be-green
commented
3 years ago Here's an example that helps clarify. The asymptotics look fine, but right now if you have (for example) equally spaced knots for the quantiles, the splines start to look like a Runge phenomenon demonstration.
beta = c(1, -3)
X = MASS::mvrnorm(n, mu = c(0.2, 0.3), Sigma = matrix(c(1, 0.2, 0.2, 2), ncol = 2))
y = X %*% beta + rnorm(n)
arg1 <- X[,1]
arg2 <- X[,2]
dat = data.frame(out = y, one = arg1, two = arg2)
dat2 <- data.frame(out = NA, one = mean(first), two = mean(second))
fit <- qs(out ~ one + two, data = dat, quantiles = seq_quant(0,1,by = 0.1))
de <- distributional_effects(fit, newdata = data.frame(one = 0.05251955,
two = 0.2959033))
plot(de, newdata = mean(X), what = "pdf")Compare N = 100 to N = 1,000 to N = 10,0000 at the same forecasting coordinates:
N = 100
N = 1,000
N = 10,000
N = 100,000
We might want to a) provide guidance on selecting quantiles to fit with the model for distributional effect estimate (e.g. Chebyshev Polynomials) b) incorporate some kind of penalization to the spline, because right now they are really overfitting in the small samples.
Right now the spline code has sharp breaks where it meets the tails. This isn't always wrong, but I think it happens more often than it should. Perhaps there is a way to resolve this issue via penalized splines (so that they don't get too wiggly). Or perhaps we could do some kind of weighting deal where the splines take into account the tail density more directly when they are being fitted, rather than just the knot at the quantile.