ridge {fastmatrix}

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, x = FALSE, y = FALSE, contrasts = NULL, ...)

Arguments

formula

an object of class "formula" (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 NAs. 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.

x, y

logicals. If TRUE the corresponding components of the fit (the model matrix, the response) are returned.

contrasts

an optional list. See the contrasts.arg of model.matrix.default.

...

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:

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

HKB estimate 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.

See Also

lm, ols

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)