# Various regression techniques for the 'cars' data
# Y = braking distance, X = speed
# Data already in R; called 'cars'

x = cars$speed
y = cars$dist

par(mfrow=c(2,2))  # Sets the plotting function to give a 2 by 2 panel of plots

## Two Least Squares fits
plot(x, y, xlab="speed", ylab="dist", title(sub="LS fit"))

# Fit a straight line
fit1 = lm(dist ~ speed, data = cars)
yhat = predict.lm(fit1)
lines(x,yhat)


# Fit a quadratic
fit2 = lm(dist ~ speed + I(speed^2), data = cars) #omit the I() - what happens?
yhat = predict.lm(fit2)
lines(x,yhat)

legend(x=2, y=125, legend = paste("lin.",1:2," = ", round(as.numeric(fit1$coef),2)))
legend(x=2, y=100, legend = paste("quad.",1:3," = ", round(as.numeric(fit2$coef),2)))


## Two L1 - fits  
## Here the sums of the ABSOLUTE VALUES of the residuals (not their SQUARES) is minimized
## A special package for this has to be loaded:
# First go, in the menu, to Packages -> Set CRAN mirror (use HTTP mirrors, then Canada (BC))
# Then Packages -> Install Packages -> quantreg

library(quantreg)

plot(x, y, xlab="speed", ylab="dist", title(sub="L1 fit"))

# Fit a straight line
fit3 = rq(dist ~ speed, data = cars)
yhat = predict(fit3)
lines(x,yhat)

# Fit a quadratic
fit4 = rq(dist ~ speed + I(speed^2), data = cars)
yhat = predict(fit4)
lines(x,yhat)

legend(x=2, y=125, legend = paste("lin.",1:2," = ", round(as.numeric(fit3$coef),2)))
legend(x=2, y=100, legend = paste("quad.",1:3," = ", round(as.numeric(fit4$coef),2)))


## Smoothing spline fit (will be discussed later)
plot(x, y, xlab="speed", ylab="dist", title(sub="spline fit"))

fit5 = smooth.spline(x,y)
yhat = predict(fit5, x=cars$speed)$y
lines(x, yhat)

## Loess fit (will be discussed later)
plot(x, y, xlab="speed", ylab="dist", title(sub="loess fit"))

fit6 = loess(y~x)
yhat = predict(fit6)
lines(x, yhat)