Iteration History

The MODEL Procedure

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: the number of usable observations
Objective: the corrected objective function value
Trace(S): the trace of the S matrix
subit: the number of subiterations required to find a ${\lambda}$ or a damping factor that reduces the objective function
R: 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): If the projected Jacobian crossproducts matrix is singular, the rank of the X'X matrix is output.

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 X'X matrix and the augmented X'X matrix after a sweep operation (Goodnight 1979) has been performed on it. An example of the output from:

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

is shown in Figure 14.27.

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

Figure 14.27: XPX and I Options Output

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

$[ \matrix{ X'X & X'r \cr r'X & r'r \cr } ]$

The column labeled "Residual" in the output is the vector X'r, which is the gradient of the objective function. The diagonal scalar value r'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 X'X matrix to get:

$[ \matrix{ (X'X)^{-1} & (X'X)^{-1}X'r \cr (X'r)'(X'X)^{-1} & r'r-(X'r)'(X'X)^{-1}X'r\cr } ]$

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 14.28.

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

Figure 14.28: ITALL Option Cross-Products Matrix Output

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