nag_regsn_ridge (g02kbc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_regsn_ridge (g02kbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagg02.h>
void  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+Xβ+ε ,
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~T X~+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, nag_regsn_ridge (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 DT D+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~T X~+hI -1 X~T X~ X~T X~+hI-1 ,
which, using the SVD of X~, is equivalent to the diagonal elements of the matrix:
V DT D+hI -1 DT D DT D+hI -1 VT .
Given a value of h, any or all of the following prediction criteria are available:
(a) Generalized cross-validation (GCV):
ns n-γ 2 ;
(b) Unbiased estimate of variance (UEV):
s n-γ ;
(c) Future prediction error (FPE):
1n s+ 2γs n-γ ;
(d) Bayesian information criterion (BIC):
1n s+ lognγs n-γ ;
(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:     orderNag_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 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     nIntegerInput
On entry: n, the number of observations.
Constraint: n1.
3:     mIntegerInput
On entry: the number of independent variables available in the data matrix X.
Constraint: mn.
4:     x[dim]const doubleInput
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×m when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
The i,jth 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:     pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
  • if order=Nag_ColMajor, pdxn;
  • if order=Nag_RowMajor, pdxm.
6:     isx[m]const IntegerInput
On entry: indicates which m independent variables are included in the model.
The jth variable in x will be included in the model.
Variable j is excluded.
Constraint: isx[j-1]=0 ​ or ​ 1, for j=1,2,,m.
7:     ipIntegerInput
On entry: m, the number of independent variables in the model.
  • 1ipm;
  • 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:     lhIntegerInput
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 jth ridge parameter h.
Constraint: h[j-1]0.0, for j=1,2,,lh.
11:   nep[lh]doubleOutput
On exit: nep[j-1] is the number of effective parameters, γ, in the jth model, for j=1,2,,lh.
12:   wantbNag_ParaOptionInput
On entry: defines the options for parameter estimates.
Parameter estimates are not calculated and b is not referenced.
Parameter estimates b are calculated for the original data.
Parameter estimates b~ are calculated for the standardized data.
Constraint: wantb=Nag_NoPara, Nag_OrigPara or Nag_StandPara.
13:   b[dim]doubleOutput
Note: the dimension, dim, of the array b must be at least
  • pdb×lh when wantbNag_NoPara and order=Nag_ColMajor;
  • max1,(ip+1)×pdb when wantbNag_NoPara and order=Nag_RowMajor;
  • 1 otherwise.
Where Bi,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. B1,j, for j=1,2,,lh, contains the estimate for the intercept; Bi+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:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
  • if order=Nag_ColMajor,
    • if wantbNag_NoPara, pdbip+1;
    • otherwise pdb1;
  • if order=Nag_RowMajor,
    • if wantbNag_NoPara, pdblh;
    • otherwise pdb1.
15:   wantvfNag_VIFOptionInput
On entry: defines the options for variance inflation factors.
Variance inflation factors are not calculated and the array vf is not referenced.
Variance inflation factors are calculated.
  • 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×lh when wantvfNag_NoVIF and order=Nag_ColMajor;
  • max1,ip×pdvf when wantvfNag_NoVIF and order=Nag_RowMajor;
  • 1 otherwise.
Where VFi,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], VFi,j is the value of vi, for i=1,2,,ip.
17:   pdvfIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vf.
  • if order=Nag_ColMajor,
    • if wantvfNag_NoVIF, pdvfip;
    • otherwise pdvf1;
  • if order=Nag_RowMajor,
    • if wantvfNag_NoVIF, pdvflh;
    • otherwise pdvf1.
18:   lpecIntegerInput
On entry: the number of prediction error statistics to return; set lpec0 for no prediction error estimates.
19:   pec[lpec]const Nag_PredictErrorInput
On entry: if lpec>0, pec[j-1] defines the jth prediction error, for j=1,2,,lpec; otherwise pec is not referenced.
Bayesian information criterion (BIC).
Future prediction error (FPE).
Generalized cross-validation (GCV).
Leave-one-out cross-validation (LOOCV).
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]doubleOutput
Note: the dimension, dim, of the array pe must be at least
  • pdpe×lh when lpec>0 and order=Nag_ColMajor;
  • max1,lpec×pdpe when lpec>0 and order=Nag_RowMajor;
  • 1 otherwise.
Where PEi,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; otherwise PEi,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:   pdpeIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array pe.
  • if order=Nag_ColMajor,
    • if lpec>0, pdpelpec;
    • otherwise pdpe1;
  • if order=Nag_RowMajor,
    • if lpec>0, pdpelh;
    • otherwise pdpe1.
22:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, wantb=Nag_NoPara and wantvf=Nag_NoVIF.
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, lh=value.
Constraint: if wantbNag_NoPara, pdblh;
otherwise pdb1.
On entry, wantvf=value, pdvf=value, lh=value.
Constraint: if wantvfNag_NoVIF, pdvflh;
otherwise pdvf1.
On entry, lh=value.
Constraint: lh>0.
On entry, n=value.
Constraint: n1.
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.
On entry, pdpe=value, lpec=value and lh=value.
Constraint: if lpec>0, pdpelh;
otherwise pdpe1.
ip does not equal the sum of elements in isx.
On entry, isx[i-1]0 or 1 for at least one i.
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.
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 max5×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)

nag_regsn_ridge (g02kbc) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014