Add R2 as performance statistic

[fweber144]

As suggested by @avehtari, it would be good to have $R^2$ as a performance statistic in projpred. This could be called stats = "R2" (and stat = "R2" for suggest_size()), for example. According to @avehtari, we should go for LOO - $R^2$.

There is also related code at

projpred/R/summary_funs.R

Lines 170 to 187 in bec6258

	if (stat == "r2") {
	if (!is.null(mu.bs)) {
	y <- mu.bs
	} else {
	y <- d_test$y
	}
	eloo <- mu - y
	n <- length(y)
	rd <- bayesboot::rudirichlet(4000, n)
	vary <- (rowSums(sweep(rd, 2, y^2, FUN = "*")) -
	rowSums(sweep(rd, 2, y, FUN = "*"))^2) * (n / (n - 1))
	vareloo <- (rowSums(sweep(rd, 2, eloo^2, FUN = "*")) -
	rowSums(sweep(rd, 2, eloo, FUN = "*")^2)) * (n / (n - 1))
	looR2 <- 1 - vareloo / vary
	looR2[looR2 < -1] <- -1
	looR2[looR2 > 1] <- 1
	value <- median(looR2)
	value.se <- sd(looR2)

(Note that * (n / (n - 1) can be omitted because it cancels out.) In those lines, bayesboot::rudirichlet() is used. According to @avehtari, the SE could also be calculated without a Dirichlet approach, using the formula from stan-dev/loo#205 (comment).

[fweber144]

@AlejandroCatalina: Is line looR2[looR2 < -1] <- -1 supposed to read looR2[looR2 < 0] <- 0?

[fweber144]

@avehtari: The SE formula provided in stan-dev/loo#205 (comment) refers to LOO - $R^2$. I guess it cannot be applied directly to K-fold CV, no CV (i.e., test dataset = training dataset), or a hold-out test dataset. Do you know of similar formulas for those cases?

[avehtari]

Is line looR2[looR2 < -1] <- -1 supposed to read looR2[looR2 < 0] <- 0?

The first one is intentional.

The SE formula provided in stan-dev/loo#205 (comment) refers to LOO -
. I guess it cannot be applied directly to K-fold CV

Can be used with K-fold-CV and pointwise evaluation.

no CV (i.e., test dataset = training dataset)

We have used Bayesian-R2 for that as it has some benefits in that case, but the same formula could be used, too

or a hold-out test dataset

Can be used

[avehtari]

Implemented by 0ed8391