IB Maths AI 4.9 Notes

Non-linear regression

In this section, we extend regression beyond straight lines. A linear regression uses a model of the form $y=ax+b$, but many real data sets follow curved patterns instead. In those cases, a non-linear regression model may give a better fit. Technology is used to fit these models and determine the constants.

Common regression models include:

• Quadratic: $y=ax^2+bx+c$
• Cubic: $y=ax^3+bx^2+cx+d$
• Quartic: $y=ax^4+bx^3+cx^2+dx+e$
• Logarithmic: $y=a\ln(x)+b$
• Exponential: $y=ae^{bx}+c$
• Power: $y=ax^b+c$
• Sinusoidal: $y=a\sin(bx+c)+d$

The best model depends on both the shape of the data and the context.

To fit a non-linear regression model, enter the data into your GDC and choose the required regression type. The calculator returns the constants that define the model.

For example, a quadratic regression gives the constants in $y=ax^2+bx+c$, while an exponential regression gives the constants in a form such as $y=ae^{bx}+c$. These models can then be used to predict values of $y$ for given values of $x$.

The height of a tree, in cm, is recorded over $5$ days. Form a cubic and exponential regression model from the following data.

Day $(x)$	Height $(y)$
$1$	$2.0$
$2$	$2.8$
$3$	$4.2$
$4$	$6.0$
$5$	$8.2$

A cubic regression model of the form $y=a+bx+cx^2+dx^3$ gives:

$y\approx1.5+0.6x+0.2x^2-0.01x^3$

So $a\approx1.5$, $b\approx0.6$, $c\approx0.2$, and $d\approx-0.01$.

To predict the height on day $7$, substitute $x=7$:

$y=1.5+0.6(7)+0.2(7^2)-0.01(7^3)\approx12.07$

So the predicted height on day $7$ is about $12.07$cm.

Now fit an exponential regression model. Suppose technology gives:

$y\approx1.41e^{0.30x}+0$

Since $e^{0.30x}=(e^{0.30})^x$, this can be written as $y\approx1.41(1.35)^x+0$.

So the exponential model has $a\approx1.41$, $b\approx1.35$, and $c\approx0$.

A model may fit the data well over a short interval but still become unsuitable later.

In the cubic model above, the negative cubic term means that for large values of $x$, the model will eventually predict that the tree height starts decreasing. This is not realistic if the tree is continuing to grow.

The exponential model predicts growth that becomes faster and faster forever. In reality, a tree cannot grow indefinitely at an accelerating rate. So a good fit to the given data does not automatically mean that a model is valid outside the observed range.