Data-Driven Modeling Summary

Mathematical Modeling · Axiom Academy

A comprehensive review of Unit 7: Fitting models to data, parameter estimation, model selection, and uncertainty quantification for empirical modeling. Data-driven modeling bridges theory and observation, using measured data to inform, calibrate, and validate mathematical models rather than deriving them purely from first principles. Least squares fitting provides a principled framework for finding model parameters that best explain observed data by minimizing the sum of squared residuals. Model selection balances fit and complexity, using criteria like AIC and BIC to choose models that explain data well without overfitting. Uncertainty quantification is essential, providing confidence intervals on parameters and predictions that honestly reflect the limitations of our knowledge. Core Concept: Least Squares Fitting Objective: Find parameters that minimize the sum of squared differences between observed data and model predictions. Residual: The difference r i = y i - f ( x i ; ) measures how far each observation is from the fitted curve. Cost Function: S ( ) = r i 2 is the sum of squared residuals to be minimized. Optimality: For linear models, least squares has an analytical solution; for nonlinear models, iterative numerical methods are required. Core Concept: Linear Regression Model Form: y = 0 + 1 x + , where represents random error. Normal Equations: The optimal coefficients satisfy X T X = X T y , giving a closed-form solution.

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