Biomedical Engineering Reference
In-Depth Information
scientists have to statistics (e.g., the "research methods" courses traditional in
the life and social sciences) typically deal with systems with only a few vari-
ables and with explicit assumptions of independence, or only very weak depend-
ence. The kind of modern methods we have just seen, amenable to large systems
and strong dependence, are rarely taught in such courses, or even mentioned.
Considering the shaky grasp many students have on even the basic principles of
statistical inference, this is perhaps wise. Still, it leads to even quite eminent
researchers in complexity making disparaging remarks about statistics (e.g.,
"statistical hypothesis testing, that substitute for thought"), while actually rein-
venting tools and concepts which have long been familiar to statisticians.
For their part, many statisticians tend to overlook the very
existence
of
complex systems science as a separate discipline. One may hope that the in-
creasing interest from both fields on topics such as bioinformatics and networks
will lead to greater mutual appreciation.
3.
TIME SERIES ANALYSIS
There are two main schools of time series analysis. The older one has a long
pedigree in applied statistics (46), and is prevalent among statisticians, social
scientists (especially econometricians), and engineers. The younger school, de-
veloped essentially since the 1970s, comes out of physics and nonlinear dynam-
ics. The first views time series as samples from a stochastic process, and applies
a mixture of traditional statistical tools and assumptions (linear regression, the
properties of Gaussian distributions) and the analysis of the Fourier spectrum.
The second school views time series as distorted or noisy measurements of an
underlying dynamical system, which it aims to reconstruct.
The separation between the two schools is in part due to the fact that, when
statistical methods for time series analysis were first being formalized, in the
1920s and 1930s, dynamical systems theory was literally just beginning. The
real development of nonlinear dynamics into a powerful discipline has mostly
taken place since the 1960s, by which point the statistical theory had acquired a
research agenda with a lot of momentum. In turn, many of the physicists in-
volved in experimental nonlinear dynamics in the 1980s and early 1990s were
fairly cavalier about statistical issues, and some happily reported results which
should have been left in their file-drawers.
There are welcome signs, however, that the two streams of thought are coa-
lescing. Since the 1960s, statisticians have increasingly come to realize the vir-
tues of what they call "state-space models," which are just what the physicists
have in mind with their dynamical systems. The physicists, in turn, have become
more sensitive to statistical issues, and there is even now some cross-
disciplinary work. In this section, I will try, so far as possible, to use the state-
space idea as a common framework to present both sets of methods.
