ACCEPTED SOLUTION
Accepted Solutions
I played around a bit with your data and found a sum of 6 Gaussians which fit pretty well.
I didn't use genfit() but minerr() as the results will be the same and with minerr() the whole thing is better managable. I used Mathcad 15 for this because that version is much more user friendly and and quicker reacting and converted the sheet later in P3 format - just in case you are wondering why the datavectors are defined in a different way.
You may play around with the number of summands and the guess value and i think you will get good fits with less summands, too. But you will notice that the fit is VERY sensible wrt the guess values (just change the top left 1010 in Matrix M to 1020 or to 1000 and see yourself).
I'm not sure but I guess you'll like the splined f1 in my last post more.
But as you wrote that the data is synthetic hydrograph I would imagine that the other two regressions in my previous post are describuig the event more realistic as they don't follow every peak and valley.
I also tried a polynomial fit and also a fit with two gaussians which both look very similar and good if the goal is a smooth curve. Ofcourse they are far away from those ouliers(if they are?) and I guess you would need more data points to decide which are outliers or not. A fit using a special function like logarithms, exponatial functions or combinations of them makes sense if you are searching the parameters of the underlying physics, but then you would have to provide the general form of the function derived from the physics.
As it seems that you just have the data and no more information about it I guess a polynomial fit or s spline interpolation (which runs exactly through all your datapoints) is as good as any.
Genfit() won't find the type of curve which fits best automatically but you will have to provide it, genfit() will only find the "optimal" parameters and very often is VERY sensible concerning the initial guess values you also have to provide. There are specialized programs which try to find the best type of function, one of them is "CurveExpert Professional". Not sure if they offer a trial version to download.
And yes, you can integrate and differentiate "functions" derived by interpolation routines - numerically only, of course.
See if the attached sheet helps. Think f1 is what you want to use but maybe you are even happy with linear interpolation (f0).
I played around a bit with your data and found a sum of 6 Gaussians which fit pretty well.
I didn't use genfit() but minerr() as the results will be the same and with minerr() the whole thing is better managable. I used Mathcad 15 for this because that version is much more user friendly and and quicker reacting and converted the sheet later in P3 format - just in case you are wondering why the datavectors are defined in a different way.
You may play around with the number of summands and the guess value and i think you will get good fits with less summands, too. But you will notice that the fit is VERY sensible wrt the guess values (just change the top left 1010 in Matrix M to 1020 or to 1000 and see yourself).
I'm not sure but I guess you'll like the splined f1 in my last post more.
But as you wrote that the data is synthetic hydrograph I would imagine that the other two regressions in my previous post are describuig the event more realistic as they don't follow every peak and valley.
Here are a view more regressions. They won't run through all your data points but provide a decent fit and look more like the graphs I saw on the net when searching for synthetic hydrograph. Anyway, its your choice now.
I am trying to keep up with this discussion, perhaps there are bugs perhaps there are future enhancements.
You a referring to the thread I opened here http://communities.ptc.com/message/234933#234933 ? Not sure if I did something wrong there or not. But if not, then I would either expect a similar result as without units (at least if I use base units) or an error telling me, that regressions should not be done with units. I could image that there are reasons for better not using units here, but again I am not sure and so opened the discussion to collect other opinons.
But you lost me .. how did you come up with g(t,v). I have not seen this fitting function .. I dont think
I guess you have seen this fitting function already, but I sure should have better written it in a more convential way as the parameters would have a meaning if we do that and that can help to find guess values.
You are referring to
or in a later sheet
These are simply the sums of two or (the second) an arbitrary number of gaussians which are quite usual and useful functions to try for a fit in many situations. But I was surprised that I got that nearly perfect fit with 6 gaussians and also that slight changes in the guesses have that big impact on the result.
A gaussian is a function of the form
or may be written as
where y0 is the baseline offset, A is the total area under curve from the baseline, t0 is the center of the peak, w is something like 2 sigma, approximately 0.849 times the width of the peak at half height.
Richard had posted sums of gaussians from time to time in the past here and he uses
where h, w and t0 are controlling height, width and peak position.
The way of fitting multiple gaussians using the sum, matrix and minerr was stolen from Richard by me anyway.
I shouldn't had setup the function using meaningless coefficients, though.
Probably true,
Actually, not true. Don't know why I said it in fact 
The better solution is the one I proposed all of 12 minutes later in my next post! Make CTOL and TOL relative, rather than absolute. Not only does that fix the unit issue, it makes them insensitive to the magnitude of the data, which is something that has tripped up many users. For backwards compatibility there would have to be a new worksheet option. I'm not a fan of lots of options because they tend to confuse people, but sometimes there is no better way of dealing with a problem.
But I was surprised that I got that nearly perfect fit with 6 gaussians and also that slight changes in the guesses have that big impact on the result.
Neiter one surprises me. With 6 Gaussians you have 18 parameters, but you have only 12 data points,
Richard had posted sums of gaussians from time to time in the past here and he uses
where h, w and t0 are controlling height, width and peak position.
It depends on what I want. The w in that expression is the full width at half height, which sometimes is more useful than the standard deviation.
