glmnet

From glmnet v1.8-5 by Trevor Hastie

fit a GLM with lasso or elasticnet regularization

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.

Keywords: models, regression

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
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
alpha: The elasticnet mixing parameter, with $0\le\alpha\le 1$. The penalty is defined as $$(1-\alpha)/2||\beta||_2^2+\alpha||\beta||_1.$$ 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 th
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 over
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 wis
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
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< code=""> is type.gaussian="covariance", and saves all inner-products ever computed. This can be much faster than t
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 $$1/2 RSS/nobs + \lambda*penalty,$$ and for the other models it is $$-loglik/nobs + \lambda*penalty.$$ 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 $$(1-\alpha)/2||\beta_j||_2^2+\alpha||\beta_j||_2.$$ When alpha=1 this is a group-lasso penalty, and otherwise it mixes with quadratic just like elasticnet.

Value

An object with S3 class "glmnet","*", where "*" is "elnet", "lognet", "multnet", "fishnet" (poisson), "coxnet" or "mrelnet" for the various types of models.
callthe call that produced this object
a0Intercept sequence of length length(lambda)
betaFor "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.
lambdaThe actual sequence of lambda values used
dev.ratioThe 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.
nulldevNull 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.
dfThe number of nonzero coefficients for each value of lambda. For "multnet", this is the number of variables with a nonzero coefficient for any class.
dfmatFor "multnet" and "mrelnet" only. A matrix consisting of the number of nonzero coefficients per class
dimdimension of coefficient matrix (ices)
nobsnumber of observations
npassestotal passes over the data summed over all lambda values
offseta logical variable indicating whether an offset was included in the model
jerrerror 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

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))

Documentation reproduced from package glmnet, version 1.8-5, License: GPL-2

Community examples

mayweiwang at Apr 26, 2017 glmnet v2.0-5

## first ```{r} 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) ```