NAG CL Interface
g02mcc (lars_param)
1
Purpose
g02mcc calculates additional parameter estimates following Least Angle Regression (LARS), forward stagewise linear regression or Least Absolute Shrinkage and Selection Operator (LASSO) as performed by
g02mac and
g02mbc.
2
Specification
void |
g02mcc (Integer nstep,
Integer ip,
const double b[],
Integer pdb,
const double fitsum[],
Nag_LARSTargetType ktype,
const double nk[],
Integer lnk,
double nb[],
Integer pdnb,
NagError *fail) |
|
The function may be called by the names: g02mcc, nag_correg_lars_param or nag_lars_param.
3
Description
g02mac and
g02mbc 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).
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 1. Both
g02mac and
g02mbc 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.
g02mcc 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.
4
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
5
Arguments
-
1:
– Integer
Input
-
On entry:
, the number of steps carried out in the model fitting process, as returned by
g02mac and
g02mbc.
Constraint:
.
-
2:
– Integer
Input
-
On entry:
, number of parameter estimates, as returned by
g02mac and
g02mbc.
Constraint:
.
-
3:
– const double
Input
-
Note: the dimension,
dim, of the array
b
must be at least
.
On entry:
the parameter estimates, as returned by
g02mac and
g02mbc, with
, the parameter estimate for the
th variable, for
, at the
th step of the model fitting process.
Constraint:
b should be unchanged since the last call to
g02mac or
g02mbc.
-
4:
– Integer
Input
-
On entry: the stride separating row elements in the two-dimensional data stored in the array
b.
Constraint:
.
-
5:
– const double
Input
-
On entry: summaries of the model fitting process, as returned by
g02mac and
g02mbc.
Constraint:
fitsum should be unchanged since the last call to
g02mac or
g02mbc..
-
6:
– Nag_LARSTargetType
Input
-
On entry: indicates what target values are held in
nk.
- nk holds (fractional) LARS step numbers.
- nk holds values for norm of the (scaled) parameters.
- nk holds ratios with respect to the largest (scaled) norm.
- nk holds values for the norm of the (unscaled) parameters.
- nk holds ratios with respect to the largest (unscaled) norm.
If
g02mac was called with
or
or
g02mbc 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
.
Constraint:
, , , or .
-
7:
– const double
Input
-
On entry: target values used for predicting the new set of parameter estimates.
Constraints:
- if , , for ;
- if , , for ;
- if or , , for ;
- if , , for .
-
8:
– Integer
Input
-
On entry: number of values supplied in
nk.
Constraint:
.
-
9:
– double
Output
-
Note: the dimension,
dim, of the array
nb
must be at least
.
On exit: the predicted parameter estimates, with , the parameter estimate for variable , at the point in the fitting process associated with , .
-
10:
– Integer
Input
-
On entry: the stride separating row elements in the two-dimensional data stored in the array
nb.
Constraint:
.
-
11:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_ARRAY_SIZE
-
On entry, and
Constraint: .
On entry, and .
Constraint: .
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_OUT_OF_RANGE
-
On entry, or , .
Constraint: , for all .
On entry, , , and .
Constraint: , for all .
On entry, , and
Constraint: , for all .
On entry, , and
Constraint: , for all .
- NE_REAL_ARRAY
-
b has been corrupted since the last call to
g02mac or
g02mbc.
fitsum has been corrupted since the last call to
g02mac or
g02mbc.
7
Accuracy
Not applicable.
None.
9
Example
This example performs a LARS on a set a simulated dataset with observations and independent variables.
Additional parameter estimates are obtained corresponding to a LARS step number of and . Where, for example, corresponds to the solution halfway between that obtained at step and that obtained at step .
9.1
Program Text
9.2
Program Data
9.3
Program Results