Skip to main content

differential equations - Fails to apply NonlinearModelFit on a numerically evaluated model defined as a system of ODEs


I call for your help with because I'm getting troubles to fit a model defined through a system of ODEs. The system of ODEs is as follows:


X'[t]:= m[t].X[t]
X[t_] := {h[t], r[t], rh1[t], rh2[t], rh3[t]}
m[t_] := {{-k1*r[t], 0, ki1, 0, 0}, {0, -k1*h[t], ki1, 0, 0}, {0,   k1*h[t], -(ki1 + k2), ki2, 0}, {0, 0, k2, -(ki2 + k3), ki3}

(It describes a chemical reaction consisting in three sequential steps, the first one is a bimolecular association and the following steps are unimolecular reactions). And the actual quantity I need to evaluate is:


F:= aF.X[t]

aF:={0,aR,aRH1,aRH2,aRH3}

(This quantity represents what I can measure experimentally). Well, following the directions found in the Wizard and in several forums I've tried the following.


Definition of the model:


Clear[modelo];
modelo[k1_?NumberQ, ki1_?NumberQ, k2_?NumberQ, ki2_?NumberQ, k3_?NumberQ, ki3_?NumberQ, aR_?NumberQ, aRH1_?NumberQ, aRH2_?NumberQ, aRH3_?NumberQ, Ht_?NumberQ, Rt_?NumberQ]:= (modelo[k1, ki1, k2, ki2, k3, ki3, aR, aRH1, aRH2, aRH3, Ht, Rt] = Module[{X, X0, m, vectorR, aF, sol, F}, First[{F,      {X[t_] := {h[t], r[t], rh1[t], rh2[t], rh3[t]};
   X0 := X[0] == {Ht, Rt, 0, 0, 0};       
   m[t_] := {{-k1*r[t], 0, ki1, 0, 0}, {0, -k1*h[t], ki1, 0, 
      0}, {0, k1*h[t], -(ki1 + k2), ki2, 0}, {0, 0, 
      k2, -(ki2 + k3), ki3}, {0, 0, 0, k3, -ki3}};

   aF := {0, aR, aRH1, aRH2, aRH3};
   sol := 
    NDSolve[{X'[t] == m[t].X[t], X0}, {h, r, rh1, rh2, rh3}, {t, 
      0, 500}];
   F := aF.Flatten[(X[t] /. sol)] - aR*Rt}}]])

The model so defined seems to work since, for example, it was evaluated and plotted smoothly by the following command:


aF = 0.0034;
Ht = 800;
Rt = 62; (*The last three quantities are not fitting parameters but fixed ones, whose values are known. *)

Fsim := With[{k1 = 0.0012, ki1 = 5, k2 = 0.43, ki2 = 0.0055, k3 = 0.01, ki3 = 0.0096, aRH1 = .032, aRH2 = .01, aRH3 = .012}, modelo[k1, ki1, k2, ki2, k3, ki3, aR, aRH1, aRH2, aRH3, Ht, Rt]];
LogLinearPlot[Evaluate[Fsim], {t, 0.001, 500}, PlotRange -> All, FrameLabel -> {"t(s)", "F"}]

Data and Fitting procedure:


To test that things work, I tried to fit the model to only one time series. Here is a fake data table:


data = Table[{t, .22 (1 - E^(-7.2 t)) + 0.10 (1 - E^(-0.084 t)) + 0.15 (1 - E^(-0.027 t))}, {t, 0, 500, 0.1}]

which actually proceeds from a fitting of that function to the real experimental data.


Finally, I've tried the following code to fit the model (only three parameters of it) to these data:


k1 = 0.0012;

k2 = 0.43;
ki2 = 0.0055;
k3 = 0.01;
ki3 = 0.0096;
aR = 0.0034;
Ht = 800;
Rt = 62; (*The last three quantities are not fitting parameters but fixed ones, whose values are known. *)
fit = NonlinearModelFit[data, {modelo[k1, ki1, k2, ki2, k3, ki3, aR, aRH1, aRH2, aRH3, Ht, Rt], {2 <= ki1 <= 20, 0.001 <= aRH1 <= 0.1, 0.001 <= aRH2 <= 0.1, 0.001 <= aRH3 <= 0.1}}, {{ki1, 5}, {aRH1, .032}, {aRH2, .01}, {aRH3, .012}}, t]

The problem is that it never worked, and I don't know where is the issue. It returns the following errors:




NonlinearModelFit::nrnum: The function value 1/2 ((-0.700176+<<3>>+0.0034 InterpolatingFunction[{{<<2>>}},{4,7,1,{<<1>>},{<<1>>},0,0,0,0,Automatic, },{},False},{{<<563>>}},{Developer`PackedArrayForm,{<<564>>},{<<1126>>}},{Automatic}][t])^2+(-0.700176+<<3>>+0.0034 InterpolatingFunction[{{<<2>>}},{4,7,1,{<<1>>},{<<1>>},0,0,0,0,Automatic,{},{},False},{{<<563>>}},<<1>>,{Automatic}][t])^2+(-0.700175+<<3>>+0.0034 <<1>>)^2+(<<1>>)^2+(<<1>>)^2+<<1>>^2+<<39>>+<<1>>+(<<1>>)^2+(<<1>>)^2+(<<1>>)^2+(-0.536879+<<3>>+0.0034 InterpolatingFunction[{{<<2>>}},<<3>>,{<<9>>}][t])^2+<<1>>) is not a real number at {ki1,aRH1,aRH2,aRH3} = {5.,0.032,0.01,0.012}. >>



All suggestions will be welcome.



Answer



As I find your expressions difficult to work with, let me rewrite them somewhat:


Clear[modelo];
modelo[k1_?NumberQ, ki1_?NumberQ, k2_?NumberQ, ki2_?NumberQ, k3_?NumberQ, 
       ki3_?NumberQ, aR_?NumberQ, aRH1_?NumberQ, aRH2_?NumberQ, aRH3_?NumberQ, 
       Ht_?NumberQ, Rt_?NumberQ] := 

  modelo[k1, ki1, k2, ki2, k3, ki3, aR, aRH1, aRH2, aRH3, Ht, Rt] =
  Module[{X, m,  aF,  sol},
   X[t_] := {h[t], r[t], rh1[t], rh2[t], rh3[t]};
   m[t_] := {{-k1*r[t], 0, ki1, 0, 0}, {0, -k1*h[t], ki1, 0, 0}, 
             {0, k1*h[t], -(ki1 + k2), ki2, 0}, {0, 0, k2, -(ki2 + k3), ki3}, 
             {0, 0, 0, k3, -ki3}};
   aF = {0, aR, aRH1, aRH2, aRH3};
  sol = NDSolve[ {X'[t] == m[t].X[t], X[0] == {Ht, Rt, 0, 0, 0}}, X[t], {t, 0, 500}];
  Function[{tu}, Evaluate[aF.Flatten[X[t] /. sol] - aR*Rt] /. t :> tu]]


data = Table[{t, .22 (1 - E^(-7.2 t)) + 0.10 (1 - E^(-0.084 t)) + 
                  0.15 (1 - E^(-0.027 t))}, {t, 0, 500, 1}];

k1 = 0.0012; k2 = 0.43; ki2 = 0.0055; k3 = 0.01;  ki3 = 0.0096;
aR = 0.0034; Ht = 800; Rt = 62;
fit = 
 NonlinearModelFit[data, 
   {modelo[k1, ki1, k2, ki2, k3, ki3, aR, aRH1, aRH2, aRH3, Ht, Rt][t], 
    {2 <= ki1 <= 20, 0.001 <= aRH1 <= 0.1, 0.001 <= aRH2 <= 0.1, 0.001 <= aRH3 <= 0.1}}, 
   {{ki1, 5}, {aRH1, .032}, {aRH2, .01}, {aRH3, .012}}, t,  

  Method -> {NMinimize, Method -> "NelderMead"}]

fit["BestFitParameters"]
(* {ki1 -> 7.23261, aRH1 -> 0.0364504, aRH2 -> 0.0108148,  aRH3 -> 0.0116011} *)

Comments

Post a Comment

Popular posts from this blog

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]
Post a Comment
Read more

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]
Post a Comment
Read more

plotting - Adding a thick curve to a regionplot

Suppose we have the following simple RegionPlot: f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}] Now I'm trying to change the curve defined by $y=g[x]$ into a thick black curve, while leaving all other boundaries in the plot unchanged. I've tried adding the region $y=g[x]$ and playing with the plotstyle, which didn't work, and I've tried BoundaryStyle, which changed all the boundaries in the plot. Now I'm kinda out of ideas... Any help would be appreciated! Answer With f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 You can use Epilog to add the thick line: RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}, PlotPoints -> 50, Epilog -> (Plot[g[x], {x, 0, 2}, PlotStyle -> {Black, Thick}][[1]]), PlotStyle -> {Directive[Yellow, Opacity[0.4]], Directive[Pink, Opacity[0.4]],
Post a Comment
Read more