# Non-linear regression for complex data

I have to fit experimental data to an analytical expression involving complex variables. The expression is set such that it cannot be decomposed into real and imaginary parts (in an algebraic form), e.g., $$Z(\omega) = \frac{coth\left(H \sqrt{i \omega/ \alpha_z} \right)}{\pi R^2 k_z \sqrt{i \omega/ \alpha_z}}$$

My data points are $\hat{Z}(\omega)$ (hat to distinguish experimental data from analytical variable).

Intuitively, I would expect an equivalent least-squares solution for the complex domain which minimizes the error (* means complex conjugate): $$\varepsilon = \sqrt{\frac{1}{N} \sum_i^N \left(Z(\omega_i) - \hat{Z}(\omega_i) \right) \cdot \left(Z^*(\omega_i) - \hat{Z}^*(\omega_i) \right)}$$

How do I go about it? Are there other solutions to this problem? How would a computer implementation look like, e.g., in MATLAB?

Note: $\omega$ is independent variable, $Z$, $\hat{Z}$ dependent variable, $H$, $R$ constants and $k_z$, $\alpha_z$ are the parameters to be back-computed via curve-fitting.

• What are the constants and the variables ? What are the know parameters and the unknowns ? What is exactly the list of parameters to be optimized? – JJacquelin Apr 27 '17 at 8:20
• Thanks! I revised the question. – Mistry Apr 27 '17 at 8:23
• If I well understand, $H$ and $R$ are known numerical values (real). Are the parameters to be optimized $k_z$ and $\alpha_z$ real or complex ? – JJacquelin Apr 27 '17 at 8:44
• The parameters are real as well. The expression becomes a complex variable due to $i$ appearing inside square roots. I have a bunch of similar problems and in all of them, complex nature arises due to $i$. The constants and parameters are all real. – Mistry Apr 27 '17 at 8:54
• OK. Editing an example of data would be helpful. – JJacquelin Apr 30 '17 at 15:26

NOTE : The function $F(X)$ involved in the numerical calculus is a multivaluated function that is, for an input value $X_k$ the function is subject to returns several values $F_k$ in some cases ($X$ close to $0$.).
From a few tests with simulated data, it appears that this causes some difficulties in the numerical process to determine the convenient $F_k$ among the possible ones. This depends a lot of the range of the data. No major difficulty in cases of $0.001205<X<1$ . That is why realistic numerical examples of data would have been better than simulated data in order to investigate this problem.