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| MONITORING OF SEARCH |
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MEANING: History report on the Estimation Step search
CONTEXT: NONMEM output
DISCUSSION:
NONMEM output includes a report of the history of the search under-
taken in the Estimation Step for parameter estimates. This report is
called the intermediate output from the Estimation Step, because it
consists of summaries of the progress of the search, from iteration to
iteration, and because it may be viewed as the search progresses, pro-
vided the NONMEM output file may be viewed as the search progresses.
This report can also be viewed in a special (unbuffered) file.
The search is for parameter values that minimize the value of the
OBJECTIVE FUNCTION (See minimum value of objective function). Basi-
cally, it entails the following steps.
1. The search is carried out in a different parameter space. The
parameters are transformed to unconstrained parameters (UCP). In
the transformation process a scaling occurs so that the initial
estimate of each of the UCP is 0.1. (NONMEM Users Guide I Sec-
tions C.3.5.1 "ESTIMATION" and G.3 "Messages from the Estimation
Step" refer to the UCP as "scaled transformed parameters" or STP.
NONMEM Users Guide II Section F "Rescaling" refers to the UCP as
"rescaled canonical parameters" or RCP.)
2. At the current parameter estimate the GRADIENT vector (i.e., the
vector of first partial derivatives of the objective function
with respect to the UCP) is computed. An approximate Hessian
matrix (See hessian) is also computed. An ITERATION SUMMARY,
including the current parameter estimate and the gradient vector,
may placed into the intermediate output.
With NONMEM 7, the parameter estimates are also displayed in
their natural (unscaled) space. These lines are identified as
NPARAMETR and precede the PARAMETER lines, which display the UCP
values. Note that when OMEGA (or SIGMA) have block structure,
the values in NPARAMETR are listed in upper triangular order,
whereas elsewhere in NONMEM output they are displayed in lower
triangular order. For example, suppose the NONMEM output dis-
plays a 3x3 OMEGA matrix as
OMEGA(1,1)
OMEGA(2,1) OMEGA(2,2)
OMEGA(3,1) OMEGA(3,2) OMEGA(3,3)
It will be listed in NPARAMTR as:
OMEGA(1,1) OMEGA(2,1) OMEGA(3,1) OMEGA(2,2) OMEGA(3,2) OMEGA(3,3)
This is inconsistent, but it cannot be changed; option ORDER of
the $ESTIMATION record does not affect the NPARAMTR line.
3. Using the gradient vector and Hessian matrix, a direction in
parameter space, emananting from the current parameter estimate,
is computed, and a search is undertaken along this direction for
an approximate minimum point. When this point is found, NONMEM
returns to step 2. (An ITERATION consists of the computation of
the direction, the search along the direction, and the computa-
tion of the gradient vector and Hessian matrix at the approximate
minimum point.)
4. Iteration stops when any one of the following conditions holds:
A successful (local) minimum point has been found.
The maximum number of FUNCTION EVALUATIONS allowed by the user
(MAXEVALS option of $ESTIM) is exceeded.
It was not possible to successfully locate a minimum point due to
so-called ROUNDING ERRORS.
Here is an example of intermediate output from the Estimation Step:
MONITORING OF SEARCH:
ITERATION NO.: 0 OBJECTIVE VALUE: 110.244034784025 NO. OF FUNC. EVALS.: 6
CUMULATIVE NO. OF FUNC. EVALS.: 6
NPARAMETR: 3.0000E+00 8.0000E-02 4.0000E-02 6.0000E+00 5.0000E-03
3.0000E-01 2.0000E-04 6.0000E-03 4.0000E-01 4.0008E-01
PARAMETER: 1.0000E-01 1.0000E-01 1.0000E-01 1.0000E-01 1.0000E-01
1.0000E-01 1.0000E-01 1.0000E-01 1.0000E-01 1.0000E-01
GRADIENT: 1.4640E+01 -1.6691E+01 5.6460E+01 -9.4129E+00 -1.1699E+01
1.0623E+01 -6.3485E-01 -4.7803E+01 -9.6457E+00 5.2414E+00
ITERATION NO.: 5 OBJECTIVE VALUE: 105.453694506861 NO. OF FUNC. EVALS.: 7
CUMULATIVE NO. OF FUNC. EVALS.: 45
NPARAMETR: 2.8166E+00 8.0433E-02 3.8563E-02 6.6123E+00 9.7862E-03
-3.9509E-02 2.3438E-04 8.1375E-03 4.9657E-01 4.0355E-01
PARAMETER: -5.3055E-02 1.0703E-01 5.7598E-02 1.4858E-01 1.8644E-01
-1.2545E-02 1.5794E-01 1.3452E-01 3.7714E-02 1.0431E-01
GRADIENT: 3.5103E-01 -1.3214E+01 2.8003E+01 1.1115E+00 1.9193E+00
-5.3252E+00 7.1533E-02 -2.2091E+01 -1.4424E+00 8.2329E+00
ITERATION NO.: 10 OBJECTIVE VALUE: 104.747695563098 NO. OF FUNC. EVALS.: 7
CUMULATIVE NO. OF FUNC. EVALS.: 80
NPARAMETR: 2.7362E+00 7.6935E-02 3.5982E-02 5.7362E+00 1.0840E-03
-2.9208E-01 2.3701E-04 9.4013E-03 5.1805E-01 3.8858E-01
PARAMETER: -1.1928E-01 4.9234E-02 -2.3022E-02 7.7517E-02 2.2174E-02
-9.9574E-02 1.9498E-01 1.4956E-01 -1.7173E-01 8.5414E-02
GRADIENT: -1.3933E+00 -7.4632E+00 9.3144E+00 1.0280E+00 9.6395E-01
-3.1313E+00 1.0212E+00 -5.3784E+00 -2.0828E+00 -9.1477E-01
ITERATION NO.: 15 OBJECTIVE VALUE: 104.561086502423 NO. OF FUNC. EVALS.: 7
CUMULATIVE NO. OF FUNC. EVALS.: 116
NPARAMETR: 2.7735E+00 7.8097E-02 3.6292E-02 5.5466E+00 5.1735E-03
-1.2911E-01 2.3941E-04 9.0947E-03 5.1461E-01 3.8760E-01
PARAMETER: -8.8567E-02 6.8686E-02 -1.2994E-02 6.0711E-02 1.0762E-01
-4.4762E-02 1.9028E-01 1.4643E-01 -8.4016E-02 8.4157E-02
GRADIENT: 7.7937E-03 5.7189E-03 3.0426E-02 -5.1648E-03 -1.2939E-02
1.4490E-02 7.4602E-04 -2.5227E-02 -3.1443E-03 1.5799E-02
ITERATION NO.: 19 OBJECTIVE VALUE: 104.561067398632 NO. OF FUNC. EVALS.: 0
CUMULATIVE NO. OF FUNC. EVALS.: 149
NPARAMETR: 2.7739E+00 7.8129E-02 3.6307E-02 5.5498E+00 5.2393E-03
-1.2767E-01 2.3985E-04 9.1064E-03 5.1521E-01 3.8742E-01
PARAMETER: -8.8304E-02 6.9219E-02 -1.2510E-02 6.0998E-02 1.0895E-01
-4.4249E-02 1.9095E-01 1.4652E-01 -8.3269E-02 8.4127E-02
GRADIENT: 1.2144E-04 -1.4883E-04 -6.6257E-06 -7.5745E-05 -1.6157E-05
1.1307E-04 6.4128E-05 1.1193E-04 -1.7655E-05 4.2676E-05
Note that the values of the PARAMETERs are the values of the UCP, so
that at the 0th iteration, all the PARAMETERs have the value 0.1.
The first parameter (and gradient) elements correspond to the THETA
elements which are not fixed. The remaining elements correspond to
the OMEGA and SIGMA elements which are not fixed, but not in a simple
1-1 manner unless OMEGA and SIGMA are constrained to be diagonal.
The PRINT option of the $ESTIMATION record determines how often itera-
tion summaries are printed: not at all (with PRINT=0); only for the
0th and last iterations (with PRINT=9999); for the 0th iteration, for
every 10th iteration thereafter, and for the last iteration (with
PRINT=10, as illustrated above).
If a model specification file is output, then the estimates may also
be seen in the original parameterization for those iterations whose
summaries appear in intermediate output. These estimates may be found
in file INTER in the same order and format as elsewhere in the NONMEM
output. With NONMEM 7, INTER exists after the run is finished. |
When the Estimation Step terminates, it reports its success or lack of
it, as in this example:
MINIMIZATION SUCCESSFUL
NO. OF FUNCTION EVALUATIONS USED: 149
NO. OF SIG. DIGITS IN FINAL EST.: 4.7
Each (UCP) element of the mimimum point is determined to a number of
significant digits. The number of significant digits reported is the
number of significant digits in the least-well-determined element.
The report "MINIMIZATION SUCCESSFUL" is issued when this number is no
less than the number of significant digits requested using the SIGDIG-
ITS option of the $ESTIMATION record. Note that this report alone
does not assure that a global (or even a local) minimum point has been
located; what appears to be a minimum point may be a saddle point.
Nor, if a minimum point has been located, does the report alone assure
that the objective function is not "flat" in a region of the point.
For such assurances, one also needs to implement the Covariance Step.
REFERENCES: Guide I Section C.3.5.1
REFERENCES: Guide V Section 10.4.1
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