Biomedical Engineering Reference
In-Depth Information
Regression relationships are valid only for values of the regressor variable within the
range of the original data. This is especially true for correlations. The mathematical relation-
ship that we have tentatively assumed may be valid over the original range of
x
, but it may be
unlikely to remain so as we extrapolate
d
that is, if we use values of
x
beyond that range. In
other words, as we move beyond the range of values of
x
for which data were collected, we
become less certain about the validity of the assumed model. Regression models are not
always valid for extrapolation purposes.
7.6. GENERAL REGRESSION ANALYSIS
It is evident from previous discussion that when the regression model is algebraically
linear, the associated least squares problem reduces to solving a set of linear equations.
Therefore, linear regression is popular in textbooks as well as in practice. It was often for
one to resort to variable transformations to reduce a nonlinear algebraic regression model
to a linear regression model. While it was mathematically genius in performing such a trans-
formation at a time when there is virtually no computing power, it is not acceptable concep-
tually today as computing power is vast available.
Modeling of a physical system with mathematical expressions is an important part of
research in science and engineering. When mathematical modeling is involved, there are
commonly free parameters left in model equations. Experimental data are utilized to offer
a closure to the system or problem under consideration. In order to determine the free param-
eters, one often seeks a simple equation that directly or indirectly relates to the experimental
data. For example, linear algebraic functions or equations (with respect to the free parame-
ters) are highly desired because linear regression can then be easily applied to determine
the free parameters. Because of the popularity in linear regression, experimental data are
usually converted to suit the needs for simplicity in regression. There are errors involved
in the experimental data. Conversion of the experimental data normally causes the error to
magnify. In most cases, the errors in the new derived quantities are not linearly related to
the errors in the original experimental data. To partially counter the indirect link in error,
data treatment would normally be applied by performing data smoothing prior to parameter
estimation. However, this exercise puts a high weight on data smoothing. The regression per-
formed based on the converted experimental data is not reliable because the error that is
minimized is not linearly related to the error in the original experimental data. Therefore,
the parameters obtained in such a fashion are not reliable. To increase the reliability of
such a parametric estimation, one normally turns to design or redesign a better experiment.
When the experimental error is “eliminated” or “much reduced,” the quality of the converted
(or derived) data bears less error. Therefore, the parameters estimated in this fashion can be
reliable. However, there is a limit in redesigning an experiment. The cost associated with
such an exercise is also high. Alternatively, to rectify the discrepancy in errors in the course
of regression, one may regress the experimental data directly rather than through some
derived quantities other than the original experimental data.
Regression by minimizing the variance of the experimental data around the regression
model prediction is the correct approach one should adopt. This approach in general renders
a nonlinear regression problem. Fortunately, we now know how to solve an optimization
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