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mathematical optimization - How to find all the local minima/maxima in a range


I want to find :



  • all local maxima in range

  • all local minima in range



From those points I can interpolate and combine functions upper and lower boundary. What I am really interested in, is the mean function of those boundaries.


enter image description here


Data model for this plot:


GetRLine3[MMStdata_, IO_: 1][x_: x] := ListInterpolation[#, InterpolationOrder -> IO,
    Method -> "Spline"][x] & /@ (({{#[[1]]}, #[[2]]}) & /@ # & /@ MMStdata);
data = Transpose[{# + RandomReal[]*0.1 & /@ Range[-10, 30, 0.4], 
    Tanh[#] + (Sech[2 x - 0.5]/1.5 + 1.5) /. x -> # & /@ Range[-4, 4, 0.08]}];
xLimits = {Min@#1, Max@#1} & @@ Transpose[data];
f = D[GetRLine3[{data}, 3][x], x];


Edit: As my effort:


minimums = DeleteDuplicates[Round[x /. Last[FindMinimum[f, {x, #}]] & /@ Transpose[data][[1]], 0.0001]]
minimumvalues = (f /. x -> #)[[1]] & /@ minimums;
minimumData := Transpose[{minimums, minimumvalues}];
maximums = DeleteDuplicates[Round[x /. Last[FindMaximum[f, {x, #}]] & /@ Transpose[data][[1]], 0.0001]];
maximumsvalues = (f /. x -> #)[[1]] & /@ maximums;
maximumsData := Transpose[{maximums, maximumsvalues}];

maxf = Max[{GetRLine3[{maximumsData}, 3][x], f}]
minf = Min[{GetRLine3[{minimumData}, 3][x], f}]

mf = Mean[{maxf, minf}]

This was what I was trying to make: enter image description here


I still get quite few warnings and I'm sure it's not the best solution. I don't like the DeleteDuplicates@Round@ part, but it was necessarily to get the interpolation function working.



Answer



This can be done using event location within NDSolve. I start off as below (note f is slightly modified from what you have, mostly to rescale it).


GetRLine3[MMStdata_, IO_: 1][x_: x] := 
  ListInterpolation[#, InterpolationOrder -> IO, Method -> "Spline"][
     x] & /@ (({{#[[1]]}, #[[2]]}) & /@ # & /@ MMStdata);
data = Transpose[{# + RandomReal[]*0.1 & /@ Range[-10, 30, 0.4], 

    Tanh[#] + (Sech[2 x - 0.5]/1.5 + 1.5) /. x -> # & /@ 
     Range[-4, 4, 0.08]}];

xLimits = {Min@#1, Max@#1} & @@ Transpose[data];
f = First[100*D[GetRLine3[{data}, 3][x], x]];

We'll recapture f using NDSolve, and locate the points where the derivative vanishes in the process.


vals = Reap[
    soln = y[x] /. 
      First[NDSolve[{y'[x] == Evaluate[D[f, x]], 

         y[-9.9] == (f /. x -> -9.9)}, y[x], {x, -9.9, 30}, 
        Method -> {"EventLocator", "Event" -> y'[x], 
          "EventAction" :> Sow[{x, y[x]}]}]]][[2, 1]];

Visual check:


Plot[f, {x, -9.9, 30}, 
 Epilog -> {PointSize[Medium], Red, Point[vals]}]

enter image description here


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