g02hm Method
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g02hm computes a robust estimate of the covariance matrix for user-supplied weight functions. The derivatives of the weight functions are not required.

Syntax

C#
public static void g02hm(
	G02..::.G02HM_UCV ucv,
	int indm,
	int n,
	int m,
	double[,] x,
	double[] cov,
	double[] a,
	double[] wt,
	double[] theta,
	double bl,
	double bd,
	int maxit,
	int nitmon,
	double tol,
	out int nit,
	out int ifail
)
Visual Basic (Declaration)
Public Shared Sub g02hm ( _
	ucv As G02..::.G02HM_UCV, _
	indm As Integer, _
	n As Integer, _
	m As Integer, _
	x As Double(,), _
	cov As Double(), _
	a As Double(), _
	wt As Double(), _
	theta As Double(), _
	bl As Double, _
	bd As Double, _
	maxit As Integer, _
	nitmon As Integer, _
	tol As Double, _
	<OutAttribute> ByRef nit As Integer, _
	<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void g02hm(
	G02..::.G02HM_UCV^ ucv, 
	int indm, 
	int n, 
	int m, 
	array<double,2>^ x, 
	array<double>^ cov, 
	array<double>^ a, 
	array<double>^ wt, 
	array<double>^ theta, 
	double bl, 
	double bd, 
	int maxit, 
	int nitmon, 
	double tol, 
	[OutAttribute] int% nit, 
	[OutAttribute] int% ifail
)
F#
static member g02hm : 
        ucv:G02..::.G02HM_UCV * 
        indm:int * 
        n:int * 
        m:int * 
        x:float[,] * 
        cov:float[] * 
        a:float[] * 
        wt:float[] * 
        theta:float[] * 
        bl:float * 
        bd:float * 
        maxit:int * 
        nitmon:int * 
        tol:float * 
        nit:int byref * 
        ifail:int byref -> unit

Parameters

ucv
Type: NagLibrary..::.G02..::.G02HM_UCV
ucv must return the values of the functions u and w for a given value of its argument.

A delegate of type G02HM_UCV.

indm
Type: System..::.Int32
On entry: indicates which form of the function v will be used.
indm=1
v=1.
indm≠1
v=u.
n
Type: System..::.Int32
On entry: n, the number of observations.
Constraint: n>1.
m
Type: System..::.Int32
On entry: m, the number of columns of the matrix X, i.e., number of independent variables.
Constraint: 1≤m≤n.
x
Type: array< System..::.Double ,2>[,](,)[,]
An array of size [ldx, m]
Note: ldx must satisfy the constraint: ldx≥n
On entry: x[i-1,j-1] must contain the ith observation on the jth variable, for i=1,2,…,n and j=1,2,…,m.
cov
Type: array< System..::.Double >[]()[]
An array of size [m×m+1/2]
On exit: a robust estimate of the covariance matrix, C. The upper triangular part of the matrix C is stored packed by columns (lower triangular stored by rows), that is Cij is returned in cov[j×j-1/2+i-1], i≤j.
a
Type: array< System..::.Double >[]()[]
An array of size [m×m+1/2]
On entry: an initial estimate of the lower triangular real matrix A. Only the lower triangular elements must be given and these should be stored row-wise in the array.
The diagonal elements must be ≠0, and in practice will usually be >0. If the magnitudes of the columns of X are of the same order, the identity matrix will often provide a suitable initial value for A. If the columns of X are of different magnitudes, the diagonal elements of the initial value of A should be approximately inversely proportional to the magnitude of the columns of X.
Constraint: a[j×j-1/2+j]≠0.0, for j=0,1,…,m-1.
On exit: the lower triangular elements of the inverse of the matrix A, stored row-wise.
wt
Type: array< System..::.Double >[]()[]
An array of size [n]
On exit: wt[i-1] contains the weights, wti=uzi2, for i=1,2,…,n.
theta
Type: array< System..::.Double >[]()[]
An array of size [m]
On entry: an initial estimate of the location parameter, θj, for j=1,2,…,m.
In many cases an initial estimate of θj=0, for j=1,2,…,m, will be adequate. Alternatively medians may be used as given by (g07da not in this release).
On exit: contains the robust estimate of the location parameter, θj, for j=1,2,…,m.
bl
Type: System..::.Double
On entry: the magnitude of the bound for the off-diagonal elements of Sk, BL.
Suggested value: bl=0.9.
Constraint: bl>0.0.
bd
Type: System..::.Double
On entry: the magnitude of the bound for the diagonal elements of Sk, BD.
Suggested value: bd=0.9.
Constraint: bd>0.0.
maxit
Type: System..::.Int32
On entry: the maximum number of iterations that will be used during the calculation of A.
Suggested value: maxit=150.
Constraint: maxit>0.
nitmon
Type: System..::.Int32
On entry: indicates the amount of information on the iteration that is printed.
nitmon>0
The value of A, θ and δ (see [Accuracy]) will be printed at the first and every nitmon iterations.
nitmon≤0
No iteration monitoring is printed.
tol
Type: System..::.Double
On entry: the relative precision for the final estimate of the covariance matrix. Iteration will stop when maximum δ (see [Accuracy]) is less than tol.
Constraint: tol>0.0.
nit
Type: System..::.Int32 %
On exit: the number of iterations performed.
ifail
Type: System..::.Int32 %
On exit: ifail=0 unless the method detects an error (see [Error Indicators and Warnings]).

Description

For a set of n observations on m variables in a matrix X, a robust estimate of the covariance matrix, C, and a robust estimate of location, θ, are given by
C=τ2ATA-1,
where τ2 is a correction factor and A is a lower triangular matrix found as the solution to the following equations.
zi=Axi-θ
1n ∑i= 1nwzi2zi=0
and
1n∑i=1nuzi2zi ziT -vzi2I=0,
where xi is a vector of length m containing the elements of the ith row of X,
zi is a vector of length m,
I is the identity matrix and 0 is the zero matrix.
and w and u are suitable functions.
g02hm covers two situations:
(i) vt=1 for all t,
(ii) vt=ut.
The robust covariance matrix may be calculated from a weighted sum of squares and cross-products matrix about θ using weights wti=uzi. In case (i) a divisor of n is used and in case (ii) a divisor of ∑i=1nwti is used. If w.=u., then the robust covariance matrix can be calculated by scaling each row of X by wti and calculating an unweighted covariance matrix about θ.
In order to make the estimate asymptotically unbiased under a Normal model a correction factor, τ2, is needed. The value of the correction factor will depend on the functions employed (see Huber (1981) and Marazzi (1987)).
g02hm finds A using the iterative procedure as given by Huber; see Huber (1981).
Ak=Sk+IAk-1
and
θjk=bjD1+θjk- 1,
where Sk=sjl, for j,l=1,2,…,m is a lower triangular matrix such that
sjl= -minmaxhjl/D2,-BL,BL, j>l -minmax12hjj/D2-1,-BD,BD, j=l ,
where
  • D1=∑i=1nwzi2
  • D2=∑i=1nuzi2
  • hjl=∑i=1nuzi2zijzil, for j≥l
  • bj=∑i=1nwzi2xij-bj
and BD and BL are suitable bounds.
The value of τ may be chosen so that C is unbiased if the observations are from a given distribution.
g02hm is based on routines in ROBETH; see Marazzi (1987).

References

Error Indicators and Warnings

Accuracy

On successful exit the accuracy of the results is related to the value of tol; see [Parameters]. At an iteration let
(i) d1= the maximum value of sjl
(ii) d2= the maximum absolute change in wti
(iii) d3= the maximum absolute relative change in θj
and let δ=maxd1,d2,d3. Then the iterative procedure is assumed to have converged when δ<tol.

Further Comments

The existence of A will depend upon the function u (see Marazzi (1987)); also if X is not of full rank a value of A will not be found. If the columns of X are almost linearly related, then convergence will be slow.
If derivatives of the u and w functions are available then the method used in g02hl will usually give much faster convergence.

Example

See Also