Biomedical Engineering Reference
In-Depth Information
6.2. Analytical Techniques
Naturally enough, analytical techniques are not among the tools that first
come to mind for dealing with complex systems; in fact, they often do not come
to mind at all. This is unfortunate, because a lot of intelligence has been devoted
to devising approximate analytical techniques for classes of models that include
many of those commonly used for complex systems. A general advantage of
analytical techniques is that they are often fairly insensitive to many details of
the model. Since any model we construct of a complex system is almost cer-
tainly much simpler than the system itself, a great many of its details are just
wrong. If we can extract nontrivial results insensitive to those details, we have
less reason to worry about this.
One particularly useful, yet neglected, body of approximate analytical tech-
niques relies on the fact that many complex systems models are Markovian. In
an agent-based model, for instance, the next state of an agent generally depends
only on its present state, and the present states of the agents it interacts with. If
there is a fixed interaction graph, the agents form a Markov random field on that
graph. There are now very powerful and computationally efficient methods for
evaluating many properties of Markov chains (58,142), Markov random fields
(143), and (closely related) graphical models (144)
without
simulation. The re-
cent topics of Peyton Young (145) and Sutton (146) provide nice instances of
using analytical results about Markov processes to solve models of complex
social systems, without impractical numerical experiments.
6.3. Comparisons with Data
6.3.1.
General Issues
We can only compare particular aspects of a model of a system to particular
kinds of data about that system. The most any experimental test can tell us,
therefore, is how similar the model is to the system
in that respect
. One may
think of an experimental comparison as a test for a
particular
kind of
error
, one
of the infinite number of mistakes which we could make in building a model. A
good test is one which is very likely to alert us to an error, if we have made it,
but not otherwise (50).
These ought to be things every schoolchild knows about testing hypotheses.
It is very easy, however, to blithely ignore these truisms when confronted with,
on the one hand, a system with many strongly interdependent parts, and, on the
other hand, a model that tries to mirror that complexity. We must decide which
features of the model
ought
to be similar to the system, and how similar. It is
important not only that our model be able to adequately reproduce those phe-
