g02dkc calculates the estimates of the arguments of a general linear regression model for given constraints from the singular value decomposition results.

2 Specification

#include <nag.h>

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,

{\hat{β}}_{s v d}

, 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 g02dac and g02ddc.

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

p

arguments 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}}_{s v d} \\ = & ({I - P}_{0} {(C^{T} P_{0})}^{- 1}) {\hat{β}}_{s v d} \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: the number of terms in the linear model, $p$ .

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

Constraint: $0 < iconst < ip$ .
3: $p [ip \times ip + 2 \times ip]$ – const double Input: On entry: p as returned by g02dac and g02ddc.
4: $c [ip \times tdc]$ – const double Input: Note: the $(i, j)$ th element of the matrix $C$ is stored in $c [(i - 1) \times tdc + j - 1]$ .

On entry: the iconst constraints stored by column, i.e., the $i$ th constraint is stored in the $i$ th column of c.
5: $tdc$ – Integer Input: On entry: the stride separating matrix column elements in the array c.

Constraint: $tdc \geq iconst$ .
6: $b [ip]$ – double Input/Output: On entry: the parameter estimates computed by using the singular value decomposition, ${\hat{β}}_{s v d}$ .

On exit: the parameter estimates of the arguments with the constraints imposed, ${\hat{β}}_{c}$ .
7: $rss$ – double Input: On entry: the residual sum of squares as returned by g02dac or g02ddc.

Constraint: $rss > 0.0$ .
8: $df$ – double Input: On entry: the degrees of freedom associated with the residual sum of squares as returned by g02dac or g02ddc.

Constraint: $df > 0.0$ .
9: $se [ip]$ – double Output: On exit: the standard error of the parameter estimates in b.
10: $cov [ip \times (ip + 1) / 2]$ – 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 $b [i]$ and the parameter estimate given in $b [j]$ , $j \geq i$ , is stored in $cov [j (j + 1) / 2 + i]$ , for $i = 0, 1, \dots, ip - 1$ and $j = i, \dots, ip - 1$ .
11: $fail$ – 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, $iconst = ⟨ value ⟩$ while $ip = ⟨ value ⟩$ . These arguments must satisfy $iconst < ip$ .
NE_2_INT_ARG_LT: On entry, $tdc = ⟨ value ⟩$ while $iconst = ⟨ value ⟩$ . These arguments must satisfy $tdc \geq iconst$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LE: On entry, $iconst = ⟨ value ⟩$ .
Constraint: $iconst > 0$ .
NE_INT_ARG_LT: On entry, $ip = ⟨ value ⟩$ .
Constraint: $ip \geq 1$ .
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: $df = ⟨ value ⟩$ .

On entry, rss must not be less than or equal to 0.0: $rss = ⟨ value ⟩$ .

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.

9 Further Comments

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.

g02dk: FL CL CPP AD

NAG CL Interfaceg02dkc (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 CL Interface
g02dkc (linregm_constrain)