Nothing
emp_density<-
function(dat, L=NULL) ##density with respect to Lebesgue
##on the projected unit simplex
{ndat=nrow(dat)
if(is.null(L))
{
L=max(5,floor(sqrt(ndat/100))) #200
}
Mdat=L*dat[,c(1,2)]
Dens=matrix(0,nrow=L,ncol=L)
for(k in (1:ndat))
{
i0=ceiling(Mdat[k,1])
j0=ceiling(Mdat[k,2])
Dens[i0,j0]= Dens[i0,j0]+1#*((i0+j0)<=L)
}
return(list(l_grid=L,
emp_dens=Dens/(ndat)*L^2 ))
}
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