Introduction to Non-Linear Regression
In many real-world situations, the relationship between two variables is not a straight line. Non-linear regression allows us to fit curves to data, capturing more complex patterns that linear models cannot describe.
A statistical method that models the relationship between a dependent variable and one or more independent variables using a non-linear function, fitted by minimising the sum of squared residuals.
Non-linear regression is an extension of linear regression — the core ideas (fitting a curve, measuring residuals, assessing goodness of fit) are the same, but the shape of the model is more flexible.
Think of linear regression as fitting a straight road between two cities. Non-linear regression is like mapping a winding mountain path — the destination is the same (best fit), but the route adapts to the terrain of the data.
In IB Math AI HL, you are expected to:
- Recognise which type of non-linear model is appropriate for a dataset
- Use technology to determine regression equations
- Interpret and evaluate models using the coefficient of determination
- Understand the limitations of and the danger of overfitting
Types of Non-Linear Regression Models
The five main types of non-linear regression models you need to know are:
| Model | Equation | Typical Use Case |
|---|---|---|
| Quadratic | Projectile motion, area relationships | |
| Cubic | S-shaped or multi-bend data | |
| Exponential | Population growth, radioactive decay | |
| Power | Physical scaling laws | |
| Sine | Periodic/seasonal data |
Choosing the right model depends on both the shape of the scatter plot and the real-world context:
- Quadratic — data forms a single-bend parabolic shape
- Cubic — data has two bends or an inflection point
- Exponential — data grows or decays at an accelerating rate
- Power — data shows a proportional scaling relationship
- Sine — data oscillates regularly around a mean value
Real-world examples:
- A ball thrown upward follows a quadratic path.
- The number of bacteria doubling every hour follows an exponential model: .
- Average monthly temperature in a city follows a sine model, cycling annually.
- The relationship between an animal's mass and metabolic rate often follows a power model.
Always plot your data first! A scatter plot will immediately suggest which family of curves might fit best. Never guess the model type without looking at the data.