NAG Library Function Document

nag_lars_param (g02mcc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

▸▿ 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_lars_param (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 nag_lars (g02mac) and nag_lars_xtx (g02mbc).

2 Specification

#include <nag.h>

#include <nagg02.h>

void	nag_lars_param (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)

3 Description

nag_lars (g02mac) and nag_lars_xtx (g02mbc) fit either a LARS, forward stagewise linear regression, LASSO or positive LASSO model to a vector of

n

observed values,

y = \{y_{i} : i = 1, 2, \dots, n\}

and an

n \times p

design matrix

X

, where the

j

th column of

X

is given by the

j

th independent variable

x_{j}

. 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

(p = 6)

can be seen in Figure 1. Both nag_lars (g02mac) and nag_lars_xtx (g02mbc) return the vector of

p

parameter estimates,

β_{k}

, at

K

points along this path (so

k = 1, 2, \dots, 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.

Figure 1

nag_lars_param (g02mcc) uses the piecewise linear nature of the solution path to predict the parameter estimates,

\tilde{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

L_{1}

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: $nstep$ – IntegerInput

On entry:

K

, the number of steps carried out in the model fitting process, as returned by nag_lars (g02mac) and nag_lars_xtx (g02mbc).

Constraint:

nstep \geq 0

2: $ip$ – IntegerInput

On entry:

p

, number of parameter estimates, as returned by nag_lars (g02mac) and nag_lars_xtx (g02mbc).

Constraint:

ip \geq 1

3: $b [\dim]$ – const doubleInput

Note: the dimension, dim, of the array b must be at least

pdb \times (nstep + 1)

On entry:

β

the parameter estimates, as returned by nag_lars (g02mac) and nag_lars_xtx (g02mbc), with

b [(k - 1) \times pdb + j - 1] = β_{k j}

, the parameter estimate for the

j

th variable, for

j = 1, 2, \dots, p

, at the

k

th step of the model fitting process.

Constraint: b should be unchanged since the last call to nag_lars (g02mac) or nag_lars_xtx (g02mbc).

4: $pdb$ – IntegerInput

On entry: the stride separating row elements in the two-dimensional data stored in the array b.

Constraint:

pdb \geq ip

5: $fitsum [6 \times (nstep + 1)]$ – const doubleInput

On entry: summaries of the model fitting process, as returned by nag_lars (g02mac) and nag_lars_xtx (g02mbc).

Constraint: fitsum should be unchanged since the last call to nag_lars (g02mac) or nag_lars_xtx (g02mbc)..

6: $ktype$ – Nag_LARSTargetTypeInput

On entry: indicates what target values are held in nk.

$ktype = Nag_LARS_StepNumber$: nk holds (fractional) LARS step numbers.
$ktype = Nag_LARS_ScaledNorm$: nk holds values for $L_{1}$ norm of the (scaled) parameters.
$ktype = Nag_LARS_ProportionScaledNorm$: nk holds ratios with respect to the largest (scaled) $L_{1}$ norm.
$ktype = Nag_LARS_UnscaledNorm$: nk holds values for the $L_{1}$ norm of the (unscaled) parameters.
$ktype = Nag_LARS_ProportionUnscaledNorm$: nk holds ratios with respect to the largest (unscaled) $L_{1}$ norm.

If nag_lars (g02mac) was called with

pred = Nag_LARS_None

Nag_LARS_Centered

or nag_lars_xtx (g02mbc) was called with

pred = Nag_LARS_None

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 = Nag_LARS_ScaledNorm

Nag_LARS_ProportionScaledNorm

and

ktype = Nag_LARS_UnscaledNorm

Nag_LARS_ProportionUnscaledNorm

Constraint:

ktype = Nag_LARS_StepNumber

Nag_LARS_ScaledNorm

Nag_LARS_ProportionScaledNorm

Nag_LARS_UnscaledNorm

Nag_LARS_ProportionUnscaledNorm

7: $nk [lnk]$ – const doubleInput

On entry: target values used for predicting the new set of parameter estimates.

Constraints:

if $ktype = Nag_LARS_StepNumber$ , $0 \leq nk [i - 1] \leq nstep$ , for $i = 1, 2, \dots, lnk$ ;
if $ktype = Nag_LARS_ScaledNorm$ , $0 \leq nk [i - 1] \leq fitsum [(nstep - 1) \times 6]$ , for $i = 1, 2, \dots, lnk$ ;
if $ktype = Nag_LARS_ProportionScaledNorm$ or $Nag_LARS_ProportionUnscaledNorm$ , $0 \leq nk [i - 1] \leq 1$ , for $i = 1, 2, \dots, lnk$ ;
if $ktype = Nag_LARS_UnscaledNorm$ , $0 \leq nk [i - 1] \leq {‖β_{K}‖}_{1}$ , for $i = 1, 2, \dots, lnk$ .

8: $lnk$ – IntegerInput

On entry: number of values supplied in nk.

Constraint:

lnk \geq 1

9: $nb [\dim]$ – doubleOutput

Note: the dimension, dim, of the array nb must be at least

pdnb \times lnk

On exit:

\tilde{β}

the predicted parameter estimates, with

b [(i - 1) \times pdb + j - 1] = {\tilde{β}}_{i j}

, the parameter estimate for variable

j

j = 1, 2, \dots, p

at the point in the fitting process associated with

nk [i - 1]

i = 1, 2, \dots, lnk

10: $pdnb$ – IntegerInput

On entry: the stride separating row elements in the two-dimensional data stored in the array nb.

Constraint:

pdnb \geq ip

11: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_ARRAY_SIZE: On entry, $pdb = 〈value〉$ and $ip = 〈value〉$
Constraint: $pdb \geq ip$ .

On entry, $pdnb = 〈value〉$ and $ip = 〈value〉$ .
Constraint: $pdnb \geq ip$ .
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $ip = 〈value〉$ .
Constraint: $ip \geq 1$ .

On entry, $lnk = 〈value〉$ .
Constraint: $lnk \geq 1$ .

On entry, $nstep = 〈value〉$ .
Constraint: $nstep \geq 0$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_OUT_OF_RANGE: On entry, $ktype = Nag_LARS_ProportionScaledNorm$ or $Nag_LARS_ProportionUnscaledNorm$ , $nk [〈value〉] = 〈value〉$ .
Constraint: $0 \leq nk [i] \leq 1$ for all $i$ .

On entry, $ktype = Nag_LARS_ScaledNorm$ , $nk [〈value〉] = 〈value〉$ , $nstep = 〈value〉$ and $fitsum [(nstep - 1) \times 6] = 〈value〉$ .
Constraint: $0 \leq nk [i] \leq fitsum [(nstep - 1) \times 6]$ for all $i$ .

On entry, $ktype = Nag_LARS_StepNumber$ , $nk [〈value〉] = 〈value〉$ and $nstep = 〈value〉$
Constraint: $0 \leq nk [i] \leq nstep$ for all $i$ .

On entry, $ktype = Nag_LARS_UnscaledNorm$ , $nk [〈value〉] = 〈value〉$ and ${‖β_{K}‖}_{1} = 〈value〉$
Constraint: $0 \leq nk [i] \leq {‖β_{K}‖}_{1}$ for all $i$ .
NE_REAL_ARRAY: b has been corrupted since the last call to nag_lars (g02mac) or nag_lars_xtx (g02mbc).

fitsum has been corrupted since the last call to nag_lars (g02mac) or nag_lars_xtx (g02mbc).

7 Accuracy

Not applicable.

8 Further Comments

None.

9 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

NAG Library Function Documentnag_lars_param (g02mcc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_lars_param (g02mcc)