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10 Understanding Simulation Results 211

If, on the other hand, we believe the relationships do not vary smoothly across
a modelled surface, we instead need to find unusual clusters of activity. The ability
to represent spatial clustering is of fundamental importance, for example, within
Schelling’s well-known model of segregation in the housing market (Schelling
1969). However clustering is often not so easy to demonstrate within both real
data and complex simulation outputs. The most recent techniques use, for example,
wavelets to represent the regional surfaces, and these can then be interpreted
for cluster-like properties. However, for socio-economic work amongst the best
software for cluster detection is the geographical analysis machine (GAM), which
not only assesses clustering across multiple scales but also allows assessment of
clustering in the face of variations in the density of the population at risk. For
example, it could tell us where transport network nodes were causing an increase in
sales of A, by removing regions with high sales caused by high population density
(the population “at risk” of buying A). Clusters can be mapped and their significance
assessed (Openshaw et al. 1988).
Often, simulations will be concerned with variations in the behaviour of systems,
or their constituent agents, over time. In common with physical systems, social and
economic systems are often characterised by periodic behaviour, in which similar
states recur, although typically this recurrence is much less regular than in many
physical systems. For example, economic markets appear to be characterised by
irregular cycles of prosperity and depression. Teasing apart a model can provide
nonintuitive insights into such cycles. For example, Heppenstall et al. (2006)
considered a regional network of petrol stations and showed within an agent
simulation how asymmetric cyclical variations in pricing (fast rises and slow falls),
previously thought to be entirely due to a desire across the industry to maintain
artificially high profits, could in fact be generated from more competitive profit
maximisation in combination with local monitoring of network activity. While it
is, of course, not certain these simpler processes cause the pattern in real life, the
model exploration does give researchers a new explanation for the cycles and one
that can be investigated in real petrol stations.
In trying to detect periodic behaviour, wavelets are rapidly growing in popularity
(Graps 2004). In general, one would assume that the state of the simulation can
be represented as a single variable which varies over time (let’s say the average
price of A). A wavelet analysis of either observed or model data would decompose
this trend into chunks of time at varying intervals, and in each interval the technique
identifies both a long-term trend and a short-term fluctuation. Wavelets are therefore
particularly suitable for identifying cycles within data. They are also useful as
filters for the removal of noise from data and so may be particularly helpful in
trying to compare the results from a stylised simulation model with observed data
which would typically be messy, incomplete or subject to random bias. It has been
argued that such decompositions are fundamentally helpful in establishing a basis
for forecasting (Ramsey 2002).
Wavelets are equally applicable in both two and three dimensions. For example,
they may be useful in determining the diffusion of waves across a two-dimensional
space and over time and can be used to analyse, for example, the relationship