 4 -0.08414725 -0.0479535 -0.02499832 -0.01139645 -0.003111204
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## 6 -0.08414725 -0.0479535 -0.02499832 -0.01139645 -0.003111204
## pixel166 pixel167 pixel168 pixel169 pixel170
## 1 -0.0004943044 -1.680672e-06 0 -3.734827e-07 -9.056956e-05
## 2 -0.0004943044 -1.680672e-06 0 -3.734827e-07 -9.056956e-05
## 3 -0.0004943044 -1.680672e-06 0 -3.734827e-07 -9.056956e-05
## 4 -0.0004943044 -1.680672e-06 0 -3.734827e-07 -9.056956e-05
## 5 -0.0004943044 -1.680672e-06 0 -3.734827e-07 -9.056956e-05
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## pixel171 pixel172 pixel173 pixel174 pixel175
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## 2 -0.0008956116 -0.004527638 -0.01281578 -0.02889038 0.007424743
## 3 -0.0008956116 -0.004527638 -0.01281578 -0.02889038 -0.055320355
## 4 -0.0008956116 -0.004527638 -0.01281578 0.08091354 0.913307096
## 5 -0.0008956116 -0.004527638 -0.01281578 -0.02889038 -0.012183100
## 6 -0.0008956116 -0.004527638 -0.01281578 -0.02889038 -0.055320355
## pixel176 pixel177 pixel178 pixel179 pixel180 pixel181
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## 2 0.60734332 0.8475156 0.7842840 0.7137087 0.6412893 0.5743572
## 3 -0.09461746 -0.1485628 -0.2117944 -0.2823697 -0.3194951 0.5743572
## 4 -0.02795079 -0.1485628 -0.2117944 -0.2823697 -0.3547892 -0.4217212
## 5 0.72106881 0.8435940 0.7803625 0.7097871 0.6373677 0.5704357
## 6 -0.09461746 -0.1485628 -0.2117944 0.2470420 0.6412893 0.1665141
## pixel182 pixel183 pixel184 pixel185 pixel186 pixel187
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## 5 -0.00220859 -0.0009394958 -0.00036676 -9.738562e-05 -3.361345e-06
## 6 -0.00220859 -0.0009394958 -0.00036676 -9.738562e-05 -3.361345e-06
## pixel753 pixel754 pixel755 pixel756 pixel757 pixel758 pixel759
## 1 -5.50887e-06 0 0 0 0 0 0
## 2 -5.50887e-06 0 0 0 0 0 0
## 3 -5.50887e-06 0 0 0 0 0 0
## 4 -5.50887e-06 0 0 0 0 0 0
## 5 -5.50887e-06 0 0 0 0 0 0
## 6 -5.50887e-06 0 0 0 0 0 0
## pixel760 pixel761 pixel762 pixel763 pixel764
## 1 0 -2.408964e-05 -0.0001405229 -0.0003229692 -0.0004506069
## 2 0 -2.408964e-05 -0.0001405229 -0.0003229692 -0.0004506069
## 3 0 -2.408964e-05 -0.0001405229 -0.0003229692 -0.0004506069
## 4 0 -2.408964e-05 -0.0001405229 -0.0003229692 -0.0004506069
## 5 0 -2.408964e-05 -0.0001405229 -0.0003229692 -0.0004506069
## 6 0 -2.408964e-05 -0.0001405229 -0.0003229692 -0.0004506069
## pixel765 pixel766 pixel767 pixel768 pixel769
## 1 -0.0007008403 -0.001182166 -0.001622129 -0.002014379 -0.002191503
## 2 -0.0007008403 -0.001182166 -0.001622129 -0.002014379 -0.002191503
## 3 -0.0007008403 -0.001182166 -0.001622129 -0.002014379 -0.002191503
## 4 -0.0007008403 -0.001182166 -0.001622129 -0.002014379 -0.002191503
## 5 -0.0007008403 -0.001182166 -0.001622129 -0.002014379 -0.002191503
## 6 -0.0007008403 -0.001182166 -0.001622129 -0.002014379 -0.002191503
## pixel770 pixel771 pixel772 pixel773 pixel774
## 1 -0.002658263 -0.002363959 -0.001918581 -0.001334174 -0.000859944
## 2 -0.002658263 -0.002363959 -0.001918581 -0.001334174 -0.000859944
## 3 -0.002658263 -0.002363959 -0.001918581 -0.001334174 -0.000859944
## 4 -0.002658263 -0.002363959 -0.001918581 -0.001334174 -0.000859944
## 5 -0.002658263 -0.002363959 -0.001918581 -0.001334174 -0.000859944
## 6 -0.002658263 -0.002363959 -0.001918581 -0.001334174 -0.000859944
## pixel775 pixel776 pixel777 pixel778 pixel779
## 1 -0.000459197 -0.0002314659 -7.917834e-05 -6.760037e-05 -1.120448e-05
## 2 -0.000459197 -0.0002314659 -7.917834e-05 -6.760037e-05 -1.120448e-05
## 3 -0.000459197 -0.0002314659 -7.917834e-05 -6.760037e-05 -1.120448e-05
## 4 -0.000459197 -0.0002314659 -7.917834e-05 -6.760037e-05 -1.120448e-05
## 5 -0.000459197 -0.0002314659 -7.917834e-05 -6.760037e-05 -1.120448e-05
## 6 -0.000459197 -0.0002314659 -7.917834e-05 -6.760037e-05 -1.120448e-05
## pixel780 pixel781 pixel782 pixel783
## 1 0 0 0 0
## 2 0 0 0 0
## 3 0 0 0 0
## 4 0 0 0 0
## 5 0 0 0 0
## 6 0 0 0 0We split the data into train and test(80-20).
Covariance matrix is calculated for the scaled training feature matrix.
set.seed(3)
train = sample(nrow(digits_df),33600,replace = FALSE)
train_digits = digits_scaled_x[train,]
test_digits = digits_scaled_x[-train,]
train_labels = digits_labels[train]
test_labels = digits_labels[-train]
digits_covMatrix = cov(train_digits)We run a principal component analysis using the training covariance matrix.
pca.train<-prcomp(digits_covMatrix)The percent of variance in data explained by each component is calculated by variance explained by each divided total sum of variances explained by all the 784 components.
We can see the first 20 components explain 95% of the variance in the data
varianceExplained<-as.data.frame(pca.train$sdev^2/sum(pca.train$sdev^2))
varianceExplained = cbind(c(1:784), cumsum(varianceExplained))
colnames(varianceExplained)<-c("NmbrPCs","CumVar")
head(varianceExplained,20)## NmbrPCs CumVar
## 1 1 0.2528929
## 2 2 0.4218624
## 3 3 0.5447166
## 4 4 0.6382897
## 5 5 0.7096159
## 6 6 0.7689498
## 7 7 0.8030197
## 8 8 0.8307930
## 9 9 0.8533612
## 10 10 0.8713508
## 11 11 0.8856100
## 12 12 0.8988246
## 13 13 0.9083502
## 14 14 0.9176530
## 15 15 0.9258029
## 16 16 0.9329282
## 17 17 0.9386020
## 18 18 0.9439505
## 19 19 0.9485245
## 20 20 0.9528742plot(varianceExplained$NmbrPCs, varianceExplained$CumVar,xlab = "Number of Factors", ylab = "Proportion of Variance Explained",type = "l",col = "red")
### We rotate and take only the first 20 components. ### We then do the actual dimension reduction where the matrix of 784 colums is converted to matrix of 20 columns. ### The initial feature matrix is multiplied with the loading ,atrix which has the loading of each pixel in each of the 20 factors
rotate<-pca.train$rotation[,1:20]
trainFinal = as.matrix(train_digits)%*%(rotate)
test_Final = as.matrix(test_digits) %*% (rotate)
head(trainFinal)## PC1 PC2 PC3 PC4 PC5 PC6
## 7058 1.2385661 -0.09225390 0.9562079 -2.0222803 -1.6366896 -2.38776010
## 33915 -0.0284593 2.00042451 2.5865288 -0.1607507 -0.4447979 1.07581466
## 16167 0.7353877 3.83059568 1.2945366 -2.1717318 -0.4150144 -0.68953211
## 13764 0.1178183 0.21161657 0.4720315 2.2597078 2.9712856 -0.99220279
## 25286 -1.0383362 0.06118333 0.2854307 4.7134733 0.7629267 -0.08782803
## 25382 -2.6707079 -0.86774212 -2.9825691 -3.2648362 2.2138450 1.22963629
## PC7 PC8 PC9 PC10 PC11 PC12
## 7058 -1.086845 -2.35785696 1.0284947 -0.91969981 0.04150644 1.7334591
## 33915 -1.353297 -1.73869085 -0.3257513 -1.59872638 1.00450944 1.1671824
## 16167 4.017117 1.73822168 0.8331324 -0.45076833 0.56438436 -0.2568828
## 13764 -0.319774 0.02163055 -0.1330343 -0.09426161 -1.52209771 -1.2877522
## 25286 1.651509 -0.87644116 -0.9518641 0.37243139 -0.47137806 -1.7042729
## 25382 -1.596051 1.41136509 -0.3111290 -0.27206734 1.15861635 -0.5519245
## PC13 PC14 PC15 PC16 PC17 PC18
## 7058 1.0872803 -1.3758303 -0.3787982 -0.63335862 0.5786666 0.6908834
## 33915 -0.7712481 1.2043489 -0.9393232 1.46810195 -0.6331685 0.9451100
## 16167 0.9671882 -0.2945731 0.7480981 -0.40756464 0.3775197 -0.4560801
## 13764 -0.0926215 -1.1295332 -0.7840852 0.02696351 -1.1132493 -0.2976250
## 25286 0.5682282 0.2780718 0.6329313 0.48449443 -1.0411342 0.1539926
## 25382 -0.7343438 1.6582480 0.1862017 0.16467067 0.6227072 0.7806703
## PC19 PC20
## 7058 -1.1481536 0.71808523
## 33915 -0.6084131 0.69069427
## 16167 0.6486494 -1.07547974
## 13764 0.3576962 -0.03989037
## 25286 0.8849891 -0.97155023
## 25382 0.8856386 -0.74531181We run KNN on the dimension reduced datasets
library(class)
kVsAccuracy = matrix(ncol = 2, nrow = 8)
i = 1
for (k in 3:10){
predictions = knn(train = trainFinal, test = test_Final,cl = train_labels, k = k)
kVsAccuracy[i,] = c(k,mean(predictions==test_labels))
i = i+1
}
plot(kVsAccuracy[,1],kVsAccuracy[,2],type = "l",xlab = "Value of k", ylab = "Testing Accuracy",main = "Choosing value of k based on testing accuracy")
# We get an accuracy of * 0.967381