Fit solution paths for Huber loss regression or quantile regression penalized by lasso or elastic-net over a grid of values for the regularization parameter lambda.
hqreg(X, y, method = c("huber", "quantile", "ls"),
gamma = IQR(y)/10, tau = 0.5, alpha = 1, nlambda = 100, lambda.min = 0.05, lambda,
preprocess = c("standardize", "rescale"), screen = c("ASR", "SR", "none"),
max.iter = 10000, eps = 1e-7, dfmax = ncol(X)+1, penalty.factor = rep(1, ncol(X)),
message = FALSE)The function returns an object of S3 class "hqreg", which is a list containing:
The call that produced this object.
The fitted matrix of coefficients. The number of rows is equal to the number
of coefficients, and the number of columns is equal to nlambda. An intercept is included.
A vector of length nlambda containing the number of iterations until
convergence at each value of lambda.
A logical flag for whether the number of nonzero coefficients has reached dfmax.
The sequence of regularization parameter values in the path.
Same as above.
Same as above. NULL except when method = "huber".
Same as above. NULL except when method = "quantile".
Same as above.
Same as above.
The variable screening rules are accompanied with checks of optimality
conditions. When violations occur, the program adds in violating variables and re-runs
the inner loop until convergence. nv is the number of violations.
Input matrix.
Response vector.
The loss function to be used in the model. Either "huber" (default),
"quantile", or "ls" for least squares (see Details).
The tuning parameter of Huber loss, with no effect for the other loss functions. Huber loss is quadratic for absolute values less than gamma and linear for those greater than gamma. The default value is IQR(y)/10.
The tuning parameter of the quantile loss, with no effect for the other loss functions. It represents the conditional quantile of the response to be estimated, so must be a number between 0 and 1. It includes the absolute loss when tau = 0.5 (default).
The elastic-net mixing parameter that controls the relative contribution
from the lasso and the ridge penalty. It must be a number between 0 and 1. alpha=1
is the lasso penalty and alpha=0 the ridge penalty.
The number of lambda values. Default is 100.
The smallest value for lambda, as a fraction of lambda.max, the data derived entry value. Default is 0.05.
A user-specified sequence of lambda values. Typical usage is to leave
blank and have the program automatically compute a lambda sequence based on
nlambda and lambda.min. Specifying lambda overrides this. This
argument should be used with care and supplied with a decreasing sequence instead of
a single value. To get coefficients for a single lambda, use coef or
predict instead after fitting the solution path with hqreg or performing
k-fold CV with cv.hqreg.
Preprocessing technique to be applied to the input. Either
"standardize" (default) or "rescale"(see Details). The coefficients
are always returned on the original scale.
Screening rule to be applied at each lambda that discards variables
for speed. Either "ASR" (default), "SR" or "none". "SR" stands for the strong rule,
and "ASR" for the adaptive strong rule. Using "ASR" typically requires fewer iterations
to converge than "SR", but the computing time are generally close. Note that the option
"none" is used mainly for debugging, which may lead to much longer computing time.
Maximum number of iterations. Default is 10000.
Convergence threshold. The algorithms continue until the maximum change in the
objective after any coefficient update is less than eps times the null deviance.
Default is 1E-7.
Upper bound for the number of nonzero coefficients. The algorithm exits and
returns a partial path if dfmax is reached. Useful for very large dimensions.
A numeric vector of length equal to the number of variables. Each
component multiplies lambda to allow differential penalization. Can be 0 for
some variables, in which case the variable is always in the model without penalization.
Default is 1 for all variables.
If set to TRUE, hqreg will inform the user of its progress. This argument is kept for debugging. Default is FALSE.
Congrui Yi <eric.ycr@gmail.com>
The sequence of models indexed by the regularization parameter lambda is fit
using a semismooth Newton coordinate descent algorithm. The objective function is defined
to be $$\frac{1}{n} \sum loss_i + \lambda\textrm{penalty}.$$
For method = "huber",
$$loss(t) = \frac{t^2}{2\gamma} I(|t|\le \gamma) + (|t| - \frac{\gamma}{2};) I(|t|>
\gamma)$$
for method = "quantile", $$loss(t) = t (\tau - I(t<0));$$
for method = "ls", $$loss(t) = \frac{t^2}{2}$$
In the model, "t" is replaced by residuals.
The program supports different types of preprocessing techniques. They are applied to
each column of the input matrix X. Let x be a column of X. For
preprocess = "standardize", the formula is
$$x' = \frac{x-mean(x)}{sd(x)};$$
for preprocess = "rescale",
$$x' = \frac{x-min(x)}{max(x)-min(x)}.$$
The models are fit with preprocessed input, then the coefficients are transformed back
to the original scale via some algebra. To fit a model for raw data with no preprocessing, use hqreg_raw.
Yi, C. and Huang, J. (2017)
Semismooth Newton Coordinate Descent Algorithm for
Elastic-Net Penalized Huber Loss Regression and Quantile Regression,
tools:::Rd_expr_doi("10.1080/10618600.2016.1256816")
Journal of Computational and Graphical Statistics
plot.hqreg, cv.hqreg
X = matrix(rnorm(1000*100), 1000, 100)
beta = rnorm(10)
eps = 4*rnorm(1000)
y = drop(X[,1:10] %*% beta + eps)
# Huber loss
fit1 = hqreg(X, y)
coef(fit1, 0.01)
predict(fit1, X[1:5,], lambda = c(0.02, 0.01))
# Quantile loss
fit2 = hqreg(X, y, method = "quantile", tau = 0.2)
plot(fit2)
# Squared loss
fit3 = hqreg(X, y, method = "ls", preprocess = "rescale")
plot(fit3, xvar = "norm")
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