# Motorcycle data
# Load the MASS package; data are in mcycle
library(MASS)
attach(mcycle)
plot(mcycle)

par(mfrow=c(2,1))

# Fit orthogonal polynomials
plot(mcycle)
for(k in 2:6) lines(mcycle$times, predict(lm(accel~poly(times,k)), data=mcycle), lty=k-1)
title(sub = "Polynomial fits of degrees 2,...,6 to motorcycle data")
plot(mcycle)
lines(mcycle$times, predict(lm(accel~poly(times,20)), data=mcycle))
title(sub = "Polynomial fit of degree 20 (!) to motorcycle data")

# Fit a Fourier series
t = times
omega = 2*pi/(max(t)-min(t))

fourierx = function(maxorder) {
x = cbind(sin(omega*t), cos(omega*t))
for (k in 2:maxorder) x = cbind(x, sin(k*omega*t), cos(k*omega*t))
}
plot(mcycle)
lines(times, predict(lm(accel ~ fourierx(2)), data=mcycle$times), lty=1)
lines(times, predict(lm(accel ~ fourierx(5)), data=mcycle$times), lty=2)
legend("topleft", legend = c("K=2", "K=5"), lty=c(1,2))
plot(mcycle)
lines(times, predict(lm(accel ~ fourierx(8)), data=mcycle$times), lty=1)
lines(times, predict(lm(accel ~ fourierx(11)), data=mcycle$times), lty=2)
legend("topleft", legend = c("K=8", "K=11"), lty=c(1,2))


# Fit a smoothing spline
par(mfrow=c(1,1))
fit = smooth.spline(times, accel)
plot(mcycle)
lines(fit, col=1, lty=1)
lines(smooth.spline(times, accel, df=2), col=2, lty=2)
lines(smooth.spline(times, accel, df=5), col=4, lty=4)
lines(smooth.spline(times, accel, df=60), col=6, lty=6)
legend("bottomright", legend = c("df=12.21 (GCV)", "df=2", "df=5", "df=60"), col=c(1,2,4,6), lty=c(1,2,4,6))