PCA

The Iris flower data set is a multivariate data set introduced by the British statistician. The data was collected to quantify the morphologic variation of Iris flowers of three related species. The data set consists of 50 samples from each of three species of Iris (Iris Setosa, Iris virginica, and Iris versicolor). Four features were measured from each sample: the length and the width of the sepals and petals, in centimeters.

The dataset contains a set of 150 records under 5 attributes - Petal Length, Petal Width, Sepal Length, Sepal width and Class(Species).

data("iris")
head(iris)

str(iris)

'data.frame':   150 obs. of  5 variables:
 $ Sepal.Length: num  5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ...
 $ Sepal.Width : num  3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ...
 $ Petal.Length: num  1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ...
 $ Petal.Width : num  0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ...
 $ Species     : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...

any(is.na(iris))

[1] FALSE

summary(iris)

  Sepal.Length    Sepal.Width     Petal.Length    Petal.Width          Species  
 Min.   :4.300   Min.   :2.000   Min.   :1.000   Min.   :0.100   setosa    :50  
 1st Qu.:5.100   1st Qu.:2.800   1st Qu.:1.600   1st Qu.:0.300   versicolor:50  
 Median :5.800   Median :3.000   Median :4.350   Median :1.300   virginica :50  
 Mean   :5.843   Mean   :3.057   Mean   :3.758   Mean   :1.199                  
 3rd Qu.:6.400   3rd Qu.:3.300   3rd Qu.:5.100   3rd Qu.:1.800                  
 Max.   :7.900   Max.   :4.400   Max.   :6.900   Max.   :2.500                  

partation data

set.seed(123)
ir<- sample(2,nrow(iris),replace = TRUE, prob = c(0.80,0.20))  # 80% for traning and 20% for testing the data

train <- iris[ir == 1,]
test <- iris[ir == 2,]

scatter plot and correlation coefficents.

library(psych)
pairs.panels(train[,-5],gap = 0, 
             bg = c("blue", "green", "red")[train$Species],
             pch=21)

high corelations among independent veriables lead to “multicollinearity” problems.

Let’s perform PCA : Principle Component Analysis

pca<- prcomp(train[,-5], center = TRUE, scale. = TRUE)
attributes(pca)

$names
[1] "sdev"     "rotation" "center"   "scale"    "x"       

$class
[1] "prcomp"

pca$center

Sepal.Length  Sepal.Width Petal.Length  Petal.Width 
    5.826446     3.021488     3.759504     1.201653 

mean(train$Sepal.Length)

[1] 5.826446

pca$scale

Sepal.Length  Sepal.Width Petal.Length  Petal.Width 
   0.8105316    0.4270454    1.7377945    0.7557539 

sd(train$Sepal.Length)

[1] 0.8105316

print(pca)

Standard deviations (1, .., p=4):
[1] 1.7237041 0.9309494 0.3749824 0.1468522

Rotation (n x k) = (4 x 4):
                    PC1         PC2        PC3        PC4
Sepal.Length  0.5201832 -0.37206837  0.7292637  0.2432058
Sepal.Width  -0.2937566 -0.92188195 -0.2184802 -0.1269139
Petal.Length  0.5747475 -0.03477237 -0.1603396 -0.8017153
Petal.Width   0.5592690 -0.10241915 -0.6282771  0.5310334

each principle components are normilized linear combination of original variables which are listed here.

summary(pca)

Importance of components:
                          PC1    PC2     PC3     PC4
Standard deviation     1.7237 0.9309 0.37498 0.14685
Proportion of Variance 0.7428 0.2167 0.03515 0.00539
Cumulative Proportion  0.7428 0.9595 0.99461 1.00000

orthogonality of pca

pairs.panels(pca$x, gap = 0, bg=c("blue","green", "Red")[train$Species],
             pch=21)

Bi-plot

library(devtools)
library(factoextra)
res.pca <- prcomp(train[, -5],  scale = TRUE)

fviz_pca_biplot(): Biplot of individuals of variables

fviz_pca_biplot(res.pca)

fviz_pca_biplot(res.pca, label ="var")

fviz_pca_biplot(res.pca, label ="ind")

fviz_pca_biplot(res.pca, label ="var", col.ind="cos2") +
       theme_minimal()

fviz_pca_biplot(res.pca, label="var",
               select.ind = list(contrib = 20))

prediction with PCA

traning <- predict(pca,train)
traning <- data.frame(traning,train[5])
testing <- predict(pca,test)
testing <- data.frame(testing,test$Species)

multinomial Logstic Regression with Frist Two PCAs

model1<- multinom(Species~ PC1+PC2,data = traning)

# weights:  12 (6 variable)
initial  value 132.932087 
iter  10 value 19.674294
iter  20 value 17.413814
iter  30 value 17.299839
iter  40 value 17.297242
iter  50 value 17.296292
final  value 17.294539 
converged

summary(model1)

Call:
multinom(formula = Species ~ PC1 + PC2, data = traning)

Coefficients:
           (Intercept)      PC1      PC2
versicolor   10.584590 15.26832 4.019053
virginica     3.007363 21.95663 4.228617

Std. Errors:
           (Intercept)      PC1      PC2
versicolor    273.3561 177.9735 223.6350
virginica     273.3636 177.9818 223.6359

Residual Deviance: 34.58908 
AIC: 46.58908 

confusion matrix and missclassification error - train

prd <- predict(model1, traning)
table <- table(prd, traning$Species)
table

            
prd          setosa versicolor virginica
  setosa         39          0         0
  versicolor      0         38         4
  virginica       0          5        35

calculate misclassification error

1- sum(diag(table))/sum(table)

[1] 0.07438017

misclassification error about 7.7%

confusion matrix and missclassification error - test

testing <- data.frame(testing,test[5])
prd1 <- predict(model1, testing)
table1 <- table(prd1, testing$Species)
table1

            
prd1         setosa versicolor virginica
  setosa         11          0         0
  versicolor      0          7         2
  virginica       0          0         9

1- sum(diag(table1))/sum(table1)

[1] 0.06896552

misclassification error for testing data is about 6.8%