Sunday, July 12, 2015

Linear regression

Let's try regression on the auto dataset:
regfit.full <- regsubsets(price~., auto)

It doesn't work because by default, exhaustive search is performed.
This requires lots of permutations, so you have to set really.big=TRUE if you want this.

But let's just make things simpler by carving out a subset of variables:
autosmall <- auto[,c(2,3,5,6,7,8,9,10,11)]
regfit.full <- regsubsets(price ~ . , autosmall) # the . indicates we want all variables to be considered.

regfit.fwd <- regsubsets(price~., autosmall, method="forward") # forward stepwise selection
regfit.bwd <- regsubsets(price~., autosmall, method="backward") # backward stepwise selection
coef(regfit.fwd,3) # gives the coefficients for regfit.fwd for the optimal 3-variable model

Recall that we actually need to split our dataset into a training set and test set. This is how we can do it:
train <- sample(c(TRUE, FALSE),nrow(autosmall),rep=TRUE)
# if we wanted 75% of observations to be in training set, we can do the below:
# train <- sample(c(TRUE, FALSE),nrow(autosmall),rep=TRUE, prob = c(0.75, 0.25))
test <- (!train)

# another way of allocating the training/test sets:
# train <- sample(1:nrow(autosmall),  nrow(autosmall)/2)
# test <- (-train)
# autosmall.test <- autosmall[test]
# (notice that test contains all negative numbers, so all "train" observations are excluded)

To keep things simple, we didn't do cross validation (as well as regularization).
Here's how to do cross-validation:
regfit.full <- regsubsets(price~., autosmall[train,])
reg.summary <- summary(regfit.full) # for convenience

For model selection, it'll be useful to plot how RSS (and other measures such as adjusted R-sq) varies with number of variables:

par(mfrow = c(1,2)) # sets a 1 row by 2 column plot
plot(reg.summary$rss,xlab="# of Var", ylab="RSS", type="l")
(m1 <- which.min(reg.summary$rss))
points(m1,reg.summary$rss[m1], col="red",cex=2,pch=20)
# cex refers to the dot thickness

plot(reg.summary$adjr2,xlab="# of Var", ylab="Adj R2", type="l")
(m1 <- which.max(reg.summary$adjr2))
points(m2,reg.summary$adjr2[m2], col="red",cex=2,pch=20)

plot(regfit.full, scale="r2")

Now we need to see MSE on the test set. First off, use model.matrix to generate the "X" matrix for the test set:

test.mat <- model.matrix(price~., data=autosmall[test,])

val.errors = rep(NA,8) # generates the empty MSE matrix
for (i in 1:8) {
  coefi <- coef(regfit.full, id=i) # takes the coefficients from the optimal model with i variables
  pred <- test.mat[,names(coefi)] %*% coefi # generate predicted values of each observation
  val.errors[i] <- mean((autosmall$price[test] - pred)^2) # generate MSE for i-th model
#[1]  7781131  9856769 10060398  9574022  9421108  9470527  9269711  8942316
(v <- which.min(val.errors))
#[1] 1
# so we choose model with 1 variable (weight only)
coef(regfit.full, v)
#(Intercept)      weight 
#-426.910662    2.251572

We can also write a function, which will come in useful for k-fold cross validation
predict.regsubsets <- function(object, newdata, id, ...) {
  form <- as.formula(object$call[[2]]) # takes the second item from object$call list
  # call is part of regfit.full
  # the second item within the list is the relevant variables
  mat <- model.matrix(form, newdata) # generates the required X matrix using the formula
  coefi <- coef(object, id=id) # the X variable COEFFICIENTS of the optimal model
  xvars <- names(coefi) # these are the X variables of the optimal model
  mat[,xvars] %*% coefi # generates predicted values (by looking at the relevant columns only)

# 10-fold cross validation
k = 10
folds = sample(1:k, nrow(autosmall), replace=TRUE) 
# if an observation is labelled 'i', it will be part of the i-th fold.
# e.g. an observation that is labelled 3 will be left out of the 3rd round of cross validation
cv.errors = matrix(NA,k,8, dimnames=list(NULL, paste(1:8)))
# sets up a k by 8 matrix filled with NA's. 
# columns are named 1 to 8

# armed with our new predict() method, we can:
for (j in 1:k) {
  # obtain the required coefficients <- regsubsets(price ~ . , data=autosmall[folds != j,])
  # note: if you have more than 8 variables you will need to specify nvmax in the above command
  for (i in 1:8) {
    # now, calculate MSE using the i-th fold
    pred <- predict(, autosmall[folds == j,],id=i)
    # since is of class regsubsets, the predict command that was created will work on it.
    cv.errors[j,i] <- mean((autosmall$price[folds == j] - pred)^2)
( <- apply(cv.errors,2,mean))

Let's plot MSE of the CV dataset.
plot(, type='b')

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