Which MPFIT Files Should I Download and Use?

The easiest solution is to download the entire set of routines as either a ZIP file or a gzipped TAR file, and then extract them in your IDL path. However it is possible to selectively download what you need. Bear in mind that you will always need MPFIT.PRO which is the main fitting engine.

For standard fitting of 1D curves, where the model is a compiled function, download MPFITFUN.PRO and MPFIT.PRO. For specialized peak fitting, use MPFITPEAK.PRO in combination with these two.

For fitting of 2D images or surfaces, download MPFIT2DFUN.PRO and MPFIT.PRO. For peak fitting, use MPFIT2DPEAK.PRO in combination with these two.

For a drop-in replacement for the IDL-supplied CURVEFIT in legacy code, use MPCURVEFIT.PRO along with MPFIT.PRO.

For cases where you don't have a precompiled function, either at the command line on within your program, you can use an IDL expression. Download MPFITEXPR.PRO and MPFIT.PRO.

Is There a Citation for MPFIT?

I regularly get asked if there is a way for users to cite MPFIT in their scientific publications. The answer is yes! I presented a paper about MPFIT at the ADASS XVIII conference in Quebec, Canada, in Nov, 2008, with proceedings published by the Astronomical Society of the Pacific (2009). I welcome you to cite this publication in your own works.

• Markwardt, C. B. 2009, "Non-Linear Least Squares Fitting in IDL with MPFIT," in proc. Astronomical Data Analysis Software and Systems XVIII, Quebec, Canada, ASP Conference Series, Vol. 411, eds. D. Bohlender, P. Dowler & D. Durand (Astronomical Society of the Pacific: San Francisco), p. 251-254 (ISBN: 978-1-58381-702-5; Link to ASP title listing)
  ADS Bibcode: 2009ASPC..411..251M (click for Bibtex and other citation formats)
  Arxiv preprint: arXiv:0902.2850v1
  
• Bibtex entry:

  \bibitem[Markwardt(2009)]{2009ASPC..411..251M} Markwardt, C.~B.\ 2009,
  Astronomical Data Analysis Software and Systems XVIII, 411, 251
  

• Referring to the MPFIT website. Please use:
  http://purl.com/net/mpfit
  in your citations. Right now this link redirects to my Wisconsin website, but the Wisconsin website will not exist forever. When I move the website, I will change the purl.com redirect as well.

• Moré, J. 1978, "The Levenberg-Marquardt Algorithm: Implementation and Theory," in Numerical Analysis, vol. 630, ed. G. A. Watson (Springer-Verlag: Berlin), p. 105
  (DOI: 10.1007/BFb0067690; Link to Springer title listing)
• Moré, J. & Wright, S. 1993, "Optimization Software Guide," SIAM, Frontiers in Applied Mathematics, Number 14.
  (ISBN: 978-0-898713-22-0; Link to SIAM title listing)
• MINPACK-1, Jorge More', available from netlib. http://www.netlib.org/minpack

Are there other versions of MPFIT?

Yes! MPFIT is based on the original MINPACK-1 library, written in FORTRAN, available from Netlib at http://netlib.org/minpack.

I'm also pleased to make available a C library version of MPFIT. This library has many of the same capabilities as the IDL version, including parameter bounds and choice of numerical or explicit derivatives, in a small, fast library, with simple calling interface. Of course, some IDL-specific elements such as _EXTRA and "tied" parameters don't make as much sense in C, and they are not present. See the C Version of MPFIT page for more information. The original MINPACK-1 code was translated to C by Stephen Moshier (http://moshier.net/), from which MPFIT borrows extensively.

MPFIT was translated to Python's Numeric library in 2002 by Mark Rivers, and can be found at his website. In 2009, Sergei Koposov ported that code to use the more modern Python Numpy numerical library, and the result can be found in his astrolibpy library.

How can I calculate the best-fit model?

All of the basic functions in the library return the best-fit model function using the keyword YFIT. Simply pass a named variable with this keyword and upon return the best-fit model will be in that variable.

If you change the parameter values manually, it is still simple to recompute the model function. It differs, depending on what type of model you are fitting:

Here, p is assumed to be the parameter set.

How do I fit a 2D image?

The simplest answer is to use the MPFIT2DFUN.PRO function. You supply the image, the X and Y labels for each pixel, and a 2D model function.

Can I fit a function of several variables?

This is a question when you are fitting a function of several independent variables, such as this:

    y = f(x0,x1,x2, ...; p0, p1, p2, ...)

where the xi are the independent variables and the pi are fitting parameters.

Most of the MPFIT functions do not care how many independent variables there are. In fact, the main fitting routine MPFIT does not accept any independent variables at all! Instead, they are considered to be implicit to the problem. Functions like MPFITFUN do accept an independent variable called X. [ Since MPFIT itself doesn't deal with the independent variables, MPFITFUN creates a common block with that information so that the model function can gain access to it. ]

Therefore, if you are using plain MPFIT to do your fitting, then you have the freedom to construct your model in any way you please.

If you are using MPFITFUN or MPFIT2DFUN, then your model should still be a function of a single variable X, and you should pass a single set of independent variables called X to the fitting routine. However, X can be arbitrarily complicated, so you can have it be a 2D array, the first row containing "x0" values, the second containing "x1", and so on. Thus the burden falls to your IDL model function to decode the contents of X. I recommend the 2D array approach, but you can also simply concatenate the variables in a 1D vector, or pass them by COMMON block (not recommended).

The array approach I am advocating would be something like this:

This creates a new "X" variable which is the array concatenation of all of your independent variables. Then the burden would be upon your user function to extract the relevant quantities from this array.

Help! I can't get my fitting to work!

Generally speaking, this is not something I can help with. I am probably not an expert in your field, and I am definitely not an expert in optimization. However, I think there are a few very important things that you must do.

First, you must make sure that the problem is well defined. Make sure that you know exactly what function you plan on fitting. The MPFIT functions are not psychic; they can't figure out how to solve your problem for you.

I see quite a few people who can't get their model functions right. The best kind of model function is self-contained, not depending on any outside data or common blocks. Be sure that your model function works by itself before throwing MPFIT into the mix. It is very difficult to debug the problem when the added layer of MPFIT is disguising everything. Be aware of domain problems. For example, if you will be taking the square root of a parameter, you had better constrain it to be positive.

Finally, and this can't be stressed enough, it is crucial to estimate the starting parameters as best you can. Initializing the parameters to all zeroes is actually the worst thing you can do, since then the problem becomes scale-less, and MPFIT has a much harder time deciding what to do.

Do I need to compute derivatives?

The short answer is, probably not. I have found that in most cases the automatic finite difference technique used by MPFIT is sufficient. Computing the derivatives explicitly seems to slow things down, and can even make a worse final solution. My suggestion is to try it first without computing the derivatives, and them implement derivatives as needed (and the AUTODERIVATIVE=0 keyword).

However, see the caveats below.

How can I embed MPFIT-style fitting in a widget application?

My belief is that the best solution is to use the ITERPROC feature, which in turn would call WIDGET_EVENT manually to dispatch events. Rob Dimeo has successfully integrated this approach into his dedicated peak fitting program named PAN. PAN is the winner of an honorable mention the recent IDL programming contest. Source code is available (uses objects). Nice job, Rob! Here is a small screenshot:

Why does MPFIT just return my parameters without fitting? / How do I check for processing errors?

Users occcasional report that MPFIT, or one of its drivers, return without fitting any of the parameters. This is usually an indication that MPFIT discovered an error in the user-specified parameters and returned immediately.

Users should check the values of the STATUS and ERRMSG keywords when MPFIT returns. If STATUS is less than or equal to 0, then a processing error has occurred. A more detailed description of the problem can be found by consulting the ERRMSG keyword.

A simple example of error checking using STATUS and ERRMSG might be:

  p = MPFITFUN(x, y, dy, p0, STATUS=status, ERRMSG=errmsg)
  if status LE 0 then message, errmsg

Of course, more sophisticated error handling approaches can be implemented as needed.

Why does MPFIT not converge properly?

For certain problem sets, MPFIT will not converge to the "optimal" solution. It may seem obvious to you that there is a better solution, but MPFIT does not find it. Why could this be?

Be sure you are checking for error conditions. MPFIT may be indicating an error condition which prematurely terminates the fit.

MPFIT normally uses a finite difference approximation to estimate derivatives. This means that it varies each parameter by a small amount, and measures the corresponding variation in the chi-square value. However, if your data or model are discretized for some reason then this approximation can fail, which then causes the fit not to converge. You should check for:

• Automatically-selected step size too small. In some cases, the automatically-selected step sized used to compute derivatives is too small. This usually happens when the dynamic range of the parameter is very small compared to the absolute value. Solution: set the step size manually with PARINFO.STEP or .RELSTEP, to be small enough to resolve the variations in your fitting function. Another solution is to re-write your model function to remove large constant offsets from parameters.
• Discretized parameters. Example: a tabulated model, which is evaluated on a discrete grid of parameters. Solution: Interpolate the table smoothly between parameter values; set PARINFO.STEP or .RELSTEP values to be comparable to the grid spacing.
• Discontinuous functions. Example: use of the ABS() function, or any function which has a discontinous derivative. Solution: use analytical derivatives; or use a smoother approximation to your discontinuous function.
• Undersampled data. Example: fitting of undersampled peaks. For example, consider data sampled with a grid spacing of 1 unit, which contains a peak whose ideal full-width at half-max is 0.5 unit. Since this peak is undersampled, the small parameter variations attempted by MPFIT may have no measureable effect. Solution: get data with finer sampling; adjust PARINFO.STEP or .RELSTEP to be comparable to the grid spacing.

Also, beware that PARINFO.RELSTEP will not work if the parameter value is zero. Generally speaking, all starting parameter values should be non-zero.

Following along those lines, it's also worth checking what the dynamic range of your parameters is. For example, if you are fitting a model of 'p(0)+p(1)*x' where p(0) is 1 and p(1) is 1d-12, then the fitting won't do very well. There are matrix operations inside of MPFIT that may lose precision if there is a large dynamic range in parameter values. It's better to absorb the 1d-12 factor into the fitting function, so that all of the parameters are of equivalent magnitude.

Finally, beware that if your chi-square function has local minima then MPFIT may become irretrievably stuck. MPFIT is not a global optimizer.