There are several things I thought I would mention of 'tricks' I've used to get fitting functions to work by estimating or eliminating free parameters. First off: Lsqcurvefit is not magic, it is going to match your model to the data by varying the parameters until it can find a good combination such that the data-model = 0 ( but not really zero because we are working with finite numbers so it will keep on finding a lower minimum until you run out of iterations (MaxIter, MaxFunEvals) or until you get close enough to zero (TolFun, TolX ) )
1) Eliminate scaling parameters In this example the function already has an y = x + c you already know how far it is from center so subtract c off to center it around 0 and then fit it, if you have to add c back. This goes for y also, you can subtract an average out and then fit it, then add it back to your fitted model after your done.
2) Rescale or estimate the amplitude Something like y = A*sin(x) is easy to find A with a max function (or in the case of y = A*(sin(x+c0)+c1) combination of max min and/or mean). Use this information to your advantage! Either use it to normalize the function and eliminate a scaling parameter or use it to help lsqcurvefit. You can do this by setting the initial parameter estimate or setting upper and lower bounds to help lsqcurvefit find a local minimum.
3) Use good initial guesses and set a upper and lower bound if necessary (kind of already said it)
4) Set parameters that you know and won't change, if there is a frequency parameter and its easy to find then estimate it yourself and then set it in the model and don't allow it to change
5) Don't use data that is bad or you don't want to model You can include only the data that you want to fit, sometimes portions of the function are easier to find or data that looks more like the model. I've done this with two dimensional functions with great results.
In this case I would start with by centering and scaling the data and remove the A(5) and A(3) parameters out of the data by centering it and try and fit the rest of the function which may or may not work with all of the sign(x) functions but since it is symmetrical, only fit one half of it (the negative half could go away, then you could throw away all of the abs and sinc functions that have a potential to create problems since the minimum would be flipping when it moved across this point)
F=@(A,x) A(4)*(x)+(1-int_sinc(A(6)*abs(x))).*(A(1)*sign(x)-(1-A(2))*int_sinc(A(6)*abs(x)))
F=@(A,x) A(4)*(x)+(1-int_sinc(A(6)*(x))).*(A(1)*(x)-(1-A(2))*int_sinc(A(6)*(x)))
Of course these are only suggestions and things that I've done that may help you or anyone else trying to get a good model fit.
