#### Figure for Section 2, "Errors, In and Out of Sample" # Example: linear regression through the origin, with Gaussian noise # fix n, true slope, generate data, with X uniform on [0,1] n<-20; theta<-5 x<-runif(n); y<-x*theta+rnorm(n) # empirial risk = in-sample mean-squred error empirical.risk <- function(b) { mean((y-b*x)^2) } # generalization error of a line through the origin with slope b # EXERCISE: derive this formula true.risk <- function(b) { 1 + (theta-b)^2*(0.5^2+1/12) } # Plot the in-sample risk curve(Vectorize(empirical.risk)(x),from=0,to=2*theta, xlab="regression slope",ylab="MSE risk") # R trickery: the empirical.risk() function, as written, does not behave well # when given a vector of slopes, and curve() wants its first argument to be a # function which can take a vector. Vectorize() turns its argument into a # function which can take a vector; writing the expression # Vectorize(empirical.risk)(x) # rather than just # Vectorize(empirical.risk) # helps curve() figure out where to pass its vector of points. curve(true.risk,add=TRUE,col="grey") # by contrast the true.risk() function works nicely with vectors. #### Figures for Section 3, "Over-Fitting and Model Selection" # 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") ### Figures for section 4, "Cross-Validation" housing <- read.csv("http://www.stat.cmu.edu/~cshalizi/uADA/13/hw/01/calif_penn_2011.csv") # Divide the data randomly into two (nearly) equal halves half_A <- sample(1:nrow(housing),size=nrow(housing)/2,replace=FALSE) half_B <- setdiff(1:nrow(housing),half_A) # Write out the formulas for our two linear model specifications just once small_formula = "Median_house_value ~ Median_household_income" large_formula = "Median_house_value ~ Median_household_income + Median_rooms" small_formula <- as.formula(small_formula) large_formula <- as.formula(large_formula) # Fit each model specification to each half of the data mAsmall <- lm(small_formula,data=housing,subset=half_A) mBsmall <- lm(small_formula,data=housing,subset=half_B) mAlarge <- lm(large_formula,data=housing,subset=half_A) mBlarge <- lm(large_formula,data=housing,subset=half_B) # EXERCISE: Extract the coefficients for all the models # Calculating the in-sample MSE is a repeated task, so write a function for it in.sample.mse <- function(model) { mean(residuals(model)^2) } in.sample.mse(mAsmall); in.sample.mse(mAlarge) in.sample.mse(mBsmall); in.sample.mse(mBlarge) # Calculating the MSE of a model on new data also deserves a function new.sample.mse <- function(model,half) { test <- housing[half,] predictions <- predict(model,newdata=test) return(mean((test\$Median_house_value - predictions)^2)) } # EXERCISE: is in.sample.mse(mAsmall) == new.sample.mse(mAsmall,half_A) ? # EXERCISE: should they be equal? new.sample.mse(mAsmall,half_B); new.sample.mse(mBsmall,half_A) new.sample.mse(mBsmall,half_A); new.sample.mse(mASmall,half_B)