Transcript GeoInformatics

Programming for
Geographical Information Analysis:
Advanced Skills
Online mini-lecture: Introduction to Complexity
Dr Andy Evans
Understanding the world
Geographical systems involve lots of components…
controlled by lots of variables…
and related in complicated ways.
Complex Systems
Complex Systems are made of interacting individual
components whose behaviour can be understood as organised
at some scale but not at others.
Example
City development is
plainly very complicated.
Yet, if we rank city size
by population and plot
ranks against population
we get this graph.
This was first shown by
linguist George Zipf.
O’Connor, 2009
Example
~ a log-linear distribution (inverse power law).
Rank  1 / sizea
(where a is ~1)
Is this a feature of the maths?
We might not be surprised at this, afterall the rank is
based on size.
But why can’t we have a distribution with 1 large city
and then lots of a similar size?
Example
Even more confusing when we look at the dynamic size
changes of component cities.
Rank rule consistent.
But massive changes
in rank for individual cities.
(Batty, 2006)
Complex Systems
Systems of complicated individual dynamics, but a
clear* overall pattern.
The production of these patterns is “Emergence”.
Where these patterns are largely inevitable result of
internal drivers we might talk of a “self-organising”
system.
*Note that “clear” doesn’t always mean simple.
Complexity
“Complex systems” might be classified by degree of organised
complexity (Weaver, 1948).
Usually this is quantified by some degree of order.
Systems not showing organisation might be classified by
degree of disorganised complexity.
Usually this is quantified by simplicity of statistical or
other description.
Complexity
Various attempts to capture this with one number but
they usually end up only capturing one of these aspects.
Kolmogorov algorithmic complexity:
shortest program needed to generate the data.
Complexity
Generally combining these isn’t simple.
Low structural
complexity
Low statistical
complexity
Complexity
Also sometimes the same system can have very different
complexity dependent on our quality of understanding.
Pseudo-random numbers can be described with a simple
program
But real random number might be indescribably complex.
Most measures treat randomness as either the greatest or
lowest complexity.
Do we even belief it exists?
Does it just represent our ignorance?
Complex Systems
Is complexity more generally a description of our ignorance?
Complex Systems are made of interacting individual components
whose behaviour can be understood* as organised† at some scale
but not at others #.
*
i.e. This is about our ability to perceive something.
†
Patterns – which aren’t a “real” thing anyhow, just a perceptual
convenience.
#
Our inability to see the connections between interactions at one
level and patterns at others.
Complex Systems
Increased computer connectivity, power, and storage
has allowed greater capturing of individual behaviour.
We now have tools to see the large and small scales.
What we lack are tools for seeing the connections
between them.
Complex Systems
What makes it hard to see the links?
Non-linearity: anything where the inputs are not linearly
proportional to the outputs.
Thresholds; Rule-based; Exponential; Random etc.
Non-linearity leads to “chaos” /chaotic dynamics:
Irregular dynamics where the system keeps changing
without exact repetition.
Loss of information
The usual assumption is that non-linear systems are just
too complicated to follow.
We shouldn’t assume this is true just because a
mathematician has said it.
So where is the loss of information?
Non-linearity
One of the problems with non-linear systems is that small
variations in starting conditions become large variations in final
results [quantified by Lyapunov exponents].
Measurement errors therefore become large.
Rounding errors on computers become important in modelling.
Non-linear calculations involving large numbers of objects take a
lot of processing power.
A lot of solutions to non-linear problems use mathematical
conveniences that lose information (for example, the
approximation of functions with Taylor series).
Loss of information
The use of aggregate mathematics in
models/classification stops us tracking individual objects
through models or data.
Sometimes this doesn’t matter:
Amount of money in a bank.
Sometimes it does:
People entering and leaving a crowd.
Understanding Complex Systems
But given non-linearities, how can we hope to understand such systems?
Because real systems are understandable!
Why?
They sometimes have homeostatic control and dampening.
They sometimes have self-organised criticality.
They sometimes reach equilibrium.
They sometimes have attractors.
At the moment, because of the loss of information, techniques to look at
these are the ways most complex systems are analysed.
Homeostasis and dampening
Homeostasis: the stability arising in a system through internal control.
Classically achieved through negative feedback and judicious
positive feedback.
Dampening: the reduction of fluctuations, through, for example:
Averaging of inputs.
Buffering.
The combination of negatively-correlated variables.
Other dissipative (energy-reducing) processes.
Strangely tools to identify homeostatic and dampening processes are few
and far between, despite the fact that they seem to be the one reason we
can model anything.
Catastrophe Theory
Systems can often find themselves leaping from one
state suddenly to another with little extra input.
1980’s saw the rise of Catastrophe Theory in geography
under Alan Wilson.
Scheffer et al. (2009) suggest it may be possible to
recognise this coming in real systems.
Self-organising Criticality
With SOC, internal drivers of system keep it in a state where
movement from the state engages processes to put it back into
the state, so it hangs on the edge of catastrophe.
Generally where the propagation of the noise kicking the
system is dampened by the state the system is kicked into.
Classic example is a sandpile.
When extra sand is added and the slope exceeds the angle of
friction, avalanches reduce the slope.
Equilibrium
These processes tend to lead systems to a stable state, even
when kicked slightly.
While homeostasis implies the
same state, with other controls
this isn’t always so.
Petrol price modelling after
a price change two stations.
Note higher equilibrium price.
Equilibrium
Equilibrium can be:
Static:
the system is at rest;
no energy running through the system.
Dynamic:
inputs = outputs;
the system runs in a steady state.
Attractors
Equilibria are examples of simple attractors: states that
systems converge on.
Given that:
non-linearities often lead to chaotic behaviour;
non-linearities mean that small variations in
starting conditions lead to large variations.
You’d imagine these were rare.
Attractors
You’d be wrong.
Chaotic systems rarely reach a single point without strong selforganised criticality, but...
They often move close to (but not exactly to):
a regular state;
a trajectory through several repeated states.
These states are called “strange attractors” to distinguish them
from simple attractors.
Phase space
Rather than look at changes of a system over time we can
look at the state of the system: its variable values in variable
or “phase space”*
* “state space” tends to be reserved to describe the variable
configuration of a single example of a system.
Phase space
For example, rather than plotting traffic volumes and speeds over
time on two graphs for a location, we plot the movement over
time in volume / speed space.
We can plot the whole trajectory over time.
Or sample at fixed intervals and plot as points.
Nair et al., 2001 show one state for
the traffic on the San Antonio freeway
on weekends, but an extra
“congested” state on weekdays. The
question is can the system flip into /
out of congestion through internal
dynamics?
Stroboscopic map
Where systems are periodic, we can sample at the
periodicity and plot the points.
For a non-chaotic system we’d expect the points to all
fall in one location.
For chaotic systems with strange attractors the points
fall in a varying pattern.
Poincaré map
Here we’ve removed one dimension (time).
There’s nothing to stop you doing this for other period
dimensions by putting a plain (a “Poincaré section”)
perpendicularly through the space and seeing what on the
trajectory hits it.
In general, these plots
are called Poincaré maps.
Basins of Attraction
Poincaré maps reveal the shape of
attractors.
Where there are multiple attractors an
alternative is to map the starting
conditions in phase space coloured by the
attractor they lead to.
This gives a mix of “basins of attraction”
and isolated points from which behaviour
never settles: “chaotic saddles”
Recurrence plots
If you don’t know the periodicity you need
to assess the times between the system
visiting the same kinds of states.
To do this, plot:
time on both axes,
state reassurance as black dots,
or
Euclidian distance between states as
colours.
Attractor changes
Chaotic systems can also fluctuate
dynamically between a number of
attractors.
Not only this, but the number can change
as the values of variables in the system
change.
Classic example is the population equation
known as the Logistic Map.
populationt+1 = population x r (1 – population)
r<1
1<r<2
2<r<3
3 < r < 3.57
r > 3.57
Bifurcation diagram
Population extremes
Best way to represent this change is to plot extremes
and/or stable states against change variable.
r
Problems
Still lack tools to understand multi-scale emergence.
Still lack tools to understand loss of information / track
information through the systems.
Still lack tools to assess the relationship between objects
in the system.
Let’s look at one example analysis.
Complex Geographical System
Example system: Petrol retailing.
Petrol price set by simple rules:
Cover costs.
Make maximum profit.
Undercut competition within x km.
We can look at real or modelled data.
Characteristics: Interactions
Petrol price surface
Small changes can have large impacts
A sudden increase or drop in price at one station may
result in a change in behaviour across the system.
Characteristics: Behaviour
Price
decrease
Chaotic
Petrol price surface
ButPrice
profit
decrease
decrease
and
Chaotic
increase
Price
decrease
Stable
Stable
Chaotic
Cycling (irregular)
Stable
Cycling (irregular)
Stable
Characteristics: Behaviour
Chaotic
Petrol price surface
Stable
Irregular cycling?
Can we identify the emergent behaviour, feedback loops, non-linearity?
Individual behaviour
We see simple rules
playing out.
Motorways
Taking all stations is complicated, so lets
look at a simple subset.
1
Heppenstall et al. (2005) showed that
prices at motorway stations are more
stable than non-motorway stations.
Simple, explicit network.
2
3
4
7
6
5
Quantifying
behaviour
But how do we know which
stations are inter-related?
Look at combinations in phasespace.
Station 2
Station 3
Station 2
Station 3
75.2
75.3
75.1
75.2
75.2
75.3
75.2
75.1
75.2
75.3
75.1
75.2
75.2
75.3
75.2
75.1
75.5
1
2
3
4
6
5
75.22
75.45
75.2
Price at Station 2
Price at Station 2
75.4
75.35
75.3
75.25
75.2
75.15
75.18
75.16
75.14
75.12
75.1
75.1
75.05
75
75
75.05
75.1
75.15
75.2
Price at Station 1
(a) Fixed point/Stable
75.25
75.3
75.08
75.08
75.1
75.12
75.14
75.16
7
75.18
75.2
Price at Station 1
(b) Limit Cycle/Looping
75.22
Quantifying
behaviour
Station 1
Station 2
Station 1
Station 2
74
82
74
74
76
83.4
74.3
75
75
79.2
70.1
76
77
81.9
76.1
77
1
2
3
4
7
6
90
5
85
88
84
Price (p) at Station 2
Price (p) Station 2
86
84
82
80
78
76
74
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82
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72
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70
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Price (p) Station 1
(c) Random?
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75
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Price (p) Station 1
(d) Chaotic?
78
79
Larger scale
Better to look at sets
of stations?
Look at rural to urban
transition – simple
time/price plot.
Day
But, there may be
other stations in the
mix.
Price
Diffusion of price over geographical distance.
Groups
Station_2
Station_3
Station_4
Station_5
Station_6
Station_7
75.55
75.5
75.45
75.4
Price (p)
Multiple individuals in
phase-space allows us
to group by coincidences in behaviour
75.35
75.3
75.25
75.2
75.15
75.1
75.05
75.08
75.1
75.12
75.14
75.16
75.18
75.2
Price (p)
Reaction of all the stations in the next day to
changes in price at station 1.
75.22
Issues
While there are plainly issues with large-scale analyses,
we do have the foundation of techniques that cope with
relationships at multiple scales.
Plenty to be done.
Issues
There are several major questions we’d want to ask of
geographical systems if we could.
Stability: does the system have a dynamic equilibrium or does it fluctuate?
Robustness: can the system withstand shocks and still produce sensible
outputs?
Sensitivity: will a small perturbation produce chaos?
If we can identify complexity within systems, we can:
Understand them better
Manipulate them to our own requirements
Make better predictions