# NAG Library Routine Document

## 1Purpose

g02def adds a new independent variable to a general linear regression model.

## 2Specification

Fortran Interface
 Subroutine g02def ( n, ip, q, ldq, p, wt, x, rss, tol,
 Integer, Intent (In) :: n, ip, ldq Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: wt(*), x(n), tol Real (Kind=nag_wp), Intent (Inout) :: q(ldq,ip+2), p(ip+1) Real (Kind=nag_wp), Intent (Out) :: rss Character (1), Intent (In) :: weight
#include nagmk26.h
 void g02def_ (const char *weight, const Integer *n, const Integer *ip, double q[], const Integer *ldq, double p[], const double wt[], const double x[], double *rss, const double *tol, Integer *ifail, const Charlen length_weight)

## 3Description

A linear regression model may be built up by adding new independent variables to an existing model. g02def updates the $QR$ decomposition used in the computation of the linear regression model. The $QR$ decomposition may come from g02daf or a previous call to g02def. The general linear regression model is defined by
 $y=Xβ+ε,$
 where $y$ is a vector of $n$ observations on the dependent variable, $X$ is an $n$ by $p$ matrix of the independent variables of column rank $k$, $\beta$ is a vector of length $p$ of unknown arguments, and $\epsilon$ is a vector of length $n$ of unknown random errors such that $\mathrm{var}\epsilon =V{\sigma }^{2}$, where $V$ is a known diagonal matrix.
If $V=I$, the identity matrix, then least squares estimation is used. If $V\ne I$, then for a given weight matrix $W\propto {V}^{-1}$, weighted least squares estimation is used.
The least squares estimates, $\stackrel{^}{\beta }$ of the arguments $\beta$ minimize ${\left(y-X\beta \right)}^{\mathrm{T}}\left(y-X\beta \right)$ while the weighted least squares estimates, minimize ${\left(y-X\beta \right)}^{\mathrm{T}}W\left(y-X\beta \right)$.
The parameter estimates may be found by computing a $QR$ decomposition of $X$ (or ${W}^{\frac{1}{2}}X$ in the weighted case), i.e.,
 $X=QR* or W12X=QR*,$
where ${R}^{*}=\left(\begin{array}{l}R\\ 0\end{array}\right)$ and $R$ is a $p$ by $p$ upper triangular matrix and $Q$ is an $n$ by $n$ orthogonal matrix.
If $R$ is of full rank, then $\stackrel{^}{\beta }$ is the solution to
 $Rβ^=c1,$
where $c={Q}^{\mathrm{T}}y$ (or ${Q}^{\mathrm{T}}{W}^{\frac{1}{2}}y$) and ${c}_{1}$ is the first $p$ elements of $c$.
If $R$ is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of $R$.
To add a new independent variable, ${x}_{p+1}$, $R$ and $c$ have to be updated. The matrix ${Q}_{p+1}$ is found such that ${Q}_{p+1}^{\mathrm{T}}\left[R:{Q}^{\mathrm{T}}{x}_{p+1}\right]$ (or ${Q}_{p+1}^{\mathrm{T}}\left[R:{Q}^{\mathrm{T}}{W}^{\frac{1}{2}}{x}_{p+1}\right]$) is upper triangular. The vector $c$ is then updated by multiplying by ${Q}_{p+1}^{\mathrm{T}}$.
The new independent variable is tested to see if it is linearly related to the existing independent variables by checking that at least one of the values ${\left({Q}^{\mathrm{T}}{x}_{p+1}\right)}_{\mathit{i}}$, for $\mathit{i}=p+2,\dots ,n$, is nonzero.
The new parameter estimates, $\stackrel{^}{\beta }$, can then be obtained by a call to g02ddf.
The routine can be used with $p=0$, in which case $R$ and $c$ are initialized.

## 4References

Draper N R and Smith H (1985) Applied Regression Analysis (2nd Edition) Wiley
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
McCullagh P and Nelder J A (1983) Generalized Linear Models Chapman and Hall
Searle S R (1971) Linear Models Wiley

## 5Arguments

1:     $\mathbf{weight}$ – Character(1)Input
On entry: indicates if weights are to be used.
${\mathbf{weight}}=\text{'U'}$
Least squares estimation is used.
${\mathbf{weight}}=\text{'W'}$
Weighted least squares is used and weights must be supplied in array wt.
Constraint: ${\mathbf{weight}}=\text{'U'}$ or $\text{'W'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}\ge 1$.
3:     $\mathbf{ip}$ – IntegerInput
On entry: $p$, the number of independent variables already in the model.
Constraint: ${\mathbf{ip}}\ge 0$ and ${\mathbf{ip}}<{\mathbf{n}}$.
4:     $\mathbf{q}\left({\mathbf{ldq}},{\mathbf{ip}}+2\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{ip}}\ne 0$, q must contain the results of the $QR$ decomposition for the model with $p$ arguments as returned by g02daf or a previous call to g02def.
If ${\mathbf{ip}}=0$, the first column of q should contain the $n$ values of the dependent variable, $y$.
On exit: the results of the $QR$ decomposition for the model with $p+1$ arguments:
• the first column of q contains the updated value of $c$;
• the columns $2$ to ${\mathbf{ip}}+1$ are unchanged;
• the first ${\mathbf{ip}}+1$ elements of column ${\mathbf{ip}}+2$ contain the new column of $R$, while the remaining ${\mathbf{n}}-{\mathbf{ip}}-1$ elements contain details of the matrix ${Q}_{p+1}$.
5:     $\mathbf{ldq}$ – IntegerInput
On entry: the first dimension of the array q as declared in the (sub)program from which g02def is called.
Constraint: ${\mathbf{ldq}}\ge {\mathbf{n}}$.
6:     $\mathbf{p}\left({\mathbf{ip}}+1\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: contains further details of the $QR$ decomposition used. The first ip elements of p must contain the zeta values for the $QR$ decomposition (see f08aef (dgeqrf) for details).
The first ip elements of array p are provided by g02daf or by previous calls to g02def.
On exit: the first ip elements of p are unchanged and the $\left({\mathbf{ip}}+1\right)$th element contains the zeta value for ${Q}_{p+1}$.
7:     $\mathbf{wt}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array wt must be at least ${\mathbf{n}}$ if ${\mathbf{weight}}=\text{'W'}$.
On entry: if ${\mathbf{weight}}=\text{'W'}$ , wt must contain the weights to be used.
If ${\mathbf{wt}}\left(i\right)=0.0$, the $i$th observation is not included in the model, in which case the effective number of observations is the number of observations with nonzero weights.
If ${\mathbf{weight}}=\text{'U'}$, wt is not referenced and the effective number of observations is $n$.
Constraint: if ${\mathbf{weight}}=\text{'W'}$, ${\mathbf{wt}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$.
8:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: $x$, the new independent variable.
9:     $\mathbf{rss}$ – Real (Kind=nag_wp)Output
On exit: the residual sum of squares for the new fitted model.
Note:  this will only be valid if the model is of full rank, see Section 9.
10:   $\mathbf{tol}$ – Real (Kind=nag_wp)Input
On entry: the value of tol is used to decide if the new independent variable is linearly related to independent variables already included in the model. If the new variable is linearly related then $c$ is not updated. The smaller the value of tol the stricter the criterion for deciding if there is a linear relationship.
Suggested value: ${\mathbf{tol}}=0.000001$.
Constraint: ${\mathbf{tol}}>0.0$.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Note: g02def may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ip}}\ge 0$.
On entry, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ip}}<{\mathbf{n}}$.
On entry, ${\mathbf{ldq}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldq}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{tol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tol}}>0.0$.
On entry, ${\mathbf{weight}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{weight}}=\text{'W'}$ or $\text{'U'}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{wt}}\left(〈\mathit{\text{value}}〉\right)<0.0$.
Constraint: ${\mathbf{wt}}\left(i\right)\ge 0.0$, for $i=1,2,\dots ,n$.
${\mathbf{ifail}}=3$
x variable is a linear combination of existing model terms.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The accuracy is closely related to the accuracy of f08agf (dormqr) which should be consulted for further details.

## 8Parallelism and Performance

g02def is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02def 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.

It should be noted that the residual sum of squares produced by g02def may not be correct if the model to which the new independent variable is added is not of full rank. In such a case g02ddf should be used to calculate the residual sum of squares.

## 10Example

A dataset consisting of $12$ observations is read in. The four independent variables are stored in the array x while the dependent variable is read into the first column of q. If the character variable $\mathit{mean}$ indicates that a mean should be included in the model a variable taking the value $1.0$ for all observations is set up and fitted. Subsequently, one variable at a time is selected to enter the model as indicated by the input value of $\mathit{indx}$. After the variable has been added the parameter estimates are calculated by g02ddf and the results printed. This is repeated until the input value of $\mathit{indx}$ is $0$.

### 10.1Program Text

Program Text (g02defe.f90)

### 10.2Program Data

Program Data (g02defe.d)

### 10.3Program Results

Program Results (g02defe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017