Fitting problem

Question

I have two points stored in data:

data = {{0.0166667, 2.86927*10^-12}, {0.0333333, 1.12725*10^-11}};

data[[All,1]] contains the time t.

data[[All,2]] contains the first 2 points of the so called mean squared displacement.

To calculate the particle mass m the 2 points should be fitted with the following function:

k * T / m * t^2 (* fitting function *)

k = 1.3806488*10^-23;
T = 300;
(* m = particle mass in kg, is the fitting parameter *)

Manually I get a good fit if I take m = 4.1 * 10^-13kg.

dataPlot = 
 ListLinePlot[data, PlotStyle -> {Blue}, 
  Epilog -> {PointSize[Medium], Point[data]}];
manualFitPlot = 
  Plot[k*300/(4.1*10^-13)*t^2, {t, 1/60, 2/60}, PlotStyle -> {Green}];
Show[dataPlot, manualFitPlot, PlotRange -> All]

When I try to use NonlinearModelFit a wrong result for the mass is obtained:

nlm = NonlinearModelFit[data, k*300/m*t^2, {m}, t];
nlm["ParameterTable"]

$\small \begin{array}{l|llll} \text{} & \text{Estimate} & \text{Standard Error} & \text{t-Statistic} & \text{P-Value} \\ \hline m & 0.422029 & 4.36728\times 10^{11} & 9.66343 \times 10^{-13} & 1 \\ \end{array}$

I tried different initial values for m and also rescaled the small values by multiplying with (*10^10). It did not help.

How can I solve this?

Answer 1

I wouldn't be so quick to rule out rescaling your data or working in reduced units, since this works fine:

data = {{0.0166667, 2.86927*10^-12}, {0.0333333, 1.12725*10^-11}};
k = 1.3806488*10^-23;
T = 300;
nlm = NonlinearModelFit[data, k*300/(m*10^-13)*t^2, {m}, t, 
   Method -> "NMinimize"];

(* Gives m = 4.07829 *)

I think the problem really lies with the numbers being so small that the default gradient-based fitting methods evaluate the gradient as zero in machine precision.

So NMinimize works, as does PrincipalAxis, since both involve derivative-free methods. You can look into it further in the documentation.

---

Note that you can also do it without scaling, but specifying a starting value (this time using the PrincipalAxis method, which takes two starting values).

nlm = NonlinearModelFit[data, k*300/m*t^2, {{m, 10^-14, 5*10^-13}}, t, 
         EvaluationMonitor :> Print["x = ", m], 
         Method -> "PrincipalAxis"];
(* Gives m = 4.09157*10^-13 *)
(* Also outputs the steps *)

Answer 2

Just a quick glance, you can try this though. I didn't have time to plot the residuals (or take a look at them for that matter...no pun intended) but it could be worth taking a look at. The function seems to predict your data points with relative accuracy.

Answer 3

This model is linear in 1/m, hence use LinearModelFit

lm = LinearModelFit[data, {k*300 t^2}, t, IncludeConstantBasis -> False]

Then just invert the result:

1/lm["BestFitParameters"][[1]]

that gives

4.07829*10^-13

If you give different weights to each point, you will find another result. The "best" result will be obtained with proper weighting.