AHL 4.13 (HL)—Non-linear regression

Regression estimates a relationship between explanatory and response variables. If a straight line leaves a systematic curve in the residuals, a non-linear model or transformation may represent the pattern better.

Choose a model from theory, residual behaviour and the observed range. A transformed linear model can simplify calculation, but the fitted relationship must be converted back before interpreting predictions in the original units.

A growth dataset may follow y=ab^x; plotting ln y against x can make the pattern linear and estimate a and b. A good fit inside the measured range still says little about values far outside it.

A high R² does not prove the model is causal or valid everywhere. Check residuals, outliers, units and extrapolation before trusting a forecast.

Technology may fit linear, quadratic, cubic, exponential, power and sine regressions. Compare models using residual plots, $SS_{res}=\sum(y-\hat y)^2$ and $R^2$, the proportion of response variability accounted for by that model. Smaller $SS_{res}$ or larger $R^2$ helps only for the same response data; neither overrides mechanism, residual pattern or extrapolation limits.