Biology Reference
In-Depth Information
We cannot solve the problems we have created with the same
thinking that created them.
Albert Einstein (1879-1955)
In the preceding chapters, we have discussed the need for
using mathematical models; outlined some of the
important questions involved in constructing models; and
analyzed a variety of models related to population
ecology, epidemiology, genetics, endocrinology, and
neonatology. Whatever the application, all of these
models contained a set of numerical quantities referred
to as the model parameters. For example, our first
population growth model assumed a constant net per
capita growth rate r, which was a parameter for the
model. The new infections rate
Chapter 8
a
and the recovery rate
b
we used to construct the SIS and SIR models are
parameters; the association constant K
a
used in the
hemoglobin oxygenation models is likewise a parameter.
LIGAND BINDING,
DATA FITTING, AND
LEAST-SQUARES
ESTIMATES OF MODEL
PARAMETERS
We have emphasized that every model is built upon
certain assumptions and involves a certain number of
parameters. The specific values of the model parameters
may be unknown initially or be group-dependent. For
example, it should be expected that Mexico's net per
capita growth rate is different from Sweden's, because
population growth is driven by socioeconomic, cultural,
environmental, and other factors that differ substantially.
Also, not all assumptions in a dynamic model may be
valid for all time ranges; with the unlimited population
growth model, we modified some assumptions and
improved the model.
Data-Fitting Terms, Definitions, and
Examples
A Ligand-Binding Example
A Primer for Solving Nonlinear
Equations
Weighted Least-Squares Criterion and
the Gauss-Newton Methods for
Weighted Least Squares
Model validation is a critical part of the modeling process.
In general, validation requires gathering sufficient data
through carefully designed experiments and then
applying statistical techniques to determine how well the
model describes the data. As we have seen, model
predictions generally differ from experimental
measurements, but model parameters can be estimated
from the data to provide the best fit between actual
and predicted values. For the population growth models,
we estimated model parameters using averaging
techniques aimed at obtaining the best visual fit.
Objectives of the Data-Fitting
Procedures
Appendix: Basic Matrix Arithmetic
This approach had several major limitations. First, it did
not provide any information on whether the calculated
parameter estimates could be improved, because no
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