polynomial2.R

#
# What Polynomial?
#
data <- read.csv("sparrow.csv")
attach(data)
plot(ppha,pairs,xlab="Pedestrians per ha per min",ylab="Breeding pairs per ha")

res3 <- lm(pairs~ppha+I(ppha^2)+I(ppha^3))
summary(res3)
res3b <- lm(pairs~poly(ppha,3))
summary(res3b)
plot(ppha,pairs,xlab="Pedestrians per ha per min",ylab="Breeding pairs per ha")
nd <- data.frame(ppha=seq(min(ppha),max(ppha),(max(ppha)-min(ppha))/250))
matlines(nd,predict(res3,nd),type="l",col="red",lwd=8)
matlines(nd,predict(res3b,nd),type="l",col="blue",lwd=2)

#
# Design Matrix
#
model.matrix(res3)
model.matrix(res3b)
(a <- model.matrix(res3b)[,2])
(b <- model.matrix(res3b)[,4])
crossprod(a,b)

summary(lm(pairs~poly(ppha,2)))
summary(lm(pairs~poly(ppha,5)))
summary(lm(pairs~poly(ppha,10)))

res3c <- lm(pairs~poly(ppha,3,raw=TRUE))
summary(res3c)
model.matrix(res3c)
res11 <- lm(pairs~poly(ppha,11,raw=TRUE))
summary(res11)
resorth <- lm(pairs~poly(ppha,11))
summary(resorth)

#
# Is the 11th degree polynomial better than the cubic?
#

res3 <- lm(pairs~poly(ppha,3))
res11 <- lm(pairs~poly(ppha,11))
summary(res3)$r.squared
summary(res11)$r.squared
summary(res3)$adj.r.squared
summary(res11)$adj.r.squared

anova(res3,res11)