set.seed(2) ## set.seed(1)

## Questions 1-7
gen_weights <- function(x) -0.0004*x^3 + 0.025*x^2 + 2*x + 50
age <-  sample(40:80,1000,replace=TRUE)
trueFx <-  gen_weights(age)
noise <-  rnorm(n=length(age),mean=0,sd=10)
weight <-  trueFx + noise
dat <- data.frame(age,weight,trueFx,noise)
plot(age,weight,pch=19,cex=0.4)

## Question 8-15
ksmooth1 <- ksmooth(age,weight,kernel="box",bandwidth=10)
ksmooth1
names(ksmooth1)
lines(ksmooth1$x,ksmooth1$y,col="red",lwd=3)

ksmooth2 <- ksmooth(age,weight,kernel="normal",bandwidth=10)
ksmooth2
names(ksmooth2)
lines(ksmooth2$x,ksmooth2$y,col="blue",lwd=3)

ksmooth3 <- ksmooth(age,weight,kernel="normal",bandwidth=5)
names(ksmooth3)
lines(ksmooth3$x,ksmooth3$y,col="green",lwd=3,lty=2)

ksmooth4 <- ksmooth(age,weight,kernel="normal",bandwidth=20)
names(ksmooth4)
lines(ksmooth4$x,ksmooth4$y,col="purple",lwd=3,lty=3)

legend("bottomright",cex=0.8,lwd=3,lty=c(1,1,2,3),c("box bw=10","normal bw=10","normal bw=5","normal bw=20"),col=c("red","blue","green","purple"))

## Questions 16-22
plot(age,weight,pch=19,cex=0.4)
lm1 <- lm(weight ~ age)
summary(lm1)
abline(a=lm1$coefficients[1],b=lm1$coefficients[2],col="red",lwd=3)
## Alternatively to the previous line you can use the short-cut
abline(lm1,col="red",lwd=3)

lm2 <- lm(weight ~ age + I(age^2))
lm3 <- lm(weight ~ age + I(age^2) + I(age^3))

summary(lm2)
summary(lm3)

quad <- function(x) lm2$coefficients[1] + lm2$coefficients[2]*x + lm2$coefficients[3]*x^2

cube <- function(x) lm3$coefficients[1] + lm3$coefficients[2]*x + lm3$coefficients[3]*x^2 + lm3$coefficients[4]*x^3

curve(quad, add=TRUE, col="blue", lwd=3)
curve(cube, add=TRUE, col="green", lwd=3, lty=2)

legend("bottomright",cex=0.8,lwd=3,lty=c(1,1,2),c("linear","quadratic","cubic"),col=c("red","blue","green"))

anova(lm2,lm3)