DESCRIPTION OF OUTPUT


J. A. Rupley, Tucson, Arizona



RESULTS

At various stages of the fitting, determined by the control values
prt_cycle and quad_test and by command line options, the parameter and
data arrays and other results are displayed on the crt terminal and
stored on disk.

 This output is explained here.  The description is for LDHFIT, but it
holds, except for some special values given at the end of the data
display, for any function fit by use of the simplex fitting routines.

(1) SIMPLEX:  for each vertex of the simplex, the least squares function
value (the weighted sum of squares of the residuals) is displayed, then
one or more lines of values (four values per line), giving in order the
current values of the parameters.  The number of vertices is one greater
than the number of free parameters.  Fixed parameters show the same
value for all vertices.

NOTE:  for some (most?) displays, such as the above, the count of
vertices or parameters starts with ordinal 0.

(2) CENTROID OF THE SIMPLEX:  the least squares function value is given,
then the current values of the parameters.  The centroid is the average
of the vertices of the simplex and is the preferred set of estimates of
the best-fit parameter values.  These estimates are used in calculation
of values of the dependent variable, Ycalculated(i), for each data
point, and in various calculations special for the LDHFIT model.

(3) EXIT TESTING:  immediately following the centroid listing, one line
gives the mean of the least squares function values, the root mean
square deviation of the function values from the mean, and the test
value = rms deviation/mean.  When the test value is less than the
control value exit_test, generally set at 0.00000001, fitting is
complete.

(4) RMS ERROR:  the root mean square weighted error of the fit is the
square root of the quantity:  the least squares function value at the
centroid divided by the number of data points less the number of free
parameters ( RMS ERROR = SQRT(CENTROID SSQ/(# DATA - # FREE PARAMETERS).
The rms error is useful as a test for the quality of the fit (i.e., of
the deviations between observed and calculated values of the dependent
variable).  The rms error should be no greater than the expected
uncertainty in the data.  If it is greater than expectation, the fit may
be unsatisfactory.  The rms error can also be used as a test for flyers
among the data points.  A deviation of 3 x rms error between the
observed and calculated values is sufficient to suggest dropping a data
point from the fitting.

(5) DATA AND CALCULATED VALUES:  for each data point are listed in order
the values of:  Yobserved(i), which for LDHFIT is the initial reaction
rate measured and entered into the program as data; Ycalculated(i) =
Ymodel(i), which is the initial reaction rate calculated with the model
equation and the best-fit parameter values of the centroid; Yobserved(i)
- Ycalculated(i), which is the deviation between observation and
calculation; the statistical weight of the data point, here set
generally at unity; and the values for the data point of the independent
variables, which for LDHFIT are in order the concentrations of NADH,
pyruvate, and lactate (a, b, and p).  The format of this output is given
in the function "fdatprint()".

(6) OTHER VALUES CALCULATED FROM THE BEST-FIT PARAMETERS:  following the
display of the data, various other values are displayed, as specified in
the function "fpointprint()".  For LDHFIT:  (a) the coordinates of
points that can be used to draw the lines of the primary and secondary
reciprocal plots of the steady-state kinetic analysis; (b) values of the
parameters, a variation of the listing under CENTROID.

(7) QUADRATIC FIT:  at intervals determined by the control value
quad_test, quadratic fitting of the least squares function surface is
carried out, to reconstruct the simplex and to obtain standard
deviations of the free parameters.  There is considerable screen display
that allows tracking of the process.  The file "simplex.doc" explains
this output.  The quadratic fit may fail, and often does in the early
stages of fitting.  Failure is indicated by explicit error messages or
by zero or negative values of variables.  If the quadratic fit fails,
the values of the Q MATRIX DIAGONAL and B MATRIX DIAGONAL can be helpful
in deciding what to do, e.g., whether to fix a parameter.

 (8) STANDARD DEVIATIONS FOR THE FREE PARAMETERS:  at the end of the
quadratic fit display, a table gives estimates of the standard
deviations of the free parameters.  Following this table, the vertices
of the reconstructed simplex are given.  Further minimization starts
with this simplex.  If y-pmin is less than yzero = ycentroid, the
parameter values at pmin, an improved estimate, are inserted as one of
the vertices of the reconstructed simplex.  This procedure can speed up
convergence.



MINIMIZATION

In each iteration of the Nelder-Mead method, the vertex with the highest
least squares function value is adjusted to give a lower value.  The
output displayed at each cycle allows one to track this process.

 (1) One line giving in order:  the number of the cycle, the values of
the least squares function before adjustment and after adjustment, and
the adjustment operation.

 (2) One or more lines giving the parameter values of the vertex before
adjustment.

 (3) One or more lines giving the parameter values after adjustment.

 (4) One or more lines giving the parameter values of the centroid of
the simplex.

 (5) One line giving the mean and rms deviation of the function values
of all the vertices of the simplex after the adjustment, and the test
value rms deviation/mean.

 (6) One line giving the rms error of the current best fit to the data.

 (7) Optionally, depending on the code in "fspecial()", one or more lines
of messages giving, for example, the error(s) in calculation of y-pmin during
the last cycle of quadratic fit, the next iteration for display of the
simplex (maxiter), the next iteration for quadratic fitting (quad_test),
etc.