# earth

##### Multivariate Adaptive Regression Splines

Build a regression model using the techniques in Friedman's papers "Multivariate Adaptive Regression Splines" and "Fast MARS".

See the package vignette

- Keywords
- models, regression, smooth

##### Usage

```
## S3 method for class 'formula':
earth(formula = stop("no 'formula' argument"), data = NULL,
weights = NULL, wp = NULL, subset = NULL,
na.action = na.fail,
pmethod = c("backward", "none", "exhaustive", "forward", "seqrep", "cv"),
keepxy = FALSE, trace = 0, glm = NULL, degree = 1, nprune = NULL,
ncross=1, nfold=0, stratify=TRUE,
varmod.method = "none", varmod.exponent = 1,
varmod.conv = 1, varmod.clamp = .1, varmod.minspan = -3,
Scale.y = (NCOL(y)==1), ...)
```## S3 method for class 'default':
earth(x = stop("no 'x' argument"), y = stop("no 'y' argument"),
weights = NULL, wp = NULL, subset = NULL,
na.action = na.fail,
pmethod = c("backward", "none", "exhaustive", "forward", "seqrep", "cv"),
keepxy = FALSE, trace = 0, glm = NULL, degree = 1, nprune = NULL,
ncross=1, nfold=0, stratify=TRUE,
varmod.method = "none", varmod.exponent = 1,
varmod.conv = 1, varmod.clamp = .1, varmod.minspan = -3,
Scale.y = (NCOL(y)==1), ...)

## S3 method for class 'fit':
earth(x = stop("no 'x' argument"), y = stop("no 'y' argument"),
weights = NULL, wp = NULL, subset = NULL,
na.action = na.fail,
pmethod = c("backward", "none", "exhaustive", "forward", "seqrep", "cv"),
keepxy = FALSE, trace = 0, glm = NULL, degree = 1,
penalty = if(degree > 1) 3 else 2,
nk = min(200, max(20, 2 * ncol(x))) + 1,
thresh = 0.001, minspan = 0, endspan = 0,
newvar.penalty = 0, fast.k = 20, fast.beta = 1,
linpreds = FALSE, allowed = NULL,
nprune = NULL, Object = NULL,
Scale.y = (NCOL(y)==1), Adjust.endspan = 2, Force.weights = FALSE,
Use.beta.cache = TRUE, Force.xtx.prune = FALSE,
Get.leverages = NROW(x) < 1e5, Exhaustive.tol = 1e-10, ...)

##### Arguments

- formula
- Model formula.
- data
- Data frame for
`formula`

. - x
- Matrix or dataframe containing the independent variables.
- y
- Vector containing the response variable, or, in the case of multiple responses, a matrix or dataframe whose columns are the values for each response.
- subset
- Index vector specifying which cases to use, i.e., which rows in
`x`

to use. Default is NULL, meaning all. - weights
- Case weights.
Default is NULL, meaning no case weights.
If specified,
`weights`

must have length equal to`nrow(x)`

before applying`subset`

. Zero weights are converted to a very small nonzero value. - wp
- Response weights.
Default is NULL, meaning no response weights.
If specified,
`wp`

must have an element for each column of`y`

(after`factors`

in`y`

- na.action
- NA action. Default is
`na.fail`

, and only`na.fail`

is supported. - keepxy
- Default is
`FALSE`

. Set to`TRUE`

to retain the following in the returned value:`x`

and`y`

(or`data`

),`subset`

, and`weights`

. The function - trace
- Trace
`earth`

's execution. Default is`0`

. Values:`0`

no tracing`.3`

variance model (the`varmod.method`

arg)`.5`

cross validation (the`nfold`

arg)`1`

overview - glm
- NULL (default) or a list of arguments to pass on to
`glm`

. See the documentation of`glm`

for a description of these arguments See*Generalized line* - degree
- Maximum degree of interaction (Friedman's $mi$).
Default is
`1`

, meaning build an additive model (i.e., no interaction terms). - penalty
- Generalized Cross Validation (GCV) penalty per knot.
Default is
`if(degree>1) 3 else 2`

. Simulation studies suggest values in the range of about`2`

to`4`

. The FAQ section in the vignette has some information - nk
- Maximum number of model terms before pruning, i.e., the
maximum number of terms created by the forward pass.
Includes the intercept.
The actual number of terms created by the forward pass will often be
less than
`nk`

because of o - thresh
- Forward stepping threshold.
Default is
`0.001`

. This is one of the arguments used to decide when forward stepping should terminate: the forward pass terminates if adding a term changes RSq by less than`thresh`

. - minspan
- Minimum number of observations between knots.
(This increases resistance to runs of correlated noise in the input data.)
The default
`minspan=0`

is treated specially and means calculate the`minspan`

internally, as per Fr - endspan
- Minimum number of observations before the first and after the final knot.
The default
`endspan=0`

is treated specially and means calculate the`minspan`

internally, as per the MARS paper equation 45 with $alpha$ = 0.05. S - newvar.penalty
- Penalty for adding a new variable in the forward pass
(Friedman's $gamma$, equation 74 in the MARS paper).
Default is
`0`

, meaning no penalty for adding a new variable. Useful non-zero values typically range from about`0.01`

- fast.k
- Maximum number of parent terms considered at each step of the forward pass.
(This speeds up the forward pass. See the Fast MARS paper section 3.0.)
Default is
`20`

. A value of`0`

is treated specially (as being equival - fast.beta
- Fast MARS ageing coefficient, as described in the
Fast MARS paper section 3.1.
Default is
`1`

. A value of`0`

sometimes gives better results. - linpreds
- Index vector specifying which predictors should enter linearly, as in
`lm`

. The default is`FALSE`

, meaning all predictors enter in the standard MARS fashion, i.e., in hinge functions. This d - allowed
- Function specifying which predictors can interact and how.
Default is NULL, meaning all standard MARS terms are allowed.
During the forward pass,
`earth`

calls the`allowed`

function before considering a term for inclusion; - pmethod
- Pruning method.
One of:
`backward none exhaustive forward seqrep cv`

. Default is`"backward"`

.**New in version 4.4.0:**Specify`pmethod="cv"`

to use cross-validation to select the number of terms. This - nprune
- Maximum number of terms (including intercept) in the pruned model. Default is NULL, meaning all terms created by the forward pass (but typically not all terms will remain after pruning). Use this to enforce an upper bound on the model size
- ncross
- Only applies if
`nfold>1`

. Number of cross-validations. Each cross-validation has`nfold`

folds. Default`1`

. - nfold
- Number of cross-validation folds.
Default is
`0`

, no cross validation. If greater than`1`

,`earth`

first builds a standard model as usual with all the data. It then builds`nfold`

cross-validated - stratify
- Only applies if
`nfold>1`

. Default is`TRUE`

. Stratify the cross-validation samples so that an approximately equal number of cases with a non-zero response occur in each cross validation subset. So if the res - varmod.method
- Construct a variance model.
For details, see
`varmod`

and the vignette . Use../doc/earth-varmod.pdf {Variance models in earth}`trace=.3`

to trace construction of the va - varmod.exponent
- Power transform applied to the rhs before regressing the
absolute residuals with the specified
`varmod.method`

. Default is`1`

. For example, with`varmod.method="lm"`

, if you expect the standard deviance to increase linear - varmod.conv
- Convergence criterion for the Iteratively Reweighted Least Squares used
when creating the variance model.
Iterations stop when the mean value of the coefficients of the
residual model change by less than
`varmod.conv`

percent. Default is - varmod.clamp
- The estimated standard deviation of the main model errors
is forced to be at least a small positive value,
which we call
`min.sd`

. This prevents negative or absurdly small estimated standard deviations. Clamping takes place in`predict.var`

- varmod.minspan
- Only applies when
`varmod.method="earth"`

or`"x.earth"`

. This is the`minspan`

used in the internal call to`earth`

when creating the variance model (not the main`earth`

model). Default is`-3<`

- Object
- Earth object to be updated, for use by
`update.earth`

. - Scale.y
`Scale`

`y`

in the forward pass for better numeric stability. Scaling here means subtract the mean and divide by the standard deviation. Default is`NCOL(y)==1`

, i.e., scale`y<`

- Adjust.endspan
**New in version 4.2.0.**In interaction terms,`endspan`

gets multiplied by this value. This reduces the possibility of an overfitted interaction term supported by just a few cases on the boundary of the predictor space (as sometimes seen i- Force.weights
- Default is
`FALSE`

. For testing the`weights`

argument. Force use of the code for handling weights in the`earth`

code, even if`weights=NULL`

or all the weights are the same. This will not necessarily generate - Use.beta.cache
- Default is
`TRUE`

. Using thebeta cache takes a little more memory but is faster (by 20% and often much more for large models). The beta cache uses`nk * nk * ncol(x) * sizeof(double)`

bytes. (The beta cache is an in - Force.xtx.prune
- Default is
`FALSE`

. This argument pertains to subset evaluation in the pruning pass. By default, if`y`

has a single column then`earth`

calls the`leaps`

routines; if - Get.leverages
- New in version 4.4.0.
Default is
`TRUE`

unless the model has more than 100 thousand cases. The leverages are the diagonal hat values for the linear regression of`y`

on`bx`

. The leverages are needed only for certain model - Exhaustive.tol
- Default
`1e-10`

. Applies only when`pmethod="exhaustive"`

. If the reciprocal of the condition number of`bx`

is less than`Exhaustive.tol`

,`earth`

forces`pmethod="backward"`

. See - ...
- Dots are passed on to
`earth.fit`

.

##### Value

- An object of class
`"earth"`

which is a list with the components listed below.*Term*refers to a term created during the forward pass (each line of the output from`format.earth`

is a term). Term number 1 is always the intercept. `rss`

Residual sum-of-squares (RSS) of the model (summed over all responses, if `y`

has multiple columns).`rsq`

`1-rss/tss`

. R-Squared of the model (calculated over all responses, and calculated using the`weights`

argument if it was supplied). A measure of how well the model fits the training data. Note that`tss`

is the total sum-of-squares,`sum((y - mean(y))^2)`

.`gcv`

Generalized Cross Validation (GCV) of the model (summed over all responses). The GCV is calculated using the `penalty`

argument. For details of the GCV calculation, see equation 30 in Friedman's MARS paper and`earth:::get.gcv`

.`grsq`

`1-gcv/gcv.null`

. An estimate of the predictive power of the model (calculated over all responses, and calculated using the`weights`

argument if it was supplied).`gcv.null`

is the GCV of an intercept-only model. See in the vignette.*Can*`GRSq`

be negative?`bx`

Matrix of basis functions applied to `x`

. Each column corresponds to a selected term. Each row corresponds to a row in in the input matrix`x`

, after taking`subset`

. See`model.matrix.earth`

for an example of`bx`

handling. Example`bx`

:(Intercept) h(Girth-12.9) h(12.9-Girth) h(Girth-12.9)*h(... [1,] 1 0.0 4.6 0 [2,] 1 0.0 4.3 0 [3,] 1 0.0 4.1 0 ...`dirs`

Matrix with one row per MARS term, and with with ij-th element equal to `0`

if predictor j is not in term i`-1`

if an expression of the form`h(const - xj)`

is in term i`1`

if an expression of the form`h(xj - const)`

is in term i`2`

if predictor j should enter term i linearly (either because specified by the`linpreds`

argument or because earth discovered that a knot was unnecessary). This matrix includes all terms generated by the forward pass, including those not in`selected.terms`

. Note that here the terms may not all be in pairs, because although the forward pass add terms as hinged pairs (so both sides of the hinge are available as building blocks for further terms), it also deletes linearly dependent terms before handing control to the pruning pass. Example`dirs`

:Girth Height (Intercept) 0 0 #intercept h(12.9-Girth) -1 0 #2nd term uses Girth h(Girth-12.9) 1 0 #3rd term uses Girth h(Girth-12.9)*h(Height-76) 1 1 #4th term uses Girth and Height ...`cuts`

Matrix with ij-th element equal to the cut point for predictor j in term i. This matrix includes all terms generated by the forward pass, including those not in `selected.terms`

. Note for programmers: the precedent is to use`dirs`

for term names etc. and to only use`cuts`

where cut information needed. Example`cuts`

:Girth Height (Intercept) 0 0 #intercept, no cuts h(12.9-Girth) 12.9 0 #2nd term has cut at 12.9 h(Girth-12.9) 12.9 0 #3rd term has cut at 12.9 h(Girth-12.9)*h(Height-76) 12.9 76 #4th term has two cuts ...`prune.terms`

A matrix specifying which terms appear in which pruning pass subsets. The row index of `prune.terms`

is the model size. (The model size is the number of terms in the model. The intercept is counted as a term.) Each row is a vector of term numbers for the best model of that size. An element is 0 if the term is not in the model, thus`prune.terms`

is a lower triangular matrix, with dimensions`nprune x nprune`

. The model selected by the pruning pass is at row number`length(selected.terms)`

. Example`prune.terms`

:[1,] 1 0 0 0 0 0 0 #intercept-only model [2,] 1 2 0 0 0 0 0 #best 2 term model uses terms 1,2 [3,] 1 2 4 0 0 0 0 #best 3 term model uses terms 1,2,4 [4,] 1 2 6 9 0 0 0 #and so on ...`selected.terms`

Vector of term numbers in the selected model. Can be used as a row index vector into `cuts`

and`dirs`

. The first element`selected.terms[1]`

is always 1, the intercept.`fitted.values`

Fitted values. A matrix with dimensions `nrow(y)`

x`ncol(y)`

after factors in`y`

have been expanded.`residuals`

Residuals. A matrix with dimensions `nrow(y)`

x`ncol(y)`

after factors in`y`

have been expanded.`coefficients`

Regression coefficients. A matrix with dimensions `length(selected.terms)`

x`ncol(y)`

after factors in`y`

have been expanded. Each column holds the least squares coefficients from regressing that column of`y`

on`bx`

. The first row holds the intercept coefficient(s).`rss.per.response`

A vector of the RSS for each response. Length is the number of responses, i.e., `ncol(y)`

after factors in`y`

have been expanded. The`rss`

component above is equal to`sum(rss.per.response)`

.`rsq.per.response`

A vector of the R-Squared for each response (where R-Squared is calculated using the `weights`

argument if it was supplied). Length is the number of responses.`gcv.per.response`

A vector of the GCV for each response. Length is the number of responses. The `gcv`

component above is equal to`sum(gcv.per.response)`

.`grsq.per.response`

A vector of the GRSq for each response (calculated using the `weights`

argument if it was supplied). Length is the number of responses.`rss.per.subset`

A vector of the RSS for each model subset generated by the pruning pass. Length is `nprune`

. For multiple responses, the RSS is summed over all responses for each subset. The`rss`

above is`rss.per.subset[length(selected.terms)]`

. The RSS of an intercept only-model is`rss.per.subset[1]`

.`gcv.per.subset`

A vector of the GCV for each model in `prune.terms`

. Length is`nprune`

. For multiple responses, the GCV is summed over all responses for each subset. The`gcv`

above is`gcv.per.subset[length(selected.terms)]`

. The GCV of an intercept-only model is`gcv.per.subset[1]`

.`leverages`

Diagonal of the hat matrix (from the linear regression of the response on `bx`

).`penalty,nk,thresh`

Copies of the corresponding arguments to `earth`

.`pmethod,nprune`

Copies of the corresponding arguments to `earth`

.`weights,wp`

Copies of the corresponding arguments to `earth`

.`termcond`

Reason the forward pass terminated (an integer). `call`

The call used to invoke `earth`

.`terms`

Model frame terms. This component exists only if the model was built using `earth.formula`

.`namesx`

Column names of `x`

, generated internally by`earth`

when necessary so each column of`x`

has a name. Used, for example, by`predict.earth`

to name columns if necessary.`namesx.org`

Original column names of `x`

.`levels`

Levels of `y`

if`y`

is a`factor`

`c(FALSE,TRUE)`

if`y`

is`logical`

Else NULL**The following fields appear only if**`earth`

's argument`keepxy`

is`TRUE`

.`x`

,`y`

,`data`

,`subset`

Copies of the corresponding arguments to `earth`

. Only exist if`keepxy=TRUE`

.**The following fields appear only if**`earth`

's`glm`

argument is used.`glm.list`

List of GLM models. Each element is the value returned by `earth`

's internal call to`glm`

for each response. Thus if there is a single response (or a single binomial pair, see in the vignette) this will be a one element list and you access the GLM model with*Binomial pairs*`earth.mod$glm.list[[1]]`

.`glm.coefficients`

GLM regression coefficients. Analogous to the `coefficients`

field described above but for the GLM model(s). A matrix with dimensions`length(selected.terms)`

x`ncol(y)`

after factors in`y`

have been expanded. Each column holds the coefficients from the GLM regression of that column of`y`

on`bx`

. This duplicates, for convenience, information buried in`glm.list`

.`glm.bpairs`

NULL unless there are paired binomial columns. A logical vector, derived internally by `earth`

, or a copy the`bpairs`

specified by the user in the`glm`

list. See in the vignette.*Binomial pairs***The following fields appear only if the**`nfold`

argument is greater than 1.`cv.list`

List of `earth`

models, one model for each fold (`ncross * nfold`

models). The fold models have two extra fields,`icross`

(an integer from`1`

to`ncross`

) and`ifold`

(an integer from`1`

to`nfold`

). To save memory, lengthy fields in the fold models are removed unless you use`keepxy=TRUE`

. Thelengthy fields are`$bx`

,`$fitted.values`

, and`$residuals`

.`cv.nterms`

Vector of length `ncross * nfold + 1`

. Number of MARS terms in the model generated at each cross-validation fold, with the final element being the mean of these.`cv.nvars`

Vector of length `ncross * nfold + 1`

. Number of predictors in the model generated at each cross-validation fold, with the final element being the mean of these.`cv.groups`

Specifies which cases went into which folds. Matrix with two columns and number of rows equal to the the number of cases `nrow(x)`

Elements of the first column specify the cross-validation number,`1:ncross`

. Elements of the second column specify the fold number,`1:nfold`

.`cv.rsq.tab`

Matrix with `ncross * nfold + 1`

rows and`nresponse+1`

columns, where`nresponse`

is the number of responses, i.e.,`ncol(y)`

after factors in`y`

have been expanded. The first`nresponse`

elements of a row are the`cv.rsq`

's on the out-of-fold data for each response of the model generated at that row's fold. (A`cv.rsq`

is calculated from predictions on the out-of-fold data using the best model built from the in-fold data; wherebest means the model was selected using the in-fold GCV. The R-Squareds are calculated using the`weights`

argument if it was supplied. The final column holds the row mean (a weighted mean if`wp`

if specified)). The final row holds the column means. The values in this final row is the mean`cv.rsq`

printed by`summary.earth`

. Example for a single response model (where the`mean`

column is redundant but included for uniformity with multiple response models): y mean fold1 0.909 0.909 fold2 0.869 0.869 fold3 0.952 0.952 fold4 0.157 0.157 fold5 0.961 0.961 mean 0.769 0.769 Example for a multiple response model: y1 y2 y3 mean fold1 0.915 0.951 0.944 0.937 fold2 0.962 0.970 0.970 0.968 fold3 0.914 0.940 0.942 0.932 fold4 0.907 0.929 0.925 0.920 fold5 0.947 0.987 0.979 0.971 mean 0.929 0.955 0.952 0.946`cv.class.rate.tab`

Like `cv.rsq.tab`

but is the classification rate at each fold i.e. the fraction of classes correctly predicted. Models with discrete response only. Calculated with`thresh=.5`

for binary responses. For responses with more than two levels, the final row is the overall classification rate. The other rows are the classification rates for each level (the level versus not-the-level), which are usually higher than the overall classification rate (predicting the level versus not-the-level is easier than correctly predicting one of many levels). The`weights`

argument is ignored for all cross-validation stats except R-Squareds.`cv.maxerr.tab`

Like `cv.rsq.tab`

but is the`MaxErr`

at each fold. This is the signed max absolute value at each fold. Results are aggregated for the final column and final row using the signed max absolute value. The*signed max absolute value*is defined as the maximum of the absolute difference between the predicted and observed response values, multiplied by`-1`

if the sign of that difference is negative.`cv.auc.tab`

Like `cv.rsq.tab`

but is the`AUC`

at each fold. Binomial models only.`cv.cor.tab`

Like `cv.rsq.tab`

but is the`cor`

at each fold. Poisson models only.`cv.deviance.tab`

Like `cv.rsq.tab`

but is the`MeanDev`

at each fold. Binomial models only.`cv.calib.int.tab`

Like `cv.rsq.tab`

but is the`CalibInt`

at each fold. Binomial models only.`cv.calib.slope.tab`

Like `cv.rsq.tab`

but is the`CalibSlope`

at each fold. Binomial models only.`cv.oof.rsq.tab`

Generated only if `keepxy=TRUE`

or`pmethod="cv"`

. A matrix with`ncross * nfold + 1`

rows and`max.nterms`

columns, Each element holds an out-of-fold RSq (`oof.rsq`

), calculated from predictions from the out-of-fold observations using the model built with the in-fold data. The final row is the mean over all folds. The R-Squareds are calculated using the`weights`

argument if it was supplied.`cv.infold.rsq.tab`

Generated only if `keepxy=TRUE`

. Like`cv.oof.rsq.tab`

but from predictions made on the in-fold observations.`cv.oof.fit.tab`

Generated only if the `varmod.method`

argument is used. Predicted values on the out-of-fold data. Dataframe with`nrow(data)`

rows and`ncross`

columns.**The following field appears only if the**`varmod.method`

is specified.`varmod`

An object of class `"varmod"`

. See the`varmod`

help page for a description. Only appears if the`varmod.method`

argument is used.

##### concept

- regression
- mars
- Friedman

##### References

The primary references are the Friedman papers.
Readers may find the MARS section in Hastie, Tibshirani,
and Friedman a more accessible introduction.
The Wikipedia article is recommended for an elementary introduction.
Faraway takes a hands-on approach,
using the `ozone`

data to compare `mda::mars`

with other techniques.
(If you use Faraway's examples with `earth`

instead of `mars`

, use `$bx`

instead of `$x`

, and check out the book's errata.)
Friedman and Silverman is recommended background reading for the MARS paper.
Earth's pruning pass uses code from the `leaps`

package
which is based on techniques in Miller.

Faraway (2005) *Extending the Linear Model with R*

Friedman (1991) *Multivariate Adaptive Regression Splines (with discussion)*
Annals of Statistics 19/1, 1--141
*Fast MARS*
Stanford University Department of Statistics, Technical Report 110

Friedman and Silverman (1989)
*Flexible Parsimonious Smoothing and Additive Modeling*
Technometrics, Vol. 31, No. 1.

Hastie, Tibshirani, and Friedman (2009) *The Elements of Statistical Learning (2nd ed.)*

Leathwick, J.R., Rowe, D., Richardson, J., Elith, J., & Hastie, T. (2005)
*Using multivariate adaptive regression splines to predict the distributions
of New Zealand's freshwater diadromous fish* Freshwater Biology, 50, 2034-2052

Miller, Alan (1990, 2nd ed. 2002) *Subset Selection in Regression*

Wikipedia article on MARS

##### See Also

Start with `summary.earth`

, `plot.earth`

,
`evimp`

, and `plotmo`

.

Please see the main package vignette

The vignette
`earth`

models.

##### Examples

```
earth.mod <- earth(Volume ~ ., data = trees)
plotmo(earth.mod)
summary(earth.mod, digits = 2, style = "pmax")
```

*Documentation reproduced from package earth, version 4.4.4, License: GPL-3*