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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]

Mathematica graphics


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]
 ]

Mathematica graphics


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