NAG CL Interface
g02kbc (ridge)
1
Purpose
g02kbc calculates a ridge regression, with ridge parameters supplied by you.
2
Specification
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:
where
- 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 .
Let
be the mean-centred
and
the mean-centred
. Furthermore,
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
that minimizes
where
denotes the
-norm and
is a scalar regularization or ridge parameter. For a given value of
, the parameters 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,
g02kbc uses the singular value decomposition (SVD) of
. After decomposing
into
where
and
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) for details.
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
The
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:
Given a value of
, any or all of the following prediction criteria are available:
-
(a)Generalized cross-validation (GCV):
-
(b)Unbiased estimate of variance (UEV):
-
(c)Future prediction error (FPE):
-
(d)Bayesian information criterion (BIC):
-
(e)Leave-one-out cross-validation (LOOCV),
where is the sum of squares of residuals.
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.
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:
– 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
. 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:
or .
-
2:
– Integer
Input
-
On entry: , the number of observations.
Constraint:
.
-
3:
– Integer
Input
-
On entry: the number of independent variables available in the data matrix .
Constraint:
.
-
4:
– const double
Input
-
Note: the dimension,
dim, of the array
x
must be at least
-
when ;
-
when .
the
th element of the matrix
is stored in
- when ;
- when .
On entry: the values of independent variables in the data matrix .
-
5:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
-
6:
– const Integer
Input
-
On entry: indicates which
independent variables are included in the model.
- The th variable in x will be included in the model.
- Variable is excluded.
Constraint:
, for .
-
7:
– Integer
Input
-
On entry: , the number of independent variables in the model.
Constraints:
- ;
- Exactly ip elements of isx must be equal to .
-
8:
– const double
Input
-
On entry: the values of the dependent variable .
-
9:
– Integer
Input
-
On entry: the number of supplied ridge parameters.
Constraint:
.
-
10:
– const double
Input
-
On entry: is the value of the th ridge parameter .
Constraint:
, for .
-
11:
– double
Output
-
On exit: is the number of effective parameters, , in the th model, for .
-
12:
– Nag_ParaOption
Input
-
On entry: defines the options for parameter estimates.
- Parameter estimates are not calculated and b is not referenced.
- Parameter estimates are calculated for the original data.
- Parameter estimates are calculated for the standardized data.
Constraint:
, or .
-
13:
– double
Output
-
Note: the dimension,
dim, of the array
b
must be at least
- when
and
;
- when
and
;
- otherwise.
where
appears in this document, it refers to the array element
- when ;
- when .
On exit: if
,
b contains the intercept and parameter estimates for the fitted ridge regression model in the order indicated by
isx.
, for
, contains the estimate for the intercept;
contains the parameter estimate for the
th independent variable in the model fitted with ridge parameter
, for
.
-
14:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
- if , ;
- otherwise ;
- if ,
- if ,
;
- otherwise .
-
15:
– Nag_VIFOption
Input
-
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.
Constraints:
- or ;
- if , .
-
16:
– double
Output
-
Note: the dimension,
dim, of the array
vf
must be at least
- when
and
;
- when
and
;
- otherwise.
where
appears in this document, it refers to the array element
- when ;
- when .
On exit: if , the variance inflation factors. For the
th independent variable in a model fitted with ridge parameter , is the value of , for .
-
17:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vf.
Constraints:
- if ,
- if , ;
- otherwise ;
- if ,
- if ,
;
- otherwise .
-
18:
– Integer
Input
-
On entry: the number of prediction error statistics to return; set for no prediction error estimates.
-
19:
– const Nag_PredictError
Input
-
On entry: if
,
defines the
th prediction error, for
; otherwise
pec is not referenced and may be
NULL.
- 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 , , , , or , for .
-
20:
– double
Output
-
Note: the dimension,
dim, of the array
pe
must be at least
- when
and
;
- when
and
;
- otherwise pe may be NULL.
where
appears in this document, it refers to the array element
- when ;
- when .
On exit: if
,
pe is not referenced and may be
NULL; otherwise
contains the prediction error of criterion
for the model fitted with ridge parameter
, for
and
.
-
21:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
pe.
Constraints:
- if ,
- if , ;
- otherwise ;
- if ,
- if ,
;
- otherwise pe may be NULL.
-
22:
– 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 had an illegal value.
- NE_CONSTRAINT
-
On entry, and .
Constraint: , .
- NE_ENUM_INT_2
-
On entry, and .
Constraint: if , .
On entry, and .
Constraint: if , .
On entry, , and .
Constraint: if ,
;
otherwise .
On entry, , and .
Constraint: if ,
;
otherwise .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INT_3
-
On entry, , and .
Constraint: if ,
.
- 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
.
- NE_INT_ARRAY_VAL_1_OR_2
-
On entry, or for at least one .
Constraint: or , for all .
- 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, for at least one .
Constraint: , for all .
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.
g02kbc allocates internally 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
10.2
Program Data
10.3
Program Results