NAG Library Routine Document

G02KBF

INTEGER	N, M, LDX, ISX(M), IP, LH, WANTB, LDB, WANTVF, LDVF, LPEC, LDPE, IFAIL
REAL (KIND=nag_wp)	X(LDX,M), Y(N), H(LH), NEP(LH), B(LDB,), VF(LDVF,), PE(LDPE,*)
CHARACTER(1)	PEC(LPEC)

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, G02KBF 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 Parameters

1: $N$ – INTEGERInput

On entry:

n

, the number of observations.

Constraint:

N \geq 1

2: $M$ – INTEGERInput

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

X

Constraint:

M \leq N

3: $X (LDX, M)$ – REAL (KIND=nag_wp) arrayInput

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

X

4: $LDX$ – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G02KBF is called.

Constraint:

LDX \geq N

5: $ISX (M)$ – INTEGER arrayInput

On entry: indicates which

m

independent variables are included in the model.

$ISX (j) = 1$: The $j$ th variable in X will be included in the model.
$ISX (j) = 0$: Variable $j$ is excluded.

Constraint:

ISX (j) = 0 ​ or ​ 1

, for

j = 1, 2, \dots, M

6: $IP$ – 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 $1$ .

7: $Y (N)$ – REAL (KIND=nag_wp) arrayInput

On entry: the

n

values of the dependent variable

y

8: $LH$ – INTEGERInput

On entry: the number of supplied ridge parameters.

Constraint:

LH > 0

9: $H (LH)$ – REAL (KIND=nag_wp) arrayInput

On entry:

H (j)

is the value of the

j

th ridge parameter

h

Constraint:

H (j) \geq 0.0

, for

j = 1, 2, \dots, LH

10: $NEP (LH)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

NEP (j)

is the number of effective parameters,

γ

, in the

j

th model, for

j = 1, 2, \dots, LH

11: $WANTB$ – INTEGERInput

On entry: defines the options for parameter estimates.

$WANTB = 0$: Parameter estimates are not calculated and B is not referenced.
$WANTB = 1$: Parameter estimates $b$ are calculated for the original data.
$WANTB = 2$: Parameter estimates $\tilde{b}$ are calculated for the standardized data.

Constraint:

WANTB = 0

1

2

12: $B (LDB *)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array B must be at least

LH

WANTB \neq 0

, and at least

1

otherwise.

On exit: if

WANTB \neq 0

, 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)

, for

i = 1, 2, \dots, IP

13: $LDB$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which G02KBF is called.

Constraints:

if $WANTB \neq 0$ , $LDB \geq IP + 1$ ;
otherwise $LDB \geq 1$ .

14: $WANTVF$ – INTEGERInput

On entry: defines the options for variance inflation factors.

$WANTVF = 0$: Variance inflation factors are not calculated and the array VF is not referenced.
$WANTVF = 1$: Variance inflation factors are calculated.

Constraints:

$WANTVF = 0$ or $1$ ;
if $WANTB = 0$ , $WANTVF = 1$ .

15: $VF (LDVF *)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array VF must be at least

LH

WANTVF \neq 0

, and at least

1

otherwise.

On exit: if

WANTVF = 1

, the variance inflation factors. For the

i

th independent variable in a model fitted with ridge parameter

H (j)

VF (i, j)

is the value of

v_{i}

, for

i = 1, 2, \dots, IP

16: $LDVF$ – INTEGERInput

On entry: the first dimension of the array VF as declared in the (sub)program from which G02KBF is called.

Constraints:

if $WANTVF \neq 0$ , $LDVF \geq IP$ ;
otherwise $LDVF \geq 1$ .

17: $LPEC$ – INTEGERInput

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

LPEC \leq 0

for no prediction error estimates.

18: $PEC (LPEC)$ – CHARACTER(1) arrayInput

On entry: if

LPEC > 0

PEC (j)

defines the

j

th prediction error, for

j = 1, 2, \dots, LPEC

; otherwise PEC is not referenced.

$PEC (j) ='B'$: Bayesian information criterion (BIC).
$PEC (j) ='F'$: Future prediction error (FPE).
$PEC (j) ='G'$: Generalized cross-validation (GCV).
$PEC (j) ='L'$: Leave-one-out cross-validation (LOOCV).
$PEC (j) ='U'$: Unbiased estimate of variance (UEV).

Constraint: if

LPEC > 0

PEC (j) ='B'

'F'

'G'

'L'

'U'

, for

j = 1, 2, \dots, LPEC

19: $PE (LDPE *)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array PE must be at least

LH

LPEC > 0

, and at least

1

otherwise.

On exit: if

LPEC \leq 0

, PE is not referenced; otherwise

PE (i, j)

contains the prediction error of criterion

PEC (i)

for the model fitted with ridge parameter

H (j)

, for

i = 1, 2, \dots, LPEC

and

j = 1, 2, \dots, LH

20: $LDPE$ – INTEGERInput

On entry: the first dimension of the array PE as declared in the (sub)program from which G02KBF is called.

Constraints:

if $LPEC > 0$ , $LDPE \geq LPEC$ ;
otherwise $LDPE \geq 1$ .

21: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. 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

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $H (i) < 0$ for at least one $i$ .

On entry, $LH = 〈value〉$ .
Constraint: $LH > 0$ .

On entry, $M = 〈value〉$ and $N = 〈value〉$ .
Constraint: $M \leq N$ .

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

On entry, $PEC (i)$ is invalid for at least one $i$ .

On entry, $WANTB = 〈value〉$ .
Constraint: $WANTB = 0$ , $1$ or $2$ .

On entry, $WANTVF = 〈value〉$ .
Constraint: $WANTVF = 0$ or $1$ .

$IFAIL = 2$: IP does not equal the sum of elements in ISX.

On entry, $ISX (i) \neq 0$ or $1$ for at least one $i$ .

On entry, $LDB = 〈value〉$ and $IP = 〈value〉$ .
Constraint: if $WANTB \neq 0$ , $LDB \geq IP + 1$ .

On entry, $LDPE = 〈value〉$ and $LPEC = 〈value〉$ .
Constraint: $LDPE \geq LPEC$ .

On entry, $LDVF = 〈value〉$ and $IP = 〈value〉$ .
Constraint: if $WANTVF \neq 0$ , $LDVF \geq IP$ .

On entry, $LDX = 〈value〉$ and $N = 〈value〉$ .
Constraint: $LDX \geq N$ .

$IFAIL = 3$: On entry, $WANTB = 0$ and $WANTVF = 0$ .

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

The accuracy of G02KBF is closely related to that of the singular value decomposition.

8 Parallelism and Performance

G02KBF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

G02KBF 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

G02KBF 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 Routine DocumentG02KBF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

G02KBF