Vectors and Matrices


split.screen(c(1,2))	# Divide the graphics device into two parts horizontally
par(bg="white", pty="s")	# set background color (ensures proper erasure)
screen(2)
par(pty="s")				# and square plot regions

# Vector representation

a <- c(2, 1)

screen(1)
plot(a[1],a[2], xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='x', ylab='y')	
			# point representation of a vector
abline(h=0,v=0,col="grey")				# add coordinate axes
title("Point Representation of (2, 1)")

screen(2)
plot(0, 0, xlim=c(-3,3), ylim=c(-3,3), type="n", xlab='x', ylab='y')	
			# Empty plotting frame
arrows(0, 0, a[1], a[2])	# ray representation of a vector
abline(h=0,v=0,col="grey")
title("Ray Representation of (2, 1)")

# Data as a collection of vectors

X<-read.table("H:/S5600/Data sets/ASCII/Ch02/WOMEN.txt")
X
apply(X, 2, mean)
apply(X, 2, sd)

screen(2)
screen(1)
plot(X, pch=16, xlab="Height", ylab="Weight")
title("Raw Data (Women's Heights and Weights)")


Xd <- scale(X, scale=F)		# mean-centered data (deviations)
Xd
apply(Xd, 2, mean)
apply(Xd, 2, sd)

screen(2)
plot(Xd, pch=16, xlab="Height", ylab="Weight")
title("Mean-Centered Data (Women's Heights and Weights)")
abline(h=0,v=0,col="grey")


Xs <- scale(X)		# standardized data
Xs
apply(Xs, 2, mean)
apply(Xs, 2, sd)

screen(2)
plot(Xs, pch=16, xlab="Height", ylab="Weight")
title("Standardized Data (Women's Heights and Weights)")
abline(h=0,v=0,col="grey")

# Scalar multiplication

a


screen(2)
screen(1)

plot(a[1],a[2], xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='x', ylab='y')	
arrows(0, 0, a[1], a[2])
abline(h=0,v=0,col="grey")
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)
title("Representation of (2, 1)")

r<- .6
ra <- r*a
ra

screen(2)

plot(ra[1],ra[2], xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='x', ylab='y')	
arrows(0, 0, ra[1], ra[2])
abline(h=0,v=0,col="grey")
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)
title("Point Representation of 0.6*(2, 1)")


screen(2)
screen(1)

plot(Xs, pch=16, xlab="Height", ylab="Weight", xlim=c(-2,2), ylim=c(-2,2))
title("Standardized Data")
abline(h=0,v=0,col="grey")

screen(2)
plot(r*Xs, pch=16, xlab="Height", ylab="Weight", xlim=c(-2,2), ylim=c(-2,2))
title("Rescaled Standardized Data")
abline(h=0,v=0,col="grey")

# Vector multiplication: projection

w1 <- matrix(rep(1,2),nc=1)
w1
w1^2
lw1 <- sqrt(sum(w1^2))	# length of w1, ||w1||

w1 <- w1/lw1	# unit length
w1
sqrt(sum(w1^2))


a
aw1 <- a %*% w1
aw1

screen(2)
screen(1)

plot(a[1],a[2], xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='x', ylab='y')	
abline(h=0,v=0,col="grey")
arrows(0, 0, w1[1], w1[2])
abline(0, w1[2]/w1[1])

points(aw1*w1[1], aw1*w1[2], pch=16, col="red")
segments(a[1],a[2],aw1*w1[1],aw1*w1[2],lty=3)
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)

w2 <- matrix(c(-1, 1),nc=1)
w2
w2^2
lw2 <- sqrt(sum(w2^2))	

w2 <- w2/lw2	# unit length
w2
sqrt(sum(w2^2))


a
aw2 <- a %*% w2
aw2

screen(2)

plot(a[1],a[2], xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='x', ylab='y')	
abline(h=0,v=0,col="grey")
arrows(0, 0, w2[1], w2[2])
abline(0, w2[2]/w2[1])

points(aw2*w2[1], aw2*w2[2], pch=16, col="green")
segments(a[1],a[2],aw2*w2[1],aw2*w2[2],lty=3)
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)



Xs
Z1 <- Xs %*% w1

screen(2)
screen(1)

plot(Xs, xlim=c(-2,2), ylim=c(-2,2), pch=16, xlab='x', ylab='y')	
abline(h=0,v=0,col="grey")
arrows(0, 0, w1[1], w1[2])
abline(0, w1[2]/w1[1])

points(Z1*w1[1], Z1*w1[2], pch=16, col="red")
segments(Xs[,1],Xs[,2],Z1*w1[1],Z1*w1[2],lty=3)
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)


Z2 <- Xs %*% w2

screen(2)

plot(Xs, xlim=c(-2,2), ylim=c(-2,2), pch=16, xlab='x', ylab='y')	
abline(h=0,v=0,col="grey")
arrows(0, 0, w2[1], w2[2])
abline(0, w2[2]/w2[1])

points(Z2*w2[1], Z2*w2[2], pch=16, col="green")
segments(Xs[,1],Xs[,2],Z2*w2[1],Z2*w2[2],lty=3)
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)

# Orthogonal vectors

t(w1) %*% w2

screen(2)
screen(1)

plot(0, 0, xlim=c(-3,3), ylim=c(-3,3), type="n", xlab='x', ylab='y')	
arrows(0, 0, w1[1], w1[2])
arrows(0, 0, w2[1], w2[2])
abline(h=0,v=0,col="grey")
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)

# Matrix Multiplication: Rotation

screen(1)

plot(Xs, xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='X1', ylab='X2')	
abline(h=0,v=0,col="grey")
arrows(0, 0, w1[1], w1[2])
abline(0, w1[2]/w1[1])
arrows(0, 0, w2[1], w2[2])
abline(0, w2[2]/w2[1])


points(Z1*w1[1], Z1*w1[2], col="red")
segments(Xs[,1],Xs[,2],Z1*w1[1],Z1*w1[2],lty=3)
points(Z2*w2[1], Z2*w2[2], col="green")
segments(Xs[,1],Xs[,2],Z2*w2[1],Z2*w2[2],lty=3)
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)

screen(2)

W <- cbind(w1, w2)
Z <- Xs %*% W

plot(Z, xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='Z1', ylab='Z2')
abline(h=0,v=0)
arrows(0,0,1,0)
arrows(0,0,0,1)
points(Z1, rep(0,20), col="red")
segments(Z1,rep(0,20),Z1,Z2,lty=3)
points(rep(0,20), Z2, col="green")
segments(rep(0,20),Z2,Z1,Z2,lty=3)
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)


screen(1)

plot(Xs, xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='X1', ylab='X2')	
abline(h=0,v=0,col="grey")
arrows(0, 0, w1[1], w1[2])
arrows(0, 0, w2[1], w2[2])
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)

screen(2)

plot(Z, xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='Z1', ylab='Z2')
abline(h=0,v=0,col="grey")
arrows(0,0,1,0)
arrows(0,0,0,1)
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)

screen(1)

plot(Xs, xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='X1', ylab='X2')	
abline(h=0,v=0,col="grey")
arrows(0, 0, w1[1], w1[2])
arrows(0, 0, w2[1], w2[2])
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)

ZWt <- Z %*% t(W)
points(ZWt, col="red")

r1 <- c(cos(pi/3),-sin(pi/3))		# rotate axes pi/3 clockwise
r2 <- c(sin(pi/3),cos(pi/3))
R <- cbind(r1,r2)

screen(2)
screen(1)

plot(Xs, xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='X1', ylab='X2')	
abline(h=0,v=0,col="grey")
arrows(0, 0, r1[1], r1[2])
arrows(0, 0, r2[1], r2[2])
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)

screen(2)

plot(Xs %*% R, xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='Z1', ylab='Z2')
abline(h=0,v=0,col="grey")
arrows(0,0,1,0)
arrows(0,0,0,1)
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)


# Matrix Multiplication: Streching and Shrinking

sd(Z)

Dinv <- diag(1/sd(Z))
Dinv

Zs <- Z %*% Dinv

screen(1)

plot(Z, xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='Z1', ylab='Z2')
abline(h=0,v=0,col="grey")
arrows(0,0,1,0)
arrows(0,0,0,1)
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)

screen(2)

plot(Zs, xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='Z1', ylab='Z2')
abline(h=0,v=0,col="grey")
arrows(0,0,1,0)
arrows(0,0,0,1)
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)


# Singular Value Decomposition

D <- diag(sd(Z))
D %*% Dinv

screen(2)
screen(1)

plot(Xs, xlim=c(-3,3), ylim=c(-3,3), pch=16, xlab='X1', ylab='X2')	
abline(h=0,v=0,col="grey")
arrows(0, 0, w1[1], w1[2])
arrows(0, 0, w2[1], w2[2])
symbols(rep(0,3),rep(0,3),circ=1:3,fg="grey",add=T,inches=F)

X2 <- Zs %*% D %*% t(W)

points(X2, col="red")

Xsvd <- svd(Xs)
Xsvd

U<-Xsvd$u

screen(2)
plot(U, pch=16, xlab='U1', ylab='U2')
abline(h=0,v=0,col="grey")

cov(U)

t(U) %*% U

# Matrix Computation of the covariance Matrix

Xd	# Start with mean-centered matrix

cov(Xd)

n <- nrow(Xd)

S <- 1/(n-1) * t(Xd) %*% Xd
S

# Matrix Determinant

det(S)

screen(2)
screen(1)

plot(0, 0, xlim=c(0, 290), ylim=c(0,290), type="n", xlab='S1', ylab='S2')	
abline(h=0,v=0,col="grey")
polygon(c(0,S[1,1],S[1,1]+S[1,2],S[1,2]),c(0,S[2,1],S[2,1]+S[2,2],S[2,2]),col="red")
arrows(0, 0, S[1,1], S[2,1])
arrows(0, 0, S[1,2], S[2,2])
arrows(S[1,1], S[2,1], S[1,1]+S[1,2], S[2,1]+S[2,2])
arrows(S[1,2], S[2,2], S[1,1]+S[1,2], S[2,1]+S[2,2])

Xd.svd <- svd(Xd)

d <- Xd.svd$d
d

D <- diag(d)
D

SD <- 1/(n-1) * t(D) %*% D
SD

det(SD)

prod(d^2/(n-1))

screen(2)

plot(0, 0, xlim=c(0, 290), ylim=c(0,290), type="n", xlab='S1', ylab='S2')	
abline(h=0,v=0,col="grey")
polygon(c(0,SD[1,1],SD[1,1]+SD[1,2],SD[1,2]),c(0,SD[2,1],SD[2,1]+SD[2,2],SD[2,2]),col="red")
arrows(0, 0, SD[1,1], SD[2,1])
arrows(0, 0, SD[1,2], SD[2,2])
arrows(SD[1,1], SD[2,1], SD[1,1]+SD[1,2], SD[2,1]+SD[2,2])
arrows(SD[1,2], SD[2,2], SD[1,1]+SD[1,2], SD[2,1]+SD[2,2])



Screen(2)
screen(1)	# Rescale for better visibility

plot(0, 0, xlim=c(0, 55), ylim=c(0,290), type="n", xlab='S1', ylab='S2')	
abline(h=0,v=0,col="grey")
polygon(c(0,S[1,1],S[1,1]+S[1,2],S[1,2]),c(0,S[2,1],S[2,1]+S[2,2],S[2,2]),col="red")
arrows(0, 0, S[1,1], S[2,1])
arrows(0, 0, S[1,2], S[2,2])
arrows(S[1,1], S[2,1], S[1,1]+S[1,2], S[2,1]+S[2,2])
arrows(S[1,2], S[2,2], S[1,1]+S[1,2], S[2,1]+S[2,2])


screen(2)

plot(0, 0, xlim=c(0, 255), ylim=c(0,3), type="n", xlab='S1', ylab='S2')	
abline(h=0,v=0,col="grey")
polygon(c(0,SD[1,1],SD[1,1]+SD[1,2],SD[1,2]),c(0,SD[2,1],SD[2,1]+SD[2,2],SD[2,2]),col="red")
arrows(0, 0, SD[1,1], SD[2,1])
arrows(0, 0, SD[1,2], SD[2,2])
arrows(SD[1,1], SD[2,1], SD[1,1]+SD[1,2], SD[2,1]+SD[2,2])
arrows(SD[1,2], SD[2,2], SD[1,1]+SD[1,2], SD[2,1]+SD[2,2])