enumerate_edge_basis: Enumerate Basis Functions at Generalized Edges

In hal9001: The Scalable Highly Adaptive Lasso

Enumerate Basis Functions at Generalized Edges

Description

For degrees of smoothness greater than 1, we must generate the lower order smoothness basis functions using the knot points at the "edge" of the hypercube. For example, consider f(x) = x^2 + x, which is second-order smooth, but will not be generated by purely quadratic basis functions. We also need to include the y = x function (which corresponds to first-order HAL basis functions at the left most value/edge of x).

Usage

enumerate_edge_basis(
  x,
  max_degree = 3,
  smoothness_orders = rep(0, ncol(x)),
  include_zero_order = FALSE,
  include_lower_order = FALSE
)

Arguments

x	An input matrix containing observations and covariates following standard conventions in problems of statistical learning.
max_degree	The highest order of interaction terms for which the basis functions ought to be generated. The default (NULL) corresponds to generating basis functions for the full dimensionality of the input matrix.
smoothness_orders	An integer vector of length ncol(x) specifying the desired smoothness of the function in each covariate. k = 0 is no smoothness (indicator basis), k = 1 is first order smoothness, and so on. For an additive model, the component function for each covariate will have the degree of smoothness as specified by smoothness_orders. For non-additive components (tensor products of univariate basis functions), the univariate basis functions in each tensor product have smoothness degree as specified by smoothness_orders.
include_zero_order	A logical, indicating whether the zeroth order basis functions are included for each covariate (if TRUE), in addition to the smooth basis functions given by smoothness_orders. This allows the algorithm to data-adaptively choose the appropriate degree of smoothness.
include_lower_order	A logical, like include_zero_order, except including all basis functions of lower smoothness degrees than specified via smoothness_orders.

hal9001 documentation built on Nov. 14, 2023, 5:08 p.m.