The function may be called by the names: g02hdc, nag_correg_robustm_user or nag_robust_m_regsn_user_fn.
3Description
For the linear regression model
where
is a vector of length of the dependent variable,
is an 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
(1)
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
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).
4References
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
5Arguments
1: – Nag_OrderTypeInput
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 userExternal 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.
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 userExternal Function
psi must return the value of the weight function for a given value of its argument.
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.
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).
On entry: determines the type of regression to be performed.
Huber type regression.
Mallows type regression.
Schweppe type regression.
Constraint:
, or .
7: – Nag_SigmaEstInput
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: – IntegerInput
On entry: , the number of observations.
Constraint:
.
9: – IntegerInput
On entry: , the number of independent variables.
Constraint:
.
10: – doubleInput/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: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
if ,
;
if , .
12: – doubleInput/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: – doubleInput/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.
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: – doubleOutput
On exit: the residuals from the model evaluated at final value of theta, i.e., rs contains the vector .
18: – doubleInput
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.
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: – IntegerInput
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: – IntegerInput
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).
6Error 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 .
Background information to multithreading can be found in the Multithreading documentation.
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.
9Further Comments
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.
Having input , and the weights, a Schweppe type regression is performed using Huber's function. The function BETCAL calculates the appropriate value of .