# glmnet

##### 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 = NULL, offset = NULL, alpha = 1,
nlambda = 100, lambda.min.ratio = ifelse(nobs < nvars, 0.01, 1e-04),
lambda = NULL, standardize = TRUE, intercept = TRUE,
thresh = 1e-07, dfmax = nvars + 1, pmax = min(dfmax * 2 + 20,
nvars), exclude = NULL, penalty.factor = rep(1, nvars),
lower.limits = -Inf, upper.limits = Inf, maxit = 1e+05,
type.gaussian = ifelse(nvars < 500, "covariance", "naive"),
type.logistic = c("Newton", "modified.Newton"),
standardize.response = FALSE, type.multinomial = c("ungrouped",
"grouped"), relax = FALSE, trace.it = 0, ...)
```relax.glmnet(fit, x, ..., maxp = n - 3, path = FALSE,
check.args = TRUE)

##### 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 (the second column is treated as the target class; for a factor, the last level in alphabetical order is the target class). For`family="multinomial"`

, can be a`nc>=2`

level factor, or a matrix with`nc`

columns of counts or proportions. For either`"binomial"`

or`"multinomial"`

, if`y`

is presented as a vector, it will be coerced into a factor. 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). Either a character string representing one of the built-in families, or else a

`glm()`

family object.- 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 \(0\le\alpha\le 1\). 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 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. Avoid supplying 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"`

- relax
If

`TRUE`

then for each*active set*in the path of solutions, the model is refit without any regularization. See`details`

for more information. This argument is new, and users may experience convergence issues with small datasets, especially with non-gaussian families. Limiting the value of 'maxp' can alleviate these issues in some cases.- trace.it
If

`trace.it=1`

, then a progress bar is displayed; useful for big models that take a long time to fit.- ...
Additional argument used in

`relax.glmnet`

. These include some of the original arguments to 'glmnet', and each must be named if used.- fit
For

`relax.glmnet`

a fitted 'glmnet' object- maxp
a limit on how many relaxed coefficients are allowed. Default is 'n-3', where 'n' is the sample size. This may not be sufficient for non-gaussian familes, in which case users should supply a smaller value. This argument can be supplied directly to 'glmnet'.

- path
Since

`glmnet`

does not do stepsize optimization, the Newton algorithm can get stuck and not converge, especially with relaxed fits. With`path=TRUE`

, each relaxed fit on a particular set of variables is computed pathwise using the original sequence of lambda values (with a zero attached to the end). Not needed for Gaussian models, and should not be used unless needed, since will lead to longer compute times. Default is`path=FALSE`

. appropriate subset of variables- check.args
Should

`relax.glmnet`

make sure that all the data dependent arguments used in creating 'fit' have been resupplied. Default is 'TRUE'.

##### 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.

From version 4.0 onwards, glmnet supports both the original built-in families,
as well as *any* family object as used by `stats:glm()`

.
The built in families are specifed via a character string. For all families,
the object produced 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 (using 1/n rather than 1/(n-1) formula)
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 coefficients for any predictor
variables with zero variance are set to zero for all values of lambda.
Two useful additional families 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. A small detail in the Cox model: if death times are tied
with censored times, we assume the censored times occurred just
*before* the death times in computing the Breslow approximation; if
users prefer the usual convention of *after*, they can add a small
number to all censoring times to achieve this effect.

Version 4.0 and later allows for the family argument to be a S3 class `"family"`

object
(a list of functions and expressions).
This opens the door to a wide variety of additional models. For example
`family=binomial(link=cloglog)`

or `family=negative.binomial(theta=1.5)`

(from the MASS library).
Note that the code runs faster for the built-in families.

If `relax=TRUE`

a duplicate sequence of models is produced, where each active set in the
elastic-net path is refit without regularization. The result of this is a
matching `"glmnet"`

object which is stored on the original object in a
component named `"relaxed"`

, and is part of the glmnet output.
Generally users will not call `relax.glmnet`

directly, unless the
original 'glmnet' object took a long time to fit. But if they do, they must
supply the fit, and all the original arguments used to create that fit. They
can limit the length of the relaxed path via 'maxp'.

##### Value

An object with S3 class `"glmnet","*" `

, where `"*"`

is
`"elnet"`

, `"lognet"`

, `"multnet"`

, `"fishnet"`

(poisson), `"coxnet"`

or `"mrelnet"`

for the various types of
models. If the model was created with `relax=TRUE`

then this class has
a prefix class of `"relaxed"`

.

the call that produced this object

Intercept sequence of length `length(lambda)`

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.

The actual sequence of `lambda`

values used. When `alpha=0`

, the largest lambda reported does not quite
give the zero coefficients reported (`lambda=inf`

would in principle).
Instead, the largest `lambda`

for `alpha=0.001`

is used, and the
sequence of `lambda`

values is derived from this.

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.

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.

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.

For `"multnet"`

and `"mrelnet"`

only. A
matrix consisting of the number of nonzero coefficients per class

dimension of coefficient matrix (ices)

number of observations

total passes over the data summed over all lambda values

a logical variable indicating whether an offset was included in the model

error flag, for warnings and errors (largely for internal debugging).

If `relax=TRUE`

, this
additional item is another glmnet object with different values for
`beta`

and `dev.ratio`

##### References

Friedman, J., Hastie, T. and Tibshirani, R. (2008)
*Regularization Paths for Generalized Linear Models via Coordinate
Descent*, https://web.stanford.edu/~hastie/Papers/glmnet.pdf
*Journal of Statistical Software, Vol. 33(1), 1-22 Feb 2010*
https://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* https://www.jstatsoft.org/v39/i05/ Tibshirani,
Robert, Bien, J., Friedman, J., Hastie, T.,Simon, N.,Taylor, J. and
Tibshirani, Ryan. (2012) *Strong Rules for Discarding Predictors in
Lasso-type Problems, JRSSB vol 74*,
https://statweb.stanford.edu/~tibs/ftp/strong.pdf *Stanford
Statistics Technical Report* https://arxiv.org/abs/1707.08692
Hastie, T., Tibshirani, Robert, Tibshirani, Ryan (2019) *Extended
Comparisons of Best Subset Selection, Forward Stepwise Selection, and the
Lasso*
*Glmnet webpage with four vignettes*
https://glmnet.stanford.edu

##### See Also

`print`

, `predict`

, `coef`

and `plot`

methods,
and the `cv.glmnet`

function.

##### Examples

```
# NOT RUN {
# 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
# Relaxed
fit1r = glmnet(x, y, relax = TRUE) # can be used with any model
# multivariate gaussian
y = matrix(rnorm(100 * 3), 100, 3)
fit1m = glmnet(x, y, family = "mgaussian")
plot(fit1m, type.coef = "2norm")
# binomial
g2 = sample(c(0,1), 100, replace = TRUE)
fit2 = glmnet(x, g2, family = "binomial")
fit2n = glmnet(x, g2, family = binomial(link=cloglog))
fit2r = glmnet(x,g2, family = "binomial", relax=TRUE)
fit2rp = glmnet(x,g2, family = "binomial", relax=TRUE, path=TRUE)
# 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 = 0.3, 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 * 0.85, 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 4.0-2, 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) ```