naginterfaces.library.correg.robustm_​user¶

naginterfaces.library.correg.robustm_user(psi, psip0, beta, indw, isigma, x, y, wgt, theta, sigma, chi=None, tol=5e-05, eps=5e-06, maxit=50, nitmon=0, data=None, io_manager=None)[source]

robustm_user performs bounded influence regression (-estimates) using an iterative weighted least squares algorithm.

For full information please refer to the NAG Library document for g02hd

https://www.nag.com/numeric/nl/nagdoc_29/flhtml/g02/g02hdf.html

Parameters
psicallable retval = psi(t, data=None)

must return the value of the weight function for a given value of its argument.

Parameters
tfloat

The argument for which must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of the weight function evaluated at .

psip0float

The value of .

betafloat

If , must specify the value of .

For Huber and Schweppe type regressions, is the th percentile of the standard Normal distribution (see stat.inv_cdf_normal).

For Mallows type regression is the solution to

where is the standard Normal cumulative distribution function (see specfun.cdf_normal).

If , must specify the value of .

where is the standard normal density, i.e., .

If , is not referenced.

indwint

Determines the type of regression to be performed.

Huber type regression.

Mallows type regression.

Schweppe type regression.

isigmaint

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.

xfloat, array-like, shape

The values of the matrix, i.e., the independent variables. must contain the th element of , for , for .

If , during calculations the elements of will be transformed as described in Notes.

Before exit the inverse transformation will be applied.

As a result there may be slight differences between the input and the output .

yfloat, array-like, shape

The data values of the dependent variable.

must contain the value of for the th observation, for .

If , during calculations the elements of will be transformed as described in Notes.

Before exit the inverse transformation will be applied.

As a result there may be slight differences between the input and the output .

wgtfloat, array-like, shape

The weight for the th observation, for .

If , during calculations elements of will be transformed as described in Notes.

Before exit the inverse transformation will be applied.

As a result there may be slight differences between the input and the output .

If , the th observation is not included in the analysis.

If , is not referenced.

thetafloat, array-like, shape

Starting values of the parameter vector . These may be obtained from least squares regression. Alternatively if and or if and approximately equals the standard deviation of the dependent variable, , then , for may provide reasonable starting values.

sigmafloat

A starting value for the estimation of . should be approximately the standard deviation of the residuals from the model evaluated at the value of given by on entry.

chiNone or callable retval = chi(t, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

If , must return the value of the weight function for a given value of its argument.

The value of must be non-negative.

Parameters
tfloat

The argument for which must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of the weight function evaluated at .

tolfloat, optional

The relative precision for the final estimates. Convergence is assumed when both the relative change in the value of and the relative change in the value of each element of are less than .

It is advisable for to be greater than .

epsfloat, optional

A relative tolerance to be used to determine the rank of . See linsys.real_gen_solve for further details.

If or , machine precision will be used in place of .

A reasonable value for is where this value is possible.

maxitint, optional

The maximum number of iterations that should be used during the estimation.

A value of should be adequate for most uses.

nitmonint, optional

Determines the amount of information that is printed on each iteration.

No information is printed.

On the first and every iterations the values of , and the change in during the iteration are printed.

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

dataarbitrary, optional

User-communication data for callback functions.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns
xfloat, ndarray, shape

Unchanged, except as described above.

yfloat, ndarray, shape

Unchanged, except as described above.

wgtfloat, ndarray, shape

Unchanged, except as described above.

thetafloat, ndarray, shape

The M-estimate of , for .

kint

The column rank of the matrix .

sigmafloat

The final estimate of if or the value assigned on entry if .

rsfloat, ndarray, shape

The residuals from the model evaluated at final value of , i.e., contains the vector .

nitint

The number of iterations that were used during the estimation.

Raises
NagValueError
(errno )

On entry, and .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

Value given by function : .

The value of must be non-negative.

(errno )

Estimated value of is zero.

(errno )

Iterations to solve the weighted least squares equations failed to converge.

(errno )

The function has failed to converge in iterations.

(errno )

Having removed cases with zero weight, the value of , i.e., no degree of freedom for error. This error will only occur if .

Warns
NagAlgorithmicWarning
(errno )

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 Further Comments. The rank of the equations is given by . If the matrix just fails the test for nonsingularity then the result = 7 and is possible (see linsys.real_gen_solve).

Notes

For the linear regression model

 where y is a vector of length n of the dependent variable, X is an n×m matrix of independent variables of column rank k, θ is a vector of length m of unknown parameters, and ϵ is a vector of length n of unknown errors with var (ϵi)=σ2,

robustm_user calculates the M-estimates given by the solution, , to the equation

 where ri is the ith residual, i.e., the ith element of the vector r=y−X^θ, ψ is a suitable weight function, wi are suitable weights such as those that can be calculated by using output from robustm_wts(), and σ may be estimated at each iteration by the median absolute deviation of the residuals ^σ=medi([|ri|]/β1)

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 linsys.real_gen_solve. 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.

robustm_user is based on routines in ROBETH, see Marazzi (1987).

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