NAG CL Interface
g02dkc (linregm_constrain)
1
Purpose
g02dkc calculates the estimates of the arguments of a general linear regression model for given constraints from the singular value decomposition results.
2
Specification
void |
g02dkc (Integer ip,
Integer iconst,
const double p[],
const double c[],
Integer tdc,
double b[],
double rss,
double df,
double se[],
double cov[],
NagError *fail) |
|
The function may be called by the names: g02dkc, nag_correg_linregm_constrain or nag_regsn_mult_linear_tran_model.
3
Description
g02dkc computes the estimates given a set of linear constraints for a general linear regression model which is not of full rank. It is intended for use after a call to
g02dac or
g02ddc.
In the case of a model not of full rank the functions use a singular value decomposition (SVD) to find the parameter estimates,
, and their variance-covariance matrix. Details of the SVD are made available, in the form of the matrix
:
as described by
g02dac and
g02ddc.
Alternative solutions can be formed by imposing constraints on the arguments. If there are arguments and the rank of the model is , then constraints will have to be imposed to obtain a unique solution.
Let
be a
by
matrix of constraints, such that
then the new parameter estimates
are given by:
where
is the identity matrix, and the variance-covariance matrix is given by:
provided
exists.
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20(3) 2–25
Searle S R (1971) Linear Models Wiley
5
Arguments
-
1:
– Integer
Input
-
On entry: the number of terms in the linear model, .
Constraint:
.
-
2:
– Integer
Input
-
On entry: the number of constraints to be imposed on the arguments, .
Constraint:
.
-
3:
– const double
Input
-
On entry:
p as returned by
g02dac and
g02ddc.
-
4:
– const double
Input
-
Note: the th element of the matrix is stored in .
On entry: the
iconst constraints stored by column, i.e., the
th constraint is stored in the
th column of
c.
-
5:
– Integer
Input
-
On entry: the stride separating matrix column elements in the array
c.
Constraint:
.
-
6:
– double
Input/Output
-
On entry: the parameter estimates computed by using the singular value decomposition, .
On exit: the parameter estimates of the arguments with the constraints imposed, .
-
On entry: the residual sum of squares as returned by
g02dac or
g02ddc.
Constraint:
.
-
8:
– double
Input
-
On entry: the degrees of freedom associated with the residual sum of squares as returned by
g02dac or
g02ddc.
Constraint:
.
-
9:
– double
Output
-
On exit: the standard error of the parameter estimates in
b.
-
10:
– double
Output
-
On exit: the upper triangular part of the variance-covariance matrix of the
ip parameter estimates given in
b. They are stored packed by column, i.e., the covariance between the parameter estimate given in
and the parameter estimate given in
,
, is stored in
, for
and
.
-
11:
– 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_GE
-
On entry, while . These arguments must satisfy .
- NE_2_INT_ARG_LT
-
On entry, while . These arguments must satisfy .
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_INT_ARG_LE
-
On entry, .
Constraint: .
- NE_INT_ARG_LT
-
On entry, .
Constraint: .
- NE_MAT_NOT_FULL_RANK
-
Matrix
c does not give a model of full rank.
- NE_REAL_ARG_LE
-
On entry,
df must not be less than or equal to 0.0:
.
On entry,
rss must not be less than or equal to 0.0:
.
7
Accuracy
It should be noted that due to rounding errors an argument that should be zero when the constraints have been imposed may be returned as a value of order machine precision.
8
Parallelism and Performance
g02dkc is not threaded in any implementation.
g02dkc is intended for use in situations in which dummy (0-1) variables have been used such as in the analysis of designed experiments when you do not wish to change the arguments of the model to give a full rank model. The function is not intended for situations in which the relationships between the independent variables are only approximate.
10
Example
Data from an experiment with four treatments and three observations per treatment are read in. A model, including the mean term, is fitted by
g02dac and the results printed. The constraint that the sum of treatment effects is zero is then read in and the parameter estimates with this constraint imposed are computed by
g02dkc and printed.
10.1
Program Text
10.2
Program Data
10.3
Program Results