Calculates a robust estimation of a correlation matrix, user-supplied weight function plus derivatives

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NAG[g02hlc] NAG[nag_robust_m_corr_user_fn] - Calculates a robust estimation of a correlation matrix, user-supplied weight function plus derivatives

Calling Sequence

g02hlc(ucv, indm, x, cov, a, wt, theta, nitmon, tol, nit, 'n'=n, 'm'=m, 'bl'=bl, 'bd'=bd, 'maxit'=maxit, 'outfile'=outfile, 'comm'=comm, 'fail'=fail)

nag_robust_m_corr_user_fn(. . .)

Parameters

ucv - procedure;

ucv must return the values of the functions and and their derivatives for a given value of its argument.

Its specification is:

ucv(t, u, ud, w, wd, comm)

t - float;

On entry: the argument for which the functions and must be evaluated.

u - assignable;

Note: On exit the variable u will have a value of type float.

On exit: the value of the function at the point t.

ud - assignable;

Note: On exit the variable ud will have a value of type float.

On exit: the value of the derivative of the function at the point t.

w - assignable;

Note: On exit the variable w will have a value of type float.

On exit: the value of the function at the point t.

wd - assignable;

Note: On exit the variable wd will have a value of type float.

On exit: the value of the derivative of the function at the point t.

comm - table;

A Maple table, which should be generated using NAG[Nag_Comm], corresponding to the Nag_Comm structure.

The following fields may be accessed via NAG[GetOptions]:

p - Maple

Before calling nag_robust_m_corr_user_fn (g02hlc) this field may be initialized for use by ucv when called from nag_robust_m_corr_user_fn (g02hlc).

indm - integer;

On entry: indicates which form of the function will be used.

x - Matrix(1..dim1, 1..m, datatype=float[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: must contain the th observation on the th variable, for and .

cov - Vector(1.., datatype=float[8]);

Note: the dimension, dim, of the array cov must be at least .

On exit: contains a robust estimate of the covariance matrix, . The upper triangular part of the matrix is stored packed by columns (lower triangular stored by rows), is returned in , .

a - Vector(1.., datatype=float[8]);

Note: the dimension, dim, of the array a must be at least .

On entry: an initial estimate of the lower triangular real matrix . Only the lower triangular elements must be given and these should be stored row-wise in the array.

The diagonal elements must be , and in practice will usually be . If the magnitudes of the columns of are of the same order, the identity matrix will often provide a suitable initial value for . If the columns of are of different magnitudes, the diagonal elements of the initial value of should be approximately inversely proportional to the magnitude of the columns of .

Constraint: , for . .

On exit: the lower triangular elements of the inverse of the matrix , stored row-wise.

wt - Vector(1..n, datatype=float[8]);

On exit: contains the weights, , for .

theta - Vector(1..m, datatype=float[8]);

On entry: an initial estimate of the location argument, , for .

In many cases an initial estimate of , for , will be adequate. Alternatively medians may be used as given by g07dac (nag_median_1var).

On exit: contains the robust estimate of the location argument, , for .

nitmon - integer;

On entry: indicates the amount of information on the iteration that is printed.

The value of , and (see Section [Accuracy]) will be printed at the first and every nitmon iterations.

No iteration monitoring is printed.

tol - float;

On entry: the relative precision for the final estimates of the covariance matrix. Iteration will stop when maximum (see Section [Accuracy]) is less than tol.

Constraint: . .

nit - assignable;

Note: On exit the variable nit will have a value of type integer.

On exit: the number of iterations performed.

'n'=n - integer; (optional)

On entry: , the number of observations.

Constraint: . .

'm'=m - integer; (optional)

On entry: , the number of columns of the matrix , i.e., number of independent variables.

Constraint: . .

'bl'=bl - float; (optional)

On entry: the magnitude of the bound for the off-diagonal elements of , .

Suggested value: . (default: )

Constraint: . .

'bd'=bd - float; (optional)

On entry: the magnitude of the bound for the diagonal elements of , .

Suggested value: . (default: )

Constraint: . .

'maxit'=maxit - integer; (optional)

On entry: the maximum number of iterations that will be used during the calculation of .

Suggested value: . (default: )

Constraint: . .

'outfile'=outfile - character; (optional)

On entry: The name of a file to which intermediate or diagnostic output should be appended. If a value is not provided for this parameter then the behaviour of this routine is platform dependent. Usually all output will be suppressed, however on some platforms output will be produced and will be displayed in the Maple session.

'comm'=comm - table; (optional)

A Maple table, which should be generated using NAG[Nag_Comm], corresponding to the Nag_Comm structure.

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_robust_m_corr_user_fn (g02hlc) calculates a robust estimate of the covariance matrix for user-supplied weight functions and their derivatives.

Description

For a set of observations on variables in a matrix , a robust estimate of the covariance matrix, , and a robust estimate of location, , are given by:

where is a correction factor and is a lower triangular matrix found as the solution to the following equations.

and

where is a vector of length containing the elements of the th row of ,

is a vector of length ,

is the identity matrix and 0 is the zero matrix,

and and are suitable functions.

nag_robust_m_corr_user_fn (g02hlc) covers two situations:

for all ,

The robust covariance matrix may be calculated from a weighted sum of squares and cross-products matrix about using weights . In case (i) a divisor of is used and in case (ii) a divisor of sum(wt[i], i = 1 .. n) is used. If , then the robust covariance matrix can be calculated by scaling each row of by and calculating an unweighted covariance matrix about .

In order to make the estimate asymptotically unbiased under a Normal model a correction factor, , is needed. The value of the correction factor will depend on the functions employed (see Huber (1981) and Marazzi (1987a)).

nag_robust_m_corr_user_fn (g02hlc) finds using the iterative procedure as given by Huber.

and

theta[j[k]] = b[j]/D[1]+theta[j[k-1]]

where , for is a lower triangular matrix such that:

s[jl]={[[-min[max(h[jl],/,D[3],,,-,BL),,,BL] ,j>l],[,],[-min[max((h[jj],/,(2,D[3],-,D[4],/,D[2])),,,-,BD),,,BD] ,j=l]]

where

D[1]=(∑){w,(||z[i]||[2]),+,1/m,w',(||z[i]||[2]),||z[i]||[2]}

D[2]=(∑){1/p,(u',(||z[i]||[2]),||z[i]||[2],+,2,u,(||z[i]||[2])),||z[i]||[2],-,v',(||z[i]||[2])}||z[i]||[2]

$D[3]=1/(m+2)(∑){1/m,(u',(||z[i]||[2]),||z[i]||[2],+,2,u,(||z[i]||[2])),+,u,(||z[i]||[2])}(||z[i]||[2])^2$

$D[4]=(∑){1/m,u,(||z[i]||[2]),(||z[i]||[2])^2,-,v,((||z[i]||[2])^2)}$

h[j*l] = sum(u(LinearAlgebra[Norm](z[i], 2))*z[i*j]*z[i*l], i = 1 .. n) , for

h[jj]=(∑)u(||z[i]||[2])(z[ij]^2,-,(||z[ij]||[2])^2,/,m)

b[j]=(∑)w(||z[i]||[2])(x[ij],-,b[j])

and and are suitable bounds.

nag_robust_m_corr_user_fn (g02hlc) is based on routines in ROBETH; see Marazzi (1987a).

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument had an illegal value.

"NE_CONST_COL"

Column of x has constant value.

"NE_CONVERGENCE"

Iterations to calculate weights failed to converge.

"NE_FUN_RET_VAL"

value returned by : .

"NE_INT"

On entry, . Constraint: .

"NE_INT_2"

On entry, , . 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 consult NAG for assistance.

"NE_NOT_CLOSE_FILE"

Cannot close file .

"NE_NOT_WRITE_FILE"

Cannot open file for writing.

"NE_REAL"

On entry, . Constraint: .

"NE_ZERO_DIAGONAL"

On entry, diagonal element of a is 0.0.

"NE_ZERO_SUM"

The sum is zero.

Accuracy

On successful exit the accuracy of the results is related to the value of tol; see Section [Parameters]. At an iteration let

the maximum value of

the maximum absolute change in

the maximum absolute relative change in

and let . Then the iterative procedure is assumed to have converged when .

	Further Comments
	The existence of will depend upon the function (see Marazzi (1987a)); also if is not of full rank a value of will not be found. If the columns of are almost linearly related, then convergence will be slow.

Examples

ucv := proc(t, u::evaln, ud::evaln, w::evaln, wd::evaln, comm)
  local t2, cu, cw:
  cu := comm['p'][1]:
  assign(u, 1.0):
  assign(ud, 0.0):
  if (t <> 0.0) then
    t2 := t * t:
    if (t2 > cu) then
      assign(u, cu / t2):
      assign(ud, -eval(u) * 2.0 / t):
    end if:
  end if:
  # w function and derivative
  cw := comm['p'][2]:
  if (t > cw) then
      assign(w, cw / t):
      assign(wd, -eval(w) / t):
  else
      assign(w, 1.0):
      assign(wd, 0.0):
  end if:
end proc:
indm := 1:
n := 10:
m := 3:
bl := 0.9:
bd := 0.9:
maxit := 50:
nitmon := 0:
tol := 5e-05:
comm := NAG:-Nag_Comm():
comm['p'] := Vector([4.0, 2.0], datatype=float[8]):
x := Matrix([[3.4, 6.9, 12.2], [6.4, 2.5, 15.1], [4.9, 5.5, 14.2], [7.3, 1.9, 18.2], [8.800000000000001, 3.6, 11.7], [8.4, 1.3, 17.9], [5.3, 3.1, 15], [2.7, 8.1, 7.7], [6.1, 3, 21.9], [5.3, 2.2, 13.9]], datatype=float[8]):
cov := Vector(6, datatype=float[8]):
a := Vector([1, 0, 1, 0, 0, 1], datatype=float[8]):
wt := Vector(10, datatype=float[8]):
theta := Vector([0, 0, 0], datatype=float[8]):
NAG:-g02hlc(ucv, indm, x, cov, a, wt, theta, nitmon, tol, nit, 'n' = n, 'm' = m, 'bl' = bl, 'bd' = bd, 'maxit' = maxit, 'comm' = comm):

cov;

wt;

theta;

nit;

	See Also
	Huber P J (1981) Robust Statistics Wiley Marazzi A (1987a) Weights for bounded influence regression in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 3 Institut Universitaire de Médecine Sociale et Préventive, Lausanne g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.