2277:. 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".
1849:). The two basis functions in the pair are identical except that a different side of a mirrored hinge function is used for each function. Each new basis function consists of a term already in the model (which could perhaps be the intercept term) multiplied by a new hinge function. A hinge function is defined by a variable and a knot, so to add a new basis function, MARS must search over all combinations of the following:
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968:
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2283:(BMARS) uses the same model form, but builds the model using a Bayesian approach. It may arrive at different optimal MARS models because the model building approach is different. The result of BMARS is typically an ensemble of posterior samples of MARS models, which allows for probabilistic prediction.
2030:
A further constraint can be placed on the forward pass by specifying a maximum allowable degree of interaction. Typically only one or two degrees of interaction are allowed, but higher degrees can be used when the data warrants it. The maximum degree of interaction in the first MARS example above is
2018:
is the number of hinge-function knots, so the formula penalizes the addition of knots. Thus the GCV formula adjusts (i.e. increases) the training RSS to penalize more complex models. We penalize flexibility because models that are too flexible will model the specific realization of noise in the data
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is used when the underlying form of the function is known and regression is used only to estimate the parameters of that function. MARS, on the other hand, estimates the functions themselves, albeit with severe constraints on the nature of the functions. (These constraints are necessary because
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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
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the model. To build a model with better generalization ability, the backward pass prunes the model, deleting the least effective term at each step until it finds the best submodel. Model subsets are compared using the
Generalized cross validation (GCV) criterion described below.
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963:{\displaystyle {\begin{aligned}\mathrm {ozone} =&\ 5.2\\&{}+0.93\max(0,\mathrm {temp} -58)\\&{}-0.64\max(0,\mathrm {temp} -68)\\&{}-0.046\max(0,234-\mathrm {ibt} )\\&{}-0.016\max(0,\mathrm {wind} -7)\max(0,200-\mathrm {vis} )\end{aligned}}}
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To obtain the above expression, the MARS model building procedure automatically selects which variables to use (some variables are important, others not), the positions of the kinks in the hinge functions, and how the hinge functions are combined.
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Other constraints on the forward pass are possible. For example, the user can specify that interactions are allowed only for certain input variables. Such constraints could make sense because of knowledge of the process that generated the data.
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vary, with the other variables fixed at their median values. The figure shows that wind does not affect the ozone level unless visibility is low. We see that MARS can build quite flexible regression surfaces by combining hinge functions.
2171:(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.
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This process of adding terms continues until the change in residual error is too small to continue or until the maximum number of terms is reached. The maximum number of terms is specified by the user before model building starts.
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rather than hinge functions, and they do not automatically model variable interactions. The smoother fit and lack of regression terms reduces variance when compared to MARS, but ignoring variable interactions can worsen the
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A hinge function is zero for part of its range, so can be used to partition the data into disjoint regions, each of which can be treated independently. Thus for example a mirrored pair of hinge functions in the expression
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No regression modeling technique is best for all situations. The guidelines below are intended to give an idea of the pros and cons of MARS, but there will be exceptions to the guidelines. It is useful to compare MARS to
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This expression models air pollution (the ozone level) as a function of the temperature and a few other variables. Note that the last term in the formula (on the last line) incorporates an interaction between
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The forward pass adds terms in pairs, but the backward pass typically discards one side of the pair and so terms are often not seen in pairs in the final model. A paired hinge can be seen in the equation for
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fashion, but a key aspect of MARS is that because of the nature of hinge functions, the search can be done quickly using a fast least-squares update technique. Brute-force search can be sped up by using a
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Building MARS models often requires little or no data preparation. The hinge functions automatically partition the input data, so the effect of outliers is contained. In this respect MARS is similar to
2156:. 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.
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1451:. MARS automatically selects variables and values of those variables for knots of the hinge functions. Examples of such basis functions can be seen in the middle three lines of the ozone formula.
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The resulting fitted function is continuous, unlike recursive partitioning, which can give a more realistic model in some situations. (However, the model is not smooth or differentiable).
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The backward pass has an advantage over the forward pass: at any step it can choose any term to delete, whereas the forward pass at each step can only see the next pair of terms.
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2256:(commonly called CART). MARS can be seen as a generalization of recursive partitioning that allows for continuous models, which can provide a better fit for numerical data.
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3) a product of two or more hinge functions. These basis functions can model interaction between two or more variables. An example is the last line of the ozone formula.
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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".
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One might assume that only piecewise linear functions can be formed from hinge functions, but hinge functions can be multiplied together to form non-linear functions.
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MARS then repeatedly adds basis function in pairs to the model. At each step it finds the pair of basis functions that gives the maximum reduction in sum-of-squares
637:
2478:(1993). "Estimating Functions of Mixed Ordinal and Categorical Variables Using Adaptive Splines". In Stephan Morgenthaler; Elvezio Ronchetti; Werner Stahel (eds.).
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With MARS models, as with any non-parametric regression, parameter confidence intervals and other checks on the model cannot be calculated directly (unlike
2231:(GLMs) can be incorporated into MARS models by applying a link function after the MARS model is built. Thus, for example, MARS models can incorporate
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2280:
338:). We thus turn to MARS to automatically build a model taking into account non-linearities. MARS software constructs a model from the given
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MARS is suitable for handling large datasets, and implementations run very quickly. However, recursive partitioning can be faster than MARS.
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MARS models are simple to understand and interpret. Compare the equation for ozone concentration above to, say, the innards of a trained
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score in the special case where errors are
Gaussian, or where the squared error loss function is used. GCV was introduced by Craven and
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is a constant coefficient. For example, each line in the formula for ozone above is one basis function multiplied by its coefficient.
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and these variables will be unclear and not easily visible by plotting. We can use MARS to discover that non-linear relationship.
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The backward pass compares the performance of different models using
Generalized Cross-Validation (GCV), a minor variant on the
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MARS builds a model in two phases: the forward and the backward pass. This two-stage approach is the same as that used by
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468:{\displaystyle {\begin{aligned}{\widehat {y}}=&\ 25\\&{}+6.1\max(0,x-13)\\&{}-3.1\max(0,13-x)\end{aligned}}}
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matrix is just a single column. Given these measurements, we would like to build a model which predicts the expected
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1) a constant 1. There is just one such term, the intercept. In the ozone formula above, the intercept term is 5.2.
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MARS models can make predictions very quickly, as they only require evaluating a linear function of the predictors.
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implementations do not allow missing values in predictors, but free implementations of regression trees (such as
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One constraint has already been mentioned: the user can specify the maximum number of terms in the forward pass.
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is estimated from the data. The figure on the right shows a plot of this function: a line giving the predicted
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MARS starts with a model which consists of just the intercept term (which is the mean of the response values).
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This section introduces MARS using a few examples. We start with a set of data: a matrix of input variables
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and extended by
Friedman for MARS; lower values of GCV indicate better models. The formula for the GCV is
1995:(effective number of parameters) = (number of mars terms) + (penalty) · ((number of Mars terms) − 1 ) / 2
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may be non-linear (look at the red dots relative to the regression line at low and high values of
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Earth – Multivariate adaptive regression splines in Orange (Python machine learning library)
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in the first MARS example above; there are no complete pairs retained in the ozone example.
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To calculate the coefficient of each term, MARS applies a linear regression over the terms.
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Several free and commercial software packages are available for fitting MARS-type models.
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once again shown as red dots. The predicted response is now a better fit to the original
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2495:"Estimating Functions of Mixed Ordinal and Categorical Variables Using Adaptive Splines"
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which also partitions the data into disjoint regions, although using a different method.
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1598:. The figure on the right shows a mirrored pair of hinge functions with a knot at 3.1.
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that reduces the number of parent terms considered at each step ("Fast MARS").
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that automatically models nonlinearities and interactions between variables.
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where RSS is the residual sum-of-squares measured on the training data and
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Denison, D. G. T.; Holmes, C. C.; Mallick, B. K.; Smith, A. F. M. (2002).
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3) all values of each variable (for the knot of the new hinge function).
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and this is done below. (Recursive partitioning is also commonly called
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notation used in this article, hinge functions are often represented by
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31:
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Denison, D. G. T.; Mallick, B. K.; Smith, A. F. M. (1 December 1998).
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linear graph shown for the simple MARS model in the previous section.
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and related techniques must be used for validating the model instead.
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486:
The figure on the right shows a plot of this function: the predicted
2427:, Stanford University Department of Statistics, Technical Report 110
2338:
Friedman, J. H. (1991). "Multivariate
Adaptive Regression Splines".
2212:) do allow missing values using a technique called surrogate splits.
2081:. Statements consisting only of original research should be removed.
2698:
external links, and converting useful links where appropriate into
2627:
Denison D.G.T., Holmes C.C., Mallick B.K., and Smith A.F.M. (2004)
2624:(has a chapter on MARS and discusses some tweaks to the algorithm)
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1890:
1240:{\displaystyle {\widehat {f}}(x)=\sum _{i=1}^{k}c_{i}B_{i}(x).}
662:
In this simple example, we can easily see from the plot that
2876:
Bayesian methods for nonlinear classification and regression
2810:
Bayesian
Methods for Nonlinear Classification and Regression
2630:
Bayesian
Methods for Nonlinear Classification and Regression
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2) all variables (to select one for the new basis function)
534:
to take into account non-linearity. The kink is produced by
2480:
New
Directions in Statistical Data Analysis and Robustness
670:(and might perhaps guess that y varies with the square of
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from
Salford Systems. Based on Friedman's implementation.
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is the number of observations (the number of rows in the
538:. The hinge functions are the expressions starting with
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MARS has automatically produced a kink in the predicted
2801:
ARESLab: Adaptive Regression Splines toolbox for Matlab
1466:
A mirrored pair of hinge functions with a knot at x=3.1
659:). Hinge functions are described in more detail below.
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instead of just the systematic structure of the data.
2007:) but can be increased by the user if they so desire.
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An example MARS expression with multiple variables is
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2576:Hastie T., Tibshirani R., and Friedman J.H. (2009)
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2645:Statistical learning from a regression perspective
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2167:MARS (like recursive partitioning) does automatic
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2682:may not follow Knowledge's policies or guidelines
2003:is typically 2 (giving results equivalent to the
1991:The effective number of parameters is defined as
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2821:Earth – Multivariate adaptive regression splines
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1672:{\displaystyle 6.1\max(0,x-13)-3.1\max(0,13-x)}
1250:The model is a weighted sum of basis functions
674:). However, in general there will be multiple
1872:The search at each step is usually done in a
1815:
54:technique and can be seen as an extension of
2486:
1044:The figure on the right plots the predicted
2441:
1444:{\displaystyle \max(0,{\text{constant}}-x)}
1401:{\displaystyle \max(0,x-{\text{constant}})}
1972:· (1 − (effective number of parameters) /
2718:Learn how and when to remove this message
2610:Heping Zhang and Burton H. Singer (2010)
2351:
2097:Learn how and when to remove this message
1365:function. A hinge function has the form
326:indicates that the relationship between
73:, and a vector of the observed responses
2492:
2474:
2337:
2242:discovering a model from the data is an
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1355:takes one of the following three forms:
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477:
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36:multivariate adaptive regression splines
18:Multivariate adaptive regression splines
2613:Recursive Partitioning and Applications
214:{\displaystyle {\widehat {y}}=-37+5.1x}
14:
2897:
2662:
2579:The Elements of Statistical Learning
2437:
2435:
2433:
2152:MARS can handle both continuous and
2046:
2035:); in the ozone example it is two.
1884:
977:Variable interaction in a MARS model
482:A simple MARS model of the same data
2505:from the original on April 11, 2022
2446:. New York, NY: Springer New York.
2134:MARS models are more flexible than
1833:
666:has a non-linear relationship with
27:Non-parametric regression technique
24:
2607:(has an example using MARS with R)
2570:
2493:Friedman, Jerome H. (1991-06-01).
2442:Kuhn, Max; Johnson, Kjell (2013).
2250:without constraints on the model.)
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2596:Extending the Linear Model with R
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2051:
2042:
2031:one (i.e. no interactions or an
1692:Hinge functions are also called
1071:{\displaystyle \mathrm {ozone} }
2262:. Unlike MARS, GAMs fit smooth
2223:Extensions and related concepts
2014:(number of Mars terms − 1 ) / 2
1150:MARS builds models of the form
1102:{\displaystyle \mathrm {wind} }
1006:{\displaystyle \mathrm {wind} }
678:, and the relationship between
2878:. Chichester, England: Wiley.
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2780:package. Not Friedman's MARS.
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2403:
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1958:leave-one-out cross-validation
1923:{\displaystyle {\widehat {y}}}
1812:means take the positive part.
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1476:A key part of MARS models are
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1130:{\displaystyle \mathrm {vis} }
1034:{\displaystyle \mathrm {vis} }
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519:, with the original values of
508:{\displaystyle {\widehat {y}}}
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315:, with the original values of
304:{\displaystyle {\widehat {y}}}
275:{\displaystyle {\widehat {y}}}
246:{\displaystyle {\widehat {y}}}
13:
1:
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1940:Cross-validation (statistics)
64:
2005:Akaike information criterion
1954:Akaike information criterion
1948:Akaike information criterion
1934:Generalized cross validation
322:The data at the extremes of
7:
2444:Applied Predictive Modeling
2287:
2260:Generalized additive models
2077:the claims made and adding
1564:{\displaystyle \max(0,c-x)}
1517:{\displaystyle \max(0,x-c)}
10:
2926:
2792:package for Bayesian MARS.
2304:Rational function modeling
1937:
1852:1) existing terms (called
1819:
1816:The model building process
1704:functions. Instead of the
1594:is a constant, called the
1469:
2616:, 2nd edition. Springer,
2582:, 2nd edition. Springer,
2452:10.1007/978-1-4614-6849-3
2235:to predict probabilities.
2229:Generalized linear models
1889:The forward pass usually
586:{\displaystyle \max(a,b)}
52:non-parametric regression
2905:Nonparametric regression
2532:Statistics and Computing
2340:The Annals of Statistics
1348:{\displaystyle B_{i}(x)}
1279:{\displaystyle B_{i}(x)}
2590:(has a section on MARS)
2544:10.1023/A:1008824606259
140:Here there is only one
2362:10.1214/aos/1176347963
2254:Recursive partitioning
2162:recursive partitioning
2130:article for details).
2128:recursive partitioning
2112:recursive partitioning
1956:that approximates the
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1828:recursive partitioning
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978:
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632:{\displaystyle a>b}
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215:
171:for the above data is
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2854:STATISTICA Data Miner
2239:Non-linear regression
1938:Further information:
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1717:{\displaystyle \max }
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1470:Further information:
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1306:{\displaystyle c_{i}}
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551:{\displaystyle \max }
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2688:improve this article
2314:Spline interpolation
2309:Segmented regression
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1805:{\displaystyle _{+}}
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1772:{\displaystyle _{+}}
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2840:Commercial software
2700:footnote references
2476:Friedman, Jerome H.
2233:logistic regression
1822:Stepwise regression
319:shown as red dots.
44:regression analysis
2833:for Bayesian MARS.
2812:for Bayesian MARS.
2593:Faraway J. (2005)
2399:CRAN Package earth
2169:variable selection
2062:possibly contains
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2885:978-0-471-49036-4
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2653:978-0-387-77500-5
2643:Berk R.A. (2008)
2639:978-0-471-49036-4
2622:978-1-4419-6823-4
2605:978-1-58488-424-8
2588:978-0-387-84857-0
2319:Spline regression
2294:Linear regression
2183:linear regression
2136:linear regression
2107:
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2099:
2064:original research
1917:
1885:The backward pass
1856:in this context)
1587:{\displaystyle c}
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652:{\displaystyle b}
606:{\displaystyle a}
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50:in 1991. It is a
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2910:Machine learning
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2786:function in the
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2187:Cross-validation
2154:categorical data
2116:regression trees
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2571:Further reading
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2525:"Bayesian MARS"
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42:) is a form of
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2659:External links
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2538:(4): 337–346.
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2143:neural network
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1146:The MARS model
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163:A linear model
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2797:Matlab code:
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2734:Free software
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2482:. Birkhauser.
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2281:Bayesian MARS
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2147:random forest
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2060:This article
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2043:Pros and cons
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2020:
2013:
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2011:
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1968:GCV = RSS / (
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56:linear models
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37:
33:
19:
2875:
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2809:
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2714:
2708:October 2016
2705:
2690:by removing
2677:
2647:, Springer,
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2629:
2612:
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2535:
2531:
2518:
2507:. Retrieved
2498:
2488:
2479:
2470:
2443:
2424:
2416:
2405:
2394:
2343:
2339:
2333:
2246:that is not
2119:
2115:
2108:
2093:
2087:October 2016
2084:
2061:
2037:
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2029:
2026:
2017:
2009:
2000:
1998:
1990:
1985:
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1896:
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1871:
1867:
1864:
1861:
1858:
1854:parent terms
1853:
1851:
1840:
1837:
1825:
1698:hockey stick
1691:
1688:
1682:creates the
1681:
1600:
1595:
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1526:
1477:
1475:
1453:
1362:
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1149:
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684:
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169:linear model
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152:for a given
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39:
35:
29:
2860:ADAPTIVEREG
2346:(1): 1–67.
2023:Constraints
2010:Note that
1874:brute-force
346:as follows
2899:Categories
2744:packages:
2509:2022-04-11
2386:0765.62064
2325:References
2248:well-posed
2175:variance).
2126:; see the
2071:improve it
1820:See also:
65:The basics
32:statistics
2862:from SAS.
2777:polspline
2692:excessive
2633:, Wiley,
2552:1573-1375
2425:Fast MARS
2348:CiteSeerX
2202:polspline
2185:models).
2075:verifying
1988:matrix).
1915:^
1879:heuristic
1790:⋅
1751:−
1735:±
1702:rectifier
1684:piecewise
1661:−
1640:−
1631:−
1553:−
1506:−
1433:−
1388:−
1186:∑
1167:^
940:−
916:−
884:−
858:−
837:−
819:−
787:−
769:−
500:^
453:−
432:−
414:−
367:^
296:^
267:^
238:^
197:−
188:^
144:, so the
2826:py-earth
2772:polymars
2560:12570055
2503:Archived
2288:See also
1891:overfits
1843:residual
1429:constant
1392:constant
527:values.
2817:Python
2768:package
2756:package
2686:Please
2678:use of
2599:, CRC,
2423:(1993)
2378:1091842
2370:2241837
2268:splines
2138:models.
2069:Please
2001:penalty
1830:trees.
1286:. Each
639:, else
558:(where
515:versus
311:versus
2882:
2831:pyBASS
2651:
2637:
2620:
2603:
2586:
2558:
2550:
2458:
2384:
2376:
2368:
2350:
2275:TSMARS
2200:, and
1999:where
1946:, and
1779:where
1574:where
722:
378:
2753:earth
2748:earth
2556:S2CID
2528:(PDF)
2366:JSTOR
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