An estimator function is given by:

     n²(x) = s( x; p₁,... )

indicating that the square of the estimator function is presented as both a function of x and an additional set of parameters.

Given a set of empirical data pairs (x, f(x)), it is required that values of the parameters which minimize the error function:

     SUM( n(x) - f(x) )²

where the sum is over all the data pairs.

It is necessary (although not sufficient) that the simultaneous zero of the error derivative functions:

     k_i = SUM( 2 * ( n(x) - f(x) ) * dn(x)/dp_i )

be found.

Given an initial estimation for the parameters, the parameter estimation can be updated by:

     P = P - M_ij^-1 * K

where P is the vector of parameter estimations, M_ij is the matrix of partial derivatives of the error derivative functions and K is the transposed vector of error derivative function values.

The individual matrix terms can calculated by:

     M_ij = SUM( 2 * ( n(x) - f(x) * d²n(x)/dp_i dp_j + 2 * dn(x)/dp_i * dn(x)/dp_j )

DERIVATIVES

From the form of the estimator function, the first derivative values may be calculated by:

dn(x)/dp_i = ( 0.5/n(x) ) * ds(x;p₁,...)/dp_i

leading to the second derivative calculation:

d²n(x)/dp_i dp_j = ( 0.5/n(x) ) * ( d²s(x;p₁,...)/dp_i dp_j - 2 * dn(x)/dp_i * dn(x)/dp_j )