Biomedical Engineering Reference
In-Depth Information
quality of the regression. The integral method showed a correlation coefficient of 0.9991
and the differential method showed a lower correlation coefficient of 0.9124. Since both
methods produce very similar kinetic parameters, one may regard the kinetic parameters
as correct. In other words, the scattering of the data around the correlation is due to “exper-
imental error.” The inconsistency here is that based on the integral method, Fig. 7.7 shows
very little “experimental error.” On the other hand, Fig. 7.8 shows significant “experi-
mental error.” However, we are talking about the same set of experimental data. This
“conclusion” is definitely incorrect. One usually considers the error as correlation error.
In this case, the integral method is more convincing than the differential method as the
“error” is smaller. On the other hand, Fig. 7.8 might be used to expose potential problems
with the proposed model. Therefore, both methods were traditionally used before
computers were common.
To resolve the error difference, let us look at the difference in “error” represented by
Figs 7.7 and 7.8 . One can safely say that the “errors” indicated either in Fig. 7.7 or Fig. 7.8
directly reflect the errors might have been contained in Table 7.3 . This is obvious as the exper-
imental data (in Tab l e 7 . 3 ) are on the ( C C , t ) plane. The measurements are in C C , not y or z .
Therefore, comparing Figs 7.7 and 7.8 is not a fair comparison: they are different. The “errors”
indicated in Figs
7.7
and 7.8
are
reflections
of
the
errors propagated to
0
1
C A 0 3
58
C C
and z ¼ D C C
Dt
@
A
y ¼ ln
through C C . There is a slight reduction of error from
C A 0 55
58
C C
C C to y , but a magnification of error from C C to z . In addition, x 1.5 and x 2 in Fig. 7.8 contains
error as well, although reduced based on C C . Table 7.6 shows the “errors” if the kinetic
parameters are given by n
0.001340. One can observe from Table 7.6 that the
error structure is not uniformly transferred from C C to y or z . Therefore, minimizing
the error in y or z does not directly correspond to a uniform error minimization for C C , the
original data that contain error. As a result, the parameters obtained from neither the integral
method nor the differential method is of optimum values that directly reflect
¼
2 and k f ¼
the
experimental data.
7.8.4. A General Approach of Parametric Estimation for Differential Models
One can conclude from the previous discussions that since the experimental data is given
as a time series in C C , we will need to minimize the variance of the experimental data C C
around the regression model predictions. To this end, the integral method ( Integral Methods )
may be redesigned to achieve this task by not looking for a linear regression model, but
explicit in the dependent variable. However, the regression model is, in general, a nonlinear
differential equation of C C with respect to time t . Therefore, we will need to solve the regres-
sion model numerically for C C .
Aviable approach is to use a general differential equation solver (or integrator) to compute
the value of C C for a given time series and then minimize the variance associated with pre-
dicted and experimental values of C C . There are many integrators available both commer-
cially and on the web (for example, adaptive Runge e Kutta methods based and
extrapolation methods based integrators of particular utility). We shall introduce one that
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