Environmental Engineering Reference
In-Depth Information
relation to the main objective of the forecasting
system design - namely the minimization of the
multi-step prediction errors. In the illustrative and
much simpler single-input, single-site example
given later in this chapter, however, the hyper-
parameters are 'tuned' manually in order to better
explain their effect on the forecasts.
State and output updating
In the above KF equations (Equations 9.4a
and 9.4b), the model parameters that define the
state space matrices F
k
and G
k
in Equation 9.3 are
known initially from the model identification and
estimation analysis based on an estimation data-
set. However, by embedding the model equations
within the KF algorithm, we have introduced ad-
ditional, unknown parameters, normally termed
'hyper-parameters' to differentiate them from the
model parameters.
2
In this example, these hyper-
parameters are the variance
s
Parameter updating
Within a hydrological context, Time-Variable
Parameter (TVP) parameter estimation methods
have been implemented recursively in two main
ways: first, by joint state-parameter estimation
using either state augmentation (as in the EKF),
or by the use of parallel, interactive filters (e.g.
Todini 1978; Liu and Gupta 2007); and second, by
estimating the parameters separately from the
state estimation (e.g. Young 1984, 1999a, 1999b,
2002; Romanowicz et al. 2006; Lin and Beck 2007).
Here, we will consider only the second cate-
gory, where the TVPs are estimated separately to
the state variables. This decision is based on two
factors: first, separate estimation provides a more
flexible approach to real-time updating; and sec-
ond, it is supported by conclusions reached in one
of the most recent of the above publications (Lin
and Beck 2007). Beck and his co-workers have
previously favoured a specific, improved formula-
tion of the EKF approach, but they have now
developed a powerful Recursive Prediction Error
(RPE) algorithm inspired by the ideas of
Ljung (1979), who carried out early research aimed
at overcoming some of the 'notorious difficulties
of workingwith the EKF as a parameter estimator'.
On the basis of experience, over many years, Lin
and Beck (2007) conclude that 'as a parameter
estimator, the RPE algorithm has many advan-
tages over the conventional EKF'.
The RPE approach of Lin and Beck has some
similarity with an alternative method of separate
TVP estimation, the Refined Instrumental Vari-
able (RIV) algorithm (see, e.g., Young 1984, 2008).
Both are optimal maximum likelihood estimation
algorithms but are not limited in this sense; both
can be applied to both discrete-time and continu-
ous-time models; and finally, both model the
k
and the elements of
the covariance matrix Q
k
. In practical terms, it is
normally sufficient to assume that Q
k
is purely
diagonal in form, where the diagonal elements
specify the nature of the stochastic inputs to the
state equations and so define the level of uncer-
tainty in the evolution of each state (in the later
illustrative tutorial example, the 'quick' and
'slow' water flow states). The inherent state adap-
tion of the KF arises from the presence of the Q
k
hyper-parameters, since these allow the estimates
of the state variables to be adjusted to allow for
presence and effect of the unmeasured stochastic
disturbances that naturally affect any real system.
Clearly, the hyper-parameters have to be esti-
mated in some manner on the basis of the data.
One well-known approach is to exploit Maximum
Likelihood (ML) estimation based on Prediction
Error Decomposition (see Schweppe 1965, 1973;
Young 1999b). Another, used in the quite complex,
multi-input, multi-site catchment network of the
River Severn forecasting system (Romanowicz
et al. 2006), is to optimize the hyper-parameters
by minimizing the variance of the multi-step-
ahead forecasting errors. In effect, this optimizes
the memory of the recursive estimation and fore-
casting algorithm (see, e.g., Young and Pedre-
gal 1999) in relation to the rainfall-flow or water
level data. The main advantage of this latter ap-
proach is, of course, that the integrated model-
forecasting algorithm is optimized directly in
2
Of course this differentiation is rather arbitrary since themodel
is inherently stochastic and so these parameters are simply
additional parameters introduced to define the stochastic inputs
to the model when it is formulated in this state space form.
