NAG CL Interface
g02qgc (quantile_linreg)
Note: this function uses optional parameters to define choices in the problem specification and in the details of the algorithm. If you wish to use default
settings for all of the optional parameters, you need only read Sections 1 to 10 of this document. If, however, you wish to reset some or all of the settings please refer to Section 11 for a detailed description of the algorithm, to Section 12 for a detailed description of the specification of the optional parameters and to Section 13 for a detailed description of the monitoring information produced by the function.
1
Purpose
g02qgc performs a multiple linear quantile regression. Parameter estimates and, if required, confidence limits, covariance matrices and residuals are calculated.
g02qgc may be used to perform a weighted quantile regression. A simplified interface for
g02qgc is provided by
g02qfc.
2
Specification
void |
g02qgc (Nag_OrderType order,
Nag_IncludeIntercept intcpt,
Integer n,
Integer m,
const double dat[],
Integer pddat,
const Integer isx[],
Integer ip,
const double y[],
const double wt[],
Integer ntau,
const double tau[],
double *df,
double b[],
double bl[],
double bu[],
double ch[],
double res[],
const Integer iopts[],
const double opts[],
Integer state[],
Integer info[],
NagError *fail) |
|
The function may be called by the names: g02qgc, nag_correg_quantile_linreg or nag_regsn_quant_linear.
3
Description
Given a vector of
observed values,
, an
design matrix
, a column vector,
, of length
holding the
th row of
and a quantile
,
g02qgc estimates the
-element vector
as the solution to
where
is the piecewise linear loss function
, and
is an indicator function taking the value
if
and
otherwise. Weights can be incorporated by replacing
and
with
and
respectively, where
is an
diagonal matrix. Observations with zero weights can either be included or excluded from the analysis; this is in contrast to least squares regression where such observations do not contribute to the objective function and are therefore always dropped.
g02qgc uses the interior point algorithm of
Portnoy and Koenker (1997), described briefly in
Section 11, to obtain the parameter estimates
, for a given value of
.
Under the assumption of Normally distributed errors,
Koenker (2005) shows that the limiting covariance matrix of
has the form
where
and
is a function of
, as described below. Given an estimate of the covariance matrix,
, lower (
) and upper (
) limits for an
confidence interval can be calculated for each of the
parameters, via
where
is the
percentile of the Student's
distribution with
degrees of freedom, where
is the rank of the cross-product matrix
.
Four methods for estimating the covariance matrix,
, are available:
-
(i)Independent, identically distributed (IID) errors
Under an assumption of IID errors the asymptotic relationship for
simplifies to
where
is the sparsity function.
g02qgc estimates
from the residuals,
and a bandwidth
.
-
(ii)Powell Sandwich
Powell (1991) suggested estimating the matrix
by a kernel estimator of the form
where
is a kernel function and
satisfies
and
. When the Powell method is chosen,
g02qgc uses a Gaussian kernel (i.e.,
) and sets
where
is a bandwidth,
and
are, respectively, the standard deviation and the
and
quantiles for the residuals,
.
-
(iii)Hendricks–Koenker Sandwich
Koenker (2005) suggested estimating the matrix
using
where
is a bandwidth and
denotes the parameter estimates obtained from a quantile regression using the
th quantile. Similarly with
.
-
(iv)Bootstrap
The last method uses bootstrapping to either estimate a covariance matrix or obtain confidence intervals for the parameter estimates directly. This method therefore does not assume Normally distributed errors. Samples of size are taken from the paired data (i.e., the independent and dependent variables are sampled together). A quantile regression is then fitted to each sample resulting in a series of bootstrap estimates for the model parameters, . A covariance matrix can then be calculated directly from this series of values. Alternatively, confidence limits, and , can be obtained directly from the and sample quantiles of the bootstrap estimates.
Further details of the algorithms used to calculate the covariance matrices can be found in
Section 11.
All three asymptotic estimates of the covariance matrix require a bandwidth,
. Two alternative methods for determining this are provided:
-
(i)Sheather–Hall
for a user-supplied value ,
-
(ii)Bofinger
g02qgc allows optional parameters to be supplied via the
iopts and
opts arrays (see
Section 12 for details of the available options).
If the default values for these optional parameters are sufficient then
iopts and
opts can be set to
NULL, otherwise prior
to calling
g02qgc the optional parameter arrays,
must be initialized by calling
g02zkc with
optstr set to
. If bootstrap confidence limits are required (
) then one of the random number initialization functions
g05kfc (for a repeatable analysis) or
g05kgc (for an unrepeatable analysis) must also have been previously called.
4
References
Koenker R (2005) Quantile Regression Econometric Society Monographs, Cambridge University Press, New York
Mehrotra S (1992) On the implementation of a primal-dual interior point method SIAM J. Optim. 2 575–601
Nocedal J and Wright S J (2006) Numerical Optimization (2nd Edition) Springer Series in Operations Research, Springer, New York
Portnoy S and Koenker R (1997) The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute error estimators Statistical Science 4 279–300
Powell J L (1991) Estimation of monotonic regression models under quantile restrictions Nonparametric and Semiparametric Methods in Econometrics Cambridge University Press, Cambridge
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:
– Nag_IncludeIntercept
Input
-
On entry: indicates whether an intercept will be included in the model. The intercept is included by adding a column of ones as the first column in the design matrix,
.
- An intercept will be included in the model.
- An intercept will not be included in the model.
Constraint:
or .
-
3:
– Integer
Input
-
On entry: the total number of observations in the dataset. If no weights are supplied, or no zero weights are supplied or observations with zero weights are included in the model then . Otherwise the number of observations with zero weights.
Constraint:
.
-
4:
– Integer
Input
-
On entry: , the total number of variates in the dataset.
Constraint:
.
-
5:
– const double
Input
-
Note: where
appears in this document, it refers to the array element
- when ;
- when .
On entry: the
th value for the
th variate, for
and
, must be supplied in
The design matrix
is constructed from
dat,
isx and
intcpt.
-
6:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
dat.
Constraints:
- if , ;
- otherwise .
-
7:
– const Integer
Input
-
On entry: indicates which independent variables are to be included in the model.
- The th variate, supplied in dat, is not included in the regression model.
- The th variate, supplied in dat, is included in the regression model.
Constraints:
- or , for ;
- if , exactly values of isx must be set to ;
- if , exactly ip values of isx must be set to .
-
8:
– Integer
Input
-
On entry:
, the number of independent variables in the model, including the intercept, see
intcpt, if present.
Constraints:
- ;
- if , ;
- if , .
-
9:
– const double
Input
-
On entry: , the observations on the dependent variable.
-
10:
– const double
Input
-
Note: the dimension,
dim, of the array
wt
must be at least
- , when ;
- otherwise is not referenced and may be NULL.
On entry: optionally, the diagonal elements of the weight matrix
.
If weights are not provided then
wt must be set to
NULL.
When
- If , 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. If , the values of res will be set to zero for observations with zero weights.
- All observations are included in the model and the effective number of observations is n, i.e., .
Constraints:
- the effective number of observations ;
- , for all .
-
11:
– Integer
Input
-
On entry: the number of quantiles of interest.
Constraint:
.
-
12:
– const double
Input
-
On entry: the vector of quantiles of interest. A separate model is fitted to each quantile.
Constraint:
where
is the
machine precision returned by
X02AJC, for
.
-
13:
– double *
Output
-
On exit: the degrees of freedom given by , where is the effective number of observations and is the rank of the cross-product matrix .
-
14:
– double
Input/Output
-
Note: where appears in this document, it refers to the array element
.
On entry: if
,
must hold an initial estimates for
, for
and
. If
,
b need not be set.
On exit:
, for
, contains the estimates of the parameters of the regression model,
, estimated for
.
If
,
will contain the estimate corresponding to the intercept and
will contain the coefficient of the
th variate contained in
dat, where
is the
th nonzero value in the array
isx.
If
,
will contain the coefficient of the
th variate contained in
dat, where
is the
th nonzero value in the array
isx.
-
15:
– double
Output
-
Note: the dimension,
dim, of the array
bl
must be at least
when
.
where appears in this document, it refers to the array element
.
On exit: if
,
contains the lower limit of an
confidence interval for
, for
and
.
If
,
bl is not referenced and can be set to
NULL.
The method used for calculating the interval is controlled by the optional parameters and . The size of the interval, , is controlled by the optional parameter .
-
16:
– double
Output
-
Note: the dimension,
dim, of the array
bu
must be at least
when
.
where appears in this document, it refers to the array element
.
On exit: if
,
contains the upper limit of an
confidence interval for
, for
and
.
If
,
bu is not referenced and can be set to
NULL.
The method used for calculating the interval is controlled by the optional parameters and . The size of the interval, is controlled by the optional parameter .
-
17:
– double
Output
-
Note: the dimension,
dim, of the array
ch
must be at least
- if
and , ;
- if
, or and , .
where appears in this document, it refers to the array element .
On exit: depending on the supplied optional parameters,
ch will either not be referenced, hold an estimate of the upper triangular part of the covariance matrix,
, or an estimate of the upper triangular parts of
and
.
If
or
,
ch is not referenced.
If
or
and
,
ch is not referenced.
Otherwise, for
and
:
- If , holds an estimate of the covariance between and .
- If , holds an estimate of the th element of and holds an estimate of the th element of , for .
The method used for calculating and is controlled by the optional parameter .
In cases where
ch is not going to be referenced it can be set to
NULL.
-
18:
– double
Output
-
Note: the th element of the matrix is stored in .
On exit: if
,
holds the (weighted) residuals,
, for
, for
and
.
If
and
, the value of
res will be set to zero for observations with zero weights.
If
,
res is not referenced and can be set to
NULL.
-
19:
– const Integer
Communication Array
-
Note: the dimension,
, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
iopts in the previous call to
g02zkc.
On entry: if the default values of the optional parameters are sufficient,
iopts can be set to
NULL, otherwise the optional parameter array, as initialized by a call to
g02zkc must be supplied.
-
20:
– const double
Communication Array
-
Note: the dimension,
, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
opts in the previous call to
g02zkc.
On entry: if the default values of the optional parameters are sufficient,
opts can be set to
NULL, otherwise the optional parameter array, as initialized by a call to
g02zkc must be supplied.
-
21:
– Integer
Communication Array
-
Note: the dimension,
, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
state in the previous call to
nag_rand_init_repeatable (g05kfc) or
nag_rand_init_nonrepeatable (g05kgc).
If
,
state contains information about the selected random number generator. Otherwise
state is not referenced and can be set to
NULL.
-
22:
– Integer
Output
-
On exit:
holds additional information concerning the model fitting and confidence limit calculations when
.
Code | Warning |
| Model fitted and confidence limits (if requested) calculated successfully |
| The function did not converge. The returned values are based on the estimate at the last iteration. Try increasing whilst calculating the parameter estimates or relaxing the definition of convergence by increasing . |
| A singular matrix was encountered during the optimization. The model was not fitted for this value of . |
| Some truncation occurred whilst calculating the confidence limits for this value of . See Section 11 for details. The returned upper and lower limits may be narrower than specified. |
| The function did not converge whilst calculating the confidence limits. The returned limits are based on the estimate at the last iteration. Try increasing . |
| Confidence limits for this value of could not be calculated. The returned upper and lower limits are set to a large positive and large negative value respectively as defined by the optional parameter . |
It is possible for multiple warnings to be applicable to a single model. In these cases the value returned in
info is the sum of the corresponding individual nonzero warning codes.
-
23:
– 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_ARRAY_SIZE
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INITIALIZATION
-
On entry, either the option arrays have not been initialized or they have been corrupted.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
- NE_INT_ARRAY
-
On entry, .
Constraint: or , for all .
- 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_INVALID_STATE
-
On entry,
state vector has been corrupted or not initialized.
- NE_IP_INCOMP_SX
-
On entry,
ip is not consistent with
isx or
intcpt:
,
.
- NE_NEG_WEIGHT
-
On entry, .
Constraint: , for all .
- 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_OBSERVATIONS
-
On entry, .
Constraint: .
- NE_REAL_ARRAY
-
On entry,
.
Constraint:
where
is the
machine precision returned by
X02AJC, for all
ntau.
- NW_POTENTIAL_PROBLEM
-
A potential problem occurred whilst fitting the model(s).
Additional information has been returned in
info.
7
Accuracy
Not applicable.
8
Parallelism and Performance
g02qgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02qgc 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.
g02qgc allocates internally approximately the following elements of double storage:
. If then a further
elements are required, and this increases by
if . Where possible, any user-supplied output arrays are used as workspace and so the amount actually allocated may be less. If , , and an internal copy of the input data is avoided and the amount of locally allocated memory is reduced by .
10
Example
A quantile regression model is fitted to Engels 1857 study of household expenditure on food. The model regresses the dependent variable, household food expenditure, against two explanatory variables, a column of ones and household income. The model is fit for five different values of and the covariance matrix is estimated assuming Normal IID errors. Both the covariance matrix and the residuals are returned.
10.1
Program Text
10.2
Program Data
10.3
Program Results
11
Algorithmic Details
By the addition of slack variables the minimization
(1) can be reformulated into the linear programming problem
and its associated dual
where
is a vector of
s. Setting
gives the equivalent formulation
The algorithm introduced by
Portnoy and Koenker (1997) and used by
g02qgc, uses the primal-dual formulation expressed in equations
(2) and
(4) along with a logarithmic barrier function to obtain estimates for
. The algorithm is based on the predictor-corrector algorithm of
Mehrotra (1992) and further details can be obtained from
Portnoy and Koenker (1997) and
Koenker (2005). A good description of linear programming, interior point algorithms, barrier functions and Mehrotra's predictor-corrector algorithm can be found in
Nocedal and Wright (2006).
11.1
Interior Point Algorithm
In this section a brief description of the interior point algorithm used to estimate the model parameters is presented. It should be noted that there are some differences in the equations given here – particularly
(7) and
(9) – compared to those given in
Koenker (2005) and
Portnoy and Koenker (1997).
11.1.1
Central path
Rather than optimize
(4) directly, an additional slack variable
is added and the constraint
is replaced with
, for
.
The positivity constraint on
and
is handled using the logarithmic barrier function
The primal-dual form of the problem is used giving the Lagrangian
whose central path is described by the following first order conditions
where
denotes the diagonal matrix with diagonal elements given by
, similarly with
and
. By enforcing the inequalities on
and
strictly, i.e.,
and
for all
we ensure that
and
are positive definite diagonal matrices and hence
and
exist.
Rather than applying Newton's method to the system of equations given in
(5) to obtain the step directions
and
, Mehrotra substituted the steps directly into
(5) giving the augmented system of equations
where
and
denote the diagonal matrices with diagonal elements given by
and
respectively.
11.1.2
Affine scaling step
The affine scaling step is constructed by setting
in
(5) and applying Newton's method to obtain an intermediate set of step directions
where
.
Initial step sizes for the primal (
) and dual (
) parameters are constructed as
where
is a user-supplied scaling factor. If
then the nonlinearity adjustment, described in
Section 11.1.3, is not made and the model parameters are updated using the current step size and directions.
11.1.3
Nonlinearity Adjustment
In the nonlinearity adjustment step a new estimate of
is obtained by letting
and estimating
as
This estimate, along with the nonlinear terms (
,
,
and
) from
(6) are calculated using the values of
and
obtained from the affine scaling step.
Given an updated estimate for
and the nonlinear terms the system of equations
are solved and updated values for
and
calculated.
11.1.4
Update and convergence
At each iteration the model parameters
are updated using step directions,
and step lengths
.
Convergence is assessed using the duality gap, that is, the differences between the objective function in the primal and dual formulations. For any feasible point
the duality gap can be calculated from equations
(2) and
(3) as
and the optimization terminates if the duality gap is smaller than the tolerance supplied in the optional parameter
.
11.1.5
Additional information
Initial values are required for the parameters
and
. If you have not supplied them, initial values for
are calculated from a least squares regression of
on
. This regression is carried out by first constructing the cross-product matrix
and then using a pivoted
decomposition as performed by
f08bfc. In addition, if the cross-product matrix is not of full rank, a rank reduction is carried out and, rather than using the full design matrix,
, a matrix formed from the first
-rank columns of
is used instead, where
is the pivot matrix used during the
decomposition. Parameter estimates, confidence intervals and the rows and columns of the matrices returned in the argument
ch (if any) are set to zero for variables dropped during the rank-reduction. The rank reduction step is performed irrespective of whether initial values are supplied by the user.
Once initial values have been obtained for , the initial values for and are calculated from the residuals. If then a value of is used instead, where is supplied in the optional parameter . The initial values for the and are always set to and respectively.
The solution for
in both
(7) and
(9) is obtained using a Bunch–Kaufman decomposition, as implemented in
f07mdc.
11.2
Calculation of Covariance Matrix
g02qgc supplies four methods to calculate the covariance matrices associated with the parameter estimates for
. This section gives some additional detail on three of the algorithms, the fourth, (which uses bootstrapping), is described in
Section 3.
-
(i)Independent, identically distributed (IID) errors
When assuming IID errors, the covariance matrices depend on the sparsity,
, which
g02qgc estimates as follows:
-
(a)Let denote the residuals from the original quantile regression, that is
.
-
(b)Drop any residual where is less than , supplied in the optional parameter .
-
(c)Sort and relabel the remaining residuals in ascending order, by absolute value, so that
.
-
(d)Select the first values where , for some bandwidth .
-
(e)Sort and relabel these residuals again, so that
and regress them against a design matrix with two columns () and rows given by
using quantile regression with .
-
(f)Use the resulting estimate of the slope as an estimate of the sparsity.
-
(ii)Powell Sandwich
When using the Powell Sandwich to estimate the matrix
, the quantity
is calculated. Dependent on the value of
and the method used to calculate the bandwidth (
), it is possible for the quantities
to be too large or small, compared to
machine precision (
). More specifically, when
, or
, a warning flag is raised in
info, the value is truncated to
or
respectively and the covariance matrix calculated as usual.
-
(iii)
Hendricks–Koenker Sandwich
The Hendricks–Koenker Sandwich requires the calculation of the quantity
.
As with the Powell Sandwich, in cases where
, or
, a warning flag is raised in
info, the value truncated to
or
respectively and the covariance matrix calculated as usual.
In addition, it is required that , in this method. Hence, instead of using
in the calculation of ,
is used instead, where is supplied in the optional parameter .
12
Optional Parameters
Several optional parameters in
g02qgc control aspects of the optimization algorithm, methodology used, logic or output. Their values are contained in the arrays
iopts and
opts; these must be initialized before calling
g02qgc by first calling
g02zkc with
optstr set to
.
Each optional parameter has an associated default value; to set any of them to a non-default value, use
g02zkc. The current value of an optional parameter can be queried using
g02zlc.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in
Section 12.1.
12.1
Description of the Optional Parameters
For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
- the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted);
- a parameter value,
where the letters , and denote options that take character, integer and real values respectively;
- the default value, where the symbol is a generic notation for machine precision (see X02AJC).
Keywords and character values are case and white space insensitive.
Band Width Alpha | | Default |
A multiplier used to construct the parameter
used when calculating the Sheather–Hall bandwidth (see
Section 3), with
. Here,
is the
.
Constraint:
.
Band Width Method | | Default |
The method used to calculate the bandwidth used in the calculation of the asymptotic covariance matrix
and
if
,
or
(see
Section 3).
Constraint:
or .
This parameter should be set to something larger than the biggest value supplied in
dat and
y.
Constraint:
.
Bootstrap Interval Method | | Default |
If
,
controls how the confidence intervals are calculated from the bootstrap estimates.
- intervals are calculated. That is, the covariance matrix, is calculated from the bootstrap estimates and the limits calculated as where is the percentage point from a Student's distribution on degrees of freedom, is the effective number of observations and is given by the optional parameter .
- Quantile intervals are calculated. That is, the upper and lower limits are taken as the and quantiles of the bootstrap estimates, as calculated using g01amc.
Constraint:
or .
Bootstrap Iterations | | Default |
The number of bootstrap samples used to calculate the confidence limits and covariance matrix (if requested) when .
Constraint:
.
Bootstrap Monitoring | | Default |
If and , the parameter estimates for each of the bootstrap samples are displayed. This information is sent to the unit number specified by .
Constraint:
or .
Calculate Initial Values | | Default |
If
then the initial values for the regression parameters,
, are calculated from the data. Otherwise they must be supplied in
b.
Constraint:
or .
This special keyword is used to reset all optional parameters to their default values.
Drop Zero Weights | | Default |
If a weighted regression is being performed and then observations with zero weight are dropped from the analysis. Otherwise such observations are included.
Constraint:
or .
, the tolerance used when calculating the covariance matrix and the initial values for
and
. For additional details see
Section 11.2 and
Section 11.1.5 respectively.
Constraint:
.
Interval Method | | Default |
The value of
controls whether confidence limits are returned in
bl and
bu and how these limits are calculated. This parameter also controls how the matrices returned in
ch are calculated.
- No limits are calculated and bl, bu and ch are not referenced.
- The Powell Sandwich method with a Gaussian kernel is used.
- The Hendricks–Koenker Sandwich is used.
- The errors are assumed to be identical, and independently distributed.
- A bootstrap method is used, where sampling is done on the pair . The number of bootstrap samples is controlled by the parameter and the type of interval constructed from the bootstrap samples is controlled by .
Constraint:
, , , or .
Iteration Limit | | Default |
The maximum number of iterations to be performed by the interior point optimization algorithm.
Constraint:
.
Matrix Returned | | Default |
The value of
controls the type of matrices returned in
ch. If
, this parameter is ignored and
ch is not referenced. Otherwise:
- No matrices are returned and ch is not referenced.
- The covariance matrices are returned.
- If or , the matrices and are returned. Otherwise no matrices are returned and ch is not referenced.
The matrices returned are calculated as described in
Section 3, with the algorithm used specified by
. In the case of
the covariance matrix is calculated directly from the bootstrap estimates.
Constraint:
, or .
If then the duality gap is displayed at each iteration of the interior point optimization algorithm. In addition, the final estimates for are also displayed.
The monitoring information is sent to the unit number specified by .
Constraint:
or .
QR Tolerance | | Default |
The tolerance used to calculate the rank, , of the cross-product matrix, . Letting be the orthogonal matrix obtained from a decomposition of , then the rank is calculated by comparing with .
If the cross-product matrix is rank deficient, the parameter estimates for the
columns with the smallest values of
are set to zero, along with the corresponding entries in
bl,
bu and
ch, if returned. This is equivalent to dropping these variables from the model. Details on the
decomposition used can be found in
f08bfc.
Constraint:
.
Return Residuals | | Default |
If
, the residuals are returned in
res. Otherwise
res is not referenced.
Constraint:
or .
The scaling factor used when calculating the affine scaling step size (see equation
(8)).
Constraint:
.
Significance Level | | Default |
, the size of the confidence interval whose limits are returned in
bl and
bu.
Constraint:
.
Convergence tolerance. The optimization is deemed to have converged if the duality gap is less than
(see
Section 11.1.4).
Constraint:
.
Unit Number | |
Output sent to stdout
|
The unit number to which any monitoring information is sent. See
x04acc for details on how to assign a file to a unit number. If no unit number is specified then any monitoring information will be sent to standard output (
stdout).
Constraint:
.
13
Description of Monitoring Information
See the description of the optional parameter .