# spatial_bisquare: Spatial Bisquare Basis In stcos: Space-Time Change of Support

## Description

Spatial bisquare basis on point data.

## Usage

 1 spatial_bisquare(dom, knots, w) 

## Arguments

 dom Points \bm{u}_1, …, \bm{u}_n to evaluate. See "Details". knots Knots \bm{c}_1, …, \bm{c}_r for the basis. See "Details". w 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 two-dimensional POINT entries are expected in st_geometry(dom). Otherwise, dom will be interpreted as an n \times 2 numeric matrix.

• If an sf or sfc object is provided for knots, r two-dimensional POINT entries are expected in st_geometry(knots). Otherwise, knots will be interpreted as an r \times 2 numeric matrix.

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

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

\varphi_j(\bm{u}) = ≤ft[ 1 - \frac{\Vert\bm{u} - \bm{c}_j \Vert^2}{w^2} \right]^2 \cdot I(\Vert \bm{u} - \bm{c}_j \Vert ≤q w)

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( \varphi_1(\bm{u}_i), …, \varphi_r(\bm{u}_i) \Big).

Other bisquare: areal_spacetime_bisquare(), areal_spatial_bisquare(), spacetime_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 set.seed(1234) # Create knot points seq_x = seq(0, 1, length.out = 3) seq_y = seq(0, 1, length.out = 3) knots = expand.grid(x = seq_x, y = seq_y) knots_sf = st_as_sf(knots, coords = c("x","y"), crs = NA, agr = "constant") # Points to evaluate x = runif(50) y = runif(50) pts = data.frame(x = x, y = y) dom = st_as_sf(pts, coords = c("x","y"), crs = NA, agr = "constant") rad = 0.5 spatial_bisquare(cbind(x,y), knots, rad) spatial_bisquare(dom, knots, rad) # Plot the knots and the points at which we evaluated the basis plot(knots[,1], knots[,2], pch = 4, cex = 1.5, col = "red") points(x, y, cex = 0.5) # 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, rad * sin(tseq) + seq_y) lines(coords, col = "red")