Biomedical Engineering Reference
In-Depth Information
G(
B
)(1 -
B
)
d
y
t
= R(
B
)
w
t
,
[35]
As mentioned above (ยง3.1), ARMA and ARIMA models can be recast in state
space terms, so that our
y
is a noisy measurement of a hidden
x
(60). For these
models, both the dynamics and the observation functions are linear, that is,
x
t
+1
=
A
x
t
+ F
t
and
y
t
=
B
x
t
+ I
t
, for some matrices
A
and
B
. The matrices can be deter-
mined from the R and G parameters, though the relation is a bit too involved to
give here.
3.3.1.
Applicability of Linear Statistical Models
It is often possible to describe a nonlinear dynamical system through an
effective linear statistical model, provided the nonlinearities are cooperative
enough to appear as noise (61). It is an under-appreciated fact that this is at least
sometimes true even of turbulent flows (62,63); the generality of such an ap-
proach is not known. Certainly, if you care only about predicting a time series,
and not about its structure, it is always a good idea to try a linear model first,
even if you
know
that the real dynamics are highly nonlinear.
3.3.2.
Extensions
While standard linear models are more flexible than one might think, they
do have their limits, and recognition of this has spurred work on many exten-
sions and variants. Here I briefly discuss a few of these.
Long Memory
. The correlations of standard ARMA and ARIMA models
decay fairly rapidly, in general exponentially;
t S
r where U
c
is the corre-
lation time. For some series, however, U
c
is effectively infinite, and S(
t
)
t
-
B
for
some exponent B. These are
long-memory processes
, because they remain sub-
stantially correlated over very long times. These can still be accommodated
within the ARIMA framework, formally, by introducing the idea of
fractional
differencing, or, in continuous time, fractional derivatives (64,53). Often long-
memory processes are self-similar, which can simplify their statistical estima-
tion (65).
Volatility
. All ARMA and even ARIMA models assume constant variance.
If the variance is itself variable, it can be worthwhile to model it.
Autoregres-
sive conditionally heteroscedastic
(ARCH) models assume a fixed mean value
for
y
r
, but a variance which is an auto-regression on
y
t
2
.
Generalized ARCH
(GARCH) models expand the regression to include the (unobserved) earlier
variances. ARCH and GARCH models are especially suitable for processes that
display
clustered volatility
, periods of extreme fluctuation separated by
stretches of comparative calm.
tt
/
()
,
c
