Is it feasible to curve fit to a staircase function?

Question

Taking as a starting point the advice given in this post I am trying to fit data to a staircase function as follows:

model = {h (1/2 + ((x - a)/w) - ArcTan[Tan[π (-(1/2) + ((x - a)/w))]]/π), 60 <= a <= 90, 50 <= h <= 150, 50 <= w <= 150};
nlm2 = FindFit[ptsech, model, {a, h, w}, x, Method -> "NMinimize"]

but now I'm wondering is it even possible to fit such data to a discontinous function? though it approximates the data as I wish. The problem is that code it not producing correct values for h, w and a even when manually bringing these values close. How can it be modified to work?

ptsech={{0, 0}, {30., 0}, {54.366, 0.00901776}, {67.0222, 
0.477768}, {70.7722, 4.22777}, {71.241, 16.884}, {71.25, 
41.25}, {71.25, 71.25}, {71.259, 95.616}, {71.7278, 
108.272}, {75.4778, 112.022}, {88.134, 112.491}, {112.5, 
112.5}, {142.5, 112.5}, {166.866, 112.509}, {179.522, 
112.978}, {183.272, 116.728}, {183.741, 129.384}, {183.75, 
153.75}, {183.75, 183.75}, {183.759, 208.116}, {184.228, 
220.772}, {187.978, 224.522}, {200.634, 224.991}, {225., 
225.}, {255., 225.}, {279.366, 225.009}, {292.022, 
225.478}, {295.772, 229.228}, {296.241, 241.884}, {296.25, 
266.25}, {296.25, 296.25}, {296.259, 320.616}, {296.728, 
333.272}, {300.478, 337.022}, {313.134, 337.491}, {337.5, 
337.5}, {367.5, 337.5}}

the problem is the constraint 50 <= w <= 50. Change it into e.g. 50 <= w <= 150 and it'll work just fine. — AccidentalFourierTransform, Feb 15 '20 at 17:11
@AccidentalFourierTransform to be fair to myself the contraints I used were much wider; the 50 was an editing error. So that alone did not actually cause the problem. The solution lies in reducing the data sample which is not obvious and not necessarily the only solution. — onepound, Feb 16 '20 at 10:24
I see. If you edit your post to fix that, I'll vote to reopen. — AccidentalFourierTransform, Feb 16 '20 at 12:42

score 4 · Accepted Answer · answered Feb 16 '20 at 03:14

ptsech = {{0, 0}, {30., 0}, {54.366, 0.00901776}, {67.0222, 
    0.477768}, {70.7722, 4.22777}, {71.241, 16.884}, {71.25, 41.25}, {71.25, 
    71.25}, {71.259, 95.616}, {71.7278, 108.272}, {75.4778, 112.022}, {88.134,
     112.491}, {112.5, 112.5}, {142.5, 112.5}, {166.866, 112.509}, {179.522, 
    112.978}, {183.272, 116.728}, {183.741, 129.384}, {183.75, 
    153.75}, {183.75, 183.75}, {183.759, 208.116}, {184.228, 
    220.772}, {187.978, 224.522}, {200.634, 224.991}, {225., 225.}, {255., 
    225.}, {279.366, 225.009}, {292.022, 225.478}, {295.772, 
    229.228}, {296.241, 241.884}, {296.25, 266.25}, {296.25, 
    296.25}, {296.259, 320.616}, {296.728, 333.272}, {300.478, 
    337.022}, {313.134, 337.491}, {337.5, 337.5}, {367.5, 337.5}};

{xmin, xmax} = MinMax@ptsech[[All, 1]];

As pointed out in a comment by AccidentalFourierTransform, the range of w should be widened.

model = {h (1/2 + ((x - a)/w) - 
      ArcTan[Tan[π (-(1/2) + ((x - a)/w))]]/π), 60 <= a <= 90, 
   50 <= h <= 150, 50 <= w <= 150};

nlm2 = FindFit[ptsech, model, {a, h, w}, x, Method -> "NMinimize"]

(* {a -> 70.8647, h -> 102.825, w -> 112.674} *)

Plot[Evaluate[model /. nlm2], {x, xmin, xmax},
 PlotRange -> {-10, 350},
 Epilog -> {Red, AbsolutePointSize[3],
   Point[ptsech]}]

Assuming that the near vertical runs are causing the offsets, reduce the data set prior to fitting.

ptsechr = Union[ptsech,
   SameTest -> (Abs[#1[[1]] - #2[[1]]] < 1 &)];

nlm2r = FindFit[ptsechr, model, {a, h, w}, x, Method -> "NMinimize"]

(* {a -> 71.5147, h -> 112.658, w -> 113.258} *)

Plot[Evaluate[model /. nlm2r], {x, xmin, xmax},
 Epilog -> {Red, AbsolutePointSize[3],
   Point[ptsech]}]

The offset has been corrected.

Is it feasible to curve fit to a staircase function?

1 Answers1