void	g02ddc (Integer n, Integer ip, const double q[], Integer tdq, double rss, double df, double b[], double se[], double cov[], Nag_Boolean svd, Integer rank, double p[], double tol, NagError *fail)

The function may be called by the names: g02ddc, nag_correg_linregm_update or nag_regsn_mult_linear_upd_model.

3 Description

A general linear regression model fitted by g02dac may be adjusted by adding or deleting an observation using g02dcc, adding a new independent variable using g02dec or deleting an existing independent variable using g02dfc. These functions compute the vector

c

and the upper triangular matrix

R

. g02ddc takes these basic results and computes the regression coefficients,

\hat{β}

, their standard errors and their variance-covariance matrix.

R

is of full rank, then

\hat{β}

is the solution to:

R \hat{β} = c_{1},

where

c_{1}

is the first

p

elements of

c

R

is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of

R

R = Q_{*} (\begin{matrix} D & 0 \\ 0 & 0 \end{matrix}) P^{T}

where

D

is a

k \times k

diagonal matrix with nonzero diagonal elements,

k

being the rank of

R

, and

Q_{*}

and

P

are

p \times p

orthogonal matrices. This gives the solution

\hat{β} = P_{1} D^{−1} Q_{*_{1}}^{T} c_{1}

P_{1}

being the first

k

columns of

P

, i.e.,

P = (P_{1} P_{0})

and

Q_{*_{1}}

being the first

k

columns of

Q_{*}

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})

This will be only one of the possible solutions. Other estimates may be obtained by applying constraints to the arguments. These solutions can be obtained by calling g02dkc after calling g02ddc. Only certain linear combinations of the arguments will have unique estimates, these are known as estimable functions. These can be estimated using g02dnc.

The residual sum of squares required to calculate the standard errors and the variance-covariance matrix can either be input or can be calculated if additional information on

c

for the whole sample is provided.

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: $n$ – Integer Input: On entry: number of observations.

Constraint: $n \geq 1$ .
2: $ip$ – Integer Input: On entry: the number of terms in the regression model, $p$ .

Constraint: $ip \geq 1$ .
3: $q [n \times tdq]$ – const double Input: Note: the $(i, j)$ th element of the matrix $Q$ is stored in $q [(i - 1) \times tdq + j - 1]$ .

On entry: q must be the array q as output by g02dcc, g02dec or g02dfc. If on entry $rss \leq 0.0$ then all n elements of $c$ are needed. This is provided by functions g02dec or g02dfc.
4: $tdq$ – Integer Input: On entry: the stride separating matrix column elements in the array q.

Constraint: $tdq \geq ip + 1$ .
5: $rss$ – double * Input/Output: On entry: either the residual sum of squares or a value less than or equal to $0.0$ to indicate that the residual sum of squares is to be calculated by the function.

On exit: if $rss \leq 0.0$ on entry, then on exit rss will contain the residual sum of squares as calculated by g02ddc.
If rss was positive on entry, then it will be unchanged.
6: $df$ – double * Output: On exit: the degrees of freedom associated with the residual sum of squares.
7: $b [ip]$ – double Output: On exit: the estimates of the $p$ arguments, $\hat{β}$ .
8: $se [ip]$ – double Output: On exit: the standard errors of the $p$ arguments given in b.
9: $cov [ip \times (ip + 1) / 2]$ – double Output: On exit: the upper triangular part of the variance-covariance matrix of the $p$ 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 $j = i, \dots, ip - 1$ .
10: $svd$ – Nag_Boolean * Output: On exit: if a singular value decomposition has been performed, then $svd = Nag_TRUE$ , otherwise $svd = Nag_FALSE$ .
11: $rank$ – Integer * Output: On exit: the rank of the independent variables.
If $svd = Nag_FALSE$ , $rank = ip$ .

If $svd = Nag_TRUE$ , rank is an estimate of the rank of the independent variables.

rank is calculated as the number of singular values greater than $tol \times$ (largest singular value). It is possible for the singular value decomposition to be carried out but rank to be returned as ip.
12: $p [ip \times ip + 2 \times ip]$ – double Output: On exit: p contains details of the singular value decomposition if used.
If $svd = Nag_FALSE$ , p is not referenced.

If $svd = Nag_TRUE$ , the first ip elements of p will not be referenced, the next ip values contain the singular values. The following $ip \times ip$ values contain the matrix $P^{*}$ stored by rows.
13: $tol$ – double Input: On entry: the value of tol is used to decide if the independent variables are of full rank and, if not, what is the rank of the independent variables. The smaller the value of tol the stricter the criterion for selecting the singular value decomposition. If $tol = 0.0$ , then the singular value decomposition will never be used, this may cause run time errors or inaccuracies if the independent variables are not of full rank.

Suggested value: $tol = 0.000001$ .

Constraint: $tol \geq 0.0$ .
14: $fail$ – 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_LT: On entry, $n = ⟨ value ⟩$ while $ip = ⟨ value ⟩$ . These arguments must satisfy $n \geq ip$ .

On entry, $tdq = ⟨ value ⟩$ while $ip + 1 = ⟨ value ⟩$ . These arguments must satisfy $tdq \geq ip + 1$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_DOF_LE_ZERO: The degrees of freedom for error are less than or equal to $0$ . In this case the estimates, $\hat{β}$ , are returned but not the standard errors or covariances.
NE_INT_ARG_LT: On entry, $ip = ⟨ value ⟩$ .
Constraint: $ip \geq 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .
NE_REAL_ARG_LT: On entry, tol must not be less than 0.0: $tol = ⟨ value ⟩$ .
NE_SVD_NOT_CONV: The singular value decomposition has failed to converge. This is an unlikely error exit.

7 Accuracy

The accuracy of the results will depend on the accuracy of the input

R

matrix, which may lose accuracy if a large number of observations or variables have been dropped.

8 Parallelism and Performance

g02ddc is not threaded in any implementation.

9 Further Comments

None.

10 Example

A dataset consisting of 12 observations and four independent variables is input and a regression model fitted by calls to g02dec. The arguments are then calculated by g02ddc and the results printed.

g02dd: FL CL CPP AD

NAG CL Interfaceg02ddc (linregm_​update)

▸▿ 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
g02ddc (linregm_update)