`ridge`

Ridge regression

Description

Fit a linear model by ridge regression, returning an object of class "ridge" .

Usage

ridge(formula, data, subset, lambda = 1.0, method = "GCV", ngrid = 200, tol = 1e-07,
  maxiter = 50, na.action, model = FALSE, x = FALSE, y = FALSE, contrasts = NULL, ...)

Arguments

Argument	Description
`formula`	an object of class `"` (or one that can be coerced to that class): a symbolic description of the model to be fitted.
`data`	an optional data frame, list or environment (or object coercible by `as.data.frame` to a data frame) containing the variables in the model. If not found in `data` , the variables are taken from `environment(formula)` , typically the environment from which `ridge` is called.
`subset`	an optional vector specifying a subset of observations to be used in the fitting process.
`na.action`	a function which indicates what should happen when the data contain `NA` s. The default is set by the `na.action` setting of `options` , and is `na.fail` if that is unset.
`lambda`	a scalar or vector of ridge constants. A value of 0 corresponds to ordinary least squares.
`method`	the method for choosing the ridge parameter lambda. If `method = "none"` , then lambda is 'fixed'. If `method = "GCV"` (the default) then the ridge parameter is chosen automatically using the generalized cross validation (GCV) criterion. For `method = "grid"` , optimal value of lambda is selected computing the GCV criterion over a grid. If `method = "MSE"` the optimal ridge parameter is selected minimizing the mean squared estimation error criterion, this is the `ORPS1` subroutine by Lee (1987).
`ngrid`	number of elements in the grid used to compute the GCV criterion. Only required if `method = "grid"` and `lambda` is a scalar.
`tol`	tolerance for the optimization of the GCV criterion. Default is `1e-7` .
`maxiter`	maximum number of iterations. The default is 50.
`model, x, y`	logicals. If `TRUE` the corresponding components of the fit (the model frame, the model matrix, the response) are returned.
`contrasts`	an optional list. See the `contrasts.arg` of `model.matrix.default` .
`list()`	additional arguments to be passed to the low level regression fitting functions (not implemented).

Details

ridge function fits in linear ridge regression without scaling or centering the regressors and the response. In addition, If an intercept is present in the model, its coefficient is penalized.)

Value

A list with the following components:

Value	Description
`dims`	dimensions of model matrix.
`coefficients`	a named vector of coefficients.
`scale`	a named vector of coefficients.
`fitted.values`	the fitted mean values.
`residuals`	the residuals, that is response minus fitted values.
`RSS`	the residual sum of squares.
`edf`	the effective number of parameters.
`GCV`	vector (if method = "grid") of GCV values.
`HKB`	HKBestimate of the ridge constant.
`LW`	LW estimate of the ridge constant.
`lambda`	vector (if `method = "grid"`) of lambda values; otherwise, for methods `method = "none"`, `"GCV"` or `"MSE"`, the value of ridge parameter used by the algorithm.
`optimal`	value of lambda with the minimum GCV (only relevant if `method = "grid"`).
`iterations`	number of iterations performed by the algorithm (only relevant if `method = "MSE"`).
`call`	the matched call.
`terms`	the terms object used.
`contrasts`	(only where relevant) the contrasts used.
`y`	if requested, the response used.
`x`	if requested, the model matrix used.
`model`	if requested, the model frame used.

References

Golub, G.H., Heath, M., Wahba, G. (1979). Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21 , 215-223.

Hoerl, A.E., Kennard, R.W., Baldwin, K.F. (1975). Ridge regression: Some simulations. Communication in Statistics 4 , 105-123.

Hoerl, A.E., Kennard, R.W. (1970). Ridge regression: Biased estimation of nonorthogonal problems. Technometrics 12 , 55-67.

Lawless, J.F., Wang, P. (1976). A simulation study of ridge and other regression estimators. Communications in Statistics 5 , 307-323.

Lee, T.S (1987). Algorithm AS 223: Optimum ridge parameter selection. Applied Statistics 36 , 112-118.

Examples

z <- ridge(GNP.deflator ~ ., data = longley, lambda = 4, method = "grid")
z # ridge regression on a grid over seq(0, 4, length = 200)

z <- ridge(GNP.deflator ~ ., data = longley)
z # ridge parameter selected using GCV (default)

ridge