Iteration History

The options ITPRINT, ITDETAILS, XPX, I, and ITALL specify a detailed listing of each iteration of the minimization process.

ITPRINT Option

The ITPRINT information is selected whenever any iteration information is requested.

The following information is displayed for each iteration:

N

is the number of usable observations.

Objective

is the corrected objective function value.

Trace(S)

is the trace of the S matrix.

subit

is the number of subiterations required to find a ${\lambda }$ or a damping factor that reduces the objective function.

R

is the R convergence measure.

The estimates for the parameters at each iteration are also printed.

ITDETAILS Option

The additional values printed for the ITDETAILS option are:

Theta

is the angle in degrees between ${\Delta }$, the parameter change vector, and the negative gradient of the objective function.

Phi

is the directional derivative of the objective function in the ${\Delta }$ direction scaled by the objective function.

Stepsize

is the value of the damping factor used to reduce ${\Delta }$ if the Gauss-Newton method is used.

Lambda

is the value of ${\lambda }$ if the Marquardt method is used.

Rank(XPX)

is the rank of the ${\mb {X} ’\mb {X} }$ matrix (output if the projected Jacobian crossproducts matrix is singular).

The definitions of PPC and R are explained in the section Convergence Criteria. When the values of PPC are large, the parameter associated with the criteria is displayed in parentheses after the value.

XPX and I Options

The XPX and the I options select the printing of the augmented ${\mb {X} ’\mb {X} }$ matrix and the augmented ${\mb {X} ’\mb {X} }$ matrix after a sweep operation (Goodnight, 1979) has been performed on it. An example of the output from the following statements is shown in Figure 19.36.

proc model data=test2;
   y1 = a1 * x2 * x2 - exp( d1*x1);
   y2 = a2 * x1 * x1 + b2 * exp( d2*x2);
   fit y1 y2 / itall XPX I ;
run;

Figure 19.36: XPX and I Options Output

The MODEL Procedure
OLS Estimation

Cross Products for System At OLS Iteration 0
  a1 d1 a2 b2 d2 Residual
a1 1839468 -33818.35 0.0 0.00 0.000000 3879959
d1 -33818 1276.45 0.0 0.00 0.000000 -76928
a2 0 0.00 42925.0 1275.15 0.154739 470686
b2 0 0.00 1275.2 50.01 0.003867 16055
d2 0 0.00 0.2 0.00 0.000064 2
Residual 3879959 -76928.14 470686.3 16055.07 2.329718 24576144

XPX Inverse for System At OLS Iteration 0
  a1 d1 a2 b2 d2 Residual
a1 0.000001 0.000028 0.000000 0.0000 0.00 2
d1 0.000028 0.001527 0.000000 0.0000 0.00 -9
a2 0.000000 0.000000 0.000097 -0.0025 -0.08 6
b2 0.000000 0.000000 -0.002455 0.0825 0.95 172
d2 0.000000 0.000000 -0.084915 0.9476 15746.71 11931
Residual 1.952150 -8.546875 5.823969 171.6234 11930.89 10819902


The first matrix, labeled Cross Products, for OLS estimation is

\begin{eqnarray*}  \left[ \begin{matrix}  \Strong{X} ’\Strong{X}   &  \Strong{X} ’\Strong{r}   \\ \Strong{r} ’\Strong{X}   &  \Strong{r} ’\Strong{r}   \end{matrix} \right] \nonumber \end{eqnarray*}

The column labeled Residual in the output is the vector ${\mb {X} ’\mb {r} }$, which is the gradient of the objective function. The diagonal scalar value $\mb {r} {’}\mb {r} $ is the objective function uncorrected for degrees of freedom. The second matrix, labeled XPX Inverse, is created through a sweep operation on the augmented ${\mb {X} ’\mb {X} }$ matrix to get:

\begin{eqnarray*}  \left[ \begin{matrix}  (\Strong{X} ’\Strong{X} )^{-1}   &  (\Strong{X} ’\Strong{X} )^{-1}\Strong{X} ’\Strong{r}   \\ (\Strong{X} ’\Strong{r} )’(\Strong{X} ’\Strong{X} )^{-1}   &  \Strong{r} ’\Strong{r} -(\Strong{X} ’\Strong{r} )’(\Strong{X} ’\Strong{X} )^{-1}\Strong{X} ’\Strong{r}   \end{matrix} \right] \nonumber \end{eqnarray*}

Note that the residual column is the change vector used to update the parameter estimates at each iteration. The corner scalar element is used to compute the R convergence criteria.

ITALL Option

The ITALL option, in addition to causing the output of all of the preceding options, outputs the S matrix, the inverse of the S matrix, the CROSS matrix, and the swept CROSS matrix. An example of a portion of the CROSS matrix for the preceding example is shown in Figure 19.37.

Figure 19.37: ITALL Option Crossproducts Matrix Output

The MODEL Procedure
OLS Estimation

Crossproducts Matrix At OLS Iteration 0
  1 @PRED.y1/@a1 @PRED.y1/@d1 @PRED.y2/@a2 @PRED.y2/@b2 @PRED.y2/@d2 RESID.y1 RESID.y2
1 50.00 6409 -239.16 1275.0 50.00 0.003803 14700 16053
@PRED.y1/@a1 6409.08 1839468 -33818.35 187766.1 6409.88 0.813934 3879959 4065028
@PRED.y1/@d1 -239.16 -33818 1276.45 -7253.0 -239.19 -0.026177 -76928 -85084
@PRED.y2/@a2 1275.00 187766 -7253.00 42925.0 1275.15 0.154739 420583 470686
@PRED.y2/@b2 50.00 6410 -239.19 1275.2 50.01 0.003867 14702 16055
@PRED.y2/@d2 0.00 1 -0.03 0.2 0.00 0.000064 2 2
RESID.y1 14699.97 3879959 -76928.14 420582.9 14701.77 1.820356 11827102 12234106
RESID.y2 16052.76 4065028 -85083.68 470686.3 16055.07 2.329718 12234106 12749042