Biomedical Engineering Reference
In-Depth Information
3.1. The State-Space Picture
The
state
is a vector-valued function of time,
x
t
. In discrete time, this
evolves according to some map,
x
t
+1
F
(
x
t
,
t
,F
t
),
[10]
where the map
F
is allowed to depend on time
t
and a sequence of independent
random variables F
t
. In continuous time, we do not specify the evolution of the
state directly, but rather the rates of change of the components of the state,
dx
=
Fxt
(,, ).
t
F
[11]
dt
Since our data are generally taken in discrete time, I will restrict myself to con-
sidering that case from now on; almost everything carries over to continuous
time naturally. The evolution of
x
is, so to speak, self-contained, or more pre-
cisely Markovian: all the information needed to determine the future is con-
tained in the
present
state
x
t
, and earlier states are irrelevant. (This is basically
how physicists
define
"state" (46).) Indeed, it is often reasonable to assume that
F
is independent of time, so that the dynamics are
autonomous
(in the terminol-
ogy of dynamics) or
homogeneous
(in that of statistics). If we could look at the
series of states, then, we would find it had many properties which made it very
convenient to analyze.
Sadly, however, we do not observe the state
x
; what we observe or measure
is
y
, which is generally a noisy, nonlinear function of the state:
y
t
=
h
(
x
t
,I
t
),
where I
t
is measurement noise. Whether
y
, too, has the convenient properties
depends on
h
, and usually
y
is
not
convenient. Matters are made more compli-
cated by the fact that we do not, in typical cases, know the observation function
h
, nor the state-dynamics
F
, nor even, really, what space
x
lives in. The goal of
time-series methods is to make educated guess about all these things, so as to
better predict and understand the evolution of temporal data.
In the ideal case, simply from a knowledge of
y
, we would be able to iden-
tify the state space, the dynamics, and the observation function. As a matter of
pure mathematical possibility, this can be done for essentially arbitrary time
series (48,49). Nobody, however, knows how to do this with complete generality
in practice. Rather, one makes certain assumptions about, say, the state space,
which are strong enough that the remaining details can be filled in using
y
. Then
one checks the result for accuracy and plausibility, i.e., for the kinds of errors
which would result from breaking those assumptions (50).
Subsequent parts of this section describe classes of such methods. First,
however, I describe some of the general properties of time series, and general
measurements which can be made upon them.
