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.
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)
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")
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.
Response type (see above). Either a character string representing
one of the built-in families, or else a glm() family object.
observation weights. Can be total counts if responses are proportion matrices. Default is 1 for each observation
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.
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.
The number of lambda values - default is 100.
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.
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.
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".
Should intercept(s) be fitted (default=TRUE) or set to zero (FALSE)
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.
Limit the maximum number of variables in the model. Useful for
very large nvars, if a partial path is desired.
Limit the maximum number of variables ever to be nonzero
Indices of variables to be excluded from the model. Default is none. Equivalent to an infinite penalty factor (next item).
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.
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
Vector of upper limits for each coefficient; default
Inf. See lower.limits
Maximum number of passes over the data for all lambda values; default is 10^5.
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.
If "Newton" then the exact hessian is used
(default), while "modified.Newton" uses an upper-bound on the
hessian, and can be faster.
This is for the family="mgaussian"
family, and allows the user to standardize the response variables
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"
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.
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.
For relax.glmnet a fitted 'glmnet' object
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'.
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
Should relax.glmnet make sure that all the data
dependent arguments used in creating 'fit' have been resupplied. Default is
'TRUE'.
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
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'.
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
print, predict, coef and plot methods,
and the cv.glmnet function.
# 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))
# }
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