Environmental Engineering Reference
In-Depth Information
course, used widely for modelling dynamic systems in the
sciences, social sciences and engineering.
7.2 Philosophies of science
and modelling
In considering questions of complexity and simplicity in
mathematical modelling, it is important to note how the
mathematical modelling of natural systems has developed
over the past few centuries. In this regard, Young (2002)
points out that two main approaches to mathematical
modelling can be discerned in the history of science;
approaches that, not surprisingly, can be related to the
more general deductive and inductive approaches to
scientific inference that have been identified by philoso-
phers of science from Francis Bacon (1620; see also
Montague, 1854) to Karl Popper (1959) and Thomas
Kuhn (1962):
7.3 Statistical identification, estimation
and validation
The statistical approach to modelling assumes that the
model is stochastic: in other words, no matter how good
the model and how low the noise on the observational data
happen to be, a certain level of uncertainty will remain
after modelling has been completed. Consequently, full
stochastic modelling requires that this uncertainty, which
is associated with both the model parameters and the
stochastic inputs, should be quantified in some manner
as an inherent part of the modelling analysis.
In the statistical, time-series literature, such a stochastic
modelling procedure is normally considered in two main
stages: identification of an appropriate, identifiable model
structure; and estimation (optimization, calibration) of
the parameters that characterize this structure, using some
form of estimation or optimization (see also Chapters 3
and 8). Normally, if the data provision makes it possible,
a further stage of validation (or conditional validation:
see later) is defined, in which the ability of the model
to explain the observed data is evaluated on data sets
different to those used in the model identification and
estimation stages. In this section, we outline these three
stages in order to set the scene for the later analysis. This
discussion is intentionally brief, however, since the topic
is so large that a comprehensive review is not possible in
the present context.
The hypothetico-deductive approach. Here, the
a priori
conceptual model structure is effectively
a theory of behaviour based on the perception of
the environmental scientist/modeller and is strongly
conditioned by assumptions that derive from current
environmental science paradigms.
•
The inductive approach. Here, theoretical preconcep-
tions are avoided as much as possible in the initial stages
of the analysis. In particular, the model structure is not
prespecified by the modeller but, wherever possible, it is
inferred directly from the observational data in relation
to a more general class of models. Only then is the
model interpreted in a physically meaningful manner,
most often (but not always) within the context of the
current scientific paradigms.
•
The DBM approach to modelling is of this latter
inductive type and it forms the basis for the research
described in the rest of this paper. Previous publi-
cations (Young, 1978; Beck, 1983; Young
et al
., 1996;
Young, 1998; Young and Ratto, 2008 and the references
therein) map the evolution of this DBM philosophy
and its methodological underpinning in considerable
detail. As these references demonstrate, DBM models
can be of various kinds depending upon the nature of
the system under study. In the context of the present
paper, however, they take the form of linear and non-
linear, stochastic transfer function (TF) representations
of environmental systems. Such TFs can be formulated
in continuous or discrete-time terms, and are simply
convenient 'shorthand' representations of, respectively,
differential equation models or their discrete-time, dif-
ference equation equivalents. Both types of model are, of
7.3.1 Structureandorder identification
In the DBM approach to modelling, the identification
stage is considered as an essential prelude to the later stages
of model building. It usually involves the identification of
the most appropriate model order, as defined in dynamic
system terms. However, the model structure itself can be
the subject of the analysis if this is also considered to be
ill-defined. In the DBM approach, for instance, the nature
of linearity and nonlinearity in the model is not assumed a
priori (unless there are good reasons for such assumptions
based on previous data-based modelling studies). Rather
it is identified from the data using nonparametric and
parametric statistical estimation methods based on a
suitable, generic model class. Once a suitable model
structure has been defined within this class, there are
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