# Examples of training data # 20 standard-Gaussian X's x = rnorm(20) # Quadratic Y's y = 7*x^2 - 0.5*x + rnorm(20) # Initial plot of training data plus true regression curve plot(x,y) curve(7*x^2-0.5*x,col="grey",add=TRUE) # Fit polynomials and add them to the plot # Fit a constant (0th order polynomial), plot it y.0 = lm(y ~ 1) # We'd get the same constant by just doing mean(y), but fitting it as a # regression model means functions like residuals() and predict() are # available for use later, the same as our other models abline(h=y.0\$coefficients[1]) # Get evenly spaced points for pretty plotting of other models d = seq(min(x),max(x),length.out=200) # Fit polynomials of order 1 to 9 # It would be nicer if we let this run from 0 to 9, but R doesn't allow us # to do a polynomial of degree 0 for (degree in 1:9) { fm = lm(y ~ poly(x,degree)) # Store the results in models called y.1, y.2, through y.9 # The assign/paste trick here is often useful assign(paste("y",degree,sep="."), fm) # Plot them, with different line types lines(d, predict(fm,data.frame(x=d)),lty=(degree+1)) } # Calculate and plot in-sample errors mse.q = vector(length=10) for (degree in 0:9) { # The get() function is the inverse to assign() fm = get(paste("y",degree,sep=".")) mse.q[degree+1] = mean(residuals(fm)^2) } plot(0:9,mse.q,type="b",xlab="polynomial degree",ylab="mean squared error", log="y") # Plot the old curves with testing data x.new = rnorm(2e4) y.new = 7*x.new^2 - 0.5*x.new + rnorm(2e4) plot(x.new,y.new,xlab="x",ylab="y",pch=24,cex=0.1,col="blue") curve(7*x^2-0.5*x,col="grey",add=TRUE) abline(h=y.0\$coefficients[1]) d = seq(from=min(x.new),to=max(x.new),length.out=200) for (degree in 1:9) { fm = get(paste("y",degree,sep=".")) lines(d, predict(fm,data.frame(x=d)),lty=(degree+1)) } points(x,y) # Calculate and plot the out-of-sample errors gmse.q = vector(length=10) for (degree in 0:9) { # The get() function is the inverse to assign() fm = get(paste("y",degree,sep=".")) predictions = predict(fm,data.frame(x=x.new)) resids = y.new - predictions gmse.q[degree+1] = mean(resids^2) } plot(0:9,mse.q,type="b",xlab="polynomial degree", ylab="mean squared error",log="y",ylim=c(min(mse.q),max(gmse.q))) lines(0:9,gmse.q,lty=2,col="blue") points(0:9,gmse.q,pch=24,col="blue")