Biomedical Engineering Reference
In-Depth Information
The natural combination of the two types of model is the
autoregressive
moving average model
, ARMA(
p,q
):
p
q
y
=+
B
G
y
++
w
R
w
.
[25]
t
k
t
k
t
k
t
k
k
1
k
=
1
This combines the oscillations of the AR models with the correlated driving
noise of the MA models. An AR(
p
) model is the same as an ARMA(
p
,0) model,
and likewise an MA(
q
) model is an ARMA(0,
q
) model.
It is convenient, at this point in our exposition, to introduce the notion of the
back-shift operator
B
,
By
t
=
y
t
-1
,
[26]
and the
AR and MA polynomials
,
p
G
()
z
=
1
G
z
k
,
[27]
k
k
=
1
q
=+
,
k
R
()
z
1
R
z
[28]
k
k
=
1
respectively. Then, formally speaking, in an ARMA process is
G(
B
)
y
t
= R(
B
)
w
t
.
[29]
The advantage of doing this is that one can determine many properties of an
ARMA process by algebra on the polynomials. For instance, two important
properties we want a model to have are
invertibility
and
causality
. We say that
the model is invertible if the sequence of noise variables
w
t
can be determined
uniquely from the observations
y
t
; in this case we can write it as an MA()
model. This is possible just when R(
z
) has no roots inside the unit circle. Simi-
larly, we say the model is causal if it can be written as an AR() model, without
reference to any
future
values. When this is true, G(
z
) also has no roots inside the
unit circle.
If we have a causal, invertible ARMA model, with known parameters, we
can work out the sequence of noise terms, or
innovations
w
t
associated with our
measured values
y
t
. Then, if we want to forecast what happens past the end of
our series, we can simply extrapolate forward, getting predictions
y
++
etc.
Conversely, if we knew the innovation sequence, we could determine the pa-
rameters G and R. When both are unknown, as is the case when we want to fit a
model, we need to determine them jointly (55). In particular, a common proce-
ˆ
,
ˆ
,
T
1
T
2
