data_sim() supplies analytic reference functions for mgcv::scasm().
All covariates are uniform on [0, 1]. Univariate templates return f1, the
true function, and its analytic derivatives d1 and d2. The response
contains sampling variation; the derivative columns describe the truth.
Univariate templates
Model suffix after sc_ | Function | xt in s(x, bs = "sc", xt = ...) |
increasing | plogis(8 * (x - 0.5)) | "m+" |
decreasing | 1 - plogis(8 * (x - 0.5)) | "m-" |
convex | (x - 0.5)^2 | "c+" |
concave | 1 - (x - 0.5)^2 | "c-" |
positive | 0.2 + exp(-16 * (x - 0.5)^2) | "+", with no intercept |
increasing_convex | exp(2 * x) / exp(2) | c("m+", "c+") |
increasing_concave | log1p(4 * x) / log(5) | c("m+", "c-") |
decreasing_convex | exp(-2 * x) | c("m-", "c+") |
decreasing_concave | 1 - x^2 | c("m-", "c-") |
Remove the intercept for the positivity recipe so the constrained smooth represents the whole function. An unconstrained intercept combined with a centred smooth does not express the same positivity requirement.
Several terms and distribution parameters
sc_additive adapts mgcv's example. Its raw terms are gw_f0(x0),
2 * plogis(20 * (x1 - 0.4)), gw_f2(x2) and 10 * (x3 - 0.55)^4.
The true predictor is their sum minus 1, divided by 4. The second term is
increasing and the fourth is convex. Use predictors = function(d) d$f
with family = poisson() to sample mgcv's Poisson example.
sc_tensor uses f = x + z + x * z, increasing in both covariates. Its
dx and dz columns give the true partial derivatives.
sc_location_scale uses log mean sin(pi * x) / 2 and log dispersion
-2 + z, returned in f and f_scale. It defaults to gammals() and
records derivatives in d1, d2 and d_scale. Constraints act on the
specified predictors; in particular the concavity concerns the log mean.
The recipes use the default bs = 0 for fitting without bootstrap
replicates. To obtain native constraint-respecting bootstrap coefficient
draws, fit with a positive bs, such as bs = 100.
Examples
library("mgcv")
d <- data_sim("sc_increasing", n = 200, seed = 2)
m <- scasm(y ~ s(x, bs = "sc", xt = "m+", k = 8), data = d)
d <- data_sim("sc_positive", n = 200, seed = 2)
m <- scasm(y ~ s(x, bs = "sc", xt = "+", k = 8) - 1, data = d)
# \donttest{
d <- data_sim("sc_additive", n = 240, family = poisson(), seed = 2,
predictors = function(d) d$f)
m <- scasm(y ~ s(x0, k = 6) + s(x1, bs = "sc", xt = "m+", k = 6) +
s(x2, k = 6) + s(x3, bs = "sc", xt = "c+", k = 6),
family = poisson(), data = d)
d <- data_sim("sc_tensor", seed = 2)
m <- scasm(y ~ te(x, z, bs = c("sc", "sc"), xt = list("m+", "m+"),
k = c(6, 6)), data = d)
d <- data_sim("sc_location_scale", seed = 2)
m <- scasm(list(y ~ s(x, bs = "sc", xt = "c-"),
~s(z, bs = "sc", xt = "m+")), family = gammals(), data = d)
# }