naginterfaces.library.correg.lars_​param

naginterfaces.library.correg.lars_param(b, fitsum, ktype, nk)[source]

lars_param calculates additional parameter estimates following Least Angle Regression (LARS), forward stagewise linear regression or Least Absolute Shrinkage and Selection Operator (LASSO) as performed by lars() and lars_xtx().

For full information please refer to the NAG Library document for g02mc

https://www.nag.com/numeric/nl/nagdoc_27.1/flhtml/g02/g02mcf.html

Parameters
bfloat, array-like, shape

the parameter estimates, as returned by lars() and lars_xtx(), with , the parameter estimate for the th variable, for , at the th step of the model fitting process.

fitsumfloat, array-like, shape

Summaries of the model fitting process, as returned by lars() and lars_xtx().

ktypeint

Indicates what target values are held in .

holds (fractional) LARS step numbers.

holds values for norm of the (scaled) parameters.

holds ratios with respect to the largest (scaled) norm.

holds values for the norm of the (unscaled) parameters.

holds ratios with respect to the largest (unscaled) norm.

If lars() was called with or or lars_xtx() was called with then the model fitting routine did not rescale the independent variables, , prior to fitting the model and, therefore, there is no difference between or and or .

nkfloat, array-like, shape

Target values used for predicting the new set of parameter estimates.

Returns
nbfloat, ndarray, shape

the predicted parameter estimates, with , the parameter estimate for variable , at the point in the fitting process associated with , .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

has been corrupted since the last call to lars() or lars_xtx().

(errno )

has been corrupted since the last call to lars() or lars_xtx().

(errno )

On entry, .

Constraint: , , , or .

(errno )

On entry, , and

Constraint: , for all .

(errno )

On entry, , , and .

Constraint: , for all .

(errno )

On entry, or , .

Constraint: , for all .

(errno )

On entry, , and

Constraint: , for all .

(errno )

On entry, .

Constraint: .

Notes

lars() and lars_xtx() fit either a LARS, forward stagewise linear regression, LASSO or positive LASSO model to a vector of observed values, and an design matrix , where the th column of is given by the th independent variable . The models are fit using the LARS algorithm of Efron et al. (2004).

[figure omitted]

The full solution path for all four of these models follow a similar pattern where the parameter estimate for a given variable is piecewise linear. One such path, for a LARS model with six variables can be seen in Figure [label omitted]. Both lars() and lars_xtx() return the vector of parameter estimates, , at points along this path (so ). Each point corresponds to a step of the LARS algorithm. The number of steps taken depends on the model being fitted. In the case of a LARS model, and each step corresponds to a new variable being included in the model. In the case of the LASSO models, each step corresponds to either a new variable being included in the model or an existing variable being removed from the model; the value of is, therefore, no longer bound by the number of parameters. For forward stagewise linear regression, each step no longer corresponds to the addition or removal of a variable;, therefore, the number of possible steps is often markedly greater than for a corresponding LASSO model.

lars_param uses the piecewise linear nature of the solution path to predict the parameter estimates, , at a different point on this path. The location of the solution can either be defined in terms of a (fractional) step number or a function of the norm of the parameter estimates.

References

Efron, B, Hastie, T, Johnstone, I and Tibshirani, R, 2004, Least Angle Regression, The Annals of Statistics (Volume 32) (2), 407–499

Hastie, T, Tibshirani, R and Friedman, J, 2001, The Elements of Statistical Learning: Data Mining, Inference and Prediction, Springer (New York)

Tibshirani, R, 1996, Regression Shrinkage and Selection via the Lasso, Journal of the Royal Statistics Society, Series B (Methodological) (Volume 58) (1), 267–288

Weisberg, S, 1985, Applied Linear Regression, Wiley