I am fitting data to a system of non linear ODEs to estimate model parameters using Matlab lsqcurvefit.
In this fitting the fit depends so much on the initial guesses that I use for the lsqcurvefit .
For example, if I use x0=5 as a initial guess I will get residual norm of 30, where as if I choose x0=5.2 I get a residual norm of 1.5.
1) What does residual norm (resnorm) in Matlab represent? is it the sum of the squared errors? Is there a way to decide what range of value for resnorm is acceptable.
2) When the fit depends so much on initial guess, is there a way to deal with these problems? How would I know whether a better fit can be obtained from a different initial guess?
3) In using lsqcurvefit is it required to check whether the residuals are normally distributed?
lsqcurvefit fits your data in the least squares sense. Thus it all comes down to the minimisation, and as your data is non-linear, you do not have any guarantee, that this is the global minimum nor that it is unique.
E.g. Consider the function sin(x), which x-value minimises this function, well all x=2*pi*n + 3/2*pi for n=0,1,2,... but your numerical method can only return one solution, which will then depend on your initial guess.
To further elaborate on the problem. The simplest (in my opinion) minimisation algorithm is known as the steepest descent. it uses the idea, known from calculus, that the steepest descent is in the direction of the minus the gradient. Thus it finds the gradient in the suggested point takes a step in negative that direction (scaled by some stepsize) and continues to do this until the step/derivative is significantly small.
However, even if you consider the function cos(3 pi x)/x from 0.5 to infinity, which does have a unique global minima in 1, you only find it if your guess lies in between 0.7 and 1.3 (or so). All other guesses will find their respective local minima.
With this we can answer your questions:
1) resnorm is the 2-norm of the residuals. What would it mean that specific norm would be acceptable? The algorithm is looking for a minimum, if you already are at a minimum, what would it mean to continue the search?
2) Not in an (pseudo) exact sense. What is typically done is to either use your knowledge to come up with a sensible initial guess. If this is not possible, you simply have to repeatedly make random initial guesses and then keep the best.
3) Depends on what you want to do, if you want to make statistical tests, which depends on the residuals being normally distributed, then YES. If you are solely interested in fitting the function with the lowest residual norm, then NO.