# Regression through linear Fourier coefficient fitting?

Basically suppose on was given an unknown function/data and expected to write a function so that $$Y=f(X)$$, this can be done by linear regression in simple cases very easily. However, suppose that the out put $$Y$$ was an extreme chaotic curves, then the curve fitting become problematic, and the regression, in practice, largely dependent on the people's feeling/choice. Although there's several complete basis so that the expansion can be reduced much less, there's still question about weather one be able to fit the data after considering large sets of equations/combinations of functions.

I recently thought of the cases in calculus where we use polynomial to reexamine the original function. In corresponding, on can obtain sets of lists of function in Fourier domain, where the functions were represented by sets of coefficients. Thus, instead of fitting the data by comparing non arbitrary or even problematic fitting, one might be able to simply fit the function in Fourier domain in terms of very simple linear coefficient fitting.

My questions were:

1. Is that possible?

2. Are there other methods to fit the non chaotic function in large sample and combinations?

• The coefficients of a Fourier expansion can also be fitted by linear least squares with nonlinear basis functions. It is not really clear what you mean chaotic and non-chaotic functions. It would be better if you could provide some plots. Also, you need to specify what exactly you mean by a large sample? 1000 observations, 10000 or 1000000 and more? – MachineLearner Feb 28 at 11:03
• @MachineLearner I knew that, but in practice, the least square was either estimated by decent(which involves bunch of questions such as local minimum, singular e.t.c even the stochastic one won't guarantee to work well), or matrix multiplication, which, in practice, fails after 20-30 predictors for very chaotic data because of the machine's calculation limitation(matrix singular or accuracy). A sample here means a strings of large data, can assume 1e3, 1e5 e.t.c. Least square is really not a good idea to approximate chaotic data with extreme dynamics.(Don't think about T.A.) – user9976437 Feb 28 at 15:13