Principal component analysis
Hiba
02/03/2023
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#principal component
#Finf first two principal component
library(MASS)
s=matrix(c(13.86,6.969,1.077,2.318,6.969,4.735,0.562,1.469,1.077,0.562,0.143,0.217,2.318,1.469,0.217,0.570),nrow=4,ncol=4,byrow=TRUE)
s## [,1] [,2] [,3] [,4]
## [1,] 13.860 6.969 1.077 2.318
## [2,] 6.969 4.735 0.562 1.469
## [3,] 1.077 0.562 0.143 0.217
## [4,] 2.318 1.469 0.217 0.570e=eigen(s)$vector
e## [,1] [,2] [,3] [,4]
## [1,] 0.86628065 0.49404141 0.05937011 0.04422787
## [2,] 0.47013930 -0.84395345 0.23789638 -0.10058294
## [3,] 0.06838293 0.01449239 -0.45834017 -0.88602372
## [4,] 0.15443651 -0.20846016 -0.85428613 0.45043194lambda=eigen(s)$value
lambda## [1] 18.14040617 1.00861991 0.11625288 0.04272104e1=e[,1]
e1## [1] 0.86628065 0.47013930 0.06838293 0.15443651e2=e[,2]
e2## [1] 0.49404141 -0.84395345 0.01449239 -0.20846016#Find % of variance explained by two pc's
lambda1=lambda[1]
pc1=lambda1/(sum(lambda))#variance explained by first principal component
pc1## [1] 0.939528lambda2=lambda[2]
pc2=lambda2/sum(lambda)#variance explained by 2nd principal component
pc2## [1] 0.05223845#pricipal component analysis of iris data
data(iris)
iris## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## 1 5.1 3.5 1.4 0.2 setosa
## 2 4.9 3.0 1.4 0.2 setosa
## 3 4.7 3.2 1.3 0.2 setosa
## 4 4.6 3.1 1.5 0.2 setosa
## 5 5.0 3.6 1.4 0.2 setosa
## 6 5.4 3.9 1.7 0.4 setosa
## 7 4.6 3.4 1.4 0.3 setosa
## 8 5.0 3.4 1.5 0.2 setosa
## 9 4.4 2.9 1.4 0.2 setosa
## 10 4.9 3.1 1.5 0.1 setosa
## 11 5.4 3.7 1.5 0.2 setosa
## 12 4.8 3.4 1.6 0.2 setosa
## 13 4.8 3.0 1.4 0.1 setosa
## 14 4.3 3.0 1.1 0.1 setosa
## 15 5.8 4.0 1.2 0.2 setosa
## 16 5.7 4.4 1.5 0.4 setosa
## 17 5.4 3.9 1.3 0.4 setosa
## 18 5.1 3.5 1.4 0.3 setosa
## 19 5.7 3.8 1.7 0.3 setosa
## 20 5.1 3.8 1.5 0.3 setosa
## 21 5.4 3.4 1.7 0.2 setosa
## 22 5.1 3.7 1.5 0.4 setosa
## 23 4.6 3.6 1.0 0.2 setosa
## 24 5.1 3.3 1.7 0.5 setosa
## 25 4.8 3.4 1.9 0.2 setosa
## 26 5.0 3.0 1.6 0.2 setosa
## 27 5.0 3.4 1.6 0.4 setosa
## 28 5.2 3.5 1.5 0.2 setosa
## 29 5.2 3.4 1.4 0.2 setosa
## 30 4.7 3.2 1.6 0.2 setosa
## 31 4.8 3.1 1.6 0.2 setosa
## 32 5.4 3.4 1.5 0.4 setosa
## 33 5.2 4.1 1.5 0.1 setosa
## 34 5.5 4.2 1.4 0.2 setosa
## 35 4.9 3.1 1.5 0.2 setosa
## 36 5.0 3.2 1.2 0.2 setosa
## 37 5.5 3.5 1.3 0.2 setosa
## 38 4.9 3.6 1.4 0.1 setosa
## 39 4.4 3.0 1.3 0.2 setosa
## 40 5.1 3.4 1.5 0.2 setosa
## 41 5.0 3.5 1.3 0.3 setosa
## 42 4.5 2.3 1.3 0.3 setosa
## 43 4.4 3.2 1.3 0.2 setosa
## 44 5.0 3.5 1.6 0.6 setosa
## 45 5.1 3.8 1.9 0.4 setosa
## 46 4.8 3.0 1.4 0.3 setosa
## 47 5.1 3.8 1.6 0.2 setosa
## 48 4.6 3.2 1.4 0.2 setosa
## 49 5.3 3.7 1.5 0.2 setosa
## 50 5.0 3.3 1.4 0.2 setosa
## 51 7.0 3.2 4.7 1.4 versicolor
## 52 6.4 3.2 4.5 1.5 versicolor
## 53 6.9 3.1 4.9 1.5 versicolor
## 54 5.5 2.3 4.0 1.3 versicolor
## 55 6.5 2.8 4.6 1.5 versicolor
## 56 5.7 2.8 4.5 1.3 versicolor
## 57 6.3 3.3 4.7 1.6 versicolor
## 58 4.9 2.4 3.3 1.0 versicolor
## 59 6.6 2.9 4.6 1.3 versicolor
## 60 5.2 2.7 3.9 1.4 versicolor
## 61 5.0 2.0 3.5 1.0 versicolor
## 62 5.9 3.0 4.2 1.5 versicolor
## 63 6.0 2.2 4.0 1.0 versicolor
## 64 6.1 2.9 4.7 1.4 versicolor
## 65 5.6 2.9 3.6 1.3 versicolor
## 66 6.7 3.1 4.4 1.4 versicolor
## 67 5.6 3.0 4.5 1.5 versicolor
## 68 5.8 2.7 4.1 1.0 versicolor
## 69 6.2 2.2 4.5 1.5 versicolor
## 70 5.6 2.5 3.9 1.1 versicolor
## 71 5.9 3.2 4.8 1.8 versicolor
## 72 6.1 2.8 4.0 1.3 versicolor
## 73 6.3 2.5 4.9 1.5 versicolor
## 74 6.1 2.8 4.7 1.2 versicolor
## 75 6.4 2.9 4.3 1.3 versicolor
## 76 6.6 3.0 4.4 1.4 versicolor
## 77 6.8 2.8 4.8 1.4 versicolor
## 78 6.7 3.0 5.0 1.7 versicolor
## 79 6.0 2.9 4.5 1.5 versicolor
## 80 5.7 2.6 3.5 1.0 versicolor
## 81 5.5 2.4 3.8 1.1 versicolor
## 82 5.5 2.4 3.7 1.0 versicolor
## 83 5.8 2.7 3.9 1.2 versicolor
## 84 6.0 2.7 5.1 1.6 versicolor
## 85 5.4 3.0 4.5 1.5 versicolor
## 86 6.0 3.4 4.5 1.6 versicolor
## 87 6.7 3.1 4.7 1.5 versicolor
## 88 6.3 2.3 4.4 1.3 versicolor
## 89 5.6 3.0 4.1 1.3 versicolor
## 90 5.5 2.5 4.0 1.3 versicolor
## 91 5.5 2.6 4.4 1.2 versicolor
## 92 6.1 3.0 4.6 1.4 versicolor
## 93 5.8 2.6 4.0 1.2 versicolor
## 94 5.0 2.3 3.3 1.0 versicolor
## 95 5.6 2.7 4.2 1.3 versicolor
## 96 5.7 3.0 4.2 1.2 versicolor
## 97 5.7 2.9 4.2 1.3 versicolor
## 98 6.2 2.9 4.3 1.3 versicolor
## 99 5.1 2.5 3.0 1.1 versicolor
## 100 5.7 2.8 4.1 1.3 versicolor
## 101 6.3 3.3 6.0 2.5 virginica
## 102 5.8 2.7 5.1 1.9 virginica
## 103 7.1 3.0 5.9 2.1 virginica
## 104 6.3 2.9 5.6 1.8 virginica
## 105 6.5 3.0 5.8 2.2 virginica
## 106 7.6 3.0 6.6 2.1 virginica
## 107 4.9 2.5 4.5 1.7 virginica
## 108 7.3 2.9 6.3 1.8 virginica
## 109 6.7 2.5 5.8 1.8 virginica
## 110 7.2 3.6 6.1 2.5 virginica
## 111 6.5 3.2 5.1 2.0 virginica
## 112 6.4 2.7 5.3 1.9 virginica
## 113 6.8 3.0 5.5 2.1 virginica
## 114 5.7 2.5 5.0 2.0 virginica
## 115 5.8 2.8 5.1 2.4 virginica
## 116 6.4 3.2 5.3 2.3 virginica
## 117 6.5 3.0 5.5 1.8 virginica
## 118 7.7 3.8 6.7 2.2 virginica
## 119 7.7 2.6 6.9 2.3 virginica
## 120 6.0 2.2 5.0 1.5 virginica
## 121 6.9 3.2 5.7 2.3 virginica
## 122 5.6 2.8 4.9 2.0 virginica
## 123 7.7 2.8 6.7 2.0 virginica
## 124 6.3 2.7 4.9 1.8 virginica
## 125 6.7 3.3 5.7 2.1 virginica
## 126 7.2 3.2 6.0 1.8 virginica
## 127 6.2 2.8 4.8 1.8 virginica
## 128 6.1 3.0 4.9 1.8 virginica
## 129 6.4 2.8 5.6 2.1 virginica
## 130 7.2 3.0 5.8 1.6 virginica
## 131 7.4 2.8 6.1 1.9 virginica
## 132 7.9 3.8 6.4 2.0 virginica
## 133 6.4 2.8 5.6 2.2 virginica
## 134 6.3 2.8 5.1 1.5 virginica
## 135 6.1 2.6 5.6 1.4 virginica
## 136 7.7 3.0 6.1 2.3 virginica
## 137 6.3 3.4 5.6 2.4 virginica
## 138 6.4 3.1 5.5 1.8 virginica
## 139 6.0 3.0 4.8 1.8 virginica
## 140 6.9 3.1 5.4 2.1 virginica
## 141 6.7 3.1 5.6 2.4 virginica
## 142 6.9 3.1 5.1 2.3 virginica
## 143 5.8 2.7 5.1 1.9 virginica
## 144 6.8 3.2 5.9 2.3 virginica
## 145 6.7 3.3 5.7 2.5 virginica
## 146 6.7 3.0 5.2 2.3 virginica
## 147 6.3 2.5 5.0 1.9 virginica
## 148 6.5 3.0 5.2 2.0 virginica
## 149 6.2 3.4 5.4 2.3 virginica
## 150 5.9 3.0 5.1 1.8 virginica#describe data in exam
#Step 1: Normalize the data
#step2:covariance matrix
l=iris[1:4]
c=cov(l)#covariance matrix of iris data
c## Sepal.Length Sepal.Width Petal.Length Petal.Width
## Sepal.Length 0.6856935 -0.0424340 1.2743154 0.5162707
## Sepal.Width -0.0424340 0.1899794 -0.3296564 -0.1216394
## Petal.Length 1.2743154 -0.3296564 3.1162779 1.2956094
## Petal.Width 0.5162707 -0.1216394 1.2956094 0.5810063#Method-1 by princomb
k=princomp(l)#we get PCA
#Method-2 usind eigen vectors and eigen values
e=eigen(c)$vector
lambda=eigen(c)$value
e## [,1] [,2] [,3] [,4]
## [1,] 0.36138659 -0.65658877 -0.58202985 0.3154872
## [2,] -0.08452251 -0.73016143 0.59791083 -0.3197231
## [3,] 0.85667061 0.17337266 0.07623608 -0.4798390
## [4,] 0.35828920 0.07548102 0.54583143 0.7536574lambda## [1] 4.22824171 0.24267075 0.07820950 0.02383509e1=e[,1]
e2=e[,2]
e1## [1] 0.36138659 -0.08452251 0.85667061 0.35828920e2## [1] -0.65658877 -0.73016143 0.17337266 0.07548102summary(k)## Importance of components:
## Comp.1 Comp.2 Comp.3 Comp.4
## Standard deviation 2.0494032 0.49097143 0.27872586 0.153870700
## Proportion of Variance 0.9246187 0.05306648 0.01710261 0.005212184
## Cumulative Proportion 0.9246187 0.97768521 0.99478782 1.000000000#variance of 1st principal component
x1=c(24.9,25.4,22.9,24.9,24.3,24.5,21.2,25,19.2,24.3)
x2=c(84.2,77.2,79,83.2,85,84,89.1,86.2,65.2,84.2)
x3=c(66.1,54.9,53.2,63.5,66.9,63.2,70.2,66.2,54.2,67.2)
x4=c(124.34,98.79,71.88,115.20,154.4,143.56,175.2,164.3,69.5,145.5)
data=data.frame(x1,x2,x3,x4)
data## x1 x2 x3 x4
## 1 24.9 84.2 66.1 124.34
## 2 25.4 77.2 54.9 98.79
## 3 22.9 79.0 53.2 71.88
## 4 24.9 83.2 63.5 115.20
## 5 24.3 85.0 66.9 154.40
## 6 24.5 84.0 63.2 143.56
## 7 21.2 89.1 70.2 175.20
## 8 25.0 86.2 66.2 164.30
## 9 19.2 65.2 54.2 69.50
## 10 24.3 84.2 67.2 145.50#Method-1
j=princomp(data)
summary(j)## Importance of components:
## Comp.1 Comp.2 Comp.3 Comp.4
## Standard deviation 35.99774 3.533591330 2.1395462 1.327446993
## Proportion of Variance 0.98568 0.009497692 0.0034820 0.001340355
## Cumulative Proportion 0.98568 0.995177645 0.9986596 1.000000000