Enumerate Basis Functions

Generate basis functions for all covariates and interaction terms thereof up to a specified order/degree.

enumerate_basis(
  x,
  max_degree = NULL,
  smoothness_orders = rep(0, ncol(x)),
  include_zero_order = FALSE,
  include_lower_order = FALSE,
  num_knots = NULL
)

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.

num_knots

A vector of length max_degree, which determines how granular the knot points to generate basis functions should be for each degree of basis function. The first entry of num_knots determines the number of knot points to be used for each univariate basis function. More generally, The kth entry of num_knots determines the number of knot points to be used for the kth degree basis functions. Specifically, for a kth degree basis function, which is the tensor product of k univariate basis functions, this determines the number of knot points to be used for each univariate basis function in the tensor product.

Value

A list of basis functions generated for all covariates and interaction thereof up to a pre-specified degree.

Examples

# \donttest{
gendata <- function(n) {
  W1 <- runif(n, -3, 3)
  W2 <- rnorm(n)
  W3 <- runif(n)
  W4 <- rnorm(n)
  g0 <- plogis(0.5 * (-0.8 * W1 + 0.39 * W2 + 0.08 * W3 - 0.12 * W4))
  A <- rbinom(n, 1, g0)
  Q0 <- plogis(0.15 * (2 * A + 2 * A * W1 + 6 * A * W3 * W4 - 3))
  Y <- rbinom(n, 1, Q0)
  data.frame(A, W1, W2, W3, W4, Y)
}
set.seed(1234)
data <- gendata(100)
covars <- setdiff(names(data), "Y")
X <- as.matrix(data[, covars, drop = FALSE])
basis_list <- enumerate_basis(X)
# }