Non-linear regression models

IB Mathematics Applications and Interpretation HL· 45 min read

1. Common Non-Linear Regression Models★★☆☆☆⏱ 15 min

Many real-world bivariate relationships between an independent variable $x$ and dependent variable $y$ are not straight-line linear. Linear regression will produce a poor fit for these curved trends, so we use non-linear regression to match the pattern of the data.

📘 Definition

Non-linear regression

A statistical technique that models the relationship between two variables as a non-linear function, to better describe curved trends in observed data.

Example:

Population growth over time, radioactive decay, and the relationship between the volume and radius of a sphere are all non-linear.

Exponential growth/decay: $y = a b^{x}$
Power model: $y = a x^{b}$
Logarithmic model: $y = a + b ln x$
Reciprocal model: $y = a + \frac{b}{x}$

📐 Worked Example

Identify which standard non-linear model matches the description: The stopping distance $d$ of a car after braking is proportional to the square of its speed $v$ .

1
Translate the proportional relationship to a mathematical equation:
2
$d = k v^2$
3
Match the equation to standard non-linear model forms. Here, $y = d$ , $x = v$ , $a = k$ , and $b = 2$ .
4
This matches the power model form $y = a x^{b}$ , so this is a power non-linear relationship.

2. Linearizing Non-Linear Models★★★☆☆⏱ 20 min

To use standard linear regression to find the parameters of non-linear models, we transform the non-linear equation into the standard linear form $Y = m X + c$ , where $Y$ and/or $X$ are transformations of the original $y$ and $x$ variables.

Model	Original Equation	Transformed Form	$Y$	$X$	Slope $m$	Intercept $c$
Exponential	$y = a b^{x}$	$lo g y = lo g a + (lo g b) x$	$lo g y$	$x$	$lo g b$	$lo g a$
Power	$y = a x^{b}$	$lo g y = lo g a + b lo g x$	$lo g y$	$lo g x$	$b$	$lo g a$
Logarithmic	$y = a + b ln x$	$y = a + b ln x$	$y$	$ln x$	$b$	$a$

📐 Worked Example

Linearize the exponential decay model $A = A_{0} e^{- λ t}$ into linear form for regression.

1
Take the natural logarithm of both sides of the equation:
2
$\ln A = \ln\left(A_0 e^{-\lambda t}\right)$
3
Apply logarithm product and power rules to simplify:
4
$\ln A = \ln A_0 + \ln\left(e^{-\lambda t}\right) = \ln A_0 - \lambda t$
5
Match to linear form $Y = m X + c$ : $Y = ln A$ , $X = t$ , slope $m = - λ$ , intercept $c = ln A_{0}$ . This is the required linearized form.

3. Interpreting Parameters and Prediction★★★☆☆⏱ 20 min

Once we have the linear regression for the transformed data, we reverse the transformation to get the parameters of the original non-linear model. We can then interpret these parameters in context and use the model to predict $y$ values for given $x$ .

📐 Worked Example

A linear regression fitted to transformed log-log (base 10) data for a power model gives the equation $lo g y = 0.6 + 1.8 lo g x$ . Find the original power model and interpret the exponent.

1
Exponentiate both sides with base 10 to reverse the logarithm transformation:
2
$10^{\log y} = 10^{0.6 + 1.8\log x}$
3
Simplify using exponent rules: $1 0^{a + b} = 1 0^{a} \times 1 0^{b}$ and $a^{b c} = (a^{c})^{b}$
4
$y = 10^{0.6} \times (10^{\log x})^{1.8} = 10^{0.6} x^{1.8}$
5
Calculate $a = 1 0^{0.6} \approx 3.98$ , so the original power model is:
6
$y \approx 3.98 x^{1.8}$
7
Interpretation: $y$ is proportional to $x$ raised to the power 1.8, so a 10-fold increase in $x$ leads to a $1 0^{1.8} \approx 63$ -fold increase in $y$ .

✓ Quick check

Check your understanding of parameter calculation:

For an exponential model, transformed linear equation is $ln y = 1.5 + 0.4 x$ . What is the original model?
- $y = 1.5 e^{0.4 x}$
- $y = e^{1.5} e^{0.4 x}$
- $y = 0.4 e^{1.5 x}$
Reveal answer
1 —
Correct: The intercept $c = ln a$ , so $a = e^{c} = e^{1.5}$ , giving $y = e^{1.5} e^{0.4 x}$ .

4. Common Pitfalls

Wrong move:

Using the intercept of the linearized model directly as $a$ in the original model

Why:

For exponential and power models, the intercept is $lo g a$ , not $a$ itself, so this gives an incorrect parameter value

Correct move:

Always exponentiate the intercept to get $a$ for exponential and power models after linear regression

Wrong move:

Mixing bases of logarithms during transformation

Why:

Using natural log for transformation but base 10 for exponentiation (or vice versa) gives wrong final parameters

Correct move:

Use the same logarithm base consistently throughout all working steps

Wrong move:

Extrapolating predictions far outside the range of observed data

Why:

Non-linear models grow or decay very quickly outside the data range, leading to highly unreliable predictions

Correct move:

Only predict within the range of the original observed data, and explicitly note any extrapolation as unreliable

Wrong move:

Assuming a high $r^{2}$ for transformed data guarantees a good fit for the original model

Why:

$r^{2}$ measures linear fit of transformed data, not the goodness of fit of the original non-linear curve

Correct move:

Always plot the original data and the fitted non-linear curve to visually confirm the model fits the data

5. Quick Reference Cheatsheet

Model Type	Original Equation	Transformed Linear Form	Original Parameters (from $Y = m X + c$ )
Exponential $y = a b^{x}$	$y = a b^{x}$	$lo g y = lo g a + (lo g b) x$	$a = 1 0^{c}$ , $b = 1 0^{m}$ (base 10 log)
Exponential $y = a b^{x}$	$y = a b^{x}$	$ln y = ln a + (ln b) x$	$a = e^{c}$ , $b = e^{m}$ (natural log)
Power $y = a x^{b}$	$y = a x^{b}$	$lo g y = lo g a + b lo g x$	$a = 1 0^{c}$ , $b = m$
Logarithmic $y = a + b ln x$	$y = a + b ln x$	$y = a + b ln x$	$a = c$ , $b = m$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 2
Fit exponential model to population data
2022 · 2
Predict using fitted power regression model
2023 · 1
Transform logarithmic model to linear form

Going deeper

syllabusOfficial IB AI HL SyllabusTopic 5: Statistics and probability reference

What's Next

Non-linear regression is one of the most useful statistical tools for IB AI HL, and you will often use it to model data in your internal assessment project. Mastering linearization techniques builds a foundation for more advanced statistical modeling, and understanding how to transform non-linear relationships connects your knowledge of logarithms from algebra to applied statistics. After this module, you can learn how to test if your non-linear model is a good fit using residual analysis, or explore time series analysis which commonly uses exponential non-linear models for growth and decay forecasting.