# 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).

Other bisquare: areal_spacetime_bisquare(), areal_spatial_bisquare(), spatial_bisquare()
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 set.seed(1234) # Create knot points seq_x = seq(0, 1, length.out = 3) seq_y = seq(0, 1, length.out = 3) seq_t = seq(0, 1, length.out = 3) knots = expand.grid(x = seq_x, y = seq_y, t = seq_t) knots_sf = st_as_sf(knots, coords = c("x","y","t"), crs = NA, dim = "XYM", agr = "constant") # Points to evaluate x = runif(50) y = runif(50) t = sample(1:3, size = 50, replace = TRUE) pts = data.frame(x = x, y = y, t = t) dom = st_as_sf(pts, coords = c("x","y","t"), crs = NA, dim = "XYM", agr = "constant") rad = 0.5 spacetime_bisquare(cbind(x,y,t), knots, w_s = rad, w_t = 1) spacetime_bisquare(dom, knots_sf, w_s = rad, w_t = 1) # Plot the (spatial) knots and the points at which we evaluated the basis plot(knots[,1], knots[,2], pch = 4, cex = 1.5, col = "red") text(x, y, labels = t, cex = 0.75) # Draw a circle representing the basis' radius around one of the knot points tseq = seq(0, 2*pi, length=100) coords = cbind(rad * cos(tseq) + seq_x[2], rad * sin(tseq) + seq_y[2]) lines(coords, col = "red")