The MODEL Procedure

Convergence Criteria

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 ${\bSigma }$ is used (that is, IT3SLS), two convergence criteria are used. The termination values can be specified with the CONVERGE=(p,s) option in the FIT statement. If the second value, s, is not specified, it defaults to p. The criterion labeled S (described later in the section) 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

\[ \frac{(r'({\bS }^{-1}{\otimes }{\bW }){\bX }({\bX }'({\bS }^{-1} {\otimes }{\bW }){\bX })^{-1}{\bX }'({\bS }^{-1}{\otimes }{\bW })r)}{(r'({\bS }^{-1}{\otimes }{\bW })r)} \]

where 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 normalizing 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.

In the multivariate case, R is somewhat more complicated but is designed to go to 0 as the gradient of the objective becomes small and can still be given the previous interpretations for the aggregation of the equations weighted by 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 kth iteration, PPC is the maximum over the parameters

\[ \frac{\left|\theta ^{k+1}_{i}- \theta ^{k}_{i}\right|}{\left|\theta ^{k}_{i}\right|+10^{-6}} \]

where ${ \theta ^{k}_{i}}$ is the current value of the ith parameter and ${\theta ^{k+1}_{i} }$ 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 kth iteration, RPC is the maximum over i of

\[ \frac{\left|\theta ^{k}_{i}- \theta ^{k-1}_{i}\right|}{\left|\theta ^{k-1}_{i}\right|+10^{-6}} \]

where $\theta ^{k}_{i}$ is the current value of the ith parameter and $\theta ^{k-1}_{i}$ 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:

\[ \frac{\left|O^{k}-O^{k-1}\right|}{\left|O^{k-1}\right|+10^{-6}} \]

where O$^{k-1}$ is the value of the objective function (O$^{k}$ ) from the previous iteration.

S

measures the relative change in the S matrix. S is computed as the maximum over i, j of

\[ \frac{\left|S^{k}_{ij}- S^{k-1}_{ij}\right|}{\left|S^{k-1}_{ij}\right|+10^{-6}} \]

where ${ S^{k-1 }}$ 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 shown in Figure 26.25.

Figure 26.25: Convergence Criteria Output

Final Convergence Criteria
R 0.000737
PPC(b) 0.003943
RPC(b) 0.00968
Object 4.784E-6
Trace(S) 0.533325
Objective Value 0.522214 




            

            


            
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.