The MODEL Procedure

Iteration History

Subsections:

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 26.36. 

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 26.36: 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 26.37. 
               

               
  

               

                  
Figure 26.37: 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