Biomedical Engineering Reference
In-Depth Information
TABLE 7.6
“Errors” of the Data and Various Transformations in
Figs 7.7 and 7.8
t
i
C
C,i
C
C
y
i
Fig. 7.8
ε
C
ε
y
000 0 0 0
t
i
x
2,
i
x
2
z
i
Z
ε
x
2
ε
z
41
1.18 1.436068
L
0.256
*
0.2163
L
0.0546
20.5 27.02
26.0362
0.99 0.02878 0.034885
L
0.00610
48
1.38 1.641458
L
0.261
0.2587
L
0.0585
44.5 23.29
21.8960
1.39 0.02857 0.029338
L
0.00077
55
1.63 1.836816
L
0.207
0.3145
L
0.0489
51.5 22.08
20.8266
1.26 0.03571 0.027905
0.00781
75
2.24 2.344248
L
0.104
0.4668
L
0.0287
65.0 19.79
18.9188
0.87 0.03050 0.025349
0.00515
96
2.75 2.805283
L
0.055
0.6165
L
0.0178
85.5 16.83
16.3693
0.46 0.02429 0.021933
0.00235
127 3.31 3.373303
L
0.063
0.8140
L
0.0251
111.5 14.04
13.6474
0.39 0.01807 0.018286
L
0.00022
146 3.76 3.665667
0.094
1.009
0.0443 136.5
11.42
11.4757
L
0.06 0.02368 0.015376
0.00831
162 3.81 3.883856
L
0.074
1.033
L
0.0371 154.0 10.14
10.1723
L
0.04 0.00313 0.013630
L
0.01050
180 4.11
4.102335
0.008
1.194
0.0045 171.0
9.239
9.05287
0.19 0.01667 0.012130
0.00454
194 4.31 4.254627
0.055
1.318
0.0359 187.0
7.964
8.11548
L
0.15 0.01429 0.010874
0.00341
212 4.45 4.430265
0.020
1.415
0.0143 203.0
7.101
7.27782
L
0.18 0.00778 0.009751
L
0.00197
267 4.86 4.851535
0.008
1.773
0.0089 239.5
5.710
5.68303
0.03 0.00746 0.007615
L
0.00016
318 5.15 5.124676
0.025
2.139
0.0380 292.5
3.951
3.97739
L
0.03 0.00569 0.005329
0.00036
368 5.32 5.315579
0.004
2.441
0.0091 343.0
2.802
2.83619
L
0.04 0.00340 0.003800
L
0.00040
379 5.35 5.349688
0.000
2.505
0.0007 373.5
2.304
2.31376
L
0.01 0.00273 0.003100
L
0.00037
410 5.42 5.433392
L
0.014
2.673
-0.0357 394.5
2.055
2.01158
0.04 0.00226 0.002695
L
0.00044
* The bold numbers are errors (of model compared with the data).
is based on extrapolation and the symmetric two-step midpoint method with stiffness
removal, the GBS approach (Hairer et al. 1987).
The easiest way is to perform the regression analysis on a spreadsheet such as Excel
and
Quattro Pro
. As these spreadsheet programs have built-in optimization routines, allow the
interface of user-defined visual basic routines and have the graphical capabilities to bring
the regression analysis visually at every step of the way. We will show the use of the Excel
to solve the same kinetic parametric estimation problem dealt in the previous sections.
We have developed a general-purpose visual basic routine for solving a set of ordinary
differential equations named ODExLIMS to be served as an integrator to the differential
regression model (interested reader can e-mail
sliu@esf.edu
for a copy of the visual basic
routine).
Figure 7.9
shows the setup of the problem initially on an Excel
worksheet. We first load
the ODExLIMS.xls on to Excel
. Delete the contents on the worksheet. We input the
measured data, time, initial guess of
k
f
and
n
; input the formulas for calculating the residual
squared and the variance of the experimental data around the model predictions, and saved
the file as kinetic.xls. We also generate a graph of
C
C
versus
t
with the experimental data
shown as circles and the predictions as a continuous line. Presently, we have not computed
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