NAG FL Interface
g02mcf (lars_​param)

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1 Purpose

g02mcf calculates additional parameter estimates following Least Angle Regression (LARS), forward stagewise linear regression or Least Absolute Shrinkage and Selection Operator (LASSO) as performed by g02maf and g02mbf.

2 Specification

Fortran Interface
Subroutine g02mcf ( nstep, ip, b, ldb, fitsum, ktype, nk, lnk, nb, ldnb, ifail)
Integer, Intent (In) :: nstep, ip, ldb, ktype, lnk, ldnb
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: b(ldb,*), fitsum(6,nstep+1), nk(lnk)
Real (Kind=nag_wp), Intent (Inout) :: nb(ldnb,*)
C Header Interface
#include <nag.h>
void  g02mcf_ (const Integer *nstep, const Integer *ip, const double b[], const Integer *ldb, const double fitsum[], const Integer *ktype, const double nk[], const Integer *lnk, double nb[], const Integer *ldnb, Integer *ifail)
The routine may be called by the names g02mcf or nagf_correg_lars_param.

3 Description

g02maf and g02mbf fit either a LARS, forward stagewise linear regression, LASSO or positive LASSO model to a vector of n observed values, y = {yi:i=1,2,,n} and an n×p design matrix X, where the jth column of X is given by the jth independent variable xj. The models are fit using the LARS algorithm of Efron et al. (2004).
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 −1 0 1 2 3 4 0 20 40 60 80 100 120 140 160 180 200 220 Parameter Estimates (βkj) ||βk||1 bk1 βk1 bk2 βk2 bk3 βk3 bk4 βk4 bk5 βk5 bk6 βk6
Figure 1
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 (p=6) can be seen in Figure 1. Both g02maf and g02mbf return the vector of p parameter estimates, βk, at K points along this path (so k=1,2,,K). 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, K=p 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 K 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.
g02mcf uses the piecewise linear nature of the solution path to predict the parameter estimates, β~, 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 L1 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: nstep Integer Input
On entry: K, the number of steps carried out in the model fitting process, as returned by g02maf and g02mbf.
Constraint: nstep0.
2: ip Integer Input
On entry: p, number of parameter estimates, as returned by g02maf and g02mbf.
Constraint: ip1.
3: b(ldb,*) Real (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least nstep+1.
On entry: β the parameter estimates, as returned by g02maf and g02mbf, with b(j,k)=βkj, the parameter estimate for the jth variable, for j=1,2,,p, at the kth step of the model fitting process.
Constraint: b should be unchanged since the last call to g02maf or g02mbf.
4: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which g02mcf is called.
Constraint: ldbip.
5: fitsum(6,nstep+1) Real (Kind=nag_wp) array Input
On entry: summaries of the model fitting process, as returned by g02maf and g02mbf.
Constraint: fitsum should be unchanged since the last call to g02maf or g02mbf..
6: ktype Integer Input
On entry: indicates what target values are held in nk.
ktype=1
nk holds (fractional) LARS step numbers.
ktype=2
nk holds values for L1 norm of the (scaled) parameters.
ktype=3
nk holds ratios with respect to the largest (scaled) L1 norm.
ktype=4
nk holds values for the L1 norm of the (unscaled) parameters.
ktype=5
nk holds ratios with respect to the largest (unscaled) L1 norm.
If g02maf was called with pred=0 or 1 or g02mbf was called with pred=0 then the model fitting routine did not rescale the independent variables, X, prior to fitting the model and, therefore, there is no difference between ktype=2 or 3 and ktype=4 or 5.
Constraint: ktype=1, 2, 3, 4 or 5.
7: nk(lnk) Real (Kind=nag_wp) array Input
On entry: target values used for predicting the new set of parameter estimates.
Constraints:
  • if ktype=1, 0nk(i)nstep, for i=1,2,,lnk;
  • if ktype=2, 0nk(i)fitsum(1,nstep), for i=1,2,,lnk;
  • if ktype=3 or 5, 0nk(i)1, for i=1,2,,lnk;
  • if ktype=4, 0nk(i)βK1, for i=1,2,,lnk.
8: lnk Integer Input
On entry: number of values supplied in nk.
Constraint: lnk1.
9: nb(ldnb,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array nb must be at least lnk.
On exit: β~ the predicted parameter estimates, with b(j,i)=β~ij, the parameter estimate for variable j, j=1,2,,p at the point in the fitting process associated with nk(i), i=1,2,,lnk.
10: ldnb Integer Input
On entry: the first dimension of the array nb as declared in the (sub)program from which g02mcf is called.
Constraint: ldnbip.
11: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value −1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases g02mcf may return useful information.
ifail=11
On entry, nstep=value.
Constraint: nstep0.
ifail=21
On entry, ip=value.
Constraint: ip1.
ifail=31
b has been corrupted since the last call to g02maf or g02mbf.
ifail=41
On entry, ldb=value and ip=value
Constraint: ldbip.
ifail=51
fitsum has been corrupted since the last call to g02maf or g02mbf.
ifail=61
On entry, ktype=value.
Constraint: ktype=1, 2, 3, 4 or 5.
ifail=71
On entry, ktype=1, nk(value)=value and nstep=value
Constraint: 0nk(i)nstep, for all i.
ifail=72
On entry, ktype=2, nk(value)=value, nstep=value and fitsum(1,nstep)=value.
Constraint: 0nk(i)fitsum(1,nstep), for all i.
ifail=73
On entry, ktype=3 or 5, nk(value)=value.
Constraint: 0nk(i)1, for all i.
ifail=74
On entry, ktype=4, nk(value)=value and βK1=value
Constraint: 0nk(i)βK1, for all i.
ifail=81
On entry, lnk=value.
Constraint: lnk1.
ifail=101
On entry, ldnb=value and ip=value.
Constraint: ldnbip.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g02mcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02mcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example performs a LARS on a set a simulated dataset with 20 observations and 6 independent variables.
Additional parameter estimates are obtained corresponding to a LARS step number of 0.2,1.2,3.2,4.5 and 5.2. Where, for example, 4.5 corresponds to the solution halfway between that obtained at step 4 and that obtained at step 5.

10.1 Program Text

Program Text (g02mcfe.f90)

10.2 Program Data

Program Data (g02mcfe.d)

10.3 Program Results

Program Results (g02mcfe.r)