Multivariate adaptive regression splines

In statistics, multivariate adaptive regression splines (MARS) is a form of regression analysis introduced by Jerome H. Friedman in 1991. It is a non-parametric regression technique and can be seen as an extension of linear models that automatically models nonlinearities and interactions between variables.

The term "MARS" is trademarked and licensed to Salford Systems. In order to avoid trademark infringements, many open source implementations of MARS are called "Earth".

This section introduces MARS using a few examples. We start with a set of data: a matrix of input variables x, and a vector of the observed responses y, with a response for each row in x. For example, the data could be:

Here there is only one independent variable, so the x matrix is just a single column. Given these measurements, we would like to build a model which predicts the expected y for a given x.

A linear model for the above data is

The hat on the ${\displaystyle {\hat {y}}}$ indicates that ${\displaystyle {\hat {y}}}$ is estimated from the data. The figure on the right shows a plot of this function: a line giving the predicted ${\displaystyle {\hat {y}}}$ versus x, with the original values of y shown as red dots.

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