10
$\begingroup$

I have several large data sets which follow the following pattern: A position is measured, a force is applied until a new equilibrium is found.
I'd like to find a fit for the position, at least at the plateaus, and preferably of the inter lying section, which in this case approaches a line.
I tried fitting the data with Clip, and with Piecewise. 

nlm = NonlinearModelFit[v40s1000h,Piecewise[{{a, x < A}, {b, x > B}}], {a, b, A, B, c, d}, x]

This creates a decent fit only if I specify the values for A and B, but then I have to estimate those values for each data set manually. It also doesn't really work to just add NMinimize, or add the piecewise part for the middle bit.
Is There anything else I can try?

Setting the A to 200 and B to 250 works decently.

http://s000.tinyupload.com/?file_id=35616536027018518052 << file

Improve this question
$\endgroup$
8
  • $\begingroup$ Could you post a link to the dataset v40s1000h. An idea would be to create the first few terms of a Fourier expansion and fit that within the domain of interest. $\endgroup$
    – gpap
    Commented Apr 10, 2014 at 10:08
  • $\begingroup$ @Feyre I don't understand, you're saying that you need to manually specify A,B, but in your example you are fitting the parameters A,B. $\endgroup$
    – sam84
    Commented Apr 10, 2014 at 10:09
  • 1
    $\begingroup$ How about fitting a sigmoid? $\endgroup$
    – Peltio
    Commented Apr 10, 2014 at 10:55
  • 2
    $\begingroup$ I think NonlinearModelFit works my minimizing some cost function (e.g. total squared error), but the cost function can't be evaluated if your piecewise function doesn't return a value for some values of x. Have you tried using something like {(a+b)/2,A<x<B} as a fallback? Alternatively, you could come up with your own cost function and just optimize that. $\endgroup$
    – Niki Estner
    Commented Apr 10, 2014 at 11:06
  • $\begingroup$ well, these are guesses I made, and they fit, the images was made with those guesses, which does work, but I'd have to guess for each separate file. Fitting with just the parameters A and B yields just a single straight line, and it tries to fit them around ~13x. gpap, Peltio I haven't had too much maths yet, I know basic Fourier series and transforms, but that's it. Could you give more of an idea (or a link) on how to try this in Mathematica? $\endgroup$
    – Feyre
    Commented Apr 10, 2014 at 11:10

1 Answer 1

Reset to default
15
$\begingroup$

As usual, it's a matter of choosing a better starting values for the parameters.

I started writing this answer before the data file was uploaded, so here's some synthetic data:

data = Table[
   Interpolation[{{0, 1.1*^-6}, {200, 1.1*^-6}, {250, 9.5*^-7}, {500, 
       9.5*^-7}}, x, InterpolationOrder -> 1] + 
    2*^-8 RandomVariate[NormalDistribution[]], {x, 0, 500}];
ListPlot[data]

enter image description here

As @nikie suggested in the comments, you should define your model to return something in the intermediate values. I'll use a linear transition because that seems to match your data.

f[x_] := Piecewise[{{a, x < xa}, {a + (b - a) (x - xa)/(xb - xa), xa <= x < xb}, {b, xb <= x}}]

Now we do the fit:

fit = NonlinearModelFit[data, f[x], {a, b, xa, xb}, x];
fit // Normal // InputForm
(* Piecewise[{{1., x < 1.}, {1.016964195743597*^-6, 1. <= x}}, 0] *)

No good. Why? The documentation for NonlinearModelFit says "give starting values when parameters are far from the default value 1". Let's try that.

fit = NonlinearModelFit[data, f[x], {a, b, {xa, 100}, {xb, 200}}, x];
fit // Normal // InputForm
(* Piecewise[{{1.1004729741760495*^-6, x < 200.73035367387615}, 
  {1.1004729741760495*^-6 - 2.914483888454594*^-9*(-200.73035367387615 + x), 
   200.73035367387615 <= x < 253.00000699926912}, 
  {9.481339117040847*^-7, 253.00000699926912 <= x}}, 0] *)

That's better.

Show[ListPlot[data], Plot[fit[x], {x, 0, 500}, PlotStyle -> {Red, Thick}]]

enter image description here

Improve this answer
$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.