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The MODEL Procedure |

There are a number of measures that could be used as convergence or stopping criteria. PROC MODEL computes five convergence measures labeled R, S, PPC, RPC, and OBJECT.

When an estimation technique that iterates estimates of is used
(that is, IT3SLS),
two convergence criteria are used. The termination values
can be specified with the CONVERGE=(*p*,*s*) option on the
FIT statement. If the second value, *s*, is not specified, it
defaults to *p*.
The criterion labeled S (given in the following)
controls the convergence of the
**S** matrix. When S is less than *s*, the **S** matrix
has converged.
The criterion labeled R is compared to the *p* value to test convergence
of the parameters.

The R convergence measure cannot be computed accurately in the special
case of singular residuals (when all the residuals are close to 0) or
in the case of a 0 objective value. When
either the trace of the **S** matrix
computed from the current residuals (trace(S)) or the objective value is
less than the value of the SINGULAR= option, convergence is assumed.

The various convergence measures are explained in the following:

- R
- is the primary convergence measure for the parameters.
It measures the degree to which the residuals are orthogonal to the
Jacobian columns, and it approaches 0 as the gradient of the
objective function becomes small.
R is defined as the square root of
**X**is the Jacobian matrix and**r**is the residuals vector. R is similar to the relative offset orthogonality convergence criterion proposed by Bates and Watts (1981).

In the univariate case, the R measure has several equivalent interpretations:- the cosine of the angle between the residuals vector and the column space of the Jacobian matrix. When this cosine is 0, the residuals are orthogonal to the partial derivatives of the predicted values with respect to the parameters, and the gradient of the objective function is 0.
- the square root of the R
^{2}for the current linear pseudo-model in the residuals. - a norm of the gradient of the objective function, where the norming matrix is proportional to the current estimate of the covariance of the parameter estimates. Thus, using R, convergence is judged when the gradient becomes small in this norm.
- the prospective relative change in the objective function value expected from the next GAUSS step, assuming that the current linearization of the model is a good local approximation.

**S**^{-1}. - PPC
- is the prospective parameter change measure.
PPC measures the maximum relative change in the parameters implied by
the parameter-change vector computed for the next iteration.
At the
*k*th iteration, PPC is the maximum over the parameters*i*th parameter and is the prospective value of this parameter after adding the change vector computed for the next iteration. The parameter with the maximum prospective relative change is printed with the value of PPC, unless the PPC is nearly 0. - RPC
- is the retrospective parameter change measure.
RPC measures the maximum relative change in the parameters from the previous
iteration. At the
*k*th iteration, RPC is the maximum over*i*of*i*th parameter and is the previous value of this parameter. The name of the parameter with the maximum retrospective relative change is printed with the value of RPC, unless the RPC is nearly 0. - OBJECT
- measures the relative change in the objective
function value between iterations:
*O*is the value of the objective function (^{k-1}*O*) from the previous iteration.^{k} - S
- measures the relative change in the
**S**matrix. S is computed as the maximum over*i, j*of*S*^{k-1 }is the previous**S**matrix. The S measure is relevant only for estimation methods that iterate the**S**matrix.

An example of the convergence criteria output is as follows:

This output indicates
the total number of iterations required by the Gauss minimization
for all the **S** matrices was 35.
The "Trace(S)" is the trace (the sum of the diagonal elements) of
the **S** matrix computed from the current residuals.
This row is labeled MSE if there is only one equation.

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