Nonlinear Least Squares Fitting

Nonlinear least squares fitting is one of the core techniques in mathematical modeling for handling complex data relationships. When nonlinear relationships exist between observed data and the model, this method helps us find optimal parameter estimates.

Implementing nonlinear least squares fitting in MATLAB primarily relies on the Optimization Toolbox. The most commonly used functions are lsqcurvefit and lsqnonlin, which are specifically designed to solve least squares problems. These functions require pre-defining the nonlinear model function expression and providing reasonable initial parameter values. The algorithm iteratively adjusts parameters through optimization methods like Levenberg-Marquardt or trust-region-reflective algorithms to minimize the sum of squared residuals between model outputs and actual data.

During mathematical modeling, nonlinear fitting typically faces two key challenges: initial value sensitivity and local optimum problems. For the former, it's recommended to combine physical meaning or linear approximations to obtain better initial guesses. For the latter, multiple initial value attempts or global optimization algorithms can be employed. MATLAB's fitting results return parameter estimates along with their confidence intervals, which are crucial for evaluating model reliability using statistical metrics like R-squared and RMSE.

In mathematical experiments, this method is widely applied in areas such as physical law verification, economic trend prediction, and biological growth modeling. By using visualization tools to compare fitted curves with original data through plot functions like plot() and scatter(), one can intuitively assess model fit quality, providing a solid foundation for subsequent analysis including residual analysis and prediction intervals.