On modification of the mean squared error loss function for solving nonlinear heteroscedastic errors-in-variables problems

The paper considers the problem of finding the optimal parameters of a nonlinear regression model accounting for errors in both dependent and independent variables. The errors of different measurements are assumed to belong to different probability distributions with different variances. A modified mean squared error-based loss function is derived and analyzed for this case. In the computational experiment, the measurements of the laser's radiation power as a nonlinear function of the resonator's transparency are used to compare the parameters vectors minimizing the presented loss function and the classical mean squared error. The convergence of the parameters minimizing the presented loss function to the optimal parameters for the classical loss function is studied. In addition, some values of the parameters are considered to be “true” ones and are used to generate synthetic data using the physical model and Gaussian noise, which is then used to study the convergence of the parameters minimizing the presented and the classical loss function, respectively, as the function of the noise parameters.