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10.2 Aggregate Patterns and Conventional Representations
of Model Dynamics

Whether a model is based on deductive premises or inferred behaviours, any
new understanding of a given modelled system tends to be developed inductively.
Modellers examine model outputs, simplify them and then try to work out the cause
utilising a combination of hypothesis dismissal, refinement and experimentation.
For example, a modeller of a crowd of people might take all the responses of each
person over time and generate a single simple mean statistic for each person; these
might then be correlated against other model variables. If the correlation represents a
real causal connection, then varying the variables should vary the statistic. Proving
such causal relationships is not something we often have the ability to do in the
real world. During such an analysis, the simplification process is key: it is this that
reveals the patterns in our data. The questions are: how do we decide what needs
simplifying and, indeed, how simple to make it?
We can classify model results by the dimensionality of the outputs. A general
classification for social systems would be:
• Single statistical aggregations (1D)
• Time series of variables (2D)
• The spatial distributions of invariants (2D) or variables (3D)
• Spatio-temporal locations of invariants (3D) or variables (4D)
• Other behaviours in multidimensional variable space (nD)
For simplicity, this assumes that geographical spaces are essentially two-
dimensional (while recognising that physical space might also be represented along
linear features such as a high street, across networks or within a three-dimensional
topographical space for landforms or buildings). It should also be plain that in
the time dimension, models do not necessarily produce just a stream of data, but
that the data can have complex patternation. By their very nature, individual-level
models, predicated as they are on a life cycle, will never stabilise in the way a
mathematical model might (Uchmanski and Grimm 1996); instead models may run
away or oscillate, either periodically or chaotically.
Methods for aiding pattern recognition in data break down, again, by the
dimensionality of the data, but also by the dimensionality of their outputs. It is quite
possible to generate a one-number statistic for a 4D spatio-temporal distribution. In
some cases, the reduction of dimensionality is the explicit purpose of the technique,
and the aim is that patterns in one set of dimensions should be represented as closely
as possible in a smaller set of dimensions so they are easier to understand. Table 10.1
below presents a suite of techniques that cross this range (this is not meant to be an
exhaustive list; after all, pattern recognition is a research discipline of its own with
a whole body of literature including several dedicated journals).
To begin with, let us consider some examples which produce outputs in a single
dimension. In other words, techniques for generating global and regional statistics
describing the distribution of variables across space, either a physical space or a