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in estimating the probability of predicting a new observation. However, the likelihood function that is
appropriate depends on defining an appropriate structure for the modelling errors.
Underlying the development of the likelihood functions used in statistical likelihood approaches is
the idea that there is a “correct” model, focussing attention on the nature of the errors associated with
that model. Ideally, we would hope to find a model with zero bias and purely random errors with
minimum variance and no autocorrelation. The likelihood for this and more complex cases is developed in
Box 7.1. Statisticians have also suggested incorporating a
model inadequacy function
into the analysis for
the case where there is a recognisable and consistent structure in the residuals (Kennedy and O'Hagan,
2001). This is usually a form of additive function that serves as a statistical compensation for epistemic
error in the model or input data. The predictions might be improved in this way but it means adding a
non-physical model component. The rainfall-runoff modeller might better use an inadequacy function as
a guide to how a model structure might be improved. More discussion of statistical approaches to model
calibration, within the Bayesian framework, is given in Section 7.7.
Remember that all these performance or likelihood measures are aimed at providing a relative measure
of model performance. Statistical likelihoods also attempt to estimate the probability of a new observation
in an objective way. That objectivity is undermined by the epistemic nature of some of the sources of
uncertainty, so that we cannot be sure that the error characteristics in either calibration or prediction
have a stationary structure. In general, therefore, we should aim to use measures that reflect the aims of a
particular application (and any identified structure in the residuals) in an appropriate way. Even qualitative
or “soft” information of different types that cannot easily be reduced to a statistical likelihood might be
useful in model calibration (Seibert and McDonnell, 2002; Vache and McDonnell, 2006; Winsemius
et
al.
, 2009). There is no universal performance measure and, whatever choice is made, there will be an
effect on the relative goodness-of-fit estimates for different models and parameter sets, particularly if
an optimum parameter set is sought. Section 7.4 examines the techniques for finding optimal parameter
sets, after which a more flexible approach to model calibration is discussed.
7.4 Automatic Optimisation Techniques
A full description of all the available techniques for automatic optimisation is well beyond the scope of
this topic, particularly since we have already noted that the concept of the optimum parameter set may
not be a particularly useful one in rainfall-runoff modelling. In this section, we give just a brief outline of
the algorithms available. For more specifics, descriptions of different algorithms are available from Press
et al.
(1992) and Sen and Stoffa (1995); a discussion of techniques in respect of hydrological models is
given by Sorooshian and Gupta (1995).
7.4.1 Hill-Climbing Techniques
Hill-climbing techniques for parameter calibration have been an important area of research since the
start of computer modelling in the 1960s. Hill climbing from any point on the response surface requires
knowledge of the gradient of that surface so that the algorithm knows in which direction to climb. The
available techniques may be classified into two basic types.
Gradient algorithms require the gradient of the response surface to be defined analytically for every
point in the parameter space. Mathematically, this requires that an analytical expression be available for
the differential of the model output with respect to each parameter value. These methods are not generally
used with hydrological models since it is often impossible to define such differentials analytically for
complex model structures. Much more commonly used are “direct search” algorithms that search along
trial directions from the current point with the aim of finding improved objective function values. Different
