Multivariate adaptive regression spline: Difference between revisions

Revision as of 05:09, 1 September 2020

Revision as of 05:09, 1 September 2020

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where '''penalty''' is about 2 or 3 (the MARS software allows the user to preset penalty).

Note that

: (number of Mars terms − 1 ) / 2

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*MARS models are simple to understand and interpret.<ref name=":0">{{Cite book|title=Applied Predictive Modeling|last=Kuhn|first=Max|last2=Johnson|first2=Kjell|date=2013|publisher=Springer New York|isbn=9781461468486|___location=New York, NY|language=en|doi=10.1007/978-1-4614-6849-3}}</ref> Compare the equation for ozone concentration above to, say, the innards of a trained [[Artificial neural network|neural network]] or a [[random forest]].

*MARS can handle both continuous and categorical data.<ref>[[Friedman, J. H.]] (1993) ''Estimating Functions of Mixed Ordinal and Categorical Variables Using Adaptive Splines'', New Directions in Statistical Data Analysis and Robustness (Morgenthaler, Ronchetti, Stahel, eds.), Birkhauser</ref> MARS tends to be better than recursive partitioning for numeric data because hinges are more appropriate for numeric variables than the piecewise constant segmentation used by recursive partitioning.

*Building MARS models often requires little or no data preparation.<ref name=":0" />. The hinge functions automatically partition the input data, so the effect of outliers is contained. In this respect MARS is similar to [[recursive partitioning]] which also partitions the data into disjoint regions, although using a different method. (Nevertheless, as with most statistical modeling techniques, known outliers should be considered for removal before training a MARS model.{{Citation needed|date=March 2019}})

*MARS (like recursive partitioning) does automatic [[Feature selection|variable selection]] (meaning it includes important variables in the model and excludes unimportant ones). However, there can be some arbitrariness in the selection, especially when there are correlated predictors, and this can affect interpretability<ref name=":0" />

*MARS models tend to have a good bias-variance trade-off. The models are flexible enough to model non-linearity and variable interactions (thus MARS models have fairly low bias), yet the constrained form of MARS basis functions prevents too much flexibility (thus MARS models have fairly low variance).

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* [[Generalized additive model]]s. From the user's perspective GAMs are similar to MARS but (a) fit smooth [[Local regression|loess]] or polynomial [[Spline (mathematics)|splines]] instead of MARS basis functions, and (b) do not automatically model variable interactions. The fitting method used internally by GAMs is very different from that of MARS. For models that do not require automatic discovery of variable interactions GAMs often compete favorably with MARS.

* [[TSMARS]]. Time Series Mars is the term used when MARS models are applied in a time series context. Typically in this set up the predictors are the lagged time series values resulting in autoregressive spline models. These models and extensions to include moving average spline models are described in "Univariate Time Series Modelling and Forecasting using TSMARS: A study of threshold time series autoregressive, seasonal and moving average models using TSMARS".

* Similar schemes using Gaussian Processes can be found in the literature.<ref>{{Cite journal|last=Svendsen|first=Daniel Heestermans|last2=Martino|first2=Luca|last3=Camps-Valls|first3=Gustau|date=April 2020-04|title=Active emulation of computer codes with Gaussian processes – Application to remote sensing|url=https://linkinghub.elsevier.com/retrieve/pii/S0031320319304042|journal=Pattern Recognition|language=en|volume=100|pages=107103|doi=10.1016/j.patcog.2019.107103}}</ref>

== See also ==