NAG Library Function Document

nag_regsn_mult_linear_upd_model (g02ddc)

void	nag_regsn_mult_linear_upd_model (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)

3 Description

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

c

and the upper triangular matrix

R

. nag_regsn_mult_linear_upd_model (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

k

diagonal matrix with nonzero diagonal elements,

k

being the rank of

R

, and

Q_{*}

and

P

are

p

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 nag_regsn_mult_linear_tran_model (g02dkc) after calling nag_regsn_mult_linear_upd_model (g02ddc). Only certain linear combinations of the arguments will have unique estimates, these are known as estimable functions. These can be estimated using nag_regsn_mult_linear_est_func (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 – IntegerInput: On entry: number of observations.
Constraint: $n \geq 1$ .
2: ip – IntegerInput: On entry: the number of terms in the regression model, $p$ .
Constraint: $ip \geq 1$ .
3: q[ $n \times tdq$ ] – const doubleInput: 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 nag_regsn_mult_linear_addrem_obs (g02dcc), nag_regsn_mult_linear_add_var (g02dec) or nag_regsn_mult_linear_delete_var (g02dfc). If on entry $rss \leq 0.0$ then all n elements of $c$ are needed. This is provided by functions nag_regsn_mult_linear_add_var (g02dec) or nag_regsn_mult_linear_delete_var (g02dfc).
4: tdq – IntegerInput: 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 nag_regsn_mult_linear_upd_model (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] – doubleOutput: On exit: the estimates of the $p$ arguments, $\hat{β}$ .
8: se[ip] – doubleOutput: On exit: the standard errors of the $p$ arguments given in b.
9: cov[ $ip \times (ip + 1) / 2$ ] – doubleOutput: 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[ $ip \times ip + 2 \times ip$ ] – doubleOutput: 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 – doubleInput: 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 3.6 in the Essential Introduction).

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

Not applicable.

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 nag_regsn_mult_linear_add_var (g02dec). The arguments are then calculated by nag_regsn_mult_linear_upd_model (g02ddc) and the results printed.

NAG Library Function Documentnag_regsn_mult_linear_upd_model (g02ddc)

+− Contents

1 Purpose

2 Specification