GeoComputation

Gordon GreenHunter College

Department of Geography11/2006

What is GeoComputation? “…the eclectic application of computational methods and

techniques to portray spatial properties, to explain geographic phenomena, and to solve geographical problems” -Couclelis, 1998

“The application of a computational science paradigm to study a wide range of problems in geographical and earth systems.” –Openshaw, 2000

"The universe of computational techniques applicable to spatial problems" -Reed and Tuton, 1998

What is GeoComputation?

Its champions consider it a new discipline.

Others consider it just a toolbox of computationally intensive techniques.

What is GeoComputation?

Geo - emphasizes geographical and spatial aspects. Some other approaches entail spatial additions to methods that arose in non-spatial fields. GeoComputational methods tend to be inherently spatial.

Computation - equally important as the spatial aspect. The approaches tend to come out of computer science, rather than geography.

Traditional statistics tends to reduce and summarize information; GeoComputation tends to retain or generate more complexity in its operations.

What is GeoComputation? Fotheringham differentiates between Weak GC, where

the computer is used to augment existing quantitative methods; and Strong GC, where the computer "drives" the form of analysis.

Confirmatory versus exploratory techniques -- GC tends to fall in the latter category.

“The challenge of GC is to develop the ideas, the methods, the models, and the paradigms able to use the increasing computer speeds to do useful, worthwhile, innovative ad new science in a variety of geo-contexts.” (Openshaw)

What is GeoComputation?

Aspects of GeoComputation:

Multi-Agent Modeling Cellular Automata Fuzzy Modeling Genetic Programming Neural Networks

Agent-Based Models An agent-based model is one in which the basic unit of

activity is a software agent.

Usually a model will contain many agents (at least tens, occasionally many thousands).

Its outcomes are determined by the interactions of the agents.

A agent is an identifiable unit of computer program code which is autonomous and goal-directed.

(from Schelhorn, O’Sullivan, Haklay, Thurstain-Goodwin, 1999)

Agent-Based Models An early example of agent-based modeling is Boids (Reynolds,

1987), a model that describes flocking behavior.

Explaining flocking using functional models is complex.

By approaching it by modeling autonomous agents with simple rules, it was easier to create an explanatory model.

Agent models typically work by maintaining a collection of agents, and a timer that advances the state of each agent with each tick, each agent updating its state based on characteristics and location of the other agents and other entities in the landscape.

Agent models are best implemented in an object-oriented fashion

Agent-Based ModelsEach agent has a simple set

of rules defining its behavior. Creating

multiple instances of the model and adjusting the rules made it possible to model flocking behavior

concisely.

Boids animationMore elaborate example

Agent-Based Models These animations exhibit the phenomenon of

emergence, whereby complex behavior emerges from the behavior of simple components.

Another example of agent-based modeling is pedestrian modeling:

Pedestrian model – note formation of lanes. Other sample pedestrian models

Agent-Based Models Pedestrian model

implementation (Batty, 1999)

Agent-Based Models Pedestrian model

implementation (Batty, 1999)

Agent-Based Models Pedestrian model

implementation (Batty, 1999)

Agent-Based Models Pedestrian model

implementation (Batty, 1999)

Pedestrian model design (Schelhorn, O’Sullivan, Haklay, Thurstain-Goodwin, 1999)

Software for agent-based modeling

(from Najlis, Janssen, and Parker,

2001)

Agent-Based Models When to use agents? Axtell (1999) describes three basic

cases when agent-based models might be used:

1. Agent models as simulation of mathematical models – agent models can be used to implement monte-carlo simulations that can also be solved using other numerical methods. Even though the simulation can be solved in a functional manner, an agent-based implementation can act as an alternative implementation that can validate the results.

Similarly, an agent-based model can be useful as an illustration of a complex numerical model. Even though the agent-based model may not be necessary to come up with a solution, it may be very helpful in illustrating the results to a general audience.

Agent-Based Models An example of this first use can be found in the classic

problem of modeling a bank-teller line:

The queuing process is commonly simulated using a the Monte Carlo method to arrive at a distribution of waiting times.

This is can be equivalently modeled using agents that each have different arrival times and other parameters, and running the model with many agents to similarly build up the resulting distribution function.

(Axtell, 1999)

Agent-Based Models 2. Agent models as complementary to mathematical

models – a mathematical model may adequately model some aspects of a problem but not others, or may be awkward or incapable of a comprehensive solution. Or a mathematical model may be known, but so complex as to be practically insoluble.

An artificial stock market is an example of such a model. Trading agents have the choice of investments with varying stochastic levels of risk, adapt their behavior based on the results of prior trading.

Even though the basic features of such a model may be functionally describable, an agent-based model can evolve over time into a system that actually replicates some of the complexity of a real stock market. These complexities are impractical to model using a mathematical model. (Axtell, 1999)

Agent-Based Models 3. Agent models as substitutes for mathematical

models – some problems are not amenable to mathematical modeling.

Examples include the behavior of individuals in animal population, or groups of pedestrians as described earlier.

Agent-Based Models Pros:

An argument from economics, which could also be applied to geography:

“The reason why large scale computable general equilibrium problems are difficult for economists to solve is that they are using the wrong hardware and software. Economists should design their computations to mimic the real economy, using massively parallel computers and decentralized algorithms that allow competitive equilibria to arise as ‘emergent computations’...[T]he most promising way for economists to avoid the computational burdens associated with solving realistic large scale general equilibrium models is to adopt an “agent-based” modeling strategy where equilibrium prices and quantities emerge endogenously from the decentralized interactions of agents.” (Rust, 1996, in Axtell)

The quality of emergent behavior doesn’t correspond to any part of a traditional statistical analysis. Agent-based models uniquely provide the opportunity to model a process, and see what happens when it runs (Axtell).

Agent-Based Models Cons:

When dealing with agent models, we are quickly involved in a world where everything - both the agents and their environment - are designed, and the reliable scientific analysis of the real world may be compromised by the complexity of the models (paraphrased from Couclelis, Why I No Longer Work with Agents, discussing human/environment agent models).

Agent models don’t have a convenient way of measuring their accuracy, unlike statistical models. This can only be overcome by running the agent model many times, varying the parameters to discover the robustness of the results. There is a limit as to how many such variations can be executed (although this is increasing with increases in computer power) (Axtell).

Cellular Automata

Cellular automata are closely related to agent-based models.

Instead of an autonomous agents being given simple behavioral rules, the states of cells in a surface are dictated by similarly simple rules.

Those rules use the state of surrounding cells in a grid to determine the state of any given cell in the next iteration of the model.

Cellular Automata CAs develop in space and time.

Space and time are defined in discrete steps.

Cells are lined up in a string for one-dimensional automata, or arranged in a two or higher dimensional lattice for two- or higher- dimensional automata.

The number of states of each cell is finite.

The states of each cell are discrete.

All cells are identical.

The future state of each cell depends only of the current state of the cell and the states of the cells in the neighborhood, determined by rules (from Alexander Schatten).

Cellular Automata The simplest CAs are one-dimensional. For example,

here are a set of rules and the results of a few iterations (from David R. Green):

Cellular AutomataFor geographical applications, CAs are usually 2-dimensional,

often using one of these cell neighborhoods:

Cellular AutomataConway’s Game of life rules

Mathematician John Conway invented CAs in his Game of Life, first described in a 1970 Scientific American article.

A cell that is dead at the time step t, becomes alive at time t+1 if exactly three of the eight neighboring cells at time t were alive.

A cell that is alive at time t dies at time t+1 if at time t less than two (loneliness) or more than three cells (overcrowding) are alive. (Alexander Schatten)

From these very simple rules, very complex behaviors can be modeled.

Cellular AutomataSample game of life pattern (“glider”):

Cellular Automata

There are many other examples available on the web, such as:

http://www.ibiblio.org/lifepatterns/

These simple rules can be expanded to generate many, many patterns of change:

http://www.collidoscope.com/modernca/

Cellular AutomataThere are many enhancements involved in generating more complex

evolutionary patterns:

Probabilistic CAs, where, instead of having binary rules (and binary states), the changes are described by probabilities, give an increased level of control over how a CA develops. The rules can express the chance of a cell changing state, and each step of the CA involves selecting the state of each cell based on a random number falling within its probability range.

CAs can also be self-modifying, that is, they can respond to changes in the states of the grid as it develops.

Irregular grid cell sizes

Action a distance, instead of just immediately adjacent cells.

Incorporating agents within CA landscapes

Cellular Automata

These more sophisticated CAs have been used to model diverse geographic phenomena, such as:

Wildfire Formation of a Megalopolis West Nile Virus Infection Risk

#2 from Paul Torrens, http://geosimulation.org/simulating-sprawl/, #3 from Sean Ahearn.

Cellular AutomataCons (Batty and Xie, 2005):

Most CA rules have little relationship to the actual causes of the phenomenon being studied. Even if the model happens to model the process successfully, it doesn’t really prove anything, and is difficult to use as the basis of any kind of policy decision.

They tend not to model spatial interaction well.

In practice, CA cells tend not to match units of the phenomenon under study.

Fuzzy Modeling A frequent problem in geography is how to identify an

geographic entity.

Geographic phenomena tend to be described by vague terms.

For example: a study concerns the major cities in Europe. Each of the terms “major”, “city”, “in”, and “Europe” are somewhat vague.

Fuzzy set theory is an attempt to deal with the problems posed to traditional set and logic theory by vagueness.

(section paraphrased from Fisher, 2000)

Fuzzy Modeling Sorites paradox:

If a grain of sand is placed on a surface, is there a heap? If a second grain of sand is placed next to it, is there a

heap? If a third grain of sand is placed next to the second, is there

a heap? If a ten-millionth grain of sand is placed next to the

9,999,999th, is there a heap?

Heap is a vague concept. Other examples are “tall”, “near”, “far”, etc.

Many geographic values such as vegetation coverage, soil types, etc, are similarly “Sorites-susceptible”.

Fuzzy Modeling Classical sets follow the logic described in familiar Venn

diagram (examples drawn from behavior of boolean search terms):

Fuzzy ModelingThese can also be

expressed using linear graphs:

Fuzzy Modeling Membership in a set is determined by some threshold value.

This value can be subject to error, and can be assigned a probability, Those probabilities can then be used in set-based calculations:

Fuzzy Modeling Boolean set theory is used throughout most conventional

set and statistical analysis, For example: does the result of a test disprove the null hypothesis? We set a threshold value to the t test statistics and respond based on whether or not the results of the hypothesis are above or below that test. In many cases, there is actually a continuous probability of the null hypothesis being correct.

Zadeh 1965 put forward concept of fuzzy sets in response to shortcomings of boolean sets.

Fuzzy Modeling Fuzzy set membership is at the core of fuzzy set theory.

Boolean sets are encoded with 0 or 1, wherein an entity is or is not part of a set. Fuzzy set membership is defined by any value between (and including) 0 and 1.

Fuzzy Modeling Fuzzy set membership can take varied

forms:

Fuzzy Modeling These then result in boolean calculations that take

into account partial set membership:

Fuzzy Modeling Boolean operations become functions

rather than boolean values:

Fuzzy Modeling

Fuzzy Modeling

Example of fuzzy versus crisp classifications

Fuzzy ModelingThe concept of fuzziness can also be applied to polygon boundaries

Fuzzy Modeling

Key question: How do you choose the membership function (MF)? Is it linear or curved?

The “semantic import” (SI) approach estimates based on domain expertise and iterative evaluation of the results.

Fuzzy K-means approach uses iterative mathematical estimates of the best function.

Fuzzy ModelingFuzzy K-Means

Usually starts with random allocation of objects into k clusters.

The center of each cluster is calculated Objects are re-allocated based on similarity of attributes

using a similarity index, usually a distance measurement.

This process is repeated until a stable solution is reached.

Membership in each cluster is calculated as a range of from 0 to 1, instead of 1 as in “crisp” k-means (Burrough and McDonnell)

Fuzzy Modeling

Fuzzy K-Means

Fuzzy Modeling

Fuzzy modeling is often used in conjunction with genetic algorithms, which provide a way of iteratively refining the results of automated models.

The following few slides will review this before concluding the section on fuzzy modeling.

Genetic Algorithms First developed in the 70’s.

The basic idea is to model a problem with many parameters by treating those parameters like genes.

These are selected over time by iteratively applying a measure of fitness, selecting the fittest elements, and applying the process again.

Genetic AlgorithmsSteps in an evolutionary algorithm

Create a new population of alternatives using random values.

Select individuals from the population weighting towards the individual that represents the best solution so far.

Use them as the basis of a new set of alternatives, by combining them (crossover) or randomly changing them (mutation).

Continue this process until some terminating condition is met.

Genetic AlgorithmsSample “chromosome”:

A contrived simple problem:

Given the digits 0 through 9 and the operators +, -, * and /,  find a sequence that will represent a given target number. The operators will be applied sequentially from left to right as you read.

Encoding:

Genetic AlgorithmsEncoded solution:

Crossover:

Mutation: Randomly changing bits

Genetic Algorithms While not necessarily spatial per-se, a genetic approach

to model evaluation and selection can be applied to many problems that require selecting among many possible models.

Fuzzy Modeling

Using fuzzy modeling to classify coverage in

remotely sensed data involves fuzzy logic in

combination with a genetic classification

process (from TNTMips user guide):

Fuzzy Modeling

Fuzzy Modeling

The right-hand image shows the first-cut results of an unsupervised image classification. The right-hand image shows the extent to which each cell differs from the center of its assigned spectral class, so darker areas are more likely to be in the assigned class:

Fuzzy Modeling

The Fuzzy classification method uses rules of fuzzy logic, which recognize that class boundaries may be imprecise or gradational.

It creates an initial set of prototype classes, then determines a membership grade for each class for every cell.

The grades are used to adjust the class assignments and calculate new class centers, and the process repeats until the iteration limit is reached. (TNTMips user guide).

Fuzzy Modeling

Champions of fuzzy modeling consider probability theory to be a special case of fuzzy modeling.

Others consider fuzzy modeling to be unnecessary, and replaceable by Bayesian probability.

Neural Networks The last geocomputational technique we will cover is

neural networks, or “neurocomputing.”

Definition: A computational neural network (NCC), is a parallel distributed information structure consisting of a set of adaptive processing (computational) elements and a set of unidirectional data connections (Fischer and Abrahard).

These models have been inspired by neuroscience, but in no way actually model biological or neurological phenomena.

Neural Networks

Neural NetworksTypical neural network diagram:

Neural Networks A Neural Network consists of various layers.

Each layer can any number of neurons in it.

The first layer of the network is called an input layer, and it is here we apply the input.

The last layer is called the output layer, and it is from here we take the output.

A neural network can have any number of hidden layers, between the input and output layer.

In most neural network models, a neuron in one layer is connected to all neurons in the next layer (from a description by Anoop Madhusudanan).

Neural Networks Here is a diagram of the simplest possible neural

network. N1 and N2 are inputs, N3 and N4 are in a hidden layer, and N5 is the output:

Neural Networks Within each node, or “neuron”, there is a function that

determines the output of the node based on the inputs:

Input data are typically specified to take values in any range, whereas output is given in limited ranges.

Neural Networks Each input has a weight. The totals of the input constitute an

“activity level”. When the activity level reaches a certain threshold, the output changes.

The output may also change continuously with changes in input, for example following a curve. This curve represents the “transfer function,” by which inputs are converted to outputs:

Neural Networks Neural networks must be trained. Training consists of adjusting

the weighting of inputs depending on the value of outputs.

The can be accomplished with a feedback function, which for example, may take the output and feed it back to the hidden or input layers. The hidden layer may in turn then feedback to the input layer.

The feedback function adjusts the weighting so that correct results are increased with further iterations of the network. Note that weights can be negative, so neurons may be either inhibitory or excitatory.

Neural Networks For example, a neural network designed to detect the

pattern of the number four might have an input node for every pixel in a grid.

It would be “trained” (weights adjusted) using multiple images of the number four:

Neural Networks Training may be supervised or unsupervised. Most implementations are

supervised.

In supervised training, the training data consists of inputs together with expected outputs.

Neural networks work best with boolean or ordinal or real numeric data. Non-ordinal set-based data types tend not to work because they do not lend themselves to weighting.

With enough training data where the inputs and outputs are known, the network implicitly begins to model a function which can be applied to unknown inputs.

The training data selected depends on user knowledge and intuition about the problem domain. Neural networks take into account existing knowledge about a problem via the training process.

(Paraphrased from StatSoft manual).

Neural Networks

Neural networks work best when there is a lot of training data. For most practical problems, hundreds or thousands of training cases are required. The exact number depends on the nature of the problem.

Neural networks tend to be tolerant of noisy input, but there can be problems if the training data does not include outliers found in the subject data. In these cases the outliers may be ignored.

They tend to work well with regression problems (where the output will be a specific number) and with classification problems (where the output may be a boolean value or a set of values). Pattern recognition is one common application.

Neural NetworksApplications in GIS:

Image classification – neural networks can be used to optimally classify remotely sensed images.

Feature detection – the pattern-recognition strengths of neural networks could be used to identify features in remotely sensed data.

Transportation route selection – for example, route and congestion data can be used as inputs to neurons that optimize for shortest routes. Optimal routing could be the output.

Neural Networks Transportation example (from Thurston, GIS Café.com):

A is starting point and B is destination. Intersections are nodes in the network. The red square is the location of an accident.

Neural Networks The weighting of node inputs can be calibrated to the traffic

capacity of each. The accident causes the affected nodes to inhibit activation, resulting in the selection of non-affected nodes:

Related Topics

Parallel processing and object-oriented software development are closely related to GeoComputation.

The following slides give a quick overview of these related topics.

Parallel ProcessingWhat is it?

Parallel processing is using more than one CPU to perform computational tasks more quickly.

Types of parallel processing:

SIMD – Single instruction stream, multiple data stream. MIMD – Multiple instruction streams, multiple data streams.

SISD – Single instruction stream, single data stream, is the model in place in most current PCs.

MISD – Multiple instruction stream, single data stream. This is really only a theoretical construct, because it is not practical to have multiple processors manage the same data.

Parallel ProcessingSIMD – each processor runs the same program on different

data (from Johnston, introduction to HPC):

Parallel ProcessingMIMD – each processor runs different instructions on

different data:

Parallel Processing

MIMD has proven to be the most generally applicable parallel processing model.

Splitting up tasks into chunks that can be executed in parallel is widely applicable to geographic problems, where similar tasks need to be applied to different data.

This kind of processing model can be implemented on a SISD machine, and is a useful way of thinking about many geocomputational practices.

Parallel ProcessingWhen does it make sense to use it?

Some tasks are easier than others to make parallel – old cliché – nine women can’t produce a baby in one month; but 90 women can create 90 babies in nine months.

Whether or not a model can be implemented in a parallel fashion depends not only on the nature of the problem, but also on the number of iterations needed.

Problems that are inherently serial, or that would be very difficult to make parallel (parallel code tends to take several times longer to write), are not suitable

But most problems in geography are likely to be parallel (e.g., you can usually subdivide a map, and a problem, into sectors that can be processed independently).

Parallel Processing One interesting sidebar: Distributed computing or “task

farming” across multiple machines:

Parallel Processing For example, large distributed networks of volunteer machines

can be used to process large parallel processing tasks such as global climate modeling.

BOINC software for distributed volunteer computing supports such processing:

Parallel ProcessingBOINC ApplicationsClimate Prediction Model

Object-Oriented Programming

Traditional programming consists of functions that roughly approximate the notion of a mathematical function.

Problems are modeled by breaking the down into layers of functions in a process called structured decomposition.

Object oriented programming instead identifies units of behavior and data that can be grouped together to model some real-world entity.

Object-Oriented Programming

Object-Oriented Programming

Object-Oriented Programming

Parallel processing and object-oriented software development make it easy to approach computational problems in a bottom-up fashion, using simple components to model systems that would otherwise be extremely complex.

An object encapsulates its state and behaviors, making it easy to create complex systems composed of autonomous software entities.

An OOP and parallel-programming perspective applied to spatial modeling is a convenient approach for most GeoComputation techniques.