# Nonlinear regression for a log-linear model

I have data points $(x_i, y_i), i = 1, \dots, N$, and a model (log-linear) with parameters $w_j, j = 0, \dots, m$ such that $y_i=\alpha_i e^{w_0+w_1x_i+w_2x_i^2+\dots+w_mx_i^m}+ n_i$, where $n_i$ are the noise data points, supposed independent zero-mean white Gaussian.

I know that the straight way to estimate the parameters is to linearize using $\ln$, I did that but I'm not satisfied of the result. My $y_i$ and $\alpha_i$ can have null, and also very small, values (imagine sinusoids passing by zero each half-cycle) and this induces errors in the estimation. I tried to exclude these values, it worked but the performance deteriorates as the noise power level gets higher.

Is there any nonlinear regression method adapted for solving such problem?

• If someone is interested in this I found a nonlinear regression algorithm that seems to be adapted for solving this problem, it's the Levenberg-Marquardt algorithm. Sep 28, 2016 at 15:04
• The Levenberg-Marquardt algorithm does not accept constraints on the parameters. One alternative would be a Trust region algorithm for nonlinear least squares. Oct 10, 2016 at 15:51