# coefficients for 12nn priors.
           x         y
   o       0         2
  x+x    -101       111
 o+*+o -2-1012     00000
  x+x    -101       111
   o       0         2

x_c(0,-1,0,1,-2,-1,1,2,-1,0,1,0)
y_c(2,1,1,1,0,0,0,0,-1,-1,-1,-2)

X_cbind(1,x,y,x^2,y^2,x*y)

( solve(t(X)%*%X)%*%t(X) )[1,]

# lq coef's o=-1/8 x=1/8 +=1/4

X_cbind(1,x,y)

( solve(t(X)%*%X)%*%t(X) )[1,]

# coefficients for 40nn priors

                x         y
    8           0          4
   757         101        333
  64346       21012      22222
 7421247     3210123    1111111
8531 1358   4321 1234  0000 0000    
 7421247     3210123    1111111
  64346       21012      22222
   757         101        333
    8           0          4

x_scan()
              0 
           -1 0 1  
        -2 -1 0 1 2 
     -3 -2 -1 0 1 2 3  
  -4 -3 -2 -1   1 2 3 4 
     -3 -2 -1 0 1 2 3  
        -2 -1 0 1 2   
           -1 0 1   
              0    

y_scan()
            4
         3  3  3
      2  2  2  2  2
   1  1  1  1  1  1  1
0  0  0  0     0  0  0  0
  -1 -1 -1 -1 -1 -1 -1 
     -2 -2 -2 -2 -2 
        -3 -3 -3 
           -4

type_scan()
        8     
      7 5 7   
    6 4 3 4 6 
  7 4 2 1 2 4 7   
8 5 3 1   1 3 5 8      
  7 4 2 1 2 4 7   
    6 4 3 4 6    
      7 5 7     
        8     

X_cbind(1,x,y,x^2,y^2,x*y)

beta_( solve(t(X)%*%X)%*%t(X) )[1,]