# Estimating parameter using curve-fitting and model comprised of uncorrelated product of two functions

Suppose that I have a time-series dataset represented by the function $f[t,\omega,q]$, where $t$ is the time as an independent variable, $\omega$ is the frequency as an independent variable and $q_k$ is an unknown that I want to determine for each discrete $t_k$ timestep. The $q_k$ is constant over all $\omega$ for a certain $t_k$.

In other words, each timestep $t_k$ has a number of $f[t_k,\omega,q_k]$ corresponding to different $\omega$, but $q_k$ is constant for each $t_k$.

The observations are modeled by the product of two functions:

$f[t,\omega,q] = A[t,\omega]\exp(g[\omega,t,q])$

In the above, $A[t,\omega]$ is unknown, and changes with $t$ and $\omega$. The mathematical formula for $g[\omega,t,q]$ is known, but I do not know $q_k$, the parameter to be determined at every timestep $t_k$.

I can make the assumption that $A[t,\omega]$ is uncorrelated with $g[\omega,t,q]$, and I will not be able to measure $A[t,\omega]$, so I cannot make very many assumptions about it.

How would I set up a curve-fitting problem to estimate $q_k$ at each timestep $t_k$? Is there a good reference on this type of problem? What will affect the accuracy of the $q_k$ estimation, even if $A[t,\omega]$ is uncorrelated with $g[\omega,t,q]$?

Can Bayesian methods be useful in such a problem?

In addition, $A[t,\omega]$ and $g[\omega,t,q]$ and $f[t,\omega,q]$ can be complex numbers. How would I deal with this in curve-fitting?

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