It is known that a series of values of X is the sum of processes, each of which is approximated by a nonlinear function (for example, parabolas) of the same structure, but with different parameters. These processes have different time shifts. How to determine the parameters and the shift of these functions by their total series X? What are the approximation methods, the functions of software packages (including free packages or Matlab, Mathcad), packages in R? Links to articles with examples of solutions of similar problems are desirable.
X is a stochastic process but there is an opportunity to approximate its trend with a deterministic function. But I am interesting answers to both options - both stochastic and deterministic.
Looks exactly like the problem arising in spectral analysis: find parameters of many, possibly overlapping, Lorentz-shaped spectral peaks. The only difference is that the spectrum is a function of energy, not time.
Dear Marek Wojciech Gutowski, please write how this problem ("find parameters of many, possibly overlapping, Lorentz-shaped spectral peaks") is usually solved. And where can I read about it?
Dear Vitalii, my favorite software package is gnuplot (free and available for Mac, Windows and Linux). Origin (Windows only) may appear useful as well (I'm not sure). X-ray people (crystallographers) have their own tools, unknown to me but related to term Rietveld analysis.
After re-reading your question I have another thought. If your components are parabolas, i.e. second order polynomials then you are in trouble. The sum of any two such components (or even more than two) is also the second order polynomial what means that you can effectively fit exactly 3 its parameters. In other words you will have infinitely many solutions for the combination of two (or more) such functions. Why? Having 3 numbers describing the sum of two parabolas you may take (guess, assume) an arbitrary set of 3 parameters 'describing' the first of them and you will be able to compute (not fit!) the parameters describing the second component.
I agree with Marek that “looks exactly like the problem arising in spectral analysis”. But I wanted to add the following. As a model of a stochastic process X, for example, either an autoregressive model or a stochastic differential equation can be used. Their constructing can be performed on the basis of either an autocorrelation function or the spectral density of a stochastic process. Moreover, the model itself will be linear, and the stochastic process will be Gaussian, since the superposition principle is used ("X is the sum of processes").
The second question that needs to be clarified is whether the stochastic process is stationary or not.
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