Genetic Programming - GeoComputation - page 182

and direction of relationship to that found in the original dataset, then the model can be considered
to pass the sensitivity analysis test and may be classified as a rational model. The limitation with this
technique is that it does not examine multiple-variable effects on the predicted values. One tech-
nique that may be considered for such analyses is the generalised likelihood uncertainty estimation
approach (Beven and Binley, 1992).
8.3.5 S tage 5 M odel t eSting and e Valuation
Traditionally, data-driven modelling involves using a cross-validation data set that is run in paral-
lel with model development. Doing so determines the stopping point and means that models are
tested at the same time as they are developed (c.f. neural networks). This approach is not required
in GP, since evolutionary procedures and convergence usually result in a reasonable sub-optimal
solution that is less likely to be overfitted (Abrahart et al., 1999). However, this cannot be guaran-
teed (Fernando et al., 2009) and users should remain wary of such possibilities. GP model testing
is the application of evolved models to out-of-sample test period data (often termed unseen data)
after they have been developed. The results are then used to evaluate model performance, using
the same methods as were used for the development period data, but excluding sensitivity analysis
procedures, since the model structure itself remains unchanged.
The model evaluation completed for both development and test period data should then be
assembled to form lines of evidence that can then be evaluated and reported in support of the deci-
sion to accept or reject any proposed solution - and, finally, used to select a single preferred or
recommended model.
8.4 CASE STUDIES
Three GP case studies are summarised to show different ways in which symbolic regression can
be employed in, and contribute to, geographical enquiry. GeneXproTools software is used in the
first study. It provides a GEP exploration of pan evaporation in New Mexico, United States. Unique
to GEP is the use of Sub-ETs in the evolved solutions. This case study will show how these units
function both individually and as a whole. The second case study uses Eureqa software to perform
rainfall-runoff modelling for the Annapolis River in Nova Scotia, Canada. Eureqa ranks model per-
formance by measuring the complexity of each solution which means that the user can select which
model is preferred based on complexity as well as accuracy. The third case study is a summary of a
spatial interaction model that was originally presented in the first edition of this chapter. It is retained
because it facilitates a comparison between an early example of GP, programmed using Fortran, and
more contemporary GP explorations using proprietary software packages. Additionally, the final
case study involves a spatial analysis problem in human geography, in direct contrast to our more
recent meteorological and hydrological modelling examples, which form part of physical geography.
8.4.1 e StiMating P an e VaPoration
Evaporation is a natural process that occurs on the surface of a water body due to the synergistic
effects of temperature, relative humidity, wind speed and solar radiation (Fu et al., 2009). Pan
evaporation is an estimate of evaporation made using a purpose built pan installed to measure
evaporative losses ( E PA N ) (World Meteorological Organization, 2008).
The study site was Elephant Butte Reservoir, situated within the southern reaches of the Rio Grande
Valley, in Sierra County, New Mexico, United States (Figure 8.5). This 14,500 ha reservoir has a north-
south axis and holds ~2.5 × 10 9 m 3 of water at full capacity. It is situated at ~1345 m above mean sea level.
Two weather stations are positioned near the reservoir where standard meteorological measurements
and E PA N data are recorded daily: North Lake Weather Station (NLWS) (33°17′50″N, 107°11′38″W)
and South Lake Weather Station (SLWS) (33°8′45.52″N, 107°11′3.44″W). NLWS is located ~1.7 km