Biomedical Engineering Reference
In-Depth Information
dure is to work forward through the data, trying to predict the value at each time
on the basis of the past of the series; the sum of the squared differences between
these predicted values
ˆ
and the actual ones
y
t
forms the empirical loss:
T
L
=
(
y
ˆ
y
)
2
.
[30]
t
t
i
=
1
For this loss function, in particular, there are very fast standard algorithms, and
the estimates of G and R converge on their true values, provided one has the right
model order.
This leads naturally to the question of how one determines the order of
ARMA model to use, i.e., how one picks
p
and
q
. This is precisely a model se-
lection task, as discussed in §2. All methods described there are potentially ap-
plicable; cross-validation and regularization are more commonly used than
capacity control. Many software packages will easily implement selection ac-
cording to the AIC, for instance.
The power spectrum of an ARMA(
p,q
) process can be given in closed form:
q
(1
+
R
e
ik
O
)
2
T
2
k
f
()
O
=
.
[31]
k
p
=
1
2
Q
(1
+
G
e
O
k
)
2
k
k
=
1
Thus, the parameters of an ARMA process can be estimated directly from the
power spectrum, if you have a reliable estimate of the spectrum. Conversely,
different hypotheses about the parameters can be checked from spectral data.
All ARMA models are weakly stationary; to apply them to nonstationary
data one must transform the data so as to make it stationary. A common trans-
formation is
differencing
, i.e., applying operations of the form
/
y
t
=
y
t
-
y
t
-1
,
[32]
which tends to eliminate regular trends. In terms of the back-shift operator,
/
y
t
= (1 -
B
)
y
t
,
[33]
and higher-order differences are
/
d
y
t
= (1 - B)
d
y
t
.
[34]
Having differenced the data to our satisfaction, say
d
times, we then fit an
ARMA model to it. The result is an
autoregressive integrated moving average
model, ARIMA(
p,d,q
) (59), given by
