Misc

Pool-adjacent-violators algorithm
​Robert and Casella (2005) provides the following exercise related to pool-adjacent-violators algorithm.
We need to learn isotonic regression first, and it turns out that this is exactly the simply ordered case mentioned in the wiki.
Use R program to solve this problem, and the code is as follows.
pooladv <- function(f, w)
{
  n = length(f)
  lag = diff(f)
  if (sum(lag < 0) == 0) # f is isotonic
    return(f)
  while (TRUE) {
    idx = which(lag < 0)[1] + 1
    newf = (w[idx]*f[idx] + w[idx-1]*f[idx-1])/(w[idx]+w[idx-1])
    f[idx] = newf
    f[idx-1] = newf
    lag = diff(f)
    if (sum(lag < 0) == 0)
      return(f)
  }
}
​
## example
pooladv(c(23, 27, 25, 28), rep(1, 4))
# ans: 23, 26, 26, 28
We can also use another scientific programming language, Julia.
function pooladv(f::Array, w::Array)
   n = size(f, 1)
   lag = diff(f)
   if all(i -> i >= 0, lag)
        return(f)
   end
   while true
        for i = 1:(n-1)
            global idx
            idx = i+1
            lag[i] < 0 && break
        end
        newf = (w[idx]*f[idx] + w[idx-1]*f[idx-1])/(w[idx]+w[idx-1])
        f[idx] = newf
        f[idx-1] = newf
        lag = diff(f)
        if all(i -> i>=0, lag)
            return(f)
        end
   end
end
​
## example
g = pooladv([23, 27, 25, 28], ones(4))
for i in eachindex(g)
    println(g[i])
end
Tree ordering
In Robert and Casella (2005), there is another exercise about isotonic regression, which is the continuation of the previous section.
The following Julia program can be used to solve this exercise.
function treeordering(f::Array, w::Array)
    n = size(f, 1)
    lag = diff(f)
    # if f is isotonic
    if all(i -> i >= 0, lag)
        return(f)
    end
    for j = 1:n
        Aj = sum(f[1:j].*w[1:j])/sum(w[1:j])
        if Aj < f[j+1]
            return([Aj*ones(j); f[(j+1):end]])
        end
    end
end
​
## example
treeordering([18,17,12,21,16], [1,1,3,3,1])
Parallel Tempering
R vs. Julia
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