Is MARS linear or nonlinear?

It is linear in its fitted coefficients but can be nonlinear in the original predictors.

Are knots specified entirely by the user?

The adaptive fitting procedure selects them from the data under its search rules.

Can it model interactions?

Yes, when products of basis functions are allowed.

AI Glossary / AI and machine learning / Intermediate

Multivariate adaptive regression spline

Definition

A multivariate adaptive regression spline model represents a response as a weighted combination of simple piecewise functions chosen from the data. Commonly called MARS, it can capture nonlinear relationships and selected interactions while retaining an explicit mathematical form.

What it means

A straight-line model assumes the same slope throughout an input's range. Many relationships change slope: a process may respond differently above a threshold, for example. MARS builds flexible relationships from hinge functions that bend at selected points called knots.

A common hinge is max(0, x minus t), where t is a knot. It is zero on one side of the knot and grows linearly on the other. Combining hinges allows the fitted response to change its rate of increase or decrease.

The name contains multivariate because multiple predictors can be considered. A typical use still predicts one response. The model's flexibility comes from its selected basis functions, not from assuming that every variable must interact with every other one.

How it works

A forward stage adds useful basis functions and, where permitted, products of basis functions that represent interactions. This can create an intentionally large candidate model.

A backward stage removes terms to control complexity, using a criterion that balances fit with model size. Coefficients for the retained basis functions determine the final prediction. Implementations may expose controls for interaction degree, number of terms and pruning.

Cross-validation or a separate validation set helps select complexity. The entire basis-selection process belongs inside the training procedure; choosing knots using the test data would leak information into the model and make its evaluation overly optimistic.

Three practical examples

An operations analyst models a response that changes slope as workload crosses different ranges.
An engineering research team explores nonlinear relationships between measured inputs and a continuous output.
A forecasting study compares an adaptive spline model with linear regression and tree-based baselines on the same held-out data.

Think of it this way

Think of constructing a flexible ramp from straight segments joined at hinges. The ramp can change slope without requiring one highly curved piece to describe the whole path.

The analogy captures piecewise behavior but has limits. In multiple dimensions, interactions can make the surface difficult to picture. The selected hinges describe statistical fit and should not automatically be interpreted as physical thresholds or business rules.

Worked example

Evaluating a simple hinge model

Consider the illustrative model y-hat = 10 + 2x + 3 max(0, x minus 5). This is a small example of a hinge-based function, not a model fitted to actual business data.

At x = 3, the hinge is max(0, -2) = 0, so the prediction is 10 + 6 = 16. At x = 7, the hinge is 2, so the prediction is 10 + 14 + 6 = 30.

Below x = 5, the slope is 2. Above x = 5, the hinge contributes another 3 to the slope, making it 5. The function remains continuous at the knot: both sides meet at a prediction of 20.

A fitted MARS model can contain many such terms and products involving different predictors. The example explains how to read one term; it does not show that x = 5 is a meaningful causal threshold.

Illustrative case study

Fictional case: modeling turnaround time

Hawthorn Services, an invented repair center, studies how workload and job characteristics relate to turnaround time. Analysts compare a MARS model with a simple linear baseline, using later time periods for validation.

The spline model suggests a slope change at higher workloads. Staff investigate whether this corresponds to staffing practices, but they avoid treating the fitted knot as proof of a capacity limit. They also check predictions for workload ranges scarcely represented in training.

The case illustrates interpretation and validation. It does not report a real reduction in delays or establish an operational staffing policy.

Common misconceptions and mistakes

A fitted knot is not necessarily a true threshold in the underlying process. MARS is also not the same as fitting one high-degree polynomial, although both can model nonlinear relationships.

An explicit formula is not automatically simple to interpret when it contains many interaction terms. Good interpolation does not guarantee sensible extrapolation beyond the observed input range. Pruning helps control complexity but does not eliminate the need for independent testing.

Business use

MARS can be useful when a business needs more flexibility than a linear model while retaining an inspectable equation. It is especially worth testing when slope changes and limited interactions are plausible.

Compare predictive error, stability and practical interpretability with simpler alternatives. Examine whether knots and terms change substantially across resamples. Check output constraints, such as nonnegative quantities, rather than assuming the model respects them. Keep a record of the fitted basis functions and preprocessing. If users need explanations, show the relevant response curves and uncertainty, and distinguish modeled association from evidence about what an intervention would cause.

Check your understanding

In max(0, x minus 5), what value does the hinge take when x equals 4, and what happens when x equals 8?

Answer: It is zero at 4 and 3 at 8. The term contributes only on the side above its knot.

Related terms

Pruning
Cross-validation
Spline
hinge function
regression
basis function
interaction

Learn next

Learn next how an additive spline model differs from one containing interactions. Then plot a few hinge functions and their weighted sum to see how local slope changes create a flexible overall response.

Weighted sum

Sources

Primary source: Jerome Friedman, Multivariate Adaptive Regression Splines, https://www.stat.yale.edu/~lc436/08Spring665/Mars_Friedman_91.pdf ; author institution record, https://statistics.stanford.edu/technical-reports/multivariate-adaptive-regression-splines . Implementation documentation, https://contrib.scikit-learn.org/py-earth/content.html , was consulted for hinges, forward selection and pruning. The Stanford record was checked; the original Yale PDF and repository full text could not be retrieved. Formula and case are original.

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