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.

Non-Linear Regression: 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.

Analogy

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 $R^{2}$
Understand the limitations of $R^{2}$ 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	$y = a x^{2} + b x + c$	Projectile motion, area relationships
Cubic	$y = a x^{3} + b x^{2} + c x + d$	S-shaped or multi-bend data
Exponential	$y = a e^{b x}$	Population growth, radioactive decay
Power	$y = a x^{b}$	Physical scaling laws
Sine	$y = a sin (b x + c) + d$	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

Example

Real-world examples:

A ball thrown upward follows a quadratic path.
The number of bacteria doubling every hour follows an exponential model: $N = N_{0} e^{k t}$ .
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.

Exam Tip

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.

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Model

Equation

Typical Use Case

Quadratic

y = a x^{2} + b x + c

Projectile motion, area relationships

Cubic

y = a x^{3} + b x^{2} + c x + d

S-shaped or multi-bend data

Exponential

y = a e^{b x}

Population growth, radioactive decay

Power

y = a x^{b}

Physical scaling laws

Sine

y = a sin (b x + c) + d

Periodic/seasonal data

AHL 4.13—Non-linear regression

Introduction to Non-Linear Regression

Types of Non-Linear Regression Models

11 more sections in this topic

AHL 4.13—Non-linear regression

Introduction to Non-Linear Regression

Types of Non-Linear Regression Models

11 more sections in this topic