Fit using nlinfit with uncertainties on x and y accounted for in the fit

I would like to include uncertainties on x values and y values during the fit procedure such that the uncertainties are accounted for during the fitting. I am currently using the matlab function nlinfit as I am dealing with a mono-exponential decay but I am open to using another fit function if this does the job.

Can anyone tell me how to do that?

Thanks, Henriette

Answer 1

[Bjorn Gustavsson]

Have a look at the total least squares tools on the file exchange: total-least-squares-with-mixed-and-or-weighted-disturbances and total-least-squares-method - they might at least give you somewhere to start solving this type of problem (which is much more intricate than the standard least-squares fitting-problem...).

HTH

[Henriette Ravn Larsen]

Thanks, I will take a look at that

Answer 2

[Image Analyst]

Not sure what you mean. Of course all fitting routines can handle "uncertainties" in x and y in that they don't expect the actual data to lie perfectly on the fitted line. Attached is a demo of how to use fitnlm() to get exponential decay formula:

[Henriette Ravn Larsen]

Let me give you an example of what I mean.

x = linspace(0,20,100);

y = linspace(1,10,100);

AND each x-value and y-value also have a certain uncerainty to it:

u_x = randn(1, length(x)); % I used random just to put in some numbers - the uncertainties are actually found from experiement and therefore important for the data.

u_y = randn(1, length(y));

This means that there are several possible straight lines that can be put thourgh the points. If these uncertainties are big, it should result in big uncertainties on the fit-coefficientes, even if the data points lie on a perfekt striaght line.

Hope I made things more clear now :)

[Image Analyst]

No it's not clear. So you start with x and y being a straight line, then add u_x and u_y to them to get noisy versions of x and y. OK fine. Of course there are lots of lines that you can fit through the data, but using a fitting routine like fitnlm() or polyfit() will give you the one best equation that will minimize the residuals - the difference between the actual and fitted/estimated values. You could sum the absolute values of the residuals or residuals squared to get a measure of how good the fit it. There is an mse() function.

So I'm still not sure of the problem. Just put your data, even if it's noisy, into fitnlm() and it will give you the best fitting equation it can. What's wrong with that?

[Bjorn Gustavsson]

@Image Analyst, there are some additional complexities with uncertainties in both x and y - see for example the wikipedia-page on Total_least_squares. The standard fitting-functions only do the normal least-square-fitting as far as I know.

[Henriette Ravn Larsen]

@Image Analyst

I think you still misunderstand me. No, u_x and u_y is not just to add noise, they are the associated uncertainties to the x- and y-values. Those that you would also put on as errorbars. The data is worth nothing without the uncertainties. And since the uncertainty can vary a lot from datapoint to datapoint I would like to take that into account in the fitting - like a weighting, but where I actually give the uncertainties for x- and y-values.

Makes more sense now? :)