Curve Fitting Foundations

Curve Fitting: From Data to Model

A practical map of the curve-fitting process, from choosing a model to checking whether it deserves to be trusted.

Least Squares Finds the Closest Model

How squared residuals turn a cloud of observations into an estimated line or linear model.

Residual Plots Reveal How a Model Fails

How residual patterns expose curvature, unequal variance, dependence, and unusual observations.

Requires: Least Squares Finds the Closest Model

RMSE Expresses Typical Error in Response Units

How RMSE and residual standard error summarize squared prediction misses on an interpretable scale.

Requires: Least Squares Finds the Closest Model

R² Measures Explained Sample Variation

What the coefficient of determination says, what adjusted R² changes, and what neither can prove.

Requires: Least Squares Finds the Closest Model

Goodness of Fit Requires More Than R²

How residual error, R², adjusted R², RMSE, information criteria, and validation reveal different aspects of a model.

Requires: Least Squares Finds the Closest Model

Nonlinear Curve Fitting Is an Iterative Search

Why nonlinear models require starting values, repeated approximations, and careful checks for convergence.

Requires: Least Squares Finds the Closest Model

Gauss–Newton Linearizes Nonlinear Least Squares

How a local derivative model produces an efficient update for nonlinear curve-fitting parameters.

Requires: Nonlinear Curve Fitting Is an Iterative Search

Levenberg–Marquardt Balances Caution and Speed

How damping lets a nonlinear least-squares algorithm move between gradient descent and Gauss–Newton.

Requires: Gauss–Newton Linearizes Nonlinear Least Squares

ANOVA Tables Partition Variation in Regression

How sums of squares, degrees of freedom, mean squares, and the F statistic summarize a fitted model.

Requires: Least Squares Finds the Closest Model

AIC and BIC Balance Fit Against Complexity

Why information criteria compare models by rewarding likelihood and penalizing extra parameters.