#Problem 1
Y=c(2256,2340,2426,2293,2330,2368,2250,2409,2364,2379,2440,2364,2404,2317,2309,2328)
x0=rep(1,16)
x1=c(80,93,100,82,90,99,81,96,94,93,97,95,100,85,86,87)
x2=c(8,9,10,12,11,8,8,10,12,11,13,11,8,12,9,12)
X=cbind(x0,x1,x2)
X%*%t(X)
XTX=t(X)%*%X
IXTX=solve(XTX)
P=X%*%IXTX%*%t(X)
hatbeta=IXTX%*%t(X)%*%Y
X%*%hatbeta
P%*%Y
hate=P%*%y
res=y-hate
x=c(1,80,14)
haty=x%*%hatbeta


#Problem 4

install.packages("splines")
library(splines)
help(splines)
library(help = "splines")
help(bs)
x=seq(0,1,by=0.01)

BS=bs(x,knots=c(0.5),intercept=TRUE)
plot(x,BS[,1])
plot(x,pmax(0,(1-2*x)^3)) #pointwise maximum
dev.set(2)
plot(x,pmax(0,(1-2*x)^3),col="red")

pdf("BSplines.pdf")

Colors=c("red","blue","green","brown","black")
plot(x,BS[,1], col=Colors[1], type='l',ylab="",lwd=2)
for(i in 2:5)
{
  lines(x,BS[,i], col=Colors[i],lwd=2)
}

dev.off()