NAG Library Function Document
nag_regsn_mult_linear (g02dac)
1 Purpose
nag_regsn_mult_linear (g02dac) performs a general multiple linear regression when the independent variables may be linearly dependent. Parameter estimates, standard errors, residuals and influence statistics are computed. nag_regsn_mult_linear (g02dac) may be used to perform a weighted regression.
2 Specification
#include <nag.h> |
#include <nagg02.h> |
void |
nag_regsn_mult_linear (Nag_IncludeMean mean,
Integer n,
const double x[],
Integer tdx,
Integer m,
const Integer sx[],
Integer ip,
const double y[],
const double wt[],
double *rss,
double *df,
double b[],
double se[],
double cov[],
double res[],
double h[],
double q[],
Integer tdq,
Nag_Boolean *svd,
Integer *rank,
double p[],
double tol,
double com_ar[],
NagError *fail) |
|
3 Description
The general linear regression model is defined by
where
- is a vector of observations on the dependent variable,
- is an by matrix of the independent variables of column rank ,
- is a vector of length of unknown arguments, and
- is a vector of length of unknown random errors such that , where is a known diagonal matrix.
Note: the independent variables may be selected from a set of potential independent variables.
If , the identity matrix, then least squares estimation is used.
If , then for a given weight matrix , weighted least squares estimation is used.
The least squares estimates of the arguments minimize while the weighted least squares estimates minimize .
nag_regsn_mult_linear (g02dac) finds a
decomposition of
(or
in the weighted case), i.e.,
where
and
is a
by
upper triangular matrix and
is an
by
orthogonal matrix.
If
is of full rank, then
is the solution to
where
(or
) and
is the first
elements of
.
If
is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of
,
where
is a
by
diagonal matrix with nonzero diagonal elements,
being the rank of
and
and
are
by
orthogonal matrices. This gives the solution
being the first
columns of
, i.e.,
and
being the first
columns of
.
Details of the SVD are made available, in the form of the 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 using
nag_regsn_mult_linear_tran_model (g02dkc) after using nag_regsn_mult_linear (g02dac). Only certain linear combinations of the arguments will have unique estimates; these are known as estimable functions.
The fit of the model can be examined by considering the residuals, , where are the fitted values. The fitted values can be written as for an by matrix . The th diagonal element of , , gives a measure of the influence of the th value of the independent variables on the fitted regression model. The values are sometimes known as leverages. Both and are provided by nag_regsn_mult_linear (g02dac).
The output of nag_regsn_mult_linear (g02dac) also includes , the residual sum of squares and associated degrees of freedom, , the standard errors of the parameter estimates and the variance-covariance matrix of the parameter estimates.
In many linear regression models the first term is taken as a mean term or an intercept, i.e., , for . This is provided as an option. Also note that not all the potential independent variables need to be included in a model; a facility to select variables to be included in the model is provided.
Details of the
decomposition and, if used, the SVD, are made available. These allow the regression to be updated by adding or deleting an observation using
nag_regsn_mult_linear_addrem_obs (g02dcc), adding or deleting a variable using
nag_regsn_mult_linear_add_var (g02dec) and
nag_regsn_mult_linear_delete_var (g02dfc) or estimating and testing an estimable function using
nag_regsn_mult_linear_est_func (g02dnc).
4 References
Cook R D and Weisberg S (1982) Residuals and Influence in Regression Chapman and Hall
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
5 Arguments
- 1:
mean – Nag_IncludeMeanInput
On entry: indicates if a mean term is to be included.
- A mean term, (intercept), will be included in the model.
- The model will pass through the origin, zero point.
Constraint:
or .
- 2:
n – IntegerInput
On entry: the number of observations, .
Constraint:
.
- 3:
x[] – const doubleInput
-
On entry: must contain the th observation for the th potential independent variable, for and .
- 4:
tdx – IntegerInput
-
On entry: the stride separating matrix column elements in the array
x.
Constraint:
.
- 5:
m – IntegerInput
On entry: the total number of independent variables in the dataset, .
Constraint:
.
- 6:
sx[m] – const IntegerInput
-
On entry: indicates which of the potential independent variables are to be included in the model. If
, then the variable contained in the corresponding column of
x is included in the regression model.
Constraints:
- , for ;
- if , then exactly values of sx must be ;
- if , then exactly ip values of sx must be .
- 7:
ip – IntegerInput
On entry: the number of independent variables in the model, including the mean or intercept if present.
Constraints:
- if , ;
- if , .
- 8:
y[n] – const doubleInput
-
On entry: observations on the dependent variable, .
- 9:
wt[n] – const doubleInput
-
On entry: optionally, the weights to be used in the weighted regression.
If
, then the
th observation is not included in the model, in which case the effective number of observations is the number of observations with nonzero weights. The values of
res and
h will be set to zero for observations with zero weights.
If weights are not provided then
wt must be set to
NULL and the effective number of observations is
n.
Constraint:
if , , for .
-
On exit: the residual sum of squares for the regression.
- 11:
df – double *Output
-
On exit: the degrees of freedom associated with the residual sum of squares.
- 12:
b[ip] – doubleOutput
-
On exit:
, for
, contain the least squares estimates of the arguments of the regression model,
.
If
, then
will contain the estimate of the mean argument and
will contain the coefficient of the variable contained in column
of
x, where
is the
th positive value in the array
sx.
If
, then
will contain the coefficient of the variable contained in column
of
x, where
is the
th positive value in the array
sx.
- 13:
se[ip] – doubleOutput
-
On exit:
, for
, contains the standard errors of the
ip parameter estimates given in
b.
- 14:
cov[] – doubleOutput
-
On exit: the first
elements of
cov contain 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
and the parameter estimate given in
,
, is stored in
, for
and
.
- 15:
res[n] – doubleOutput
-
On exit: the (weighted) residuals, .
- 16:
h[n] – doubleOutput
-
On exit: the diagonal elements of , , the leverages.
- 17:
q[] – doubleOutput
-
Note: the th element of the matrix is stored in .
On exit: the results of the
decomposition: the first column of
q contains
, the upper triangular part of columns 2 to
contain the
matrix, the strictly lower triangular part of columns 2 to
contain details of the
matrix.
- 18:
tdq – IntegerInput
-
On entry: the stride separating matrix column elements in the array
q.
Constraint:
.
- 19:
svd – Nag_Boolean *Output
-
On exit: if a singular value decomposition has been performed then
svd will be Nag_TRUE, otherwise
svd will be Nag_FALSE.
- 20:
rank – Integer *Output
-
On exit: the rank of the independent variables.
If , .
If
,
rank is an estimate of the rank of the independent variables.
rank is calculated as the number of singular values greater than
tol (largest singular value). It is possible for the SVD to be carried out but
rank to be returned as
ip.
- 21:
p[] – doubleOutput
-
On exit: details of the
decomposition and SVD if used.
If
, only the first
ip elements of
p are used, these will contain details of the Householder vector in the
decomposition (see
Sections 2.2.1 and
3.3.6 in the f08 Chapter Introduction).
If
, the first
ip elements of
p will
contain details of the Householder vector in the
decomposition and the next
ip elements of
p contain singular values. The following
ip by
ip elements contain the matrix
stored by rows.
- 22:
tol – doubleInput
-
On entry: the value of
tol is used to decide 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
, then the singular value decomposition will never be used, this may cause run time errors or inaccurate results if the independent variables are not of full rank.
Suggested value:
.
Constraint:
.
- 23:
com_ar[] – doubleOutput
-
Note: the dimension,
dim, of the array
com_ar
must be at least
.
On exit: if
,
com_ar contains information which is needed by
nag_regsn_mult_linear_newyvar (g02dgc).
- 24:
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, while . These arguments must satisfy .
On entry, while . These arguments must satisfy .
On entry, while . These arguments must satisfy .
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BAD_PARAM
-
On entry, argument
mean had an illegal value.
- NE_BAD_SX_OR_IP
-
Either a value of
sx is
, or
ip is incompatible with
mean and
sx, or
the effective number of observations.
- NE_INT_ARG_LT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, must not be less than 0: .
- NE_REAL_ARG_LT
-
On entry,
tol must not be less than 0.0:
.
On entry, must not be less than 0.0: .
- NE_SVD_NOT_CONV
-
The singular value decomposition has failed to converge.
- NE_ZERO_DOF_RESID
-
The degrees of freedom for the residuals are zero, i.e., the designated number of arguments the effective number of observations. In this case the parameter estimates will be returned along with the diagonal elements of , but neither standard errors nor the variance-covariance matrix will be calculated.
7 Accuracy
The accuracy of this function is closely related to the accuracy of the decomposition.
8 Parallelism and Performance
Not applicable.
Function
nag_regsn_std_resid_influence (g02fac) can be used to compute standardized residuals and further measures of influence. nag_regsn_mult_linear (g02dac) requires, in particular, the results stored in
res and
h.
10 Example
For this function two examples are presented. There is a single example program for nag_regsn_mult_linear (g02dac), with a main program and the code to solve the two example problems is given in the functions ex1 and ex2.
Example 1 (ex1)
Data from an experiment with four treatments and three observations per treatment are read in. The treatments are represented by dummy variables. An unweighted model is fitted with a mean included in the model.
Example 2 (ex2)
This example program uses nag_regsn_mult_linear (g02dac) to find the coefficient of the
degree polynomial
that fits the data,
to
, in a least squares sense.
In this example nag_regsn_mult_linear (g02dac) is called with both and . The polynomial degree, the number of data points and the tolerance can be modified using the example data file.
10.1 Program Text
Program Text (g02dace.c)
10.2 Program Data
Program Data (g02dace.d)
10.3 Program Results
Program Results (g02dace.r)