# glmnet {glmnet}

### 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*

### 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 1.9-5. License: GPL-2