\documentclass{beamer}
%\usepackage{beamerthemeshadow}
%\usepackage{url,graphics,amsmath,pstricks,hyperref,times,color}
%\usepackage{graphicx}
\usepackage{Sweave}
%\usepackage{amscd}
%\usepackage[tableposition=top]{caption}
%\usepackage{ifthen}
% \usepackage[utf8]{inputenc}
\usepackage{amssymb, amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}



\def\R{\mathbb{R}}


\begin{document}
\SweaveOpts{concordance=TRUE}

\begin{frame}[fragile]
Suppose we have the following feature vectors:

<<echo = False>>=
x <- c(0, 1, 2) 

y1 <- c(1, 2, 0) 
y2 <- c(2, 1, 0) 
y3 <- c(1.5, .2, 0) 
y4 <- c(1, 1.5, 0) 
y5 <- c(0.5, 1.2, 0)


y1 <- c(2, 1, 0) 
y2 <- c(1.8, 1.1, 0) 
y3 <- c(1.5, .2, 0) 
y4 <- c(1, .6, 0) 
y5 <- c(0.8, .3, 0) 

Y<- rbind(y1, y2, y3, y4, y5)
columns <- c("f(0)", "f(1)", "f(2)")
colorY <- c("red", "blue", "purple", "chocolate", "green")
colnames(Y) <- columns 
rownames(Y) <- colorY 
Yc <- Y
Yc
@

We can plot these feature vectors (regardless of their length) in $\R^2$

\vskip -20pt

<<echo = False, fig=TRUE, height=4>>=
p1 <- cbind(x, y1)
p2 <- cbind(x, y2)
p3 <- cbind(x, y3)
p4 <- cbind(x, y4)
p5 <- cbind(x, y5)

plot(p1, type = "l", col="red3", xlab = "", ylab = "")
points(p1, cex= 2, pch=1, col="red3")
points(p2, type = "l", col="blue")
points(p2, cex= 2, pch=2, col="blue")
points(p3, type = "l", col="purple3")
points(p3, cex= 2, pch=3, col="purple3")
points(p4, type = "l", col="chocolate4")
points(p4, cex= 2, pch=4, col="chocolate4")
points(p5, type = "l", col="darkgreen")
points(p5, cex= 2, pch=5, col="darkgreen")
@
\end{frame}



\begin{frame}[fragile]
Let's take the mean of our feature vectors and plot it's graph in black:

<<echo = TRUE>>=
Y<- rbind(y1, y2, y3, y4, y5)
mean <- colMeans(Y)
XY <- cbind(x, mean)
t(XY)
@

\vskip -20pt
<<echo = FALSE, fig=TRUE, height=4>>=
plot(XY, type = "l", ylim=c(0,2), lwd=3)
points(p1, type = "l", col="red3")
points(p1, cex= 2, pch=1, col="red3")
points(p2, type = "l", col="blue")
points(p2, cex= 2, pch=2, col="blue")
points(p3, type = "l", col="purple3")
points(p3, cex= 2, pch=3, col="purple3")
points(p4, type = "l", col="chocolate4")
points(p4, cex= 2, pch=4, col="chocolate4")
points(p5, type = "l", col="darkgreen")
points(p5, cex= 2, pch=5, col="darkgreen")
@
\end{frame}

\begin{frame}[fragile]
To see how these functions (feature vectors) vary about the mean, we will subtract the mean vector from each feacture vector:



\begin{columns}
        \begin{column}{0.48\textwidth}
<<echo = FALSE, label=fig1plot, include=FALSE >>=
plot(XY, type = "l", ylim=c(0,2), lwd=3)
points(p1, type = "l", col="red3")
points(p1, cex= 2, pch=1, col="red3")
points(p2, type = "l", col="blue")
points(p2, cex= 2, pch=2, col="blue")
points(p3, type = "l", col="purple3")
points(p3, cex= 2, pch=3, col="purple3")
points(p4, type = "l", col="chocolate4")
points(p4, cex= 2, pch=4, col="chocolate4")
points(p5, type = "l", col="darkgreen")
points(p5, cex= 2, pch=5, col="darkgreen")
@

\begin{figure}
\begin{center}
<<label=fig1, fig=TRUE, echo=FALSE>>=
<<fig1plot>>
@
\end{center}
\caption{Raw Data + Mean}
\label{fig:fig1plot}
\end{figure}
  \end{column}
  
  \begin{column}{0.48\textwidth}
<<echo=FALSE, label=fig2plot, include=FALSE >>=
mp1 <- cbind(x, y1 - mean)
mp2 <- cbind(x, y2 - mean)
mp3 <- cbind(x, y3 - mean)
mp4 <- cbind(x, y4 - mean)
mp5 <- cbind(x, y5 - mean)
mpmean <- cbind(x, mean - mean)

plot(mp1, type = "l", col="red3", ylim=c(-1.3,1))
points(mp1, cex= 2, pch=1, col="red3")
points(mp2, type = "l", col="blue")
points(mp2, cex= 2, pch=2, col="blue")
points(mp3, type = "l", col="purple3")
points(mp3, cex= 2, pch=3, col="purple3")
points(mp4, type = "l", col="chocolate4")
points(mp4, cex= 2, pch=4, col="chocolate4")
points(mp5, type = "l", col="darkgreen")
points(mp5, cex= 2, pch=5, col="darkgreen")
points(mpmean, type = "l")
@

\begin{figure}
\begin{center}
<<label=fig2, fig=TRUE, echo=FALSE>>=
<<fig2plot>>
@
\end{center}
\caption{Mean Residuals}
\label{fig:fig2plot}
\end{figure}
  \end{column}
\end{columns}

@
\end{frame}



\begin{frame}[fragile]
We can also plot our feature vectors in $\R^n$ where $n$ is the number of features

<<echo = False, fig=TRUE, height=4>>=
Yc

library(dplyr)
Y1 <- data.frame(Y)
Y2 <- select(Y1, -X3)
plot(Y2[1,], cex= 2, pch=1, col="red3", xlim=c(0,2), ylim=c(0,1.5))
points(Y2[2,], cex= 2, pch=2, col="blue")
points(Y2[3,], cex= 2, pch=3, col="purple3")
points(Y2[4,], cex= 2, pch=4, col="chocolate4")
points(Y2[5,], cex= 2, pch=5, col="darkgreen")
points(Y2[6,], cex= 2, pch=10)
@
\end{frame}

\begin{frame}[fragile]
Let's translate these points by subracting off the mean:

\begin{columns}
        \begin{column}{0.48\textwidth}
<<echo = FALSE, label=fig3plot, include=FALSE >>=
plot(Y2[1,], cex= 2, pch=1, col="red3", xlim=c(0,2), ylim=c(0,1.2))
points(Y2[2,], cex= 2, pch=2, col="blue")
points(Y2[3,], cex= 2, pch=3, col="purple3")
points(Y2[4,], cex= 2, pch=4, col="chocolate4")
points(Y2[5,], cex= 2, pch=5, col="darkgreen")
points(Y2[6,], cex= 2, pch=10)
@

\begin{figure}
\begin{center}
<<label=fig3, fig=TRUE, echo=FALSE>>=
<<fig3plot>>
@
\end{center}
\caption{Raw Data + Mean}
\label{fig:fig3plot}
\end{figure}
  \end{column}
  
  \begin{column}{0.48\textwidth}
<<echo=FALSE, label=fig4plot, include=FALSE >>=
plot(Y2[1,] - mean, cex= 2, pch=1, col="red3", xlim=c(-1.42,0.58), ylim=c(-0.64,0.56))
points(Y2[2,]- mean, cex= 2, pch=2, col="blue")
points(Y2[3,]- mean, cex= 2, pch=3, col="purple3")
points(Y2[4,]- mean, cex= 2, pch=4, col="chocolate4")
points(Y2[5,]- mean, cex= 2, pch=5, col="darkgreen")
points(Y2[6,]- mean, cex= 2, pch=10)
@

\begin{figure}
\begin{center}
<<label=fig4, fig=TRUE, echo=FALSE>>=
<<fig4plot>>
@
\end{center}
\caption{Centered about the origin}
\label{fig:fig4plot}
\end{figure}
  \end{column}
\end{columns}
@
\end{frame}

\begin{frame}[fragile]
Covariance: $$Cov(X,Y) = \sum_{i=0}^N \frac{(x_i - \overline{x})(y_i - \overline{y})}{N}$$

<<echo=FALSE>>=
Y2
@
covariance matrix:
<<echo=FALSE>>=
  cov(Y2)
@

\end{frame}


\begin{frame}[fragile]
We can now apply PCA to our data:
\vskip 10pt
\begin{columns}
        \begin{column}{0.5\textwidth}
Covariance matrix:  
<<label=a, echo=FALSE>>=
cov(Y2)
E<-eigen(cov(Y2))
@

<<echo=FALSE, fig=TRUE, height=20,width=20>>=
plot(Y2[1,] - mean, cex= 2, pch=1, col="red3", xlim=c(-1,1), ylim=c(-1,1), asp=1)
points(Y2[2,]- mean, cex= 2, pch=2, col="blue")
points(Y2[3,]- mean, cex= 2, pch=3, col="purple3")
points(Y2[4,]- mean, cex= 2, pch=4, col="chocolate4")
points(Y2[5,]- mean, cex= 2, pch=5, col="darkgreen")
points(Y2[6,]- mean, cex= 2, pch=10)
arrows(0,0,E$values[1]*E$vectors[1,1],E$values[1]*E$vectors[2,1], col="red", lwd = 3)
arrows(0,0,E$values[2]*E$vectors[1,2],E$values[2]*E$vectors[2,2], col="orange", lwd = 3)
@
\end{column}
\begin{column}{0.6\textwidth}
Eigenvalues and Eigenvectors

<<echo=FALSE>>=
E
@
<<echo=TRUE>>=
E$values[1]/(E$values[1]+E$values[2])
sqrt(E$values)
@
@
\end{column}
\end{columns}
\end{frame}

\begin{frame}[fragile]
<<echo=FALSE, fig=TRUE, height=20,width=20>>=
plot(Y2[1,] - mean, cex= 4, pch=1, col="red3", xlim=c(-1,1), ylim=c(-1,1), asp=1)
points(Y2[2,]- mean, cex= 4, pch=2, col="blue")
points(Y2[3,]- mean, cex= 4, pch=3, col="purple3")
points(Y2[4,]- mean, cex= 4, pch=4, col="chocolate4")
points(Y2[5,]- mean, cex= 4, pch=5, col="darkgreen")
points(Y2[6,]- mean, cex= 4, pch=10)
arrows(0,0,E$values[1]*E$vectors[1,1],E$values[1]*E$vectors[2,1], col="red", lwd = 3)
arrows(0,0,E$values[2]*E$vectors[1,2],E$values[2]*E$vectors[2,2], col="orange", lwd = 3)
@
\end{frame}

\begin{frame}[fragile]
We can now apply PCA to our data:
\vskip 10pt
\begin{columns}
        \begin{column}{0.5\textwidth}
Covariance matrix:  
<<label=a, echo=FALSE>>=
cov(Y2)
E<-eigen(cov(Y2))
@

<<echo=FALSE, fig=TRUE, height=20,width=20>>=
plot(Y2[1,] - mean, cex= 2, pch=1, col="red3", xlim=c(-1,1), ylim=c(-1,1), asp=1)
points(Y2[2,]- mean, cex= 2, pch=2, col="blue")
points(Y2[3,]- mean, cex= 2, pch=3, col="purple3")
points(Y2[4,]- mean, cex= 2, pch=4, col="chocolate4")
points(Y2[5,]- mean, cex= 2, pch=5, col="darkgreen")
points(Y2[6,]- mean, cex= 2, pch=10)
arrows(0,0,E$values[1]*E$vectors[1,1],E$values[1]*E$vectors[2,1], col="red", lwd = 3)
arrows(0,0,E$values[2]*E$vectors[1,2],E$values[2]*E$vectors[2,2], col="orange", lwd = 3)
@
\end{column}
\begin{column}{0.6\textwidth}
Eigenvalues and Eigenvectors

<<echo=FALSE>>=
E
@
<<echo=TRUE>>=
E$values[1]/(E$values[1]+E$values[2])
sqrt(E$values)
@
@
\end{column}
\end{columns}
\end{frame}



\begin{frame}[fragile]
<<echo=TRUE>>=
Y3<- as.matrix(t(t(Y2) - c(mean[1], mean[2])))
prin_comp <- prcomp(Y3, scale=FALSE)
prin_comp
summary(prin_comp)
@
\end{frame}

\begin{frame}[fragile]
<<echo=FALSE, fig=TRUE>>=
plot(Y2[1,] - mean, cex= 2, pch=1, col="red3", xlim=c(-0.6,0.6), ylim=c(-0.6,0.6), asp=1)
points(Y2[2,]- mean, cex= 2, pch=2, col="blue")
points(Y2[3,]- mean, cex= 2, pch=3, col="purple3")
points(Y2[4,]- mean, cex= 2, pch=4, col="chocolate4")
points(Y2[5,]- mean, cex= 2, pch=5, col="darkgreen")
points(Y2[6,]- mean, cex= 2, pch=10)
arrows(0,0,E$values[1]*E$vectors[1,1],E$values[1]*E$vectors[2,1], col="red", lwd = 3)
arrows(0,0,E$values[2]*E$vectors[1,2],E$values[2]*E$vectors[2,2], col="orange", lwd = 3)

@
\end{frame}


\begin{frame}[fragile]
<<echo=FALSE>>=
prin_comp$sdev
prin_comp$rotation
prin_comp$x
t(t(Y2) - c(mean[1], mean[2]))
@
\end{frame}

\begin{frame}[fragile]

\begin{columns}
        \begin{column}{0.5\textwidth}
Covariance matrix:  
<<label=a, echo=FALSE>>=
cov(Y2)
E<-eigen(cov(Y2))
@

<<echo=FALSE, fig=TRUE, height=10,width=10>>=
plot(Y2[1,] - mean, cex= 2, pch=1, col="red3", xlim=c(-1,1), ylim=c(-1,1))
points(Y2[2,]- mean, cex= 2, pch=2, col="blue")
points(Y2[3,]- mean, cex= 2, pch=3, col="purple3")
points(Y2[4,]- mean, cex= 2, pch=4, col="chocolate4")
points(Y2[5,]- mean, cex= 2, pch=5, col="darkgreen")
points(Y2[6,]- mean, cex= 2, pch=10)
arrows(0,0,E$values[1]*E$vectors[1,1],E$values[1]*E$vectors[2,1], col="red", lwd = 3)
arrows(0,0,E$values[2]*E$vectors[1,2],E$values[2]*E$vectors[2,2], col="orange", lwd = 3)
@
\end{column}

\begin{column}{0.5\textwidth}
<<echo=FALSE, fig=TRUE>>=
# perform principal components analysis
pca <- prcomp(Y3) 

# project new data onto the PCA space
RotY<-as.matrix(Y3) %*% pca$rotation
#RotE<-E%*% pca$rotation  

plot(E$vectors[,1]*RotY[1,1], cex= 2, pch=1, col="red3", xlim=c(-3,3), ylim=c(-3,3))
points(E$vectors[,1]*RotY[2,1], cex= 2, pch=2, col="blue")
points(E$vectors[,1]*RotY[3,1], cex= 2, pch=3, col="purple3")
points(E$vectors[,1]*RotY[4,1], cex= 2, pch=4, col="chocolate4")
points(E$vectors[,1]*RotY[5,1], cex= 2, pch=5, col="darkgreen")
@
\end{column}
\end{columns}
\end{frame}

\begin{frame}[fragile]
<<echo=FALSE, fig=TRUE>>=
plot(RotY[1,], cex= 2, pch=1, col="red3", xlim=c(-3,3), ylim=c(-3,3))
points(RotY[2,], cex= 2, pch=2, col="blue")
points(RotY[3,], cex= 2, pch=3, col="purple3")
points(RotY[4,], cex= 2, pch=4, col="chocolate4")
points(RotY[5,], cex= 2, pch=5, col="darkgreen")
@
\end{frame}

\begin{frame}[fragile]
<<echo=FALSE, fig=TRUE>>=

D1<- E$vectors[,1]*RotY[1,1]
D2<- E$vectors[,1]*RotY[2,1]
D3<- E$vectors[,1]*RotY[3,1]
D4<- E$vectors[,1]*RotY[4,1]
D5<- E$vectors[,1]*RotY[5,1]

plot(D1[1], D1[2], cex= 2, pch=1, col="red3", xlim=c(-3,3), ylim=c(-3,3))
points(D2[1], D2[2], cex= 2, pch=2, col="blue")
points(D3[1], D3[2], cex= 2, pch=3, col="purple3")
points(D4[1], D4[2], cex= 2, pch=4, col="chocolate4")
points(D5[1], D5[2], cex= 2, pch=5, col="darkgreen")
@
\end{frame}


\begin{frame}[fragile]
Suppose we have the following feature vectors:

<<echo = False>>=
x <- c(0, 1, 2) 

ny1 <- cbind(t(D1),0)
ny2 <- cbind(t(D2),0)
ny3 <- cbind(t(D3),0)
ny4 <- cbind(t(D4),0)
ny5 <- cbind(t(D5),0)

nY<- rbind(ny1, ny2, ny3, ny4, ny5)
nYc <- cbind(nY, colorY)
nYc
@

We can plot these feature vectors (regardless of their length) in $\R^2$

\vskip -20pt

<<echo = False, fig=TRUE, height=4>>=
np1 <- cbind(x, t(ny1))
np2 <- cbind(x, t(ny2))
np3 <- cbind(x, t(ny3))
np4 <- cbind(x, t(ny4))
np5 <- cbind(x, t(ny5))

plot(np1, type = "l", col="red3", xlab = "", ylim = c(-1, 1))
points(np1, cex= 2, pch=1, col="red3")
points(np2, type = "l", col="blue")
points(np2, cex= 2, pch=2, col="blue")
points(np3, type = "l", col="purple3")
points(np3, cex= 2, pch=3, col="purple3")
points(np4, type = "l", col="chocolate4")
points(np4, cex= 2, pch=4, col="chocolate4")
points(np5, type = "l", col="darkgreen")
points(np5, cex= 2, pch=5, col="darkgreen")
@
\end{frame}


\begin{frame}[fragile]
Raw Data
\vskip -20pt
<<echo = False, fig=TRUE, height=4>>=
plot(p1, type = "l", col="red3", xlab = "", ylab = "")
points(p1, cex= 2, pch=1, col="red3")
points(p2, type = "l", col="blue")
points(p2, cex= 2, pch=2, col="blue")
points(p3, type = "l", col="purple3")
points(p3, cex= 2, pch=3, col="purple3")
points(p4, type = "l", col="chocolate4")
points(p4, cex= 2, pch=4, col="chocolate4")
points(p5, type = "l", col="darkgreen")
points(p5, cex= 2, pch=5, col="darkgreen")
@
\vskip -20pt
Mean Residual
\vskip -20pt
<<echo = False, fig=TRUE, height=4>>=
plot(mp1, type = "l", col="red3", ylim=c(-1,1))
points(mp1, cex= 2, pch=1, col="red3")
points(mp2, type = "l", col="blue")
points(mp2, cex= 2, pch=2, col="blue")
points(mp3, type = "l", col="purple3")
points(mp3, cex= 2, pch=3, col="purple3")
points(mp4, type = "l", col="chocolate4")
points(mp4, cex= 2, pch=4, col="chocolate4")
points(mp5, type = "l", col="darkgreen")
points(mp5, cex= 2, pch=5, col="darkgreen")
points(mpmean, type = "l")
@
\end{frame}


\begin{frame}[fragile]
Mean Residual
\vskip -20pt
<<echo = False, fig=TRUE, height=4>>=
plot(mp1, type = "l", col="red3", ylim=c(-1,1))
points(mp1, cex= 2, pch=1, col="red3")
points(mp2, type = "l", col="blue")
points(mp2, cex= 2, pch=2, col="blue")
points(mp3, type = "l", col="purple3")
points(mp3, cex= 2, pch=3, col="purple3")
points(mp4, type = "l", col="chocolate4")
points(mp4, cex= 2, pch=4, col="chocolate4")
points(mp5, type = "l", col="darkgreen")
points(mp5, cex= 2, pch=5, col="darkgreen")
points(mpmean, type = "l")
@
\vskip -20pt
PC1
\vskip -20pt
<<echo = False, fig=TRUE, height=4>>=
np1 <- cbind(x, t(ny1))
np2 <- cbind(x, t(ny2))
np3 <- cbind(x, t(ny3))
np4 <- cbind(x, t(ny4))
np5 <- cbind(x, t(ny5))

plot(np1, type = "l", col="red3", xlab = "", ylim = c(-1,1))
points(np1, cex= 2, pch=1, col="red3")
points(np2, type = "l", col="blue")
points(np2, cex= 2, pch=2, col="blue")
points(np3, type = "l", col="purple3")
points(np3, cex= 2, pch=3, col="purple3")
points(np4, type = "l", col="chocolate4")
points(np4, cex= 2, pch=4, col="chocolate4")
points(np5, type = "l", col="darkgreen")
points(np5, cex= 2, pch=5, col="darkgreen")
@
\end{frame}


\begin{frame}[fragile]
<<echo=FALSE, fig=TRUE>>=

D1<- E$vectors[,2]*RotY[1,2]
D2<- E$vectors[,2]*RotY[2,2]
D3<- E$vectors[,2]*RotY[3,2]
D4<- E$vectors[,2]*RotY[4,2]
D5<- E$vectors[,2]*RotY[5,2]

plot(D1[1], D1[2], cex= 2, pch=1, col="red3", xlim=c(-3,3), ylim=c(-3,3))
points(D2[1], D2[2], cex= 2, pch=2, col="blue")
points(D3[1], D3[2], cex= 2, pch=3, col="purple3")
points(D4[1], D4[2], cex= 2, pch=4, col="chocolate4")
points(D5[1], D5[2], cex= 2, pch=5, col="darkgreen")
@
\end{frame}


\begin{frame}[fragile]
Suppose we have the following feature vectors:

<<echo = False>>=
x <- c(0, 1, 2) 

ny1 <- cbind(t(D1),0)
ny2 <- cbind(t(D2),0)
ny3 <- cbind(t(D3),0)
ny4 <- cbind(t(D4),0)
ny5 <- cbind(t(D5),0)

nY<- rbind(ny1, ny2, ny3, ny4, ny5)
nYc <- cbind(nY, colorY)
nYc
@

We can plot these feature vectors (regardless of their length) in $\R^2$

\vskip -20pt

<<echo = False, fig=TRUE, height=4>>=
snp1 <- cbind(x, t(ny1))
snp2 <- cbind(x, t(ny2))
snp3 <- cbind(x, t(ny3))
snp4 <- cbind(x, t(ny4))
snp5 <- cbind(x, t(ny5))

plot(snp1, type = "l", col="red3", xlab = "", ylim = c(-1, 1))
points(snp1, cex= 2, pch=1, col="red3")
points(snp2, type = "l", col="blue")
points(snp2, cex= 2, pch=2, col="blue")
points(snp3, type = "l", col="purple3")
points(snp3, cex= 2, pch=3, col="purple3")
points(snp4, type = "l", col="chocolate4")
points(snp4, cex= 2, pch=4, col="chocolate4")
points(snp5, type = "l", col="darkgreen")
points(snp5, cex= 2, pch=5, col="darkgreen")
@
\end{frame}


\begin{frame}[fragile]
PC1 + PC2
\vskip -20pt
<<echo = False, fig=TRUE, height=4>>=
plot(snp1 + np1, type = "l", col="red3", xlab = "", ylim = c(-1, 1))
points(snp1 + np1, cex= 2, pch=1, col="red3")
points(snp2 + np2, type = "l", col="blue")
points(snp2 + np2, cex= 2, pch=2, col="blue")
points(snp3 + np3, type = "l", col="purple3")
points(snp3 + np3, cex= 2, pch=3, col="purple3")
points(snp4 + np4, type = "l", col="chocolate4")
points(snp4 + np4, cex= 2, pch=4, col="chocolate4")
points(snp5 + np5, type = "l", col="darkgreen")
points(snp5 + np5, cex= 2, pch=5, col="darkgreen")
@

\vskip -20pt
Mean Residual
\vskip -20pt
<<echo = False, fig=TRUE, height=4>>=
plot(mp1, type = "l", col="red3", ylim=c(-1,1))
points(mp1, cex= 2, pch=1, col="red3")
points(mp2, type = "l", col="blue")
points(mp2, cex= 2, pch=2, col="blue")
points(mp3, type = "l", col="purple3")
points(mp3, cex= 2, pch=3, col="purple3")
points(mp4, type = "l", col="chocolate4")
points(mp4, cex= 2, pch=4, col="chocolate4")
points(mp5, type = "l", col="darkgreen")
points(mp5, cex= 2, pch=5, col="darkgreen")
points(mpmean, type = "l")
@
\end{frame}

\begin{frame}[fragile]
<<echo=FALSE, fig=TRUE>>=
plot(Y2[1,] - mean, cex= 2, pch=1, col="red3", xlim=c(-0.6,0.6), ylim=c(-0.6,0.6), asp=1)
points(Y2[2,]- mean, cex= 2, pch=2, col="blue")
points(Y2[3,]- mean, cex= 2, pch=3, col="purple3")
points(Y2[4,]- mean, cex= 2, pch=4, col="chocolate4")
points(Y2[5,]- mean, cex= 2, pch=5, col="darkgreen")
points(Y2[6,]- mean, cex= 2, pch=10)
arrows(0,0,E$values[1]*E$vectors[1,1],E$values[1]*E$vectors[2,1], col="red", lwd = 3)
arrows(0,0,E$values[2]*E$vectors[1,2],E$values[2]*E$vectors[2,2], col="orange", lwd = 3)

@
\end{frame}

\begin{frame}[fragile]
PC1
\vskip -20pt
<<echo = False, fig=TRUE, height=4>>=
plot(np1, type = "l", col="red3", xlab = "", ylim = c(-1,1))
points(np1, cex= 2, pch=1, col="red3")
points(np2, type = "l", col="blue")
points(np2, cex= 2, pch=2, col="blue")
points(np3, type = "l", col="purple3")
points(np3, cex= 2, pch=3, col="purple3")
points(np4, type = "l", col="chocolate4")
points(np4, cex= 2, pch=4, col="chocolate4")
points(np5, type = "l", col="darkgreen")
points(np5, cex= 2, pch=5, col="darkgreen")
@
PC2
\vskip -20pt
<<echo = False, fig=TRUE, height=4>>=
plot(snp1, type = "l", col="red3", xlab = "", ylim = c(-1, 1))
points(snp1, cex= 2, pch=1, col="red3")
points(snp2, type = "l", col="blue")
points(snp2, cex= 2, pch=2, col="blue")
points(snp3, type = "l", col="purple3")
points(snp3, cex= 2, pch=3, col="purple3")
points(snp4, type = "l", col="chocolate4")
points(snp4, cex= 2, pch=4, col="chocolate4")
points(snp5, type = "l", col="darkgreen")
points(snp5, cex= 2, pch=5, col="darkgreen")
@
\end{frame}



\end{document}