spacetime_bisquare: Space-Time Bisquare Basis In stcos: Space-Time Change of Support

Description

Space-time bisquare basis on point data.

Usage

 1 spacetime_bisquare(dom, knots, w_s, w_t)

Arguments

 dom Space-time points (\bm{u}_1,v_1), …, (\bm{u}_n,v_n) to evaluate. See "Details". knots Spatio-temporal knots (\bm{c}_1,g_1), …, (\bm{c}_r,g_r) for the basis. See "Details". w_s Spatial radius for the basis. w_t Temporal radius for the basis.

Details

• Both dom and knots may be provided as either sf or sfc objects, or as matrices of points.

• If an sf or sfc object is provided for dom, n three-dimensional POINT entries are expected in st_geometry(dom). Otherwise, dom will be interpreted as an n \times 3 numeric matrix.

• If an sf or sfc object is provided for knots, r three-dimensional POINT entries are expected in st_geometry(knots). Otherwise, knots will be interpreted as an r \times 3 numeric matrix.

• If both dom and knots_s are given as sf or sfc objects, they will be checked to ensure a common coordinate system.

For each (\bm{u}_i,v_i), compute the basis functions

ψ_j(\bm{u},v) = ≤ft[ 2 - \frac{\Vert \bm{u} - \bm{c}_j \Vert^2}{w_s^2}- \frac{|v - g_j|^2}{w_t^2} \right]^2 \cdot I(\Vert \bm{u} - \bm{c}_j \Vert ≤q w_s) \cdot I(|v - g_j| ≤q w_t)

for j = 1, …, r.

Due to the treatment of \bm{u}_i and \bm{c}_j as points in a Euclidean space, this basis is more suitable for coordinates from a map projection than coordinates based on a globe representation.

Value

A sparse n \times r matrix whose ith row is

\bm{s}_i^\top = \Big( ψ_1(\bm{u}_i,v_i), …, ψ_r(\bm{u}_i,v_i) \Big).