Geoscience Reference
In-Depth Information
subset of the data corresponding to these events which allowed each neural network to focus on a
particular behaviour (See and Openshaw 1999). Enormous pattern classification tasks, for exam-
ple, using census-based small area statistics for consumer behaviour discrimination, have however
proven to be difficult in unconstrained settings for conventional clustering algorithmic approaches -
even when using powerful computers (Openshaw and Wymer 1995). Unsupervised categorisation
methods have also been used to develop cartograms (Henriques et al. 2009), employed in carto-
graphic generalisation (Allouche and Moulin 2005; Sester 2005) and in classifying the environmen-
tal structure of landscapes (Bryan 2006).
13.2.3 f
function
a
PProxiMation
The task of function approximation is to create a generalised model of a known or unknown func-
tion. Suppose a set of
n
training patterns (input-output pairs) {(
x
1
,
y
1
), (
x
2
,
y
2
), …, (
x
n
,
y
n
)} have been
generated from an unknown function Φ(
x
) (subject to noise). Then the task of function approxima-
tion would be to find an estimate of that unknown function. Various spatial analysis problems require
function approximation (for more details, see Fischer 2006). Examples include spatial regression
and spatial interaction modelling (Openshaw 1993, 1998; Fischer and Gopal 1994; Openshaw 1998;
Fischer and Reggiani 2004) and modelling of the stage-discharge relationship for hydrological
purposes (Sudheer and Jain 2003; Goel 2011). Spatial interpolation would be another case in point,
for example, the production of continuous surfaces from point data for subsequent use in hydrogeo-
logical applications (Rizzo and Dougherty 1994), the interpolation of temperature surfaces (Rigol
et al. 2001) and estimation of population distributions using ancillary data (Merwin et al. 2009).
13.2.4 P
rediction
/f
orecaSting
In mathematical terms, given a set of
n
samples {
y
(
t
1
),
y
(
t
2
), …,
y
(
t
n
)} in a time sequence
t
1
,
t
2
, …,
t
n
,
the task of prediction or forecasting is to estimate the sample
y
(
t
) at some future time (often
t
n
+ 1).
Time series prediction is an important task and can have a significant impact on decision-making
with respect to regional development and policymaking. A great deal of research in this area has
also been concentrated on attempts at simulating the rainfall-runoff transformation process and in
hydrological modelling more generally. A number of recent reviews have appeared that capture the
growing literature on the use of CNNs in this area, for example, Maier et al. (2010) and Abrahart
et al. (2010, 2012). Spatiotemporal modelling examples would include forecasting the spread of aids
in Ohio (Gould 1994) and predicting rainfall output generated from a space-time mathematical
model (Gholizadeh and Darand 2009). CNNs are also used extensively in transportation model-
ling for the prediction of traffic flows, travel times and accidents; see, for example, Srinivasan et al.
(2004), Van Hinsbergen et al. (2009), Akgüngör and Doğan (2009) and Chan et al. (2012).
13.2.5 o
PtiMiSation
A wide variety of problems in GC can be posed as (non-linear) spatial optimisation problems.
The goal of an optimisation algorithm is to find a solution that satisfies a set of constraints such
that an objective function is maximised or minimised. The
travelling salesman problem
(Favata
and Walker 1991; Henrique et al. 2008; Abdel-Moetty 2010) and the question of finding optimal
site locations (Benjamin et al. 1995; Guerrero et al. 2000; Kuo et al. 2002) are classic analytical
examples (both being of a nonpolynomial-complete nature) where neural networks can provide
solutions. In these cases, the solution will often depend on a number of factors or possibilities
which must all be examined and tested, with the enormous number of possible combinations
often rendering such problems insoluble using conventional methods of spatial analysis (Murnion
1996). Similar problems and constraints would also be associated with the optimisation of complex
computer simulation models. For example, neural networks have been used to determine optimal
