Multivariate Adaptive Regression Splines (MARS)

Multivariate adaptive regression splines (MARS) build a regression from pairs of hinge functions and their products, adding terms forward and pruning them backward by generalized cross-validation.

MARS(다변량 적응 회귀 스플라인)는 경첩 함수 쌍과 그 곱으로 회귀식을 만들고, 정방향으로 항을 더한 뒤 일반화 교차검증으로 가지치는 방법입니다.

I suspect thresholds and interactions among my predictors; is there a regression that finds the breakpoints automatically?

설명변수에 문턱값과 상호작용이 있을 것 같은데, 꺾이는 지점을 자동으로 찾아주는 회귀가 있나요?

MARS searches each predictor for knots where the slope changes, adds the hinge-function pair that most reduces the residual sum of squares, and then prunes terms by generalized cross-validation. The result is a piecewise-linear model with interpretable knots and, if allowed, interactions, plus a ranking of predictor importance.

When to use Multivariate Adaptive Regression Splines (MARS)

• Relationships are expected to change slope at unknown thresholds.
• Interactions between predictors should be found from the data.
• An interpretable alternative to tree ensembles is wanted for regression.

When to use something else

• Smooth curves with uncertainty bands are needed; use a generalized additive model.
• The sample is small relative to the number of predictors; use penalized linear regression.

Assumptions and requirements

• Observations are independent.
• The mean is well approximated by piecewise-linear hinge functions.
• Knot search is limited to 40 candidate quantiles per predictor in this implementation.

What the analysis reports

• Selected basis functions with knots and coefficients
• R-squared, GCV and GCV R-squared
• Variable importance (RSS increase when a variable's terms are removed)
• Number of terms before and after pruning

How to report it

A MARS model with maximum interaction degree [value] retained [number] basis functions (R2 = [value], GCV R2 = [value]); the most important predictor was [X].

Related methods

• Segmented (Piecewise) Regression
• Generalized Additive Models (GAM)
• Regression Trees

Run Multivariate Adaptive Regression Splines (MARS) on Quorant