bases (version 0.2.0)

b_tpsob: Tensor-product Sobolev space basis

Description

Generates features from a tensor-product Sobolev space basis for estimating functions in a Sobolev space with dominating mixed derivatives. Basis functions are of the form $$ \psi_{\mathbf{j}}(\mathbf{x}) = \prod_{k=1}^d \psi_{j_k}(x_k), $$ where $$ \phi_1(x) = 1 \quad\text{and}\quad \phi_j(x) = \sqrt{2}\cos(\pi (j-1) x). $$ The multi-indices \(\mathbf{j}\) are generated in a specific order to maximize statistical efficiency. All inputs are standardized to lie in the unit hypercube \([0, 1]^d\).

Usage

b_tpsob(..., p = 100, shift = NULL, scale = NULL)

Value

A matrix of tensor-product Sobolev space basis features.

Arguments

...

The variable(s) to build features for. A single data frame or matrix may be provided as well. Missing values are not allowed.

p

The number of basis functions to generate.

shift

Vector of shifts, or single shift value, to use. If provided, overrides those calculated according to stdize.

scale

Vector of scales, or single scale value, to use. If provided, overrides those calculated according to stdize.

References

Zhang, T., & Simon, N. (2023). Regression in tensor product spaces by the method of sieves. Electronic journal of statistics, 17(2), 3660.

Examples

data(quakes)

m = ridge(depth ~ b_tpsob(lat, long, p = 100), quakes)
plot(fitted(m), quakes$depth)

x = 1:150
y = as.numeric(BJsales)
m = lm(y ~ b_tpsob(x, p = 10))
plot(x, y)
lines(x, fitted(m), col="blue")