g02dkf calculates the estimates of the parameters of a general linear regression model for given constraints from the singular value decomposition results.

2 Specification

Fortran Interface

Subroutine g02dkf (

ip, iconst, p, c, ldc, b, rss, idf, se, cov, wk, ifail)

Integer, Intent (In)	::	ip, iconst, ldc, idf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	p(ipip+2ip), c(ldc,iconst), rss
Real (Kind=nag_wp), Intent (Inout)	::	b(ip)
Real (Kind=nag_wp), Intent (Out)	::	se(ip), cov(ip(ip+1)/2), wk(2ipip+ipiconst+2iconsticonst+4*iconst)

C Header Interface

#include <nag.h>

void	g02dkf_ (const Integer ip, const Integer iconst, const double p[], const double c[], const Integer ldc, double b[], const double rss, const Integer idf, double se[], double cov[], double wk[], Integer ifail)

The routine may be called by the names g02dkf or nagf_correg_linregm_constrain.

3 Description

g02dkf 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 g02daf or g02ddf.

In the case of a model not of full rank the routines use a singular value decomposition (SVD) to find the parameter estimates,

{\hat{β}}_{svd}

, and their variance-covariance matrix. Details of the SVD are made available in the form of the matrix

P^{*}

P^{*} = (\begin{matrix} D^{- 1} P_{1}^{T} \\ P_{0}^{T} \end{matrix}),

as described by g02daf and g02ddf.

Alternative solutions can be formed by imposing constraints on the parameters. If there are

p

parameters and the rank of the model is

k

, then

n_{c} = p - k

constraints will have to be imposed to obtain a unique solution.

Let

C

be a

p \times n_{c}

matrix of constraints, such that

C^{T} β = 0

then the new parameter estimates

{\hat{β}}_{c}

are given by

\begin{matrix} {\hat{β}}_{c} & = A {\hat{β}}_{svd}; \\ = (I - P_{0} {(C^{T} P_{0})}^{- 1}) {\hat{β}}_{svd}, \end{matrix}

where

I

is the identity matrix, and the variance-covariance matrix is given by

A P_{1} D^{- 2} P_{1}^{T} A^{T},

provided

{(C^{T} P_{0})}^{- 1}

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: $ip$ – Integer Input: On entry: $p$ , the number of terms in the linear model.

Constraint: $ip \geq 1$ .
2: $iconst$ – Integer Input: On entry: the number of constraints to be imposed on the parameters, $n_{c}$ .

Constraint: $0 < iconst < ip$ .
3: $p (ip \times ip + 2 \times ip)$ – Real (Kind=nag_wp) array Input: On entry: as returned by g02daf and g02ddf.
4: $c (ldc, iconst)$ – Real (Kind=nag_wp) array Input: On entry: the iconst constraints stored by column, i.e., the $i$ th constraint is stored in the $i$ th column of c.
5: $ldc$ – Integer Input: On entry: the first dimension of the array c as declared in the (sub)program from which g02dkf is called.

Constraint: $ldc \geq ip$ .
6: $b (ip)$ – Real (Kind=nag_wp) array Input/Output: On entry: the parameter estimates computed by using the singular value decomposition, ${\hat{β}}_{svd}$ .
On exit: the parameter estimates of the parameters with the constraints imposed, ${\hat{β}}_{c}$ .
7: $rss$ – Real (Kind=nag_wp) Input: On entry: the residual sum of squares as returned by g02daf or g02ddf.

Constraint: $rss > 0.0$ .
8: $idf$ – Integer Input: On entry: the degrees of freedom associated with the residual sum of squares as returned by g02daf or g02ddf.

Constraint: $idf > 0$ .
9: $se (ip)$ – Real (Kind=nag_wp) array Output: On exit: the standard error of the parameter estimates in b.
10: $cov (ip \times (ip + 1) / 2)$ – Real (Kind=nag_wp) array 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 $b (i)$ and the parameter estimate given in $b (j)$ , $j \geq i$ , is stored in $cov ((j \times (j - 1) / 2 + i))$ .
11: $wk (2 \times ip \times ip + ip \times iconst + 2 \times iconst \times iconst + 4 \times iconst)$ – Real (Kind=nag_wp) array Workspace: Note that a simple upper bound for the size of the workspace is $5 \times ip \times ip$ .
12: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $- 1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $- 1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $iconst = ⟨ value ⟩$ .
Constraint: $iconst > 0$ .

On entry, $iconst = ⟨ value ⟩$ and $ip = ⟨ value ⟩$ .
Constraint: $iconst < ip$ .

On entry, $idf = ⟨ value ⟩$ .
Constraint: $idf > 0$ .

On entry, $ip = ⟨ value ⟩$ .
Constraint: $ip \geq 1$ .

On entry, $ldc = ⟨ value ⟩$ and $ip = ⟨ value ⟩$ .
Constraint: $ldc \geq ip$ .

On entry, $rss = ⟨ value ⟩$ .
Constraint: $rss > 0.0$ .

$ifail = 2$: c does not give a model of full rank.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

It should be noted that due to rounding errors a parameter that should be zero when the constraints have been imposed may be returned as a value of order machine precision.

8 Parallelism and Performance

g02dkf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02dkf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

g02dkf 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 parameters of the model to give a full rank model. The routine 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 g02daf and the results printed. The constraint that the sum of treatment effect is zero is then read in and the parameter estimates with this constraint imposed are computed by g02dkf and printed.

g02dk: FL CL CPP AD

NAG FL Interfaceg02dkf (linregm_​constrain)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g02dkf (linregm_constrain)