Part I

Training and testing errors

Reminder: statistical (regression) models

You have some data \(X_1,\ldots,X_p,Y\): the variables \(X_1,\ldots,X_p\) are called predictors, and \(Y\) is called a response. You’re interested in the relationship that governs them

So you posit that \(Y|X_1,\ldots,X_p \sim P_\theta\), where \(\theta\) represents some unknown parameters. This is called regression model for \(Y\) given \(X_1,\ldots,X_p\). Goal is to estimate parameters. Why?

Reminder: linear regression models

The linear model is arguably the most widely used statistical model, has a place in nearly every application domain of statistics

Given response \(Y\) and predictors \(X_1,\ldots,X_p\), in a linear regression model, we posit:

\[ Y = \beta_0 + \beta_1 X_1 + \ldots + \beta_p X_p + \epsilon, \quad \text{where $\epsilon \sim N(0,\sigma^2)$} \]

Goal is to estimate parameters \(\beta_0,\beta_1,\ldots,\beta_p\). Why?

Good diagnostics, or good predictions?

Nowadays, we try to fit linear models in such a wide variety of difficult problem settings that, in many cases, we have no reason to believe the true data generating model is linear, the errors are close to Gaussian or homoskedastic, etc. Hence, a modern perspective:

The linear model is only a rough approximation, so evaluate prediction accuracy, and let this determine its usefulness

This idea, to focus on prediction, is far more general than linear models. More on this, shortly

Test error

Suppose we have training data \(X_{i1},\ldots,X_{ip},Y_i\), \(i=1,\ldots,n\) used to estimate regression coefficients \(\hat{\beta}_0,\hat{\beta}_1,\ldots,\hat{\beta}_p\)

Given new \(X_1^*,\ldots,X_p^*\) and asked to predict the associated \(Y^*\). From the estimated linear model, prediction is: \(\hat{Y}^* = \hat{\beta}_0 + \hat{\beta}_1 X_1^* + \ldots + \hat{\beta}_p X_p^*\). We define the test error, also called prediction error, by \[ \mathbb{E}(Y^* - \hat{Y}^*)^2 \] where the expectation is over every random: training data, \(X_{i1},\ldots,X_{ip},Y_i\), \(i=1,\ldots,n\) and test data, \(X_1^*,\ldots,X_p^*,Y^*\)

This was explained for a linear model, but the same definition of test error holds in general

Estimating test error

Often, we want an accurate estimate of the test error of our method (e.g., linear regression). Why? Two main purposes:

Training error

Suppose, as an estimate the test error of our method, we take the observed training error \[ \frac{1}{n} \sum_{i=1}^n (Y_i - \hat{Y}_i)^2 \]

What’s wrong with this? Generally too optimistic as an estimate of the test error—after all, the parameters \(\hat{\beta}_0,\hat{\beta}_1,,\ldots,\hat{\beta}_p\) were estimated to make \(\hat{Y}_i\) close to \(Y_i\), \(i=1,\ldots,n\), in the first place!

Also, importantly, the more complex/adaptive the method, the more optimistic its training error is as an estimate of test error

Examples

set.seed(1)
n = 30
x = sort(runif(n, -3, 3))
y = 2*x + 2*rnorm(n)
x0 = sort(runif(n, -3, 3))
y0 = 2*x0 + 2*rnorm(n)

par(mfrow=c(1,2))
xlim = range(c(x,x0)); ylim = range(c(y,y0))
plot(x, y, xlim=xlim, ylim=ylim, main="Training data")
plot(x0, y0, xlim=xlim, ylim=ylim, main="Test data")

# Training and test errors for a simple linear model
lm.1 = lm(y ~ x)
yhat.1 = predict(lm.1, data.frame(x=x))
train.err.1 = mean((y-yhat.1)^2)
y0hat.1 = predict(lm.1, data.frame(x=x0))
test.err.1 = mean((y0-y0hat.1)^2)

par(mfrow=c(1,2))
plot(x, y, xlim=xlim, ylim=ylim, main="Training data")
lines(x, yhat.1, col=2, lwd=2)
text(0, -6, label=paste("Training error:", round(train.err.1,3)))

plot(x0, y0, xlim=xlim, ylim=ylim, main="Test data")
lines(x0, y0hat.1, col=3, lwd=2)
text(0, -6, label=paste("Test error:", round(test.err.1,3)))

# Training and test errors for a 10th order polynomial regression
# (The problem is only exacerbated!)
lm.10 = lm(y ~ poly(x,10))
yhat.10 = predict(lm.10, data.frame(x=x)) 
train.err.10 = mean((y-yhat.10)^2)
y0hat.10 = predict(lm.10, data.frame(x=x0))
test.err.10 = mean((y0-y0hat.10)^2)

par(mfrow=c(1,2))
xx = seq(min(xlim), max(xlim), length=100)

plot(x, y, xlim=xlim, ylim=ylim, main="Training data")
lines(xx, predict(lm.10, data.frame(x=xx)), col=2, lwd=2)
text(0, -6, label=paste("Training error:", round(train.err.10,3)))

plot(x0, y0, xlim=xlim, ylim=ylim, main="Test data")
lines(xx, predict(lm.10, data.frame(x=xx)), col=3, lwd=2)
text(0, -6, label=paste("Test error:", round(test.err.10,3)))

Part II

Sample-splitting and cross-validation

Sample-splitting

Given a data set, how can we estimate test error? (Can’t simply simulate more data for testing.) We know training error won’t work

A tried-and-true technique with an old history in statistics: sample-splitting

Examples

dat = read.table("https://www.stat.cmu.edu/~arinaldo/Teaching/36350/F22/data/xy.dat")
head(dat)
n = nrow(dat)

# Split data in half, randomly
set.seed(0)
inds = sample(rep(1:2, length=n))
head(inds, 10)
##  [1] 2 2 1 1 2 1 1 2 2 1
table(inds)
## inds
##  1  2 
## 25 25
dat.tr = dat[inds==1,] # Training data
dat.te = dat[inds==2,] # Test data

plot(dat$x, dat$y, pch=c(21,19)[inds], main="Sample-splitting")
legend("topleft", legend=c("Training","Test"), pch=c(21,19))

# Train on the first half
lm.1 = lm(y ~ x, data=dat.tr)
lm.10 = lm(y ~ poly(x,10), data=dat.tr)

# Predict on the second half, evaluate test error
pred.1 = predict(lm.1, data.frame(x=dat.te$x))
pred.10 = predict(lm.10, data.frame(x=dat.te$x))

test.err.1 = mean((dat.te$y - pred.1)^2)
test.err.10 = mean((dat.te$y - pred.10)^2)

# Plot the results
par(mfrow=c(1,2))
xx = seq(min(dat$x), max(dat$x), length=100)

plot(dat$x, dat$y, pch=c(21,19)[inds], main="Sample-splitting")
lines(xx, predict(lm.1, data.frame(x=xx)), col=2, lwd=2)
legend("topleft", legend=c("Training","Test"), pch=c(21,19))
text(0, -6, label=paste("Test error:", round(test.err.1,3)))

plot(dat$x, dat$y, pch=c(21,19)[inds], main="Sample-splitting")
lines(xx, predict(lm.10, data.frame(x=xx)), col=3, lwd=2)
legend("topleft", legend=c("Training","Test"), pch=c(21,19))
text(0, -6, label=paste("Test error:", round(test.err.10,3)))

Cross-validation

Sample-splitting is simple, effective. But its it estimates the test error when the model/method is trained on less data (say, roughly half as much)

An improvement over sample splitting: \(k\)-fold cross-validation

A common choice is \(k=5\) or \(k=10\) (sometimes \(k=n\), called leave-one-out!)

For demonstration purposes, suppose \(n=6\) and we choose \(k=3\) parts

Data point Part Trained on Prediction
\(Y_1\) 1 2,3 \(\hat{Y}^{-(1)}_1\)
\(Y_2\) 1 2,3 \(\hat{Y}^{-(1)}_2\)
\(Y_3\) 2 1,3 \(\hat{Y}^{-(2)}_3\)
\(Y_4\) 2 1,3 \(\hat{Y}^{-(2)}_4\)
\(Y_5\) 3 1,2 \(\hat{Y}^{-(3)}_5\)
\(Y_6\) 3 1,2 \(\hat{Y}^{-(3)}_6\)

Notation: model trained on parts 2 and 3 in order to make predictions for part 1. So prediction \(\hat{Y}^{-(1)}_1\) for \(Y_1\) comes from model trained on all data except that in part 1. And so on

The cross-validation estimate of test error (also called the cross-validation error) is \[ \frac{1}{6}\Big( (Y_1-\hat{Y}^{-(1)}_1)^2 + (Y_2-\hat{Y}^{-(1)}_2)^2 + (Y_3-\hat{Y}^{-(2)}_3)^2 + \\ (Y_4-\hat{Y}^{-(2)}_4)^2 + (Y_5-\hat{Y}^{-(3)}_5)^2 + (Y_6-\hat{Y}^{-(3)}_6)^2 \Big) \]

Examples

# Split data in 5 parts, randomly
k = 5
set.seed(0)
inds = sample(rep(1:k, length=n))
head(inds, 10)
##  [1] 4 4 4 1 4 3 3 5 3 3
table(inds)
## inds
##  1  2  3  4  5 
## 10 10 10 10 10
# Now run cross-validation: easiest with for loop, running over
# which part to leave out
pred.mat = matrix(0, n, 2) # Empty matrix to store predictions
for (i in 1:k) {
  cat(paste("Fold",i,"... "))
  
  dat.tr = dat[inds!=i,] # Training data
  dat.te = dat[inds==i,] # Test data
  
  # Train our models
  lm.1.minus.i = lm(y ~ x, data=dat.tr)
  lm.10.minus.i = lm(y ~ poly(x,10), data=dat.tr)
  
  # Record predictions
  pred.mat[inds==i,1] = predict(lm.1.minus.i, data.frame(x=dat.te$x))
  pred.mat[inds==i,2] = predict(lm.10.minus.i, data.frame(x=dat.te$x))
}
## Fold 1 ... Fold 2 ... Fold 3 ... Fold 4 ... Fold 5 ...
# Compute cross-validation error, one for each model
cv.errs = colMeans((pred.mat - dat$y)^2)

# Plot the results
par(mfrow=c(1,2))
xx = seq(min(dat$x), max(dat$x), length=100)

plot(dat$x, dat$y, pch=20, col=inds+1, main="Cross-validation")
lines(xx, predict(lm.1, data.frame(x=xx)), # Note: model trained on FULL data!
      lwd=2, lty=2) 
legend("topleft", legend=paste("Fold",1:k), pch=20, col=2:(k+1))
text(0, -6, label=paste("CV error:", round(cv.errs[1],3)))

plot(dat$x, dat$y, pch=20, col=inds+1, main="Cross-validation")
lines(xx, predict(lm.10, data.frame(x=xx)), # Note: model trained on FULL data!
      lwd=2, lty=2) 
legend("topleft", legend=paste("Fold",1:k), pch=20, col=2:(k+1))
text(0, -6, label=paste("CV error:", round(cv.errs[2],3)))

# Now we visualize the different models trained, one for each CV fold
for (i in 1:k) {
  dat.tr = dat[inds!=i,] # Training data
  dat.te = dat[inds==i,] # Test data
  
  # Train our models
  lm.1.minus.i = lm(y ~ x, data=dat.tr)
  lm.10.minus.i = lm(y ~ poly(x,10), data=dat.tr)
  
  # Plot fitted models
  par(mfrow=c(1,2)); cols = c("red","gray")
  plot(dat$x, dat$y, pch=20, col=cols[(inds!=i)+1], main=paste("Fold",i))
  lines(xx, predict(lm.1.minus.i, data.frame(x=xx)), lwd=2, lty=2) 
  legend("topleft", legend=c(paste("Fold",i),"Other folds"), pch=20, col=cols)
  text(0, -6, label=paste("Fold",i,"error:",
       round(mean((dat.te$y - pred.mat[inds==i,1])^2),3)))
                          
  plot(dat$x, dat$y, pch=20, col=cols[(inds!=i)+1], main=paste("Fold",i))
  lines(xx, predict(lm.10.minus.i, data.frame(x=xx)), lwd=2, lty=2) 
  legend("topleft", legend=c(paste("Fold",i),"Other folds"), pch=20, col=cols)
  text(0, -6, label=paste("Fold",i,"error:",
       round(mean((dat.te$y - pred.mat[inds==i,2])^2),3)))
}

Part III

Statistical prediction

Shifting tides: a focus on prediction

Classically, statistics has focused in large part on inference. The tides are shifting (at least in some part), and in many modern problems, the following view is taken:

Models are only approximations; some methods need not even have underlying models; let’s evaluate prediction accuracy, and let this determine model/method usefulness

This is (in some sense) one of the basic tenets of machine learning

“Early” influential paper

versus ?

Statistical prediction machines

Some methods for predicting \(Y\) from \(X_1,\ldots,X_p\) have (in a sense) no parameters at all. Perhaps better said: they are not motivated from writing down a statistical model like \(Y|X_1,\ldots,X_p \sim P_\theta\)

We’ll call these statistical prediction machines. Admittedly: not a real term, but it’s evocative of what they are doing, and there’s no real consensus terminology. You might also see these described as:

Comment: in a broad sense, most of these methods would have been completely unthinkable before the rise of high-performance computing

\(k\)-nearest neighbors

One of the simplest prediction machines: \(k\)-nearest neighbors regression

Ask yourself: what happens when \(k=1\)? What happens when \(k=n\)?

Advantages: simple and flexible. Disadvantages: can be slow and cumbersome

From \(k\)-nearest neighbors to trees

Can think of 1-nearest neighbor predictions as being simply given by predicting a constant within each element of what is called as Voronoi tesellation: these are polyhedra that partition the predictor space

Regression trees are similar but somewhat different. In a nutshell, they use (nested) rectangles instead of polyhedra. These rectangles are fit through sequential (greedy) split-point determinations

Advantage: easier to make predictions (from split-points). Disadvantage: less flexible

From trees to boosting

Boosting is a method built on top of regression trees in a clever way. To make predictions, can think of taking predictions from a sequence of trees, and combining them with weights (coefficients)

\(\beta_1 \cdot\) \(+\) \(\beta_2 \cdot\) \(+\ldots\)

Advantage: much more flexible than a single tree. Disadvantage: not generally interpretable …

Many, many others

There are many, many other statistical prediction methods out there; examples below. If you’re interesting in learning more, take 36-462 Data Mining, or one of the Introduction to Machine Learning Courses 10-401, 10-601, 10-701

Two world views?

versus

Conclusion