How do I fit to a step function?

[nforde]

Hi,

I'm using the Levenberg-Marquard algorithm to optimize a fit to some data.  I would like to fit to a step function, but when I try to do this with simulated data, the algorithm is not at all responsive to optimizing the position of where the step occurs.  (It simply returns the initial guess position of the step location, rather than optimizing it.)  I have tried using both the step(x) and sign(x) functions but both have the same problem.  Does anyone have any experience with this? 

Cheers,

Nancy

[johnsold]

Nancy,

I have not used the Lev_Mar functions very much, but I think I recall that they use derivatives. Since the derivatives of the sign and step functions are undefined at the step, this may be part of your difficulty.

To fit to a step function try the Threshold Array function with the threshhold set to 0.5*(ArrayMax - ArrayMin). This should give you the index of the step. The Max and Min values would be first estimates of the before and after values, if the data is not too noisy.

Lynn

[altenbach]

The problem is with the partial derivatives for the position parameter. If the position changes less than the point spacing, the partial derivative with respect to that parameter is always zero.

I have quickly hacked the ABX subVI to use something larger than the x spacing of the points as increment for the partial derivative (instead of 1e-6), and it converges just fine after that if the initial estimate is reasonable. (See image). Still, I recommend to not do that!

Follow Lynn's advice. 😉

Message Edited by altenbach on 05-10-2006 01:01 PM

[nforde]

Hi Lynn & altenbach,

Thanks for your suggestions.  Yours seems quite simple to implement, Lynn, but unfortunately, while for my test case the noise is less than the step size, the signal I would ideally like to fit to is not substantially larger than the noise, so thresholding the array won't work in general. 

As to the step size, I don't see how to change it in the fitting algorithm, but I did try making the increments of x orders of magnitude smaller, so that the step occurs over a length less than 1E-6.  While the Lev_Mar algorithm now gives a step location different than that input, it is still much closer to the input guess than to the new location of the step. 

Any other ideas?

Cheers,

Nancy

[altenbach]

Can you attach some sample data?

[altenbach]

Yes, the increment trick should not work that well in general, because even though we now have a direction for the partial derivative, the magnitude is still very quantized and only relies on one or a few data points.

Is your data equally spaced in X?

[nforde]

If I use the following as my y-coordinates:  (0,0,0,0,0,2,2,2,2,2) and try to fit to the following function:

a*0.5*(sign(y-b)+1)+c

with guesses:

a=2

b=5

c=0

I get out as best-fit coefficients:

a=2.00

b=5.00

c=8.70E-5

If I now change the position of the step in the input data to position four (0,0,0,0,2,2,2,2,2,2) with the same input guesses, I get out, as best-fit coefficients:

a=2.08

b=5.00

c=0.136

I have just tried using a sigmoid function (continuous) to fit to this step data, with much better results (the step position is optimized).  Ideally, I'd like to use a step function, but there is at least one alternate solution if that fails.

Cheers,

Nancy

[nforde]

Yes, the data is equally spaced in x.  For the sample data I just quoted, the results assume a spacing of one.  The results don't change (in any meaningful way) if I use a spacing of 1E-7.  (The numbers might change, but the problem is the same, where the step location, b, does not get optimized.)

[medvall]

Hello,

You can easily do this with TOMVIEW using a derivative free solver, such as glbDirect or glcDirect.

These are global solvers so you will need to set conservative bounds on your decision variables.

Look at for example nllsQG.vi in the quickguide, then use glcDirect as the solver options.

Best wishes, Marcus