Normalized inverse distance weighting applied over a grid

matrix_distance_weighting.r

#!/usr/bin/env Rscript

n <- 5

# fill matrix of distances
d <- matrix(nrow = n, ncol = n)

d[1:n, 1] <- 0:(n-1)
d[1, 1:n] <- 0:(n-1)

d <- d / (n - 1)

for (i in 2:n) {
  for (j in 2:n) {
    d[i, j] = sqrt(d[i, 1]^2 + d[1, j]^2)
  }
}

# fill matrix of weights
w <- d
w[ , ] <- NA

for (i in 2:(n-1)) {
  for (j in 2:(n-1)) {
    r1 <- d[i, j]
    r2 <- d[(n+1)-i, j]
    r3 <- d[(n+1)-i, (n+1)-j]
    r4 <- d[i, (n+1)-j]
    w[i, j] = (1/r1) / ((1/r1) + (1/r2) + (1/r3) + (1/r4))
  }
}

# ensure the sum of applied weights is always 1
ws <- w
ws[ , ] <- NA
for (i in 2:(n-1)) {
  for (j in 2:(n-1)) {
    w1 <- w[i, j]
    w2 <- w[(n+1)-i, j]
    w3 <- w[(n+1)-i, (n+1)-j]
    w4 <- w[i, (n+1)-j]
    ws[i, j] = w1 + w2 + w3 + w4
  }
}

# distance from (0, 0)
d
#      [,1]      [,2]      [,3]      [,4]     [,5]
# [1,] 0.00 0.2500000 0.5000000 0.7500000 1.000000
# [2,] 0.25 0.3535534 0.5590170 0.7905694 1.030776
# [3,] 0.50 0.5590170 0.7071068 0.9013878 1.118034
# [4,] 0.75 0.7905694 0.9013878 1.0606602 1.250000
# [5,] 1.00 1.0307764 1.1180340 1.2500000 1.414214


# weight applied
w
#      [,1]      [,2]      [,3]      [,4] [,5]
# [1,]   NA        NA        NA        NA   NA
# [2,]   NA 0.4488813 0.3086089 0.2007458   NA
# [3,]   NA 0.3086089 0.2500000 0.1913911   NA
# [4,]   NA 0.2007458 0.1913911 0.1496271   NA
# [5,]   NA        NA        NA        NA   NA

# sum of applied weights (should be 1)
ws
#      [,1] [,2] [,3] [,4] [,5]
# [1,]   NA   NA   NA   NA   NA
# [2,]   NA    1    1    1   NA
# [3,]   NA    1    1    1   NA
# [4,]   NA    1    1    1   NA
# [5,]   NA   NA   NA   NA   NA

df <- data.frame(d = c(d), w = c(w))
with(na.omit(df), plot(d, w, type = 'p'))