How do I fit NDSolve result to experimental data? [duplicate]

Question

I have two coupled ordinary differential equations which I solve numerically in Mathematica, but now I want to fit the solution with experimental data.

I do following, (the experimental data corresponds to z[t] in my notation)

ss = 
  ParametricNDSolve[{
    z'[t] == 4*(1 - X^-1)*z[t] + 4*z[t]*y[t]/X - 8*(z[t])^2 && 
      y'[t] == γ*z[t] - γ*y[t]/X && z[0] == z0 && 
      y[0] == y0} /. {X -> 20}, 
    {z, y}, {t, 0, 5000}, {γ, z0, y0}]

I know the value of X = 20. The rest has to be determined by fitting. Also time has to be rescaled to fit with experimental data. hence in total of 4 parameters.

γ and y0 does not have any constraint apart from being positive. but x0 has to be very small (close to zero).

data = {{4.01, 0.0338}, {6.02, 0.0719}, {7.99, 0.14}, {9.99, 0.216}, {9.98, 
  0.25}, {12., 0.307}, {12., 0.35}, {14., 0.459}, {16., 0.558}, {16., 
  0.605}, {18., 0.688}, {20., 0.767}, {22., 0.848}, {24., 
  0.898}, {26., 0.932}, {28., 0.965}, {30., 0.98}, {32., 0.987}, {34.,
   0.99}, {36., 0.994}, {38., 0.983}, {40., 0.973}, {42., 
  0.968}, {44., 0.95}, {46., 0.941}, {48., 0.933}, {50., 0.918}, {52.,
   0.911}, {54., 0.901}, {56., 0.89}, {58., 0.881}, {60., 
  0.871}, {62., 0.864}, {64., 0.861}, {66., 0.854}, {68., 
  0.846}, {70., 0.845}, {72., 0.837}, {74., 0.835}, {76., 
  0.825}, {78., 0.825}, {80., 0.822}, {82., 0.817}, {84., 
  0.813}, {84., 0.82}, {86., 0.812}, {88., 0.816}, {90., 0.808}, {92.,
   0.805}, {94., 0.807}, {96., 0.802}, {98., 0.798}, {100., 
  0.799}, {102., 0.797}, {104., 0.79}, {106., 0.797}, {108., 
  0.792}, {110., 0.791}, {112., 0.788}, {114., 0.792}, {116., 
  0.786}, {118., 0.787}, {120., 0.785}, {122., 0.783}, {124., 
  0.788}, {126., 0.783}, {128., 0.781}, {130., 0.78}, {132., 
  0.78}, {134., 0.775}, {136., 0.774}, {138., 0.777}, {140., 
  0.775}, {142., 0.774}, {144., 0.77}, {146., 0.773}, {148., 
  0.772}, {150., 0.768}, {152., 0.772}, {154., 0.764}, {156., 
  0.77}, {158., 0.769}, {160., 0.771}, {162., 0.769}, {164., 
  0.769}, {166., 0.77}, {168., 0.764}, {170., 0.765}, {172., 
  0.762}, {174., 0.761}, {176., 0.762}, {178., 0.765}, {180., 
  0.764}, {182., 0.763}, {184., 0.764}, {186., 0.761}, {188., 
  0.762}, {190., 0.765}, {192., 0.76}, {194., 0.756}, {196., 
  0.764}, {198., 0.757}, {200., 0.762}, {202., 0.758}, {204., 
  0.758}, {206., 0.757}, {208., 0.753}, {210., 0.759}, {212., 
  0.757}, {214., 0.758}, {216., 0.76}, {218., 0.755}, {220., 
  0.758}, {222., 0.754}, {224., 0.758}, {226., 0.754}, {228., 
  0.755}, {230., 0.753}, {232., 0.753}}

and then I use NonlinearModelFit](http://reference.wolfram.com/mathematica/ref/NonlinearModelFit.html) to obtain a fit, but it's really bad:

fit1 = 
  NonlinearModelFit[
    datak, 
    {z[γ, z0, y0][α*t] /. ss, γ > 0, 0 < z0 < 0.05, y0 > 0, 0 < α < 0.5}, 
    {γ, z0, y0, α}, t]

Answer 1

using technique described at NonlinearModelFit does not find a good fit

I managed to get a much better with some rearrangements,

ss = ParametricNDSolve[{
      z'[t] == 4*(1 - X^-1)*z[t] + 4*z[t]*y[t]/X - 8*(z[t])^2 && 
      y'[t] == \[Gamma]*z[t] - \[Gamma]*y[t]/X && z[0] == z0 && 
      y[0] == y0} /. {X -> 3, z0 -> 0.01},
      {z, y}, {t, 0, 5000}, {y0, \[Gamma]}
     ]

data = {{4.01, 0.0338}, {6.02, 0.0719}, {7.99, 0.14}, {9.99, 0.216}, {9.98, 
  0.25}, {12., 0.307}, {12., 0.35}, {14., 0.459}, {16., 0.558}, {16., 
  0.605}, {18., 0.688}, {20., 0.767}, {22., 0.848}, {24., 
  0.898}, {26., 0.932}, {28., 0.965}, {30., 0.98}, {32., 0.987}, {34.,
   0.99}, {36., 0.994}, {38., 0.983}, {40., 0.973}, {42., 
  0.968}, {44., 0.95}, {46., 0.941}, {48., 0.933}, {50., 0.918}, {52.,
   0.911}, {54., 0.901}, {56., 0.89}, {58., 0.881}, {60., 
  0.871}, {62., 0.864}, {64., 0.861}, {66., 0.854}, {68., 
  0.846}, {70., 0.845}, {72., 0.837}, {74., 0.835}, {76., 
  0.825}, {78., 0.825}, {80., 0.822}, {82., 0.817}, {84., 
  0.813}, {84., 0.82}, {86., 0.812}, {88., 0.816}, {90., 0.808}, {92.,
   0.805}, {94., 0.807}, {96., 0.802}, {98., 0.798}, {100., 
  0.799}, {102., 0.797}, {104., 0.79}, {106., 0.797}, {108., 
  0.792}, {110., 0.791}, {112., 0.788}, {114., 0.792}, {116., 
  0.786}, {118., 0.787}, {120., 0.785}, {122., 0.783}, {124., 
  0.788}, {126., 0.783}, {128., 0.781}, {130., 0.78}, {132., 
  0.78}, {134., 0.775}, {136., 0.774}, {138., 0.777}, {140., 
  0.775}, {142., 0.774}, {144., 0.77}, {146., 0.773}, {148., 
  0.772}, {150., 0.768}, {152., 0.772}, {154., 0.764}, {156., 
  0.77}, {158., 0.769}, {160., 0.771}, {162., 0.769}, {164., 
  0.769}, {166., 0.77}, {168., 0.764}, {170., 0.765}, {172., 
  0.762}, {174., 0.761}, {176., 0.762}, {178., 0.765}, {180., 
  0.764}, {182., 0.763}, {184., 0.764}, {186., 0.761}, {188., 
  0.762}, {190., 0.765}, {192., 0.76}, {194., 0.756}, {196., 
  0.764}, {198., 0.757}, {200., 0.762}, {202., 0.758}, {204., 
  0.758}, {206., 0.757}, {208., 0.753}, {210., 0.759}, {212., 
  0.757}, {214., 0.758}, {216., 0.76}, {218., 0.755}, {220., 
  0.758}, {222., 0.754}, {224., 0.758}, {226., 0.754}, {228., 
  0.755}, {230., 0.753}, {232., 0.753}};

fit = ComplexFit[
  data, {model = \[Beta]*z[y0, \[Gamma]][\[Alpha]*t] /. ss, 
   TransformedParameter[Re, y0] > 8, 
   TransformedParameter[Re, \[Gamma]] > 5, 
   TransformedParameter[Re, \[Beta]] > 1, 
   TransformedParameter[Re, \[Alpha]] < 0.05},
   {y0, \[Gamma], \[Alpha], \[Beta]}, t, 
  "CoordinateSystem" -> "Real", 
  Method -> {NMinimize, 
    Method -> {"DifferentialEvolution", "SearchPoints" -> 4, 
      "ScalingFactor" -> 0.95, "CrossProbability" -> 0.5, 
      "PostProcess" -> {FindMinimum, Method -> "QuasiNewton"}}}]

Show[{ListPlot[data, AxesOrigin -> {0, 0}, PlotRange -> All, 
   Frame -> True, 
   FrameLabel -> {Style["Total A\[Beta] conc", 30], 
     Style["\!\(\*SubscriptBox[\(\[Tau]\), \(lag\)]\)(MCS units)", 
      30]}, ImageSize -> {600, 400}, FrameTicks -> Automatic, 
   FrameTicksStyle -> Directive[30], 
   PlotStyle -> Directive[PointSize[0.01], Red]], 
  Plot[model /. fit, {t, Min@data[[All, 1]], Max@data[[All, 1]]}, 
   PlotStyle -> Thickness[0.005], PlotRange -> All]}]