liureg {fastliu}

Fit a Liu Regression Coefficients Path

Description

liureg fits coefficients paths for Liu regression models over a grid of values for the regularization (biasing) parameter lambda. The returned object is of class liureg.

Usage

liureg(X, y, lambda = 1, scale = c("ulength", "unormal", "none"), ...)

Arguments

X

The design matrix of features. fastliu standardizes the data and includes an intercept term by default.

y

The response vector.

lambda

User-specified values of lambda. The default value is 1, which corresponds to the least squares estimator. A lambda sequence can be entered to generate multiple models.

scale

Scaling type of the design matrix. "ulength" corresponds to unit-length scaling. In this scaling the scaled design matrix is in the form of a correlation matrix. "unormal" scales the features to have unit variance (using 1/n rather than 1/(n-1) formula). "none" does not make scaling and computations are done on centered features.

...

Not used in this implementation.

Details

The sequence of Liu regression models indexed by the tuning parameter. \lambda are obtained by

\hat{\boldsymbol{\beta}}^{liu}\left(\lambda\right)= \left(\mathbf{X}^{T}\mathbf{X}+\mathbf{I}_{p}\right)^{-1} \left(\mathbf{X}^{T}\mathbf{y}+\lambda\hat{\boldsymbol{\beta}}^{ls}\right),

where \hat{\boldsymbol{\beta}}^{ls} is the ordinary least squares estimator.To obtain the models, the singular value decomposition (SVD) of the matrix \mathbf{X} is used. This SVD is done once and is used to generate all models.

Explanatory variables in the design matrix are always centered before fitting a model in the fastliu package. For scaling, two options are possible: unit-length and unit-normal scaling. In unit-length scaling, the matrix of explanatory variables has correlation form. In unit-normal scaling, the explanatory variables have zero mean and unit variance. Both Coefficient estimates based on the scaled data and in original scale are presented. The intercept of the model is not penalized and computed by \bar{y}-\bar{X}\boldsymbol{\hat{\beta}}_1, where \bar{X} is the row vector of the explanatory variables and \boldsymbol{\hat{\beta}}_1 is computed based on centered design matrix.

The returned liureg object is used for statistical testing of Liu coefficients, plotting method and computing the Liu regression related statistics.

Value

Fitted Liu regression object with the class of liureg

Author(s)

Murat Genç and Ömer Özbilen

See Also

coef(), predict(), summary(), pressliu(), residuals()

Examples

data("Hitters")
Hitters <- na.omit(Hitters)
X <- model.matrix(Salary ~ ., Hitters)[, -1]
y <- Hitters$Salary
lam <- seq(0, 1, 0.05)
liu.mod <- liureg(X, y, lam)