NAG Library Routine Document
G02KAF calculates a ridge regression, optimizing the ridge parameter according to one of four prediction error criteria.
|SUBROUTINE G02KAF (
||N, M, X, LDX, ISX, IP, TAU, Y, H, OPT, NITER, TOL, NEP, ORIG, B, VIF, RES, RSS, DF, OPTLOO, PERR, IFAIL)
||N, M, LDX, ISX(M), IP, OPT, NITER, ORIG, DF, OPTLOO, IFAIL
||X(LDX,M), TAU, Y(N), H, TOL, NEP, B(IP+1), VIF(IP), RES(N), RSS, PERR(5)
A linear model has the form:
- is an by matrix of values of a dependent variable;
- is a scalar intercept term;
- is an by matrix of values of independent variables;
- is an by matrix of unknown values of parameters;
- is an by matrix of unknown random errors such that variance of .
be the mean-centred
is scaled such that the diagonal elements of the cross product matrix
are one. The linear model now takes the form:
Ridge regression estimates the parameters
in a penalised least squares sense by finding the
is a scalar regularization or ridge parameter. For a given value of
, the parameter estimates
are found by evaluating
Note that if
the ridge regression solution is equivalent to the ordinary least squares solution.
Rather than calculate the inverse of (
) directly, G02KAF uses the singular value decomposition (SVD) of
. After decomposing
are orthogonal matrices and
is a diagonal matrix, the parameter estimates become
A consequence of introducing the ridge parameter is that the effective number of parameters,
, in the model is given by the sum of diagonal elements of
see Moody (1992)
Any multi-collinearity in the design matrix
may be highlighted by calculating the variance inflation factors for the fitted model. The
th variance inflation factor,
, is a scaled version of the multiple correlation coefficient between independent variable
and the other independent variables,
, and is given by
variance inflation factors are calculated as the diagonal elements of the matrix:
which, using the SVD of
, is equivalent to the diagonal elements of the matrix:
Although parameter estimates are calculated by using , it is usual to report the parameter estimates associated with . These are calculated from , and the means and scalings of . Optionally, either or may be calculated.
The method can adopt one of four criteria to minimize while calculating a suitable value for
||Generalized cross-validation (GCV):
||Unbiased estimate of variance (UEV):
||Future prediction error (FPE):
||Bayesian information criterion (BIC):
is the sum of squares of residuals. However, the function returns all four of the above prediction errors regardless of the one selected to minimize the ridge parameter,
. Furthermore, the function will optionally return the leave-one-out cross-validation (LOOCV) prediction error.
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
- 1: N – INTEGERInput
On entry: , the number of observations.
- 2: M – INTEGERInput
On entry: the number of independent variables available in the data matrix .
- 3: X(LDX,M) – REAL (KIND=nag_wp) arrayInput
On entry: the values of independent variables in the data matrix .
- 4: LDX – INTEGERInput
: the first dimension of the array X
as declared in the (sub)program from which G02KAF is called.
- 5: ISX(M) – INTEGER arrayInput
: indicates which
independent variables are included in the model.
- The th variable in X will be included in the model.
- Variable is excluded.
, for .
- 6: IP – INTEGERInput
On entry: , the number of independent variables in the model.
- Exactly IP elements of ISX must be equal to .
- 7: TAU – REAL (KIND=nag_wp)Input
: singular values less than TAU
of the SVD of the data matrix
will be set equal to zero.
- 8: Y(N) – REAL (KIND=nag_wp) arrayInput
On entry: the values of the dependent variable .
- 9: H – REAL (KIND=nag_wp)Input/Output
On entry: an initial value for the ridge regression parameter ; used as a starting point for the optimization.
is the optimized value of the ridge regression parameter
- 10: OPT – INTEGERInput
: the measure of prediction error used to optimize the ridge regression parameter
. The value of OPT
must be set equal to one of:
- Generalized cross-validation (GCV);
- Unbiased estimate of variance (UEV)
- Future prediction error (FPE)
- Bayesian information criteron (BIC).
, , or .
- 11: NITER – INTEGERInput/Output
On entry: the maximum number of iterations allowed to optimize the ridge regression parameter .
: the number of iterations used to optimize the ridge regression parameter
- 12: TOL – REAL (KIND=nag_wp)Input
: iterations of the ridge regression parameter
will halt when consecutive values of
lie within TOL
- 13: NEP – REAL (KIND=nag_wp)Output
On exit: the number of effective parameters, , in the model.
- 14: ORIG – INTEGERInput
On entry: if , the parameter estimates are calculated for the original data; otherwise and the parameter estimates are calculated for the standardized data.
- 15: B() – REAL (KIND=nag_wp) arrayOutput
: contains the intercept and parameter estimates for the fitted ridge regression model in the order indicated by ISX
. The first element of B
contains the estimate for the intercept;
contains the parameter estimate for the
th independent variable in the model, for
- 16: VIF(IP) – REAL (KIND=nag_wp) arrayOutput
: the variance inflation factors in the order indicated by ISX
. For the
th independent variable in the model,
is the value of
- 17: RES(N) – REAL (KIND=nag_wp) arrayOutput
On exit: is the value of the th residual for the fitted ridge regression model, for .
On exit: the sum of squares of residual values.
- 19: DF – INTEGEROutput
: the degrees of freedom for the residual sum of squares RSS
- 20: OPTLOO – INTEGERInput
On entry: if , the leave-one-out cross-validation estimate of prediction error is calculated; otherwise no such estimate is calculated and .
- 21: PERR() – REAL (KIND=nag_wp) arrayOutput
: the first four elements contain, in this order, the measures of prediction error: GCV, UEV, FPE and BIC.
If , is the LOOCV estimate of prediction error; otherwise is not referenced.
- 22: IFAIL – INTEGERInput/Output
must be set to
. 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
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
Maximum number of iterations used.
|or||, , or ;|
|or|| or ;|
|or|| or |
|or|| or ;|
|or||An element of or ;|
|or||IP does not equal the sum of elements in ISX.|
SVD failed to converge.
Internal error. Check all array sizes and calls to G02KAF. Please contact NAG
G02KAF allocates internally elements of double precision storage.
This example reads in data from an experiment to model body fat, and a ridge regression is calculated that optimizes GCV prediction error.
9.1 Program Text
Program Text (g02kafe.f90)
9.2 Program Data
Program Data (g02kafe.d)
9.3 Program Results
Program Results (g02kafe.r)