Geoscience Reference
In-Depth Information
Box 8.1 Adaptive Gain Parameter Estimation for Real-Time Forecasting
Consider the simple first-order transfer function model:
y
t
=
G
t
b
o
+
b
1
z
−1
1
u
t
−
ı
(B8.1.1)
−
a
1
z
−1
where
y
t
is the predicted output variable,
u
t
−
ı
is the input variable (effective rainfall or upstream
river level or flow),
z
is the backward difference operator (see Box 4.1),
a
1
,
b
o
and
b
1
are
parameters fixed by previous calibration and
G
t
is a time variable
gain
.
The time variable gain can be estimated in real time by the methods of time variable param-
eter estimation developed in Young (1984; see also Young (2011a) and Box 4.3). The algorithm
is a form of recursive filtering of the gain parameter
G
in that the gain at time
t
,
G
t
, depends on
G
t
−1
at the previous time step, together with a function of the current prediction error
y
t
−
y
t
where
.
The algorithm has the form of predictor-corrector equations as follows:
Predictor step:
y
t
is the observed flow at time
t
G
t
|
t
−1
=
G
t
−1
(B8.1.2)
P
t
|
t
−1
=
P
t
−1
+
Q
(B8.1.3)
Corrector step:
y
t
−
G
t
|
t
−1
y
t
P
t
|
t
−1
y
t
G
t
=
G
t
|
t
−1
+
(B8.1.4)
2
t
1
+
P
t
|
t
−1
y
P
t
|
t
−1
y
t
2
1
P
t
=
P
t
|
t
−1
−
(B8.1.5)
+
P
t
|
t
−1
y
2
t
The degree to which the time variable gain
G
t
is allowed to change between time steps is
controlled by the value of
Q
which is a noise variance ratio (NVR). The higher the value of
Q,
is allowed to
decay away. Thus a large value of NVR means that there is a short memory and less filtering of
the changes in
the faster the memory effect of previous values of
G
at times,
t
−
1
,t
−
2
,...
was allowed to
vary with the magnitude of the prediction error so that the larger the prediction error at a time
step, the more rapidly
G.
In the Dumfries study by Lees
et al.
(1994) the value of
Q
was allowed to change. Romanowicz
et al.
(2008) have shown how
a representation of a heteroscedastic noise variance ratio can be included in this approach.
One advantage of this type of methodology is that an estimate of the uncertainty in the
predictions can be calculated and updated recursively. If we define the
G
n
step ahead prediction
error as
t
+
n
=
y
t
+
n
−
y
t
+
n
(B8.1.6)
then the variance of
n
is given by
1
2
t
2
t
+
n
var
(
n
)
=
+
P
t
+
n
y
(B8.1.7)
where
2
t
is the current forecast error variance which is estimated from the further recursion
+
P
t
e
1
t
=
2
2
t
−
2
t
−
2
t
−
(B8.1.8)
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