naginterfaces.library.correg.glm_​normal

naginterfaces.library.correg.glm_normal(x, isx, y, link='I', mean='M', wt=None, s=0.0, a=0.0, v=None, tol=0.0, maxit=10, iprint=0, eps=0.0, io_manager=None)

glm_normal fits a generalized linear model with normal errors.

For full information please refer to the NAG Library document for g02ga

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02gaf.html

Parameters

xfloat, array-like, shape $(n,m)$

$x[\text{i}-1,\text{j}-1]$ must contain the $\text{i}$th observation for the $\text{j}$th independent variable, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

isxint, array-like, shape $\left(m\right)$

Indicates which independent variables are to be included in the model.

If $isx[j-1]>0$, the variable contained in the $j$th column of $x$ is included in the regression model.

yfloat, array-like, shape $\left(n\right)$

The observations on the dependent variable, $y_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

linkstr, length 1, optional

Indicates which link function is to be used.

$link=\text{'E'}$

An exponent link is used.

$link=\text{'I'}$

An identity link is used.

$link=\text{'L'}$

A log link is used.

$link=\text{'S'}$

A square root link is used.

$link=\text{'R'}$

A reciprocal link is used.

meanstr, length 1, optional

Indicates if a mean term is to be included.

$mean=\text{'M'}$

A mean term, intercept, will be included in the model.

$mean=\text{'Z'}$

The model will pass through the origin, zero-point.

wtNone or float, array-like, shape $\left(n\right)$, optional

If provided $wt$ must contain the weights to be used with the model.

If $wt[i-1]=0.0$, 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 $wt$ is not provided the effective number of observations is $n$.

sfloat, optional

The scale parameter for the model, ${\sigma }^2$.

If $s=0.0$, the scale parameter is estimated with the function using the residual mean square.

afloat, optional

If $link=\text{'E'}$, $a$ must contain the power of the exponential.

If $link\ne \text{'E'}$, $a$ is not referenced.

vNone or float, array-like, shape $(n,\text{ip}+7)$, optional

If $v\text{is not}None$, $v[\text{i}-1,6]$, for $\text{i}=1,2,\dots ,n$, must contain the offset values $o_{\text{i}}$.

All other values need not be set.

tolfloat, optional

Indicates the accuracy required for the fit of the model.

The iterative weighted least squares procedure is deemed to have converged if the absolute change in deviance between interactions is less than $tol\times (1.0+\text{current residual sum of squares})$.

This is approximately an absolute precision if the residual sum of squares is small and a relative precision if the residual sum of squares is large.

If $0.0\le tol<\text{machine precision}$, glm_normal will use $10\times \text{machine precision}$.

maxitint, optional

The maximum number of iterations for the iterative weighted least squares.

If $maxit=0$, a default value of $10$ is used.

iprintint, optional

Indicates if the printing of information on the iterations is required.

$iprint\le 0$

There is no printing.

$iprint>0$

Every $iprint$ iteration, the following is printed:

the deviance,

the current estimates,

and if the weighted least squares equations are singular, then this is indicated.

When printing occurs the output is directed to the file object associated with the advisory I/O unit (see FileObjManager).

epsfloat, optional

The value of $eps$ 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 $eps$ the stricter the criterion for selecting the singular value decomposition.

If $0.0\le eps<\text{machine precision}$, the function will use machine precision instead.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

sfloat

If on input $s=0.0$, $s$ contains the estimated value of the scale parameter, ${\hat{\sigma }}^2$.

If on input $s\ne 0.0$, $s$ is unchanged on exit.

rssfloat

The residual sum of squares for the fitted model.

idfint

The degrees of freedom associated with the residual sum of squares for the fitted model.

bfloat, ndarray, shape $\left(\text{ip}\right)$

The estimates of the parameters of the generalized linear model, $\hat{\beta }$.

If $mean=\text{'M'}$, $b\left[0\right]$ will contain the estimate of the mean parameter and $b\left[i\right]$ will contain the coefficient of the variable contained in column $j$ of $x$, where $isx[j-1]$ is the $i$th positive value in the array $isx$.

If $mean=\text{'Z'}$, $b[i-1]$ will contain the coefficient of the variable contained in column $j$ of $x$, where $isx[j-1]$ is the $i$th positive value in the array $isx$.

irankint

The rank of the independent variables.

If the model is of full rank, $irank=\text{ip}$.

If the model is not of full rank, $irank$ is an estimate of the rank of the independent variables. $irank$ is calculated as the number of singular values greater than $eps\times$ (largest singular value).

It is possible for the SVD to be carried out but for $irank$ to be returned as $\text{ip}$.

sefloat, ndarray, shape $\left(\text{ip}\right)$

The standard errors of the linear parameters.

$se[\text{i}-1]$ contains the standard error of the parameter estimate in $b[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{ip}$.

covfloat, ndarray, shape $(\text{ip}\times (\text{ip}+1)/2)$

The upper triangular part of the variance-covariance matrix of the $\text{ip}$ parameter estimates given in $b$. They are stored packed by column, i.e., the covariance between the parameter estimate given in $b[i-1]$ and the parameter estimate given in $b[j-1]$, $j\ge i$, is stored in $cov[(j\times (j-1)/2+i)-1]$.

vfloat, ndarray, shape $(n,\text{ip}+7)$

Auxiliary information on the fitted model.

$v[i-1,0]$	contains the linear predictor value, ${\eta }_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,1]$	contains the fitted value, ${\hat{\mu }}_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,2]$	is only included for consistency with other functions. $v[\text{i}-1,2]=1.0$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,3]$	contains the square root of the working weight, $w_{\text{i}}^{\frac{1}{2}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,4]$	contains the residual, $r_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,5]$	contains the leverage, $h_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,6]$	contains the offset, for $i=1,2,\dots ,n$. If $v$ is None on entry, all values will be zero.
$v[i-1,j-1]$	for $j=8,\dots ,\text{ip}+7$, contains the results of the $QR$ decomposition or the singular value decomposition.

If the model is not of full rank, i.e., $irank<\text{ip}$, the first $\text{ip}$ rows of columns $8$ to $\text{ip}+7$ contain the $P^{\ast }$ matrix.

Raises

NagValueError

(errno $1$)

On entry, $\text{ip}=⟨value⟩$.

Constraint: $\text{ip}\ge 1$.

(errno $1$)

On entry, $eps=⟨value⟩$.

Constraint: $eps\ge 0.0$.

(errno $1$)

On entry, $tol=⟨value⟩$.

Constraint: $tol\ge 0.0$.

(errno $1$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit\ge 0$.

(errno $1$)

On entry, $\text{offset}=⟨value⟩$.

Constraint: $\text{offset}=\text{'Y'}$ or $\text{'N'}$.

(errno $1$)

On entry, $link=⟨value⟩$.

Constraint: $link=\text{'E'}$, $\text{'I'}$, $\text{'L'}$, $\text{'S'}$ or $\text{'R'}$.

(errno $1$)

On entry, $a=0.0$ and $link=\text{'E'}$.

Constraint: if $link=\text{'E'}$, $a\ne 0.0$.

(errno $1$)

On entry, $s=⟨value⟩$.

Constraint: $s\ge 0.0$.

(errno $1$)

On entry, $\text{weight}=⟨value⟩$.

Constraint: $\text{weight}=\text{'W'}$ or $\text{'U'}$.

(errno $1$)

On entry, $mean=⟨value⟩$.

Constraint: $mean=\text{'M'}$ or $\text{'Z'}$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $2$)

On entry, $wt[⟨value⟩]<0.0$.

Constraint: $wt[i-1]\ge 0.0$, for $i=1,2,\dots ,n$.

(errno $3$)

On entry, $isx[⟨value⟩]<0$.

Constraint: $isx[j-1]\ge 0.0$, for $j=1,2,\dots ,m$.

(errno $3$)

On entry, $\text{ip}$ incompatible with number of nonzero values of $isx$: $\text{ip}=⟨value⟩$.

(errno $3$)

Number of requested x-variables greater than $\text{n}$.

(errno $4$)

A fitted value is at a boundary. This will only occur with $link=\text{'L'}$, $\text{'R'}$ or $\text{'E'}$. This may occur if there are small values of $y$ and the model is not suitable for the data. The model should be reformulated with, perhaps, some observations dropped.

(errno $5$)

SVD solution failed to converge.

(errno $6$)

The iterative weighted least squares has failed to converge in $maxit$ (or default $10$) iterations. The value of $maxit$ could be increased but it may be advantageous to examine the convergence using the $iprint$ option. This may indicate that the convergence is slow because the solution is at a boundary in which case it may be better to reformulate the model.

Warns

NagAlgorithmicWarning

(errno $7$)

The rank of the model has changed during the weighted least squares iterations. The estimate for $\beta$ returned may be reasonable, but you should check how the deviance has changed during iterations.

(errno $8$)

The degrees of freedom for error are $0$. A saturated model has been fitted.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

A generalized linear model with Normal errors consists of the following elements:

1. a set of $n$ observations, $y_i$, from a Normal distribution with probability density function:

   $$\frac{1}{\sqrt{2\pi }\sigma }exp(-\frac{{(y-\mu )}^2}{2{\sigma }^2})\text{,}$$

   where $\mu$ is the mean and ${\sigma }^2$ is the variance.

2. $X$, a set of $p$ independent variables for each observation, $x_1,x_2,\dots ,x_p$.

3. a linear model:

   $$\eta =\sum {\beta }_j{x}_j\text{.}$$

4. a link between the linear predictor, $\eta$, and the mean of the distribution, $\mu$, i.e., $\eta =g\left(\mu \right)$. The possible link functions are:

   1. exponent link: $\eta ={\mu }^a$, for a constant $a$,

   2. identity link: $\eta =\mu$,

   3. log link: $\eta =\log \left(\mu \right)$,

   4. square root link: $\eta =\sqrt{\mu }$,

   5. reciprocal link: $\eta =\frac{1}{\mu }$.

5. a measure of fit, the residual sum of squares $=\sum {(y_i-{\hat{\mu }}_i)}^2$.

The linear parameters are estimated by iterative weighted least squares. An adjusted dependent variable, $z$, is formed:

$$z=\eta +(y-\mu )\frac{d\eta }{d\mu }$$

and a working weight, $w$,

$$w^{-1}={\left(\frac{d\eta }{d\mu }\right)}^2\text{.}$$

At each iteration an approximation to the estimate of $\beta$, $\hat{\beta }$, is found by the weighted least squares regression of $z$ on $X$ with weights $w$.

glm_normal finds a $QR$ decomposition of $w^{\frac{1}{2}}X$, i.e., $w^{\frac{1}{2}}X=QR$ where $R$ is a $p\times p$ triangular matrix and $Q$ is an $n\times p$ column orthogonal matrix.

If $R$ is of full rank, then $\hat{\beta }$ is the solution to

$$R\hat{\beta }=Q^T{w}^{\frac{1}{2}}z\text{.}$$

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

$$\begin{array}{c}R=Q_{\ast }\left(\begin{array}{cc}D& 0\\ 0& 0\end{array}\right)P^T\text{,}\end{array}$$

where $D$ is a $k\times k$ diagonal matrix with nonzero diagonal elements, $k$ being the rank of $R$ and $w^{\frac{1}{2}}X$.

This gives the solution

$$\begin{array}{c}\hat{\beta }=P_1{D}^{-1}\left(\begin{array}{cc}{Q}_{\ast }& 0\\ 0& I\end{array}\right)Q^T{w}^{\frac{1}{2}}z\end{array}$$

$P_1$ being the first $k$ columns of $P$, i.e., $P=\left(P_1{P}_0\right)$.

The iterations are continued until there is only a small change in the residual sum of squares.

The initial values for the algorithm are obtained by taking

$$\hat{\eta }=g\left(y\right)\text{.}$$

The fit of the model can be assessed by examining and testing the residual sum of squares, in particular comparing the difference in residual sums of squares between nested models, i.e., when one model is a sub-model of the other.

Let ${RSS}_f$ be the residual sum of squares for the full model with degrees of freedom ${\nu }_f$ and let ${RSS}_s$ be the residual sum of squares for the sub-model with degrees of freedom ${\nu }_s$ then:

$$F=\frac{({RSS}_s-{RSS}_f)/({\nu }_s-{\nu }_f)}{{RSS}_f/{\nu }_f}\text{,}$$

has, approximately, an $F$-distribution with (${\nu }_s-{\nu }_f$), ${\nu }_f$ degrees of freedom.

The parameter estimates, $\hat{\beta }$, are asymptotically Normally distributed with variance-covariance matrix:

$C=R^{-1}{R^{-1}}^T{\sigma }^2$ in the full rank case,

otherwise $C=P_1{D}^{-2}{P}_1^T{\sigma }^2$

The residuals and influence statistics can also be examined.

The estimated linear predictor $\hat{\eta }=X\hat{\beta }$, can be written as $H{w}^{\frac{1}{2}}z$ for an $n\times n$ matrix $H$. The $i$th diagonal elements of $H$, $h_i$, give a measure of the influence of the $i$th values of the independent variables on the fitted regression model. These are sometimes known as leverages.

The fitted values are given by $\hat{\mu }=g^{-1}\left(\hat{\eta }\right)$.

glm_normal also computes the residuals, $r$:

$$r_i=y_i-{\hat{\mu }}_i\text{.}$$

An option allows prior weights ${\omega }_i$ to be used; this gives a model with:

$${\sigma }_i^2=\frac{{\sigma }^2}{{\omega }_i}\text{.}$$

In many linear regression models the first term is taken as a mean term or an intercept, i.e., $x_{\text{i},1}=1$, for $\text{i}=1,2,\dots ,n$; this is provided as an option.

Often only some of the possible independent variables are included in a model, the facility to select variables to be included in the model is provided.

If part of the linear predictor can be represented by a variable with a known coefficient, then this can be included in the model by using an offset, $o$:

$$\eta =o+\sum {\beta }_j{x}_j\text{.}$$

If the model is not of full rank the solution given will be only one of the possible solutions. Other estimates may be obtained by applying constraints to the parameters. These solutions can be obtained by using glm_constrain() after using glm_normal. Only certain linear combinations of the parameters will have unique estimates; these are known as estimable functions and can be estimated and tested using glm_estfunc().

Details of the SVD are made available, in the form of the matrix $P^{\ast }$:

$$\begin{array}{c}{P}^{\ast }=\left(\begin{array}{c}{D}^{-1}{P}_1^T\\ P_0^T\end{array}\right)\text{.}\end{array}$$

References

Cook, R D and Weisberg, S, 1982, Residuals and Influence in Regression, Chapman and Hall

McCullagh, P and Nelder, J A, 1983, Generalized Linear Models, Chapman and Hall

See also

naginterfaces.library.examples.correg.glm_normal_ex.main()