Nonlinear Curve Fitting

Mathematical Modeling · Axiom Academy

LESSON Nonlinear Curve Fitting - Mathematical Modeling Unit 7: Data-Driven Modeling - Mathematical Modeling What is Nonlinear Curve Fitting? Nonlinear curve fitting is the process of finding parameters for a nonlinear model that best describes observed data. Unlike linear regression, the model function is not linear in its parameters. "Find parameters that minimize the difference between model predictions and data" The model function is nonlinear in parameters No closed-form solution exists - iterative methods are required Initial parameter guesses are crucial for convergence Multiple local minima may exist A model is linear in parameters if it can be written as a linear combination of the parameters, even if the functions of x are nonlinear. Example: Polynomial regression, even with x^2 , x^3 , etc., is linear in parameters. The parameters appear in nonlinear ways - inside functions, as exponents, or multiplied together. These require iterative fitting methods! Linear models: Can be solved exactly using matrix algebra (normal equations). Nonlinear models: The derivatives with respect to parameters depend on the parameters themselves, preventing a closed-form solution. We must use iterative optimization algorithms that progressively improve parameter estimates. For fitting, we minimize the sum of squared residuals between observations and model predictions. Nonlinear Least Squares Problem Given data points (x_i, y_i) for and model : where is the residual for observation i .

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.