#excercise 8.16 page 475

# Variable explanation

# x1= Bluegill
# x2= Black crappie
# x3= Smallmouth bass
# x4= Largemouth bass
# x5= Walleye
# x6= Northern pike


#Correlation matrix of the data


#read the given matrix from the textbook
Corr <- matrix(scan(n=36),nrow=6, byrow=TRUE)
1.0000  0.4919  0.2636  0.4653 -0.2277  0.0652
0.4919  1.0000  0.3217  0.3506 -0.1917  0.2045
0.2636  0.3217  1.0000  0.4108  0.0647  0.2493
0.4653  0.3506  0.4108  1.0000 -0.2249  0.2293
-0.2277 -0.1917  0.0647 -0.2249  1.0000 -0.2144
0.0652  0.2045  0.2493  0.2293 -0.2144  1.0000




colnames(Corr)=c('X1','X2','X3','X4','X5','X6')
rownames(Corr)=c('X1','X2','X3','X4','X5','X6')

# correlations between members 
# of the centrarchid family (x1 to x4) are positive, 
# whereas all correlations between these 
# and x5 (Walleye) are either negative or very small. 
# So x5 is not in the group with the centrarchids (x1 to x4).


#Principal component variances and direction vectors
#Principal component analysis (PCA) performed on x1-x6 variables
(PCA=eigen(Corr))

#set names to the vector of eigen values
names(PCA$values)=c('X1','X2','X3','X4','X5','X6')

#set column names to the vector of eigen vectors
#set row names to the vector of eigen vectors
colnames(PCA$vectors)=c('X1','X2','X3','X4','X5','X6')
rownames(PCA$vectors)=c('X1','X2','X3','X4','X5','X6')

#PC1= -0.4744514*X1+ (-0.4730439)*X2 + (-0.3950955)X3 +(-0.4954348)*X4 +  0.2553558*X5 +(-0.2904526)*X6


# The first eigenvector shows that the direction of largest variation is 
# pointing to the common variation in the centrarchids (x1-x4),
# with a lesser weight on pike (X6) (same sign), 
# and opposite weight on walleye X5. 
# This is confirming the interpretation of the correlation matrix itself. 
# The second eigenvector focuses on the walleye catch (-0.80680989); the third, 
# on northern pike (-0.808772699). 
# The others are more difficult to interpret. 


#scree plot 
PC.variance=PCA$values

(PC.prop=PC.variance/sum(PC.variance))


plot(1:length(PC.variance),PC.variance,main="Scree Plot for game fish - all 6 variables",
     xlab="Principal Component Number", ylab="Principal Component Variance",type="b")






#now perform PCA on the first four variables


#Correlation matrix for the first four variables.
Corr4 <- Corr[1:4,1:4]


#Principal component variances and direction vectors.
(PCA1=eigen(Corr4))

#set names to the vector of eigen values
names(PCA1$values)=c('X1','X2','X3','X4')

#set column names to the vector of eigen vectors
#set row names to the vector of eigen vectors
colnames(PCA1$vectors)=c('X1','X2','X3','X4')
rownames(PCA1$vectors)=c('X1','X2','X3','X4')

PCA1
#scree plot 
PC1.variance=PCA1$values

(PC1.prop=PC1.variance/sum(PC1.variance))

plot(1:length(PC1.variance),PC1.variance,main="Scree Plot for game fish - first four variables",
     xlab="Principal Component Number", ylab="Principal Component Variance",type="b")


#Scree plot shows substantial decline in the eigenvalues from the first 
#to the remaining three. 
#The first eigenvector shows that the catches of these four 
#centrarchid species tend to vary together (all values positive and close to each other). 
#The second eigenvector exhibits a tendency for 
#smallmouth bass (X3, large negative number (-0.7602768 )) to vary separately
#from the others. 
#Smallmouth bass are arguably the most popular of these four species.
#The other eigen vectors are not as easily interpreted, but again, 
#are beyond the elbow in the scree plot.