NAG CL Interface
g02hdc (robustm_user)
1
Purpose
g02hdc performs bounded influence regression (-estimates) using an iterative weighted least squares algorithm.
2
Specification
void |
g02hdc (Nag_OrderType order,
double |
(*chi)(double t,
Nag_Comm *comm),
|
|
double |
(*psi)(double t,
Nag_Comm *comm),
|
|
double psip0,
double beta,
Nag_RegType regtype,
Nag_SigmaEst sigma_est,
Integer n,
Integer m,
double x[],
Integer pdx,
double y[],
double wgt[],
double theta[],
Integer *k,
double *sigma,
double rs[],
double tol,
double eps,
Integer maxit,
Integer nitmon,
const char *outfile,
Integer *nit,
Nag_Comm *comm,
NagError *fail) |
|
The function may be called by the names: g02hdc, nag_correg_robustm_user or nag_robust_m_regsn_user_fn.
3
Description
For the linear regression model
where |
is a vector of length of the dependent variable, |
|
is an by matrix of independent variables of column rank , |
|
is a vector of length of unknown parameters, |
and |
is a vector of length of unknown errors with var , |
g02hdc calculates the M-estimates given by the solution,
, to the equation
where |
is the th residual, i.e., the th element of the vector , |
|
is a suitable weight function, |
|
are suitable weights such as those that can be calculated by using output from g02hbc, |
and |
may be estimated at each iteration by the median absolute deviation of the residuals |
or as the solution to
for a suitable weight function
, where
and
are constants, chosen so that the estimator of
is asymptotically unbiased if the errors,
, have a Normal distribution. Alternatively
may be held at a constant value.
The above describes the Schweppe type regression. If the
are assumed to equal
for all
, then Huber type regression is obtained. A third type, due to Mallows, replaces
(1) by
This may be obtained by use of the transformations
(see
Marazzi (1987)).
The calculation of the estimates of
can be formulated as an iterative weighted least squares problem with a diagonal weight matrix
given by
The value of
at each iteration is given by the weighted least squares regression of
on
. This is carried out by first transforming the
and
by
and then using
a least squares solver.
If
is of full column rank then an orthogonal-triangular (
) decomposition is used; if not, a singular value decomposition is used.
Observations with zero or negative weights are not included in the solution.
Note: there is no explicit provision in the function for a constant term in the regression model. However, the addition of a dummy variable whose value is for all observations will produce a value of corresponding to the usual constant term.
g02hdc is based on routines in ROBETH, see
Marazzi (1987).
4
References
Hampel F R, Ronchetti E M, Rousseeuw P J and Stahel W A (1986) Robust Statistics. The Approach Based on Influence Functions Wiley
Huber P J (1981) Robust Statistics Wiley
Marazzi A (1987) Subroutines for robust and bounded influence regression in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 2 Institut Universitaire de Médecine Sociale et Préventive, Lausanne
5
Arguments
-
1:
– Nag_OrderType
Input
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– function, supplied by the user
External Function
-
If
,
chi must return the value of the weight function
for a given value of its argument. The value of
must be non-negative.
The specification of
chi is:
double |
chi (double t,
Nag_Comm *comm)
|
|
-
1:
– double
Input
-
On entry: the argument for which
chi must be evaluated.
-
2:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
chi.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
g02hdc you may allocate memory and initialize these pointers with various quantities for use by
chi when called from
g02hdc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: chi should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
g02hdc. If your code inadvertently
does return any NaNs or infinities,
g02hdc is likely to produce unexpected results.
chi is required only if
, otherwise it can be specified as a pointer with
value.
-
3:
– function, supplied by the user
External Function
-
psi must return the value of the weight function
for a given value of its argument.
The specification of
psi is:
double |
psi (double t,
Nag_Comm *comm)
|
|
-
1:
– double
Input
-
On entry: the argument for which
psi must be evaluated.
-
2:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
psi.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
g02hdc you may allocate memory and initialize these pointers with various quantities for use by
psi when called from
g02hdc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: psi should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
g02hdc. If your code inadvertently
does return any NaNs or infinities,
g02hdc is likely to produce unexpected results.
-
4:
– double
Input
-
On entry: the value of .
-
5:
– double
Input
-
On entry: if
,
beta must specify the value of
.
For Huber and Schweppe type regressions,
is the
th percentile of the standard Normal distribution (see
g01fac). For Mallows type regression
is the solution to
where
is the standard Normal cumulative distribution function (see
s15abc).
If
,
beta must specify the value of
.
where
is the standard normal density, i.e.,
.
If
,
beta is not referenced.
Constraint:
if , .
-
6:
– Nag_RegType
Input
-
On entry: determines the type of regression to be performed.
- Huber type regression.
- Mallows type regression.
- Schweppe type regression.
Constraint:
, or .
-
7:
– Nag_SigmaEst
Input
-
On entry: determines how
is to be estimated.
- is held constant at its initial value.
- is estimated by median absolute deviation of residuals.
- is estimated using the function.
Constraint:
, or .
-
8:
– Integer
Input
-
On entry: , the number of observations.
Constraint:
.
-
9:
– Integer
Input
-
On entry: , the number of independent variables.
Constraint:
.
-
10:
– double
Input/Output
-
Note: the dimension,
dim, of the array
x
must be at least
-
when ;
-
when .
where
appears in this document, it refers to the array element
- when ;
- when .
On entry: the values of the
matrix, i.e., the independent variables.
must contain the
th element of
, for
and
.
If
, during calculations the elements of
x will be transformed as described in
Section 3. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input
x and the output
x.
On exit: unchanged, except as described above.
-
11:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
-
12:
– double
Input/Output
-
On entry: the data values of the dependent variable.
must contain the value of for the th observation, for .
If
, during calculations the elements of
y will be transformed as described in
Section 3. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input
y and the output
y.
On exit: unchanged, except as described above.
-
13:
– double
Input/Output
-
On entry: the weight for the
th observation, for
.
If
, during calculations elements of
wgt will be transformed as described in
Section 3. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input
wgt and the output
wgt.
If , the th observation is not included in the analysis.
If
,
wgt is not referenced.
On exit: unchanged, except as described above.
-
14:
– double
Input/Output
-
On entry: starting values of the parameter vector
. These may be obtained from least squares regression. Alternatively if
and
or if
and
sigma approximately equals the standard deviation of the dependent variable,
, then
, for
may provide reasonable starting values.
On exit: the M-estimate of
, for .
-
15:
– Integer *
Output
-
On exit: the column rank of the matrix .
-
16:
– double *
Input/Output
-
On entry: a starting value for the estimation of
.
sigma should be approximately the standard deviation of the residuals from the model evaluated at the value of
given by
theta on entry.
Constraint:
.
On exit: the final estimate of if or the value assigned on entry if .
-
17:
– double
Output
-
On exit: the residuals from the model evaluated at final value of
theta, i.e.,
rs contains the vector
.
-
18:
– double
Input
-
On entry: the relative precision for the final estimates. Convergence is assumed when both the relative change in the value of
sigma and the relative change in the value of each element of
theta are less than
tol.
It is advisable for
tol to be greater than
.
Constraint:
.
-
19:
– double
Input
-
On entry: a relative tolerance to be used to determine the rank of
.
If
or
,
machine precision will be used in place of
tol.
A reasonable value for
eps is
where this value is possible.
-
20:
– Integer
Input
-
On entry: the maximum number of iterations that should be used during the estimation.
A value of should be adequate for most uses.
Constraint:
.
-
21:
– Integer
Input
-
On entry: determines the amount of information that is printed on each iteration.
- No information is printed.
- On the first and every nitmon iterations the values of sigma, theta and the change in theta during the iteration are printed.
-
22:
– const char *
Input
-
On entry: a null terminated character string giving the name of the file to which results should be printed. If or an empty string then the stdout stream is used. Note that the file will be opened in the append mode.
-
23:
– Integer *
Output
-
On exit: the number of iterations that were used during the estimation.
-
24:
– Nag_Comm *
-
The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
-
25:
– 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_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CHI
-
Value given by
chi function
:
.
The value of
chi must be non-negative.
- NE_CONVERGENCE_SOL
-
Iterations to solve the weighted least squares equations failed to converge.
- NE_CONVERGENCE_THETA
-
The function has failed to converge in
maxit iterations.
- NE_FULL_RANK
-
The weighted least squares equations are not of full rank. This may be due to the
matrix not being of full rank, in which case the results will be valid. It may also occur if some of the
values become very small or zero, see
Section 9. The rank of the equations is given by
k. If the matrix just fails the test for nonsingularity then the result
NE_FULL_RANK and
is possible .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_NOT_CLOSE_FILE
-
Cannot close file .
- NE_NOT_WRITE_FILE
-
Cannot open file for writing.
- NE_REAL
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_ZERO_DF
-
Having removed cases with zero weight, the value of , i.e., no degree of freedom for error. This error will only occur if .
- NE_ZERO_VALUE
-
Estimated value of
sigma is zero.
7
Accuracy
The accuracy of the results is controlled by
tol.
8
Parallelism and Performance
g02hdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02hdc 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 function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
In cases when
it is important for the value of
sigma to be of a reasonable magnitude. Too small a value may cause too many of the winsorized residuals, i.e.,
, to be zero, which will lead to convergence problems and may trigger the
NE_FULL_RANK error.
By suitable choice of the functions
chi and
psi this function may be used for other applications of iterative weighted least squares.
For the variance-covariance matrix of
see
g02hfc.
10
Example
Having input , and the weights, a Schweppe type regression is performed using Huber's function. The function BETCAL calculates the appropriate value of .
10.1
Program Text
10.2
Program Data
10.3
Program Results