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earth {earth}

Multivariate Adaptive Regression Splines
Package: 
earth
Version: 
3.2-7

Description

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

See the package vignette “../doc/earth-notes.pdfNotes on the earth package”.

Usage

 
## S3 method for class 'formula':
earth((formula = stop("no 'formula' arg"),
   data, weights = NULL, wp = NULL, scale.y = (NCOL(y)==1), subset = NULL,
   na.action = na.fail, glm = NULL, trace = 0,
   keepxy = FALSE, ncross=1, nfold=0, stratify=TRUE, ...))

## S3 method for class 'default':
earth((x = stop("no 'x' arg"), y = stop("no 'y' arg"),
    weights = NULL, wp = NULL, scale.y = (NCOL(y)==1), subset = NULL,
    na.action = na.fail, glm = NULL, trace = 0,
    keepxy = FALSE, ncross=1, nfold=0, stratify=TRUE, ...))

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

Arguments

To start off, look at the arguments formula, data, x, y, nk, degree, and trace. Many users will find that those arguments are all they need, plus in some cases keepxy, nprune, penalty, and minspan. For GLM models, use the glm argument. For cross validation, use the nfold argument.

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
Weight vector (not yet supported).
wp
Vector of 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, if any, have been expanded).
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 unless y has multiple columns.
na.action
NA action. Default is na.fail, and only na.fail is supported.
glm
NULL (default) or a list of arguments to pass on to glm. See the documentation of glm for a description of these arguments (but not all glm's arguments are supported, earth will give an error message if you use a glm argument it does not support).
See also “Generalized linear models” in the package vignette. Example:
earth(y~x, glm=list(family=binomial))
trace
Trace earth's execution. Default is 0. Values:
0 no tracing
0.5 cross validation
1 overview
2 forward pass
3 pruning
4 model mats, pruning details
5 internal details of operation
keepxy
Set to TRUE to retain the following in the returned value: x and y (or data), subset, and weights. Default is FALSE. The function update.earth and friends will use these if present instead of searching for them in the environment at the time update.earth is invoked. This argument also affects the amount of data kept when the nfold argument is used (see cv.list in the “Value” section below).

The following arguments are for the forward pass.

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. See “Termination conditions for the forward pass” in the package vignette.
The default is semi-automatically calculated from the number of predictors but may need adjusting. Use trace=1 or more to see why the forward pass terminated --- if it is "Reached max number of terms" then you could consider increasing nk.
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. A value of 0 penalizes only terms, not knots. The value -1 is treated specially to mean no penalty, so GCV=RSS/n. Simulation studies have suggested values in the range of about 2 to 4. In practice, for big datasets larger values can be useful to force a smaller model. The FAQ section in the package vignette has some information on GCVs.
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. See also “Termination conditions for the forward pass” in the package vignette.
minspan
Minimum distance between knots. The minspan argument is intended to increase resistance to runs of noise in the input data. (Note: predictor value extremes are ineligible for knots regardless of the minspan setting, as per the MARS paper equation 45.)
Use minspan=1 to consider all x values (which is good if the data are not noisy).
The default is minspan=0. This value 0 is treated specially and means calculate the minspan internally as per Friedman's MARS paper section 3.8 with alpha = 0.05. Set trace>=2 to see the calculated value.
minspan=-1 is also treated specially. It is for back compatibility, and means calculate minspan using the (incorrect) method used in versions of earth prior to 2.4-0. Using minspan=-1 instead of the default minspan=0 will usually build a model with a very similar GCV, although with slightly different knots and terms.
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 to 0.2 --- you will need to experiment.
With the default newvar.penalty=0, if two variables have nearly the same effect (e.g. they are collinear), at any step in the forward pass earth may arbitrarily select one or the other. Both variables can appear in the final model, complicating model interpretation. On the other hand with a non-zero newvar.penalty, the forward pass will be reluctant to add a new variable --- it will rather try to use a variable already in the model, if that does not affect RSq too much. The resulting final model may be easier to interpret, if you are lucky. There will often be a small performance hit (a worse GCV).
fast.k
Maximum number of parent terms considered at each step of the forward pass. Friedman invented this parameter to speed up the forward pass (see the Fast MARS paper section 3.0). Default is 20. Values of 0 is treated specially (as being equivalent to infinity), meaning no Fast MARS. Typical values, apart from 0, are 20, 10, or 5. In general, with a lower fast.k (say 5), earth is faster; with a higher fast.k, or with fast.k disabled (set to 0), earth builds a better model. However it is not unusual because of random variation to get a slightly better model with a lower fast.k.
fast.beta
Fast MARS aging 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. This does not say that a predictor must enter the model; only that if it enters, it enters linearly.
The default is FALSE, meaning all predictors enter in the standard MARS fashion, i.e., in hinge functions.
A predictor's index in linpreds is the column number in the input matrix x after factors have been expanded. Examples are given in the package vignette.
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; the term can go into the model only if the allowed function returns TRUE. See the package vignette for details.

The following arguments are for cross validation.

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 models, measuring R-Squared on the out-of-fold (left out) data each time. The final cross validation R-Squared (cv.rsq) is the mean of these fold R-Squareds.
The above process of building nfold models is repeated ncross times (by default, once).
Further statistics are calculated if keepxy=TRUE or if a binomial or poisson model (the glm argument). For details, see “Cross validation” in the package vignette.
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 response y is logical, the TRUEs will be spread evenly across folds. If the response is a factor, there will be an approximately equal number of each factor level in each fold (because a multilevel factor response gets expanded to columns of zeros and ones, see “Factors” in the package vignette). We say “approximately equal” because the number of occurrences of a factor level may not be exactly divisible by the number of folds.

The following arguments are for the pruning pass.

pmethod
Pruning method. One of: backward none exhaustive forward seqrep.
Default is "backward".
Use none to retain all the terms created by the forward pass.
If y has multiple columns, then only backward or none is allowed.
Pruning can take a while if exhaustive is chosen and the model is big (more than about 30 terms). The current version of the leaps package used during pruning does not allow user interrupts (i.e., you have to kill your R session to interrupt; in Windows use the Task Manager or from the command line use taskkill).
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 (that is less than nk), or to reduce exhaustive search time with pmethod="exhaustive".

The following arguments are for internal or advanced use.

Object
Earth object to be updated, for use by update.earth.
Get.crit
Criterion function for model selection during pruning. By default a function that returns the GCV. See “The pruning pass” in the package vignette.
Eval.model.subsets
Function to evaluate model subsets --- see notes in source code.
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 y has multiple columns then earth calls EvalSubsetsUsingXtx. The leaps routines are more accurate but do not support multiple responses (leaps is based on the QR decomposition and EvalSubsetsUsingXtx is based on the inverse of X'X). Setting Force.xtx.prune=TRUE forces use of EvalSubsetsUsingXtx, even if y has a single column.
Use.beta.cache
Default is TRUE. Using the “beta cache” takes 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. Set Use.beta.cache=FALSE to save memory. (The beta cache is an innovation in this implementation of MARS and does not appear in Friedman's papers. It is not related to the fast.beta argument. Certain regression coefficients in the forward pass can be saved and re-used, thus saving recalculation time.)
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 “XHAUST returned error code -999” in the package vignette.
...
Dots are passed on to earth.fit.

Values

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). 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, gcv.null is the GCV of an intercept-only model). See “Can GRSq be negative?” in the package vignette for some discussion.
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

  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 enters term i linearly (the linpreds argument).

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(Girth-12.9)              1  0 #2nd term uses Girth h(12.9-Girth)             -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(Girth-12.9)               12.9   0  #2nd term has cut at 12.9 h(12.9-Girth)               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 ...
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.
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 ...
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. 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. Length is the number of responses.
rss.per.subset
A vector of the RSS (measured on the training data) 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].
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).
penalty
The GCV penalty used during pruning. A copy of earth's penalty argument.
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
wp
Copy of the wp argument to earth.

The following fields appear only if earth's argument keepxy is TRUE.

x
y
data
subset
weights
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 “Binomial pairs” in the package vignette) this will be a one element list and you access the GLM model with my.earth.model$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 “Binomial pairs” in the package vignette.

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). To save memory, lengthy fields in the fold models are removed unless you use keepxy=TRUE. The “lengthy fields” are $bx, $fitted.values, and $residuals. The fold models have two extra fields, icross (the cross-validation index, 1:ncross) and ifold (the fold index, 1:nfold).
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; where “best” means the model was selected using the in-fold GCV.) 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.oof.rsq.tab
Matrix with ncross * nfold + 1 rows and max.nterms columns, Only calculated and kept if keepxy=TRUE. 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.
cv.infold.rsq.tab
Like cv.oof.rsq.tab but from predictions made on the in-fold observations.
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 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).
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.

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 http://www.maths.bath.ac.uk/~jjf23

Friedman (1991) Multivariate Adaptive Regression Splines (with discussion) Annals of Statistics 19/1, 1--141 http://www.salfordsystems.com/doc/MARS.pdf

Friedman (1993) Fast MARS Stanford University Department of Statistics, Technical Report 110 http://www.milbo.users.sonic.net/earth/Friedman-FastMars.pdf, http://www-stat.stanford.edu/research/index.html

Friedman and Silverman (1989) Flexible Parsimonious Smoothing and Additive Modeling Technometrics, Vol. 31, No. 1. http://links.jstor.org/sici?sici=0040-1706%28198902%2931%3A1%3C3%3AFPSAAM%3E2.0.CO%3B2-Z

Hastie, Tibshirani, and Friedman (2009) The Elements of Statistical Learning (2nd ed.) http://www-stat.stanford.edu/~hastie/pub.htm

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 http://www-stat.stanford.edu/~hastie/pub.htm, http://www.botany.unimelb.edu.au/envisci/about/staff/elith.html

Miller, Alan (1990, 2nd ed. 2002) Subset Selection in Regression http://www.cmis.csiro.au/Alan_Miller/index.html

Wikipedia article on MARS http://en.wikipedia.org/wiki/Multivariate_adaptive_regression_splines

Note

Please see the package vignette “../doc/earth-notes.pdfNotes on the earth package”. The vignette can also be downloaded from http://www.milbo.org/doc/earth-notes.pdf.

See Also

Start with summary.earth, plot.earth, evimp, and plotmo.

Please see the package vignette “../doc/earth-notes.pdfNotes on the earth package”. The vignette can also be downloaded from http://www.milbo.org/doc/earth-notes.pdf.

Examples

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

Author(s)

Stephen Milborrow, derived from mda::mars by Trevor Hastie and Robert Tibshirani.

The approach used for GLMs was motivated by work done by Jane Elith and John Leathwick (a representative paper is given below).

The evimp function uses ideas from Max Kuhn's caret package http://cran.r-project.org/web/packages/caret/index.html.

Parts of Thomas Lumley's leaps package has been incorporated into earth, so earth can directly access Alan Miller's Fortran functions.

Users are encouraged to send feedback --- use milbo AT sonic PERIOD net http://www.milbo.users.sonic.net.

Documentation reproduced from package earth, version 3.2-7. License: GPL-3