NAG CL Interface
g02kbc (ridge)

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1 Purpose

g02kbc calculates a ridge regression, with ridge parameters supplied by you.

2 Specification

#include <nag.h>
void  g02kbc (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)
The function may be called by the names: g02kbc, nag_correg_ridge or nag_regsn_ridge.

3 Description

A linear model has the form:
y = c+Xβ+ε ,  
where
Let X~ be the mean-centred X and y~ the mean-centred y. Furthermore, X~ is scaled such that the diagonal elements of the cross product matrix X~TX~ are one. The linear model now takes the form:
y~ = X~ β~ + ε .  
Ridge regression estimates the parameters β~ in a penalised least squares sense by finding the b~ that minimizes
X~b~-y~ 2 + h b~ 2 ,h>0 ,  
where · denotes the 2-norm and h is a scalar regularization or ridge parameter. For a given value of h, the parameters estimates b~ are found by evaluating
b~ = (X~TX~+hI)-1 X~T y~ .  
Note that if h=0 the ridge regression solution is equivalent to the ordinary least squares solution.
Rather than calculate the inverse of (X~TX~+hI) directly, g02kbc uses the singular value decomposition (SVD) of X~. After decomposing X~ into UDVT where U and V are orthogonal matrices and D is a diagonal matrix, the parameter estimates become
b~ = V (DTD+hI)-1 DUT 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
DT D (DTD+hI)-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 jth variance inflation factor, vj, is a scaled version of the multiple correlation coefficient between independent variable j and the other independent variables, Rj, and is given by
vj = 1 1-Rj ,j=1,2,,m .  
The m variance inflation factors are calculated as the diagonal elements of the matrix:
(X~TX~+hI)-1 X~T X~ (X~TX~+hI)-1 ,  
which, using the SVD of X~, is equivalent to the diagonal elements of the matrix:
V (DTD+hI)-1 DT D (DTD+hI)-1 VT .  
Given a value of h, any or all of the following prediction criteria are available:
  1. (a)Generalized cross-validation (GCV):
    ns (n-γ) 2 ;  
  2. (b)Unbiased estimate of variance (UEV):
    s n-γ ;  
  3. (c)Future prediction error (FPE):
    1n (s+ 2γs n-γ ) ;  
  4. (d)Bayesian information criterion (BIC):
    1n (s+ log(n)γs n-γ ) ;  
  5. (e)Leave-one-out cross-validation (LOOCV),
where s is the sum of squares of residuals.
Although parameter estimates b~ are calculated by using X~, it is usual to report the parameter estimates b associated with X. These are calculated from b~, and the means and scalings of X. Optionally, either b~ or 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_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: n Integer Input
On entry: n, the number of observations.
Constraint: n1.
3: m Integer Input
On entry: the number of independent variables available in the data matrix X.
Constraint: mn.
4: x[dim] const double Input
Note: the dimension, dim, of the array x must be at least
  • max(1,pdx×m) when order=Nag_ColMajor;
  • max(1,n×pdx) when order=Nag_RowMajor.
the (i,j)th element of the matrix X is stored in
  • x[(j-1)×pdx+i-1] when order=Nag_ColMajor;
  • x[(i-1)×pdx+j-1] when order=Nag_RowMajor.
On entry: the values of independent variables in the data matrix X.
5: pdx Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxn;
  • if order=Nag_RowMajor, pdxm.
6: isx[m] const Integer Input
On entry: indicates which m independent variables are included in the model.
isx[j-1]=1
The jth 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,,m.
7: ip Integer Input
On entry: m, the number of independent variables in the model.
Constraints:
  • 1ipm;
  • Exactly ip elements of isx must be equal to 1.
8: y[n] const double Input
On entry: the n values of the dependent variable y.
9: lh Integer Input
On entry: the number of supplied ridge parameters.
Constraint: lh>0.
10: h[lh] const double Input
On entry: h[j-1] is the value of the jth ridge parameter h.
Constraint: h[j-1]0.0, for j=1,2,,lh.
11: nep[lh] double Output
On exit: nep[j-1] is the number of effective parameters, γ, in the jth model, for j=1,2,,lh.
12: wantb Nag_ParaOption Input
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 b~ are calculated for the standardized data.
Constraint: wantb=Nag_NoPara, Nag_OrigPara or Nag_StandPara.
13: b[dim] double Output
Note: the dimension, dim, of the array b must be at least
  • pdb×lh when wantbNag_NoPara and order=Nag_ColMajor;
  • max(1,(ip+1)×pdb) when wantbNag_NoPara and order=Nag_RowMajor;
  • 1 otherwise.
where B(i,j) appears in this document, it refers to the array element
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On exit: if wantbNag_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,,lh, contains the estimate for the intercept; B(i+1,j) contains the parameter estimate for the ith independent variable in the model fitted with ridge parameter h[j-1], for i=1,2,,ip.
14: pdb Integer Input
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 wantbNag_NoPara, pdbip+1;
    • otherwise pdb1;
  • if order=Nag_RowMajor,
    • if wantbNag_NoPara, pdblh;
    • otherwise pdb1.
15: wantvf Nag_VIFOption Input
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] double Output
Note: the dimension, dim, of the array vf must be at least
  • pdvf×lh when wantvfNag_NoVIF and order=Nag_ColMajor;
  • max(1,ip×pdvf) when wantvfNag_NoVIF and order=Nag_RowMajor;
  • 1 otherwise.
where VF(i,j) appears in this document, it refers to the array element
  • vf[(j-1)×pdvf+i-1] when order=Nag_ColMajor;
  • vf[(i-1)×pdvf+j-1] when order=Nag_RowMajor.
On exit: if wantvf=Nag_WantVIF, the variance inflation factors. For the ith independent variable in a model fitted with ridge parameter h[j-1], VF(i,j) is the value of vi, for i=1,2,,ip.
17: pdvf Integer Input
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 wantvfNag_NoVIF, pdvfip;
    • otherwise pdvf1;
  • if order=Nag_RowMajor,
    • if wantvfNag_NoVIF, pdvflh;
    • otherwise pdvf1.
18: lpec Integer Input
On entry: the number of prediction error statistics to return; set lpec0 for no prediction error estimates.
19: pec[lpec] const Nag_PredictError Input
On entry: if lpec>0, pec[j-1] defines the jth prediction error, for j=1,2,,lpec; otherwise pec is not referenced and may be NULL.
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 or Nag_EUV, for j=1,2,,lpec.
20: pe[dim] double Output
Note: the dimension, dim, of the array pe must be at least
  • pdpe×lh when lpec>0 and order=Nag_ColMajor;
  • lpec×pdpe when lpec>0 and order=Nag_RowMajor;
  • otherwise pe may be NULL.
where PE(i,j) appears in this document, it refers to the array element
  • pe[(j-1)×pdpe+i-1] when order=Nag_ColMajor;
  • pe[(i-1)×pdpe+j-1] when order=Nag_RowMajor.
On exit: if lpec0, pe is not referenced and may be NULL; 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,,lpec and j=1,2,,lh.
21: pdpe Integer Input
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, pdpelpec;
    • otherwise pdpe1;
  • if order=Nag_RowMajor,
    • if lpec>0, pdpelh;
    • otherwise pe may be NULL.
22: fail 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_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONSTRAINT
On entry, wantb=Nag_NoPara and wantvf=Nag_NoVIF.
Constraint: wantb=Nag_NoPara, wantvf=Nag_WantVIF.
NE_ENUM_INT_2
On entry, pdb=value and ip=value.
Constraint: if wantbNag_NoPara, pdbip+1.
On entry, pdvf=value and ip=value.
Constraint: if wantvfNag_NoVIF, pdvfip.
On entry, wantb=value, pdb=value and lh=value.
Constraint: if wantbNag_NoPara, pdblh;
otherwise pdb1.
On entry, wantvf=value, pdvf=value and lh=value.
Constraint: if wantvfNag_NoVIF, pdvflh;
otherwise pdvf1.
NE_INT
On entry, lh=value.
Constraint: lh>0.
On entry, n=value.
Constraint: n1.
NE_INT_2
On entry, m=value and n=value.
Constraint: mn.
On entry, pdpe=value and lpec=value.
Constraint: pdpelpec.
On entry, pdx=value and m=value.
Constraint: pdxm.
On entry, pdx=value and n=value.
Constraint: pdxn.
NE_INT_3
On entry, pdpe=value, lpec=value and lh=value.
Constraint: if lpec>0, pdpelh.
NE_INT_ARG_CONS
On entry, ip is not equal to the sum of elements in isx.
Constraint: exactly ip elements of isx must be equal to 1.
NE_INT_ARRAY_VAL_1_OR_2
On entry, isx[j-1]0 or 1 for at least one j.
Constraint: isx[j-1]=0 or 1, for all j.
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_REAL_ARRAY_CONS
On entry, h[j-1]<0 for at least one j.
Constraint: h[j-1]0.0, for all j.

7 Accuracy

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

8 Parallelism and Performance

g02kbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

g02kbc allocates internally max(5×(n-1),2×ip×ip)+(n+3)×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.

10.1 Program Text

Program Text (g02kbce.c)

10.2 Program Data

Program Data (g02kbce.d)

10.3 Program Results

Program Results (g02kbce.r)