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glmnet {glmnet}

fit a GLM with lasso or elasticnet regularization
Package: 
glmnet
Version: 
1.9-5

Description

Fit a generalized linear model via penalized maximum likelihood. The regularization path is computed for the lasso or elasticnet penalty at a grid of values for the regularization parameter lambda. Can deal with all shapes of data, including very large sparse data matrices. Fits linear, logistic and multinomial, poisson, and Cox regression models.

Usage

glmnet(x, y, family=c("gaussian","binomial","poisson","multinomial","cox","mgaussian"),
    weights, offset=NULL, alpha = 1, nlambda = 100,
    lambda.min.ratio = ifelse(nobs

Arguments

x
input matrix, of dimension nobs x nvars; each row is an observation vector. Can be in sparse matrix format (inherit from class "sparseMatrix" as in package Matrix; not yet available for family="cox")
y
response variable. Quantitative for family="gaussian", or family="poisson" (non-negative counts). For family="binomial" should be either a factor with two levels, or a two-column matrix of counts or proportions. For family="multinomial", can be a nc>=2 level factor, or a matrix with nc columns of counts or proportions. For family="cox", y should be a two-column matrix with columns named 'time' and 'status'. The latter is a binary variable, with '1' indicating death, and '0' indicating right censored. The function Surv() in package survival produces such a matrix. For family="mgaussian", y is a matrix of quantitative responses.
family
Response type (see above)
weights
observation weights. Can be total counts if responses are proportion matrices. Default is 1 for each observation
offset
A vector of length nobs that is included in the linear predictor (a nobs x nc matrix for the "multinomial" family). Useful for the "poisson" family (e.g. log of exposure time), or for refining a model by starting at a current fit. Default is NULL. If supplied, then values must also be supplied to the predict function.
alpha
The elasticnet mixing parameter, with 0≤α≤ 1. The penalty is defined as (1-α)/2||β||_2^2+α||β||_1. alpha=1 is the lasso penalty, and alpha=0 the ridge penalty.
nlambda
The number of lambda values - default is 100.
lambda.min.ratio
Smallest value for lambda, as a fraction of lambda.max, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero). The default depends on the sample size nobs relative to the number of variables nvars. If nobs > nvars, the default is 0.0001, close to zero. If nobs < nvars, the default is 0.01. A very small value of lambda.min.ratio will lead to a saturated fit in the nobs <   nvars case. This is undefined for "binomial" and "multinomial" models, and glmnet will exit gracefully when the percentage deviance explained is almost 1.
lambda
A user supplied lambda sequence. Typical usage is to have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Supplying a value of lambda overrides this. WARNING: use with care. Do not supply a single value for lambda (for predictions after CV use predict() instead). Supply instead a decreasing sequence of lambda values. glmnet relies on its warms starts for speed, and its often faster to fit a whole path than compute a single fit.
standardize
Logical flag for x variable standardization, prior to fitting the model sequence. The coefficients are always returned on the original scale. Default is standardize=TRUE. If variables are in the same units already, you might not wish to standardize. See details below for y standardization with family="gaussian".
intercept
Should intercept(s) be fitted (default=TRUE) or set to zero (FALSE)
thresh
Convergence threshold for coordinate descent. Each inner coordinate-descent loop continues until the maximum change in the objective after any coefficient update is less than thresh times the null deviance. Defaults value is 1E-7.
dfmax
Limit the maximum number of variables in the model. Useful for very large nvars, if a partial path is desired.
pmax
Limit the maximum number of variables ever to be nonzero
exclude
Indices of variables to be excluded from the model. Default is none. Equivalent to an infinite penalty factor (next item).
penalty.factor
Separate penalty factors can be applied to each coefficient. This is a number that multiplies lambda to allow differential shrinkage. Can be 0 for some variables, which implies no shrinkage, and that variable is always included in the model. Default is 1 for all variables (and implicitly infinity for variables listed in exclude). Note: the penalty factors are internally rescaled to sum to nvars, and the lambda sequence will reflect this change.
lower.limits
Vector of lower limits for each coefficient; default -Inf. Each of these must be non-positive. Can be presented as a single value (which will then be replicated), else a vector of length nvars
upper.limits
Vector of upper limits for each coefficient; default Inf. See lower.limits
maxit
Maximum number of passes over the data for all lambda values; default is 10^5.
type.gaussian
Two algorithm types are supported for (only) family="gaussian". The default when nvar<500 is type.gaussian="covariance", and saves all inner-products ever computed. This can be much faster than type.gaussian="naive", which loops through nobs every time an inner-product is computed. The latter can be far more efficient for nvar >>     nobs situations, or when nvar > 500.
type.logistic
If "Newton" then the exact hessian is used (default), while "modified.Newton" uses an upper-bound on the hessian, and can be faster.
standardize.response
This is for the family="mgaussian" family, and allows the user to standardize the response variables
type.multinomial
If "grouped" then a grouped lasso penalty is used on the multinomial coefficients for a variable. This ensures they are all in our out together. The default is "ungrouped"

Details

The sequence of models implied by lambda is fit by coordinate descent. For family="gaussian" this is the lasso sequence if alpha=1, else it is the elasticnet sequence. For the other families, this is a lasso or elasticnet regularization path for fitting the generalized linear regression paths, by maximizing the appropriate penalized log-likelihood (partial likelihood for the "cox" model). Sometimes the sequence is truncated before nlambda values of lambda have been used, because of instabilities in the inverse link functions near a saturated fit. glmnet(...,family="binomial") fits a traditional logistic regression model for the log-odds. glmnet(...,family="multinomial") fits a symmetric multinomial model, where each class is represented by a linear model (on the log-scale). The penalties take care of redundancies. A two-class "multinomial" model will produce the same fit as the corresponding "binomial" model, except the pair of coefficient matrices will be equal in magnitude and opposite in sign, and half the "binomial" values. Note that the objective function for "gaussian" is and for the other models it is -loglik/nobs + λ*penalty. Note also that for "gaussian", glmnet standardizes y to have unit variance before computing its lambda sequence (and then unstandardizes the resulting coefficients); if you wish to reproduce/compare results with other software, best to supply a standardized y. The latest two features in glmnet are the family="mgaussian" family and the type.multinomial="grouped" option for multinomial fitting. The former allows a multi-response gaussian model to be fit, using a "group -lasso" penalty on the coefficients for each variable. Tying the responses together like this is called "multi-task" learning in some domains. The grouped multinomial allows the same penalty for the family="multinomial" model, which is also multi-responsed. For both of these the penalty on the coefficient vector for variable j is (1-α)/2||β_j||_2^2+α||β_j||_2. When alpha=1 this is a group-lasso penalty, and otherwise it mixes with quadratic just like elasticnet.

Values

An object with S3 class "glmnet","*", where "*" is "elnet", "lognet", "multnet", "fishnet" (poisson), "coxnet" or "mrelnet" for the various types of models.

call
the call that produced this object
a0
Intercept sequence of length length(lambda)
beta
For "elnet", "lognet", "fishnet" and "coxnet" models, a nvars x       length(lambda) matrix of coefficients, stored in sparse column format ("CsparseMatrix"). For "multnet" and "mgaussian", a list of nc such matrices, one for each class.
lambda
The actual sequence of lambda values used
dev.ratio
The fraction of (null) deviance explained (for "elnet", this is the R-square). The deviance calculations incorporate weights if present in the model. The deviance is defined to be 2*(loglike_sat - loglike), where loglike_sat is the log-likelihood for the saturated model (a model with a free parameter per observation). Hence dev.ratio=1-dev/nulldev.
nulldev
Null deviance (per observation). This is defined to be 2*(loglike_sat -loglike(Null)); The NULL model refers to the intercept model, except for the Cox, where it is the 0 model.
df
The number of nonzero coefficients for each value of lambda. For "multnet", this is the number of variables with a nonzero coefficient for any class.
dfmat
For "multnet" and "mrelnet" only. A matrix consisting of the number of nonzero coefficients per class
dim
dimension of coefficient matrix (ices)
nobs
number of observations
npasses
total passes over the data summed over all lambda values
offset
a logical variable indicating whether an offset was included in the model
jerr
error flag, for warnings and errors (largely for internal debugging).

References

Friedman, J., Hastie, T. and Tibshirani, R. (2008) Regularization Paths for Generalized Linear Models via Coordinate Descent, http://www.stanford.edu/~hastie/Papers/glmnet.pdf
Journal of Statistical Software, Vol. 33(1), 1-22 Feb 2010
http://www.jstatsoft.org/v33/i01/
Simon, N., Friedman, J., Hastie, T., Tibshirani, R. (2011) Regularization Paths for Cox's Proportional Hazards Model via Coordinate Descent, Journal of Statistical Software, Vol. 39(5) 1-13
http://www.jstatsoft.org/v39/i05/
Tibshirani, Robert., Bien, J., Friedman, J.,Hastie, T.,Simon, N.,Taylor, J. and Tibshirani, Ryan. (2010) Strong Rules for Discarding Predictors in Lasso-type Problems,
http://www-stat.stanford.edu/~tibs/ftp/strong.pdf
Stanford Statistics Technical Report

See Also

print, predict, coef and plot methods, and the cv.glmnet function.

Examples

# Gaussian
x=matrix(rnorm(100*20),100,20)
y=rnorm(100)
fit1=glmnet(x,y)
print(fit1)
coef(fit1,s=0.01) # extract coefficients at a single value of lambda
predict(fit1,newx=x[1:10,],s=c(0.01,0.005)) # make predictions
 
#multivariate gaussian
y=matrix(rnorm(100*3),100,3)
fit1m=glmnet(x,y,family="mgaussian")
plot(fit1m,type.coef="2norm")
 
#binomial
g2=sample(1:2,100,replace=TRUE)
fit2=glmnet(x,g2,family="binomial")
 
#multinomial
g4=sample(1:4,100,replace=TRUE)
fit3=glmnet(x,g4,family="multinomial")
fit3a=glmnet(x,g4,family="multinomial",type.multinomial="grouped")
#poisson
N=500; p=20
nzc=5
x=matrix(rnorm(N*p),N,p)
beta=rnorm(nzc)
f = x[,seq(nzc)]%*%beta
mu=exp(f)
y=rpois(N,mu)
fit=glmnet(x,y,family="poisson")
plot(fit)
pfit = predict(fit,x,s=0.001,type="response")
plot(pfit,y)
 
#Cox
set.seed(10101)
N=1000;p=30
nzc=p/3
x=matrix(rnorm(N*p),N,p)
beta=rnorm(nzc)
fx=x[,seq(nzc)]%*%beta/3
hx=exp(fx)
ty=rexp(N,hx)
tcens=rbinom(n=N,prob=.3,size=1)# censoring indicator
y=cbind(time=ty,status=1-tcens) # y=Surv(ty,1-tcens) with library(survival)
fit=glmnet(x,y,family="cox")
plot(fit)
 
# Sparse
n=10000;p=200
nzc=trunc(p/10)
x=matrix(rnorm(n*p),n,p)
iz=sample(1:(n*p),size=n*p*.85,replace=FALSE)
x[iz]=0
sx=Matrix(x,sparse=TRUE)
inherits(sx,"sparseMatrix")#confirm that it is sparse
beta=rnorm(nzc)
fx=x[,seq(nzc)]%*%beta
eps=rnorm(n)
y=fx+eps
px=exp(fx)
px=px/(1+px)
ly=rbinom(n=length(px),prob=px,size=1)
system.time(fit1<-glmnet(sx,y))
system.time(fit2n<-glmnet(x,y))

Author(s)

Jerome Friedman, Trevor Hastie, Noah Simon and Rob Tibshirani
Maintainer: Trevor Hastie hastie@stanford.edu

Documentation reproduced from package glmnet, version 1.9-5. License: GPL-2