list manipulation - How to fit a function to data so that the fit is always greater than or equal to the data?

b = nst[n_] := 
  Length[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, 
    n, # > 1 &]]; 
nn = 500; 
With[{stps = Array[nst, nn]}, 

  Table[Max[Take[stps, n]], {n, nn}]
]

I'm working with the following list and I am trying to find a fit so that it's always greater than the data rather then the normal fitting method used in the FindFit function:

FindFit[b, x + y*Log[z], {x, y}, z]

I like the ability to change the fitting model in the FindFit function but I can't figure out how to set it for what I want. Help would be appreciated.

Answer

Create the list b as you have shown.

nst[n_] := Length[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]]


b = With[{stps = Array[nst, nn]}, Table[Max[Take[stps, n]], {n, nn}]];

It looks like

ListPlot[b, PlotStyle -> Blue]

It is apparent that we want to locate the first point in each group of horizontal points and use that in the constraint.

Those points can be located as follows:

data = Transpose@Join[{Range[500], b}];

(* {{{1, 1}, {2, 2}, {3, 8}, ..., {500, 144}} *)

data is a list of {index, b} pairs.

Next locate the positions where there is a jump.

pos = Position[Differences[b], x_ /; x > 0] + 1

Build a list of constraints forcing the desired function to exceed the y value at those positions.

constraints = 
 Map[x + y*Log[#[[1]]] >= #[[2]] &, Extract[data, pos]]
(* {x + y Log[2] >= 2, x + y Log[3] >= 8, x + y Log[6] >= 9, 

 x + y Log[7] >= 17, x + y Log[9] >= 20, x + y Log[18] >= 21, 
 x + y Log[25] >= 24, x + y Log[27] >= 112, x + y Log[54] >= 113, 
 x + y Log[73] >= 116, x + y Log[97] >= 119, x + y Log[129] >= 122, 
 x + y Log[171] >= 125, x + y Log[231] >= 128, x + y Log[313] >= 131, 
 x + y Log[327] >= 144} *)

Use the constraints in FindFit.

solution = 
 FindFit[b, {x + y*Log[z], Sequence @@ constraints}, {x, y}, z]
(* {x -> 69.7139, y -> 12.8302} *)

Plot it to validate the solution

Show[
 ListPlot[list, PlotStyle -> Blue],
 Plot[Evaluate[x + y*Log[z] /. solution], {z, 1, 500},
  PlotStyle -> Black]
 ]