Biology Reference
In-Depth Information
In this chapter, we present one of the most popular mathematical criteria
for estimating model parameters from data: the least-squares criterion.
We begin with a mathematical introduction, describing a measure
used to determine the optimal fit of the model. We consider linear models
first, and then extend the definitions to general nonlinear models,
using models of ligand binding and hemoglobin-oxygen binding as
examples.
I. DATA-FITTING TERMS, DEFINITIONS, AND EXAMPLES
Once a data set has been acquired and a hypothesis-driven mathematical
model developed, the next step is to fit the model to the data and
obtain values for the parameters that provide the best description of the
data.
Consider a hypothetical situation in which a linear model of the form
Y
has been determined to provide a description of
a biological phenomenon and the experimental data have been
collected. The variable X is said to be the independent variable, while Y is
the response or dependent variable. One of the objectives of the fitting
process is to determine the values of a and b that will fit the data points
best. If all the data points lie on a straight line, the slope a and the
vertical intercept b of this line will provide the best fit. In practice,
however, for any set of more than two data points, it is unrealistic to
expect them all to lie on a straight line. Even if the linear model
Y
¼
aX
þ
b
¼
G
ð
a
;
b
X
Þ
;
b describes the dependence between the variables X and Y
accurately, there will be some discrepancies (if nothing else, rounding
errors are always present). Figure 8-2 illustrates this situation. The
vertical deviations from the line Y
¼
aX
þ
¼
aX
þ
b are denoted by r
1
, r
2
,
r
n
...
Y
Y
=
aX
+
b
Y
4
r
4
r
5
Y
5
r
3
Y
3
r
2
Y
2
r
1
Y
1
0
X
X
1
X
2
X
3
X
4
X
5
FIGURE 8-2.
Vertical residuals for the linear model Y ¼ aX þ b and five data points.
