NAG Library Function Document

nag_regsn_ridge (g02kbc)

nag_regsn_ridge (Nag_OrderType order, Integer n, Integer m, const double x[], Integer pdx, const Integer isx[], Integer ip, const double y[], Integer lh, const double h[], double nep[], Nag_ParaOption wantb, double b[], Integer pdb, Nag_VIFOption wantvf, double vf[], Integer pdvf, Integer lpec, const Nag_PredictError pec[], double pe[], Integer pdpe, NagError *fail)

3 Description

A linear model has the form:

y = c + X β + ε,

where

$y$ is an $n$ by $1$ matrix of values of a dependent variable;
$c$ is a scalar intercept term;
$X$ is an $n$ by $m$ matrix of values of independent variables;
$β$ is a $m$ by $1$ matrix of unknown values of parameters;
$ε$ is an $n$ by $1$ matrix of unknown random errors such that variance of ${ε = σ}^{2} I$ .

Let

\tilde{X}

be the mean-centred

X

and

\tilde{y}

the mean-centred

y

. Furthermore,

\tilde{X}

is scaled such that the diagonal elements of the cross product matrix

{\tilde{X}}^{T} \tilde{X}

are one. The linear model now takes the form:

\tilde{y} = \tilde{X} \tilde{β} + ε .

Ridge regression estimates the parameters

\tilde{β}

in a penalised least squares sense by finding the

\tilde{b}

that minimizes

{‖\tilde{X} \tilde{b} - \tilde{y}‖}^{2} + h {‖\tilde{b}‖}^{2}, h > 0,

where

‖\cdot‖

denotes the

ℓ_{2}

-norm and

h

is a scalar regularization or ridge parameter. For a given value of

h

, the parameters estimates

\tilde{b}

are found by evaluating

\tilde{b} = {({\tilde{X}}^{T} \tilde{X} + h I)}^{- 1} {\tilde{X}}^{T} \tilde{y} .

Note that if

h = 0

the ridge regression solution is equivalent to the ordinary least squares solution.

Rather than calculate the inverse of (

{\tilde{X}}^{T} \tilde{X} + h I

) directly, nag_regsn_ridge (g02kbc) uses the singular value decomposition (SVD) of

\tilde{X}

. After decomposing

\tilde{X}

into

U D V^{T}

where

U

and

V

are orthogonal matrices and

D

is a diagonal matrix, the parameter estimates become

\tilde{b} = V {(D^{T} D + h I)}^{- 1} D U^{T} \tilde{y} .

A consequence of introducing the ridge parameter is that the effective number of parameters,

γ

, in the model is given by the sum of diagonal elements of

D^{T} D {(D^{T} D + h I)}^{- 1},

see Moody (1992) for details.

Any multi-collinearity in the design matrix

X

may be highlighted by calculating the variance inflation factors for the fitted model. The

j

th variance inflation factor,

v_{j}

, is a scaled version of the multiple correlation coefficient between independent variable

j

and the other independent variables,

R_{j}

, and is given by

v_{j} = \frac{1}{1 - R_{j}}, j = 1, 2, \dots, m .

The

m

variance inflation factors are calculated as the diagonal elements of the matrix:

{({\tilde{X}}^{T} \tilde{X} + h I)}^{- 1} {\tilde{X}}^{T} \tilde{X} {({\tilde{X}}^{T} \tilde{X} + h I)}^{- 1},

which, using the SVD of

\tilde{X}

, is equivalent to the diagonal elements of the matrix:

V {(D^{T} D + h I)}^{- 1} D^{T} D {(D^{T} D + h I)}^{- 1} V^{T} .

Given a value of

h

, any or all of the following prediction criteria are available:

(a)

Generalized cross-validation (GCV):

\frac{n s}{{(n - γ)}^{2}};

(b)

Unbiased estimate of variance (UEV):

\frac{s}{n - γ};

(c)

Future prediction error (FPE):

\frac{1}{n} (s + \frac{2 γ s}{n - γ});

(d)

Bayesian information criterion (BIC):

\frac{1}{n} (s + \frac{\log (n) γ s}{n - γ});

(e)

Leave-one-out cross-validation (LOOCV),

where

s

is the sum of squares of residuals.

Although parameter estimates

\tilde{b}

are calculated by using

\tilde{X}

, it is usual to report the parameter estimates

b

associated with

X

. These are calculated from

\tilde{b}

, and the means and scalings of

X

. Optionally, either

\tilde{b}

b

may be calculated.

4 References

Hastie T, Tibshirani R and Friedman J (2003) The Elements of Statistical Learning: Data Mining, Inference and Prediction Springer Series in Statistics

Moody J.E. (1992) The effective number of parameters: An analysis of generalisation and regularisation in nonlinear learning systems In: Neural Information Processing Systems (eds J E Moody, S J Hanson, and R P Lippmann) 4 847–854 Morgan Kaufmann San Mateo CA

5 Arguments

1: order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: n – IntegerInput

On entry:

n

, the number of observations.

Constraint:

n \geq 1

3: m – IntegerInput

On entry: the number of independent variables available in the data matrix

X

Constraint:

m \leq n

4: x[ $\dim$ ] – const doubleInput

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

$\max (1, pdx \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

X

is stored in

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the values of independent variables in the data matrix

X

5: pdx – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq m$ .

6: isx[m] – const IntegerInput

On entry: indicates which

m

independent variables are included in the model.

$isx [j - 1] = 1$: The $j$ th variable in x will be included in the model.
$isx [j - 1] = 0$: Variable $j$ is excluded.

Constraint:

isx [j - 1] = 0 ​ or ​ 1

, for

j = 1, 2, \dots, m

7: ip – IntegerInput

On entry:

m

, the number of independent variables in the model.

Constraints:

$1 \leq ip \leq m$ ;
Exactly ip elements of isx must be equal to $1$ .

8: y[n] – const doubleInput

On entry: the

n

values of the dependent variable

y

9: lh – IntegerInput

On entry: the number of supplied ridge parameters.

Constraint:

lh > 0

10: h[lh] – const doubleInput

On entry:

h [j - 1]

is the value of the

j

th ridge parameter

h

Constraint:

h [j - 1] \geq 0.0

, for

j = 1, 2, \dots, lh

11: nep[lh] – doubleOutput

On exit:

nep [j - 1]

is the number of effective parameters,

γ

, in the

j

th model, for

j = 1, 2, \dots, lh

12: wantb – Nag_ParaOptionInput

On entry: defines the options for parameter estimates.

$wantb = Nag_NoPara$: Parameter estimates are not calculated and b is not referenced.
$wantb = Nag_OrigPara$: Parameter estimates $b$ are calculated for the original data.
$wantb = Nag_StandPara$: Parameter estimates $\tilde{b}$ are calculated for the standardized data.

Constraint:

wantb = Nag_NoPara

Nag_OrigPara

Nag_StandPara

13: b[ $\dim$ ] – doubleOutput

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

$pdb \times lh$ when $wantb \neq Nag_NoPara$ and $order = Nag_ColMajor$ ;
$\max (1, (ip + 1) \times pdb)$ when $wantb \neq Nag_NoPara$ and $order = Nag_RowMajor$ ;
$1$ otherwise.

Where

B (i, j)

appears in this document, it refers to the array element

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

wantb \neq Nag_NoPara

, b contains the intercept and parameter estimates for the fitted ridge regression model in the order indicated by isx.

B (1, j)

, for

j = 1, 2, \dots, lh

, contains the estimate for the intercept;

B (i + 1, j)

contains the parameter estimate for the

i

th independent variable in the model fitted with ridge parameter

h [j - 1]

, for

i = 1, 2, \dots, ip

14: pdb – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if order=Nag_ColMajor,
- if $wantb \neq Nag_NoPara$ , $pdb \geq ip + 1$ ;
- otherwise $pdb \geq 1$ ;
if order=Nag_RowMajor,
- if $wantb \neq Nag_NoPara$ , $pdb \geq lh$ ;
- otherwise $pdb \geq 1$ .

15: wantvf – Nag_VIFOptionInput

On entry: defines the options for variance inflation factors.

$wantvf = Nag_NoVIF$: Variance inflation factors are not calculated and the array vf is not referenced.
$wantvf = Nag_WantVIF$: Variance inflation factors are calculated.

Constraints:

$wantvf = Nag_NoVIF$ or $Nag_WantVIF$ ;
if $wantb = Nag_NoPara$ , $wantvf = Nag_WantVIF$ .

16: vf[ $\dim$ ] – doubleOutput

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

$pdvf \times lh$ when $wantvf \neq Nag_NoVIF$ and $order = Nag_ColMajor$ ;
$\max (1, ip \times pdvf)$ when $wantvf \neq Nag_NoVIF$ and $order = Nag_RowMajor$ ;
$1$ otherwise.

Where

VF (i, j)

appears in this document, it refers to the array element

$vf [(j - 1) \times pdvf + i - 1]$ when $order = Nag_ColMajor$ ;
$vf [(i - 1) \times pdvf + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

wantvf = Nag_WantVIF

, the variance inflation factors. For the

i

th independent variable in a model fitted with ridge parameter

h [j - 1]

VF (i, j)

is the value of

v_{i}

, for

i = 1, 2, \dots, ip

17: pdvf – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array vf.

Constraints:

if order=Nag_ColMajor,
- if $wantvf \neq Nag_NoVIF$ , $pdvf \geq ip$ ;
- otherwise $pdvf \geq 1$ ;
if order=Nag_RowMajor,
- if $wantvf \neq Nag_NoVIF$ , $pdvf \geq lh$ ;
- otherwise $pdvf \geq 1$ .

18: lpec – IntegerInput

On entry: the number of prediction error statistics to return; set

lpec \leq 0

for no prediction error estimates.

19: pec[lpec] – const Nag_PredictErrorInput

On entry: if

lpec > 0

pec [j - 1]

defines the

j

th prediction error, for

j = 1, 2, \dots, lpec

; otherwise pec is not referenced.

$pec [j - 1] = Nag_BIC$: Bayesian information criterion (BIC).
$pec [j - 1] = Nag_FPE$: Future prediction error (FPE).
$pec [j - 1] = Nag_GCV$: Generalized cross-validation (GCV).
$pec [j - 1] = Nag_LOOCV$: Leave-one-out cross-validation (LOOCV).
$pec [j - 1] = Nag_EUV$: Unbiased estimate of variance (UEV).

Constraint: if

lpec > 0

pec [j - 1] = Nag_BIC

Nag_FPE

Nag_GCV

Nag_LOOCV

Nag_EUV

, for

j = 1, 2, \dots, lpec

20: pe[ $\dim$ ] – doubleOutput

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

$pdpe \times lh$ when $lpec > 0$ and $order = Nag_ColMajor$ ;
$\max (1, lpec \times pdpe)$ when $lpec > 0$ and $order = Nag_RowMajor$ ;
$1$ otherwise.

Where

PE (i, j)

appears in this document, it refers to the array element

$pe [(j - 1) \times pdpe + i - 1]$ when $order = Nag_ColMajor$ ;
$pe [(i - 1) \times pdpe + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

lpec \leq 0

, pe is not referenced; otherwise

PE (i, j)

contains the prediction error of criterion

pec [i - 1]

for the model fitted with ridge parameter

h [j - 1]

, for

i = 1, 2, \dots, lpec

and

j = 1, 2, \dots, lh

21: pdpe – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array pe.

Constraints:

if order=Nag_ColMajor,
- if $lpec > 0$ , $pdpe \geq lpec$ ;
- otherwise $pdpe \geq 1$ ;
if order=Nag_RowMajor,
- if $lpec > 0$ , $pdpe \geq lh$ ;
- otherwise $pdpe \geq 1$ .

22: 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.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CONSTRAINT: On entry, $wantb = Nag_NoPara$ and $wantvf = Nag_NoVIF$ .
NE_ENUM_INT_2: On entry, $pdb = ⟨value⟩$ and $ip = ⟨value⟩$ .
Constraint: if $wantb \neq Nag_NoPara$ , $pdb \geq ip + 1$ .
On entry, $pdvf = ⟨value⟩$ and $ip = ⟨value⟩$ .
Constraint: if $wantvf \neq Nag_NoVIF$ , $pdvf \geq ip$ .
On entry, $wantb = ⟨value⟩$ , $pdb = ⟨value⟩$ , $lh = ⟨value⟩$ .
Constraint: if $wantb \neq Nag_NoPara$ , $pdb \geq lh$ ;
otherwise $pdb \geq 1$ .
On entry, $wantvf = ⟨value⟩$ , $pdvf = ⟨value⟩$ , $lh = ⟨value⟩$ .
Constraint: if $wantvf \neq Nag_NoVIF$ , $pdvf \geq lh$ ;
otherwise $pdvf \geq 1$ .
NE_INT: On entry, $lh = ⟨value⟩$ .
Constraint: $lh > 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
NE_INT_2: On entry, $m = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $m \leq n$ .
On entry, $pdpe = ⟨value⟩$ and $lpec = ⟨value⟩$ .
Constraint: $pdpe \geq lpec$ .
On entry, $pdx = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pdx \geq m$ .
On entry, $pdx = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdx \geq n$ .
NE_INT_3: On entry, $pdpe = ⟨value⟩$ , $lpec = ⟨value⟩$ and $lh = ⟨value⟩$ .
Constraint: if $lpec > 0$ , $pdpe \geq lh$ ;
otherwise $pdpe \geq 1$ .
NE_INT_ARG_CONS: ip does not equal the sum of elements in isx.
NE_INT_ARRAY_VAL_1_OR_2: On entry, $isx [i - 1] \neq 0$ or $1$ for at least one $i$ .
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.
NE_REAL_ARRAY_CONS: On entry, $h [i - 1] < 0$ for at least one $i$ .

7 Accuracy

The accuracy of nag_regsn_ridge (g02kbc) is closely related to that of the singular value decomposition.

8 Parallelism and Performance

nag_regsn_ridge (g02kbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_regsn_ridge (g02kbc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

nag_regsn_ridge (g02kbc) allocates internally

\max (5 \times (n - 1), 2 \times ip \times ip) + (n + 3) \times ip + n

elements of double precision storage.

10 Example

This example reads in data from an experiment to model body fat, and a selection of ridge regression models are calculated.

NAG Library Function Documentnag_regsn_ridge (g02kbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_regsn_ridge (g02kbc)