NAG CL Interface
g02kac (ridge_opt)
1
Purpose
g02kac calculates a ridge regression, optimizing the ridge parameter according to one of four prediction error criteria.
2
Specification
void |
g02kac (Nag_OrderType order,
Integer n,
Integer m,
const double x[],
Integer pdx,
const Integer isx[],
Integer ip,
double tau,
const double y[],
double *h,
Nag_PredictError opt,
Integer *niter,
double tol,
double *nep,
Nag_EstimatesOption orig,
double b[],
double vif[],
double res[],
double *rss,
Integer *df,
Nag_OptionLOO optloo,
double perr[],
NagError *fail) |
|
The function may be called by the names: g02kac, nag_correg_ridge_opt or nag_regsn_ridge_opt.
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 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,
g02kac 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:
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
:
-
(a)Generalized cross-validation (GCV):
-
(b)Unbiased estimate of variance (UEV):
-
(c)Future prediction error (FPE):
-
(d)Bayesian information criterion (BIC):
where
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.
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
.
where
appears in this document, it refers to the array element
- 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 , ;
- otherwise .
-
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:
– double
Input
-
On entry: singular values less than
tau of the SVD of the data matrix
will be set equal to zero.
Suggested value:
.
Constraint:
.
-
9:
– const double
Input
-
On entry: the values of the dependent variable .
-
10:
– double *
Input/Output
-
On entry: an initial value for the ridge regression parameter ; used as a starting point for the optimization.
Constraint:
.
On exit:
h is the optimized value of the ridge regression parameter
.
-
11:
– Nag_PredictError
Input
-
On entry: 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).
Constraint:
, , or .
-
12:
– Integer *
Input/Output
-
On entry: the maximum number of iterations allowed to optimize the ridge regression parameter .
Constraint:
.
On exit: the number of iterations used to optimize the ridge regression parameter
within
tol.
-
13:
– double
Input
-
On entry: iterations of the ridge regression parameter
will halt when consecutive values of
lie within
tol.
Constraint:
.
-
14:
– double *
Output
-
On exit: the number of effective parameters, , in the model.
-
15:
– Nag_EstimatesOption
Input
-
On entry: if , the parameter estimates are calculated for the original data; otherwise and the parameter estimates are calculated for the standardized data.
Constraint:
or .
-
16:
– double
Output
-
On exit: 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
.
-
17:
– double
Output
-
On exit: the variance inflation factors in the order indicated by
isx. For the
th independent variable in the model,
is the value of
, for
.
-
18:
– double
Output
-
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.
-
20:
– Integer *
Output
-
On exit: the degrees of freedom for the residual sum of squares
rss.
-
21:
– Nag_OptionLOO
Input
-
On entry: if , the leave-one-out cross-validation estimate of prediction error is calculated; otherwise no such estimate is calculated and .
Constraint:
or .
-
22:
– double
Output
-
On exit: 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.
-
23:
– 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_2_INT_ARG_CONS
-
On entry, ; .
Constraint: .
- 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_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, ; .
Constraint: .
On entry, ; .
Constraint: .
- NE_INT_ARG_CONS
-
On entry, .
Constraint: .
- NE_INT_ARRAY_VAL_1_OR_2
-
On entry, .
Constraint: or .
- 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
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_SVD_FAIL
-
SVD failed to converge.
- NW_TOO_MANY_ITER
-
Maximum number of iterations used.
7
Accuracy
Not applicable.
8
Parallelism and Performance
g02kac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02kac 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.
g02kac allocates internally elements of double precision storage.
10
Example
This example reads in data from an experiment to model body fat, and a ridge regression is calculated that optimizes GCV prediction error.
10.1
Program Text
10.2
Program Data
10.3
Program Results