Biomedical Engineering Reference
In-Depth Information
used instead of the unsquared absolute sum of values. This allows the residuals to
be treated as continuous differentiable quantities. If the least-square problem is not
treated analytically, it is more convenient to minimize the unsquared absolute sum
of distances, as done in the code example added in the appendix. The reason is that
squares of the residuals may produce a disproportionately large weighting of
outlying points. In such cases, weighting of the solution as implemented in (
3.385
)
may represent an adoptable approach. In addition, vertical deviation rather then
perpendicular offset is used since this permits a more practical approach. A 'dis-
advantage', however, is that interpolation (linear, polynomial, spline, etc.) of data
is required to obtain comparable quantities at consisted abscissa coordinates
(function arguments). This issue will be described as follows in more detail.
As previously mentioned, experimental and simulated data must be compared at
consistent points. In this process direct comparison is, in most cases, not a priori
possible since both data sets are likely to differ significantly. Generally, simulated
data does not represent a proper subset of the experimental data set:
D
Sim
6
D
Exp
:
Usually, more experimental data is on hand than simulation output. If sample
intervals are known, simulation output can be requested accordingly, depending on
the simulation strategy used. The simulation output intervals must coincide with
the experimental output intervals. In nonlinear static analysis employing direct
incrementation (direct user control of the time increment size, instead of an
automated incrementation schema) this may lead to serious convergence problems
during simulation. Increased simulation time or even incomplete termination of the
analysis may then result. A more feasible approach is to use variable time
increment size and to adjust the simulation data output to the experimental data via
interpolation. A crucial issue is the use of (fixed) sample points of the experimental
curve rather than of simulation output as reference points. This guaranties com-
parability of the sums of squared residuals, due to independence of simulation
convergence and thus data point numbers of simulation output.
Generally, on the basis of one data set that defines the total number of discrete
data points, appropriate values incorporating information of the other data set must
be found to objectively compare both sets and establish information regarding the
quality of agreement. In Fig.
3.34
these reference data points are denoted u
exp
i
(i = 1,…,n) comprised in the set of experimental data. If particular function
arguments of both sets are not identical u
ex
h
6¼
u
si
i
, the linear interpolation rule
(
3.389
), can be used to denote each simulated function value F
sim
i
a comparable
value F
ex
i
on the corresponding linear interpolant of the experimental curve. Using
the example illustrated in Fig.
3.34
, the linear interpolant is between the points
(F
exp
h
, u
exp
h
) and (F
exp
h
þ
1
, u
exp
h
þ
1
) in the interval [u
exp
, u
exp
h
þ
1
].
h
F
ex
i
¼
u
exp
þ
u
si
i
u
exp
h
þ
1
u
sim
i
F
exp
h
F
exp
h
þ
1
h
ð
3
:
389
Þ
u
exp
h
þ
1
u
exp
u
exp
h
þ
1
u
exp
h
h
In the figure, Fig.
3.34
, the above relation of (
3.389
), is depicted.
