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We begin with the constraint that the adjusted points are
required to satisfy the hyperplane equation, i.e.
where is as defined in Equation (2).
Defining residuals
allows this constraint
to be rewritten as

(4) 
Now, given any values of the parameters , we seek those values of
that will minimize for that choice of the , subject
to the constraint (4). Thus we require

(5) 
From (4) we have

(6) 
Multiplying each of the equations (6) by its own
undetermined multiplier and adding them all to
Equation (5), we obtain

(7) 
Since the are independent, the coefficients of in this equation must individually be zero, giving

(8) 
Substituting this result into Equation (4) and
solving for yields

(9) 
where

(10) 
This allows to be rewritten as

(11) 
This expresses in the form of a weighted sum of the
residuals as defined for a conventional regression, but with weights
that properly take account of the individual uncertainties
. Now is to be minimized with respect to the
parameters . Setting
leads to the
following set of equations analogous to the normal equations of the
conventional regression:

(12) 
Since the parameters occur in , , and
, these equations are nonlinear and cannot be solved in
closed form. However, recognizing that and are
only weakly dependent on the parameters of the hyperplane, we can
linearize Equations (12) by treating those quantities as
constants. Writing out in terms of the parameters, then, we
obtain

(13) 
Identifying the parenthesized terms in
Equation (13) as the elements of an
matrix and the righthand sides as elements of a vector
of length , this set of equations is seen as a linear system of the
form
. Solution proceeds iteratively. Starting
with an initial guess for the vector of coefficients , the
matrix and vector are evaluated and the system
is solved for the new value of . This new
value is used to reevaluate and , and the equation is
solved again. The
iteration is continued until convergence is obtained. In practice, it
is not necessary to have a good starting guess for . From
Equation (10) it can be seen that the initial choice
gives, as the result of the first iteration, the same parameters as
would be obtained if the were zero for all
except . This is the same result as would be given by the
conventional weighted regression of against the other
coordinates. Incidentally, this means that weighted averages ()
are computed correctly in one iteration. Experience has shown that
the convergence is rapid for data
sets where the fit is justified, and only a few iterations are
necessary to obtain the coefficients to accuracies that are well
within their uncertainties.
Subsections
Next: Refinement
Up: LeastSquares Fitting of a
Previous: Definition of problem
Robert Moniot
20021020